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|---|---|---|---|---|---|---|
α✝ : Type u
β : Type v
α : Type u
inst✝⁴ : Lattice α
inst✝³ : DecidableEq α
inst✝² : DecidableRel fun x x_1 => x ≤ x_1
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : IsTotal α fun x x_1 => x ≤ x_1
src✝ : Lattice α := inst✝⁴
⊢ ∀ (a b : α), min a b = if a ≤ b then a else b
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by
|
exact congr_fun₂ inf_eq_minDefault
|
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by
|
Mathlib.Order.Lattice.909_0.wE3igZl9MFbJBfv
|
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α
|
Mathlib_Order_Lattice
|
α✝ : Type u
β : Type v
α : Type u
inst✝⁴ : Lattice α
inst✝³ : DecidableEq α
inst✝² : DecidableRel fun x x_1 => x ≤ x_1
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : IsTotal α fun x x_1 => x ≤ x_1
src✝ : Lattice α := inst✝⁴
⊢ ∀ (a b : α), max a b = if a ≤ b then b else a
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by
|
exact congr_fun₂ sup_eq_maxDefault
|
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by
|
Mathlib.Order.Lattice.909_0.wE3igZl9MFbJBfv
|
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → SemilatticeSup (π i)
f : (i : ι) → π i
i : ι
a b : π i
j : ι
⊢ update f i (a ⊔ b) j = (update f i a ⊔ update f i b) j
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by
|
obtain rfl | hji := eq_or_ne j i
|
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by
|
Mathlib.Order.Lattice.1044_0.wE3igZl9MFbJBfv
|
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b
|
Mathlib_Order_Lattice
|
case inl
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → SemilatticeSup (π i)
f : (i : ι) → π i
j : ι
a b : π j
⊢ update f j (a ⊔ b) j = (update f j a ⊔ update f j b) j
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;>
|
simp [update_noteq, *]
|
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;>
|
Mathlib.Order.Lattice.1044_0.wE3igZl9MFbJBfv
|
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b
|
Mathlib_Order_Lattice
|
case inr
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → SemilatticeSup (π i)
f : (i : ι) → π i
i : ι
a b : π i
j : ι
hji : j ≠ i
⊢ update f i (a ⊔ b) j = (update f i a ⊔ update f i b) j
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;>
|
simp [update_noteq, *]
|
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;>
|
Mathlib.Order.Lattice.1044_0.wE3igZl9MFbJBfv
|
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → SemilatticeInf (π i)
f : (i : ι) → π i
i : ι
a b : π i
j : ι
⊢ update f i (a ⊓ b) j = (update f i a ⊓ update f i b) j
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by
|
obtain rfl | hji := eq_or_ne j i
|
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by
|
Mathlib.Order.Lattice.1049_0.wE3igZl9MFbJBfv
|
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b
|
Mathlib_Order_Lattice
|
case inl
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → SemilatticeInf (π i)
f : (i : ι) → π i
j : ι
a b : π j
⊢ update f j (a ⊓ b) j = (update f j a ⊓ update f j b) j
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;>
|
simp [update_noteq, *]
|
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;>
|
Mathlib.Order.Lattice.1049_0.wE3igZl9MFbJBfv
|
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b
|
Mathlib_Order_Lattice
|
case inr
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → SemilatticeInf (π i)
f : (i : ι) → π i
i : ι
a b : π i
j : ι
hji : j ≠ i
⊢ update f i (a ⊓ b) j = (update f i a ⊓ update f i b) j
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;>
|
simp [update_noteq, *]
|
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;>
|
Mathlib.Order.Lattice.1049_0.wE3igZl9MFbJBfv
|
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
f : α → β
h : ∀ (x y : α), f (x ⊓ y) = f x ⊓ f y
x y : α
hxy : x ≤ y
⊢ f x ⊓ f y = f x
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by
|
rw [← h, inf_eq_left.2 hxy]
|
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by
|
Mathlib.Order.Lattice.1099_0.wE3igZl9MFbJBfv
|
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
⊢ f (x ⊔ y) = f x ⊔ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by
|
simp only [h, hf h, sup_of_le_right]
|
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by
|
Mathlib.Order.Lattice.1111_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
f : α → β
hf : Monotone f
x y : α
h : y ≤ x
⊢ f (x ⊔ y) = f x ⊔ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
|
simp only [h, hf h, sup_of_le_left]
|
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
|
Mathlib.Order.Lattice.1111_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
f : α → β
s : Set α
x✝ y✝ : α
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeInf β
h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f (x ⊓ y) = f x ⊓ f y
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≤ y
⊢ f x ⊓ f y = f x
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by
|
rw [← h _ hx _ hy, inf_eq_left.2 hxy]
|
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by
|
Mathlib.Order.Lattice.1151_0.wE3igZl9MFbJBfv
|
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
⊢ f (x ⊔ y) = f x ⊔ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
|
cases le_total x y
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inl
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ f (x ⊔ y) = f x ⊔ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;>
|
have := hf ?_ ?_ ‹_›
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;>
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inr
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ f (x ⊔ y) = f x ⊔ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;>
|
have := hf ?_ ?_ ‹_›
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;>
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inl.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
this : f x ≤ f y
⊢ f (x ⊔ y) = f x ⊔ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inl.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
this : f x ≤ f y
⊢ f (x ⊔ y) = f x ⊔ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
assumption
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inl.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
this : f x ≤ f y
⊢ f (x ⊔ y) = f x ⊔ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
|
|
simp only [*, sup_of_le_left, sup_of_le_right]
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
|
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inl.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ x ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inl.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ x ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
assumption
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inl.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ y ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inl.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ y ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
assumption
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inr.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
this : f y ≤ f x
⊢ f (x ⊔ y) = f x ⊔ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inr.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
this : f y ≤ f x
⊢ f (x ⊔ y) = f x ⊔ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
assumption
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inr.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
this : f y ≤ f x
⊢ f (x ⊔ y) = f x ⊔ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
|
|
simp only [*, sup_of_le_left, sup_of_le_right]
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
|
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inr.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ y ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inr.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ y ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
assumption
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inr.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ x ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
case inr.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeSup β
hf : MonotoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ x ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
assumption
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
Mathlib.Order.Lattice.1163_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
f : α → β
s : Set α
x✝ y✝ : α
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeSup β
h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f (x ⊓ y) = f x ⊔ f y
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≤ y
⊢ f x ⊔ f y = f x
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by
|
rw [← h _ hx _ hy, inf_eq_left.2 hxy]
|
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by
|
Mathlib.Order.Lattice.1257_0.wE3igZl9MFbJBfv
|
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
⊢ f (x ⊔ y) = f x ⊓ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
|
cases le_total x y
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inl
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ f (x ⊔ y) = f x ⊓ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;>
|
have := hf ?_ ?_ ‹_›
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;>
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inr
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ f (x ⊔ y) = f x ⊓ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;>
|
have := hf ?_ ?_ ‹_›
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;>
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inl.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
this : f y ≤ f x
⊢ f (x ⊔ y) = f x ⊓ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inl.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
this : f y ≤ f x
⊢ f (x ⊔ y) = f x ⊓ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
assumption
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inl.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
this : f y ≤ f x
⊢ f (x ⊔ y) = f x ⊓ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
|
|
simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
|
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inl.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ x ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inl.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ x ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
assumption
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inl.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ y ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inl.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : x ≤ y
⊢ y ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
assumption
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inr.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
this : f x ≤ f y
⊢ f (x ⊔ y) = f x ⊓ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inr.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
this : f x ≤ f y
⊢ f (x ⊔ y) = f x ⊓ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
assumption
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inr.refine_3
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
this : f x ≤ f y
⊢ f (x ⊔ y) = f x ⊓ f y
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
|
|
simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
|
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inr.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ y ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inr.refine_1
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ y ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
assumption
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inr.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ x ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
case inr.refine_2
α : Type u
β : Type v
f : α → β
s : Set α
x y : α
inst✝¹ : LinearOrder α
inst✝ : SemilatticeInf β
hf : AntitoneOn f s
hx : x ∈ s
hy : y ∈ s
h✝ : y ≤ x
⊢ x ∈ s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
assumption
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
|
|
Mathlib.Order.Lattice.1269_0.wE3igZl9MFbJBfv
|
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ a ≤ a ⊔ b
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
|
change f a ≤ f (a ⊔ b)
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
|
Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f a ≤ f (a ⊔ b)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
|
rw [map_sup]
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
|
Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f a ≤ f a ⊔ f b
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
|
exact le_sup_left
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
|
Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ b ≤ a ⊔ b
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
|
change f b ≤ f (a ⊔ b)
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
|
Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f b ≤ f (a ⊔ b)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
|
rw [map_sup]
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
|
Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f b ≤ f a ⊔ f b
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
|
exact le_sup_right
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
|
Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b c : α
ha : a ≤ c
hb : b ≤ c
⊢ a ⊔ b ≤ c
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
|
change f (a ⊔ b) ≤ f c
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
|
Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b c : α
ha : a ≤ c
hb : b ≤ c
⊢ f (a ⊔ b) ≤ f c
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
|
rw [map_sup]
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
|
Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Sup α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b c : α
ha : a ≤ c
hb : b ≤ c
⊢ f a ⊔ f b ≤ f c
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
|
exact sup_le ha hb
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
|
Mathlib.Order.Lattice.1448_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ a ⊓ b ≤ a
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
exact sup_le ha hb }
#align function.injective.semilattice_sup Function.Injective.semilatticeSup
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
|
change f (a ⊓ b) ≤ f a
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
|
Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f (a ⊓ b) ≤ f a
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
exact sup_le ha hb }
#align function.injective.semilattice_sup Function.Injective.semilatticeSup
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
|
rw [map_inf]
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
|
Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f a ⊓ f b ≤ f a
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
exact sup_le ha hb }
#align function.injective.semilattice_sup Function.Injective.semilatticeSup
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
|
exact inf_le_left
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
|
Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ a ⊓ b ≤ b
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
exact sup_le ha hb }
#align function.injective.semilattice_sup Function.Injective.semilatticeSup
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
|
change f (a ⊓ b) ≤ f b
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
|
Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f (a ⊓ b) ≤ f b
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
exact sup_le ha hb }
#align function.injective.semilattice_sup Function.Injective.semilatticeSup
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
|
rw [map_inf]
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
|
Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b : α
⊢ f a ⊓ f b ≤ f b
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
exact sup_le ha hb }
#align function.injective.semilattice_sup Function.Injective.semilatticeSup
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
rw [map_inf]
|
exact inf_le_right
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
rw [map_inf]
|
Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b c : α
ha : a ≤ b
hb : a ≤ c
⊢ a ≤ b ⊓ c
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
exact sup_le ha hb }
#align function.injective.semilattice_sup Function.Injective.semilatticeSup
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
rw [map_inf]
exact inf_le_right,
le_inf := fun a b c ha hb => by
|
change f a ≤ f (b ⊓ c)
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
rw [map_inf]
exact inf_le_right,
le_inf := fun a b c ha hb => by
|
Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b c : α
ha : a ≤ b
hb : a ≤ c
⊢ f a ≤ f (b ⊓ c)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
exact sup_le ha hb }
#align function.injective.semilattice_sup Function.Injective.semilatticeSup
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
rw [map_inf]
exact inf_le_right,
le_inf := fun a b c ha hb => by
change f a ≤ f (b ⊓ c)
|
rw [map_inf]
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
rw [map_inf]
exact inf_le_right,
le_inf := fun a b c ha hb => by
change f a ≤ f (b ⊓ c)
|
Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝¹ : Inf α
inst✝ : SemilatticeInf β
f : α → β
hf_inj : Injective f
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : PartialOrder α := PartialOrder.lift f hf_inj
a b c : α
ha : a ≤ b
hb : a ≤ c
⊢ f a ≤ f b ⊓ f c
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
exact sup_le ha hb }
#align function.injective.semilattice_sup Function.Injective.semilatticeSup
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
rw [map_inf]
exact inf_le_right,
le_inf := fun a b c ha hb => by
change f a ≤ f (b ⊓ c)
rw [map_inf]
|
exact le_inf ha hb
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
rw [map_inf]
exact inf_le_right,
le_inf := fun a b c ha hb => by
change f a ≤ f (b ⊓ c)
rw [map_inf]
|
Mathlib.Order.Lattice.1470_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝² : Sup α
inst✝¹ : Inf α
inst✝ : DistribLattice β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : Lattice α := Injective.lattice f hf_inj map_sup map_inf
a b c : α
⊢ (a ⊔ b) ⊓ (a ⊔ c) ≤ a ⊔ b ⊓ c
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
exact sup_le ha hb }
#align function.injective.semilattice_sup Function.Injective.semilatticeSup
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
rw [map_inf]
exact inf_le_right,
le_inf := fun a b c ha hb => by
change f a ≤ f (b ⊓ c)
rw [map_inf]
exact le_inf ha hb }
#align function.injective.semilattice_inf Function.Injective.semilatticeInf
/-- A type endowed with `⊔` and `⊓` is a `Lattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `Lattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.lattice [Sup α] [Inf α] [Lattice β] (f : α → β)
(hf_inj : Function.Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : Lattice α :=
{ hf_inj.semilatticeSup f map_sup, hf_inj.semilatticeInf f map_inf with }
#align function.injective.lattice Function.Injective.lattice
/-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
DistribLattice α :=
{ hf_inj.lattice f map_sup map_inf with
le_sup_inf := fun a b c => by
|
change f ((a ⊔ b) ⊓ (a ⊔ c)) ≤ f (a ⊔ b ⊓ c)
|
/-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
DistribLattice α :=
{ hf_inj.lattice f map_sup map_inf with
le_sup_inf := fun a b c => by
|
Mathlib.Order.Lattice.1502_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
DistribLattice α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝² : Sup α
inst✝¹ : Inf α
inst✝ : DistribLattice β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : Lattice α := Injective.lattice f hf_inj map_sup map_inf
a b c : α
⊢ f ((a ⊔ b) ⊓ (a ⊔ c)) ≤ f (a ⊔ b ⊓ c)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
exact sup_le ha hb }
#align function.injective.semilattice_sup Function.Injective.semilatticeSup
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
rw [map_inf]
exact inf_le_right,
le_inf := fun a b c ha hb => by
change f a ≤ f (b ⊓ c)
rw [map_inf]
exact le_inf ha hb }
#align function.injective.semilattice_inf Function.Injective.semilatticeInf
/-- A type endowed with `⊔` and `⊓` is a `Lattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `Lattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.lattice [Sup α] [Inf α] [Lattice β] (f : α → β)
(hf_inj : Function.Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : Lattice α :=
{ hf_inj.semilatticeSup f map_sup, hf_inj.semilatticeInf f map_inf with }
#align function.injective.lattice Function.Injective.lattice
/-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
DistribLattice α :=
{ hf_inj.lattice f map_sup map_inf with
le_sup_inf := fun a b c => by
change f ((a ⊔ b) ⊓ (a ⊔ c)) ≤ f (a ⊔ b ⊓ c)
|
rw [map_inf, map_sup, map_sup, map_sup, map_inf]
|
/-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
DistribLattice α :=
{ hf_inj.lattice f map_sup map_inf with
le_sup_inf := fun a b c => by
change f ((a ⊔ b) ⊓ (a ⊔ c)) ≤ f (a ⊔ b ⊓ c)
|
Mathlib.Order.Lattice.1502_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
DistribLattice α
|
Mathlib_Order_Lattice
|
α : Type u
β : Type v
inst✝² : Sup α
inst✝¹ : Inf α
inst✝ : DistribLattice β
f : α → β
hf_inj : Injective f
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
src✝ : Lattice α := Injective.lattice f hf_inj map_sup map_inf
a b c : α
⊢ (f a ⊔ f b) ⊓ (f a ⊔ f c) ≤ f a ⊔ f b ⊓ f c
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align_import order.lattice from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `SemilatticeSup`: a type class for join semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeSup` via proofs that `⊔` is
commutative, associative and idempotent.
* `SemilatticeInf`: a type class for meet semilattices
* `SemilatticeSup.mk'`: an alternative constructor for `SemilatticeInf` via proofs that `⊓` is
commutative, associative and idempotent.
* `Lattice`: a type class for lattices
* `Lattice.mk'`: an alternative constructor for `Lattice` via proofs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `DistribLattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
universe u v w
variable {α : Type u} {β : Type v}
#align le_antisymm' le_antisymm
/-!
### Join-semilattices
-/
-- TODO: automatic construction of dual definitions / theorems
/-- A `SemilatticeSup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
/-- The supremum is an upper bound on the first argument -/
protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
/-- The supremum is an upper bound on the second argument -/
protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
/-- The supremum is the *least* upper bound -/
protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
#align semilattice_sup SemilatticeSup
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def SemilatticeSup.mk' {α : Type*} [Sup α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (sup_idem : ∀ a : α, a ⊔ a = a) :
SemilatticeSup α where
sup := (· ⊔ ·)
le a b := a ⊔ b = b
le_refl := sup_idem
le_trans a b c hab hbc := by
-- Porting note: dsimp doesn't work here?
-- This is the same issue as discussed at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfolding.20earlier.20fields
show a ⊔ c = c
rw [← hbc, ← sup_assoc, hab]
le_antisymm a b hab hba := by
rwa [← hba, sup_comm]
le_sup_left a b := show a ⊔ (a ⊔ b) = a ⊔ b by rw [← sup_assoc, sup_idem]
le_sup_right a b := show b ⊔ (a ⊔ b) = a ⊔ b by rw [sup_comm, sup_assoc, sup_idem]
sup_le a b c hac hbc := by
show (a ⊔ b) ⊔ c = c
rwa [sup_assoc, hbc]
#align semilattice_sup.mk' SemilatticeSup.mk'
instance instSupOrderDual (α : Type*) [Inf α] : Sup αᵒᵈ :=
⟨((· ⊓ ·) : α → α → α)⟩
instance instInfOrderDual (α : Type*) [Sup α] : Inf αᵒᵈ :=
⟨((· ⊔ ·) : α → α → α)⟩
section SemilatticeSup
variable [SemilatticeSup α] {a b c d : α}
@[simp]
theorem le_sup_left : a ≤ a ⊔ b :=
SemilatticeSup.le_sup_left a b
#align le_sup_left le_sup_left
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_left' : a ≤ a ⊔ b :=
le_sup_left
#align le_sup_left' le_sup_left'
@[simp]
theorem le_sup_right : b ≤ a ⊔ b :=
SemilatticeSup.le_sup_right a b
#align le_sup_right le_sup_right
-- Porting note: no ematch attribute
--@[ematch]
theorem le_sup_right' : b ≤ a ⊔ b :=
le_sup_right
#align le_sup_right' le_sup_right'
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
#align le_sup_of_le_left le_sup_of_le_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
#align le_sup_of_le_right le_sup_of_le_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
#align lt_sup_of_lt_left lt_sup_of_lt_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
#align lt_sup_of_lt_right lt_sup_of_lt_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
SemilatticeSup.sup_le a b c
#align sup_le sup_le
@[simp]
theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨fun h : a ⊔ b ≤ c => ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
fun ⟨h₁, h₂⟩ => sup_le h₁ h₂⟩
#align sup_le_iff sup_le_iff
@[simp]
theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_left sup_eq_left
@[simp]
theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align sup_eq_right sup_eq_right
@[simp]
theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
#align left_eq_sup left_eq_sup
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
#align right_eq_sup right_eq_sup
alias ⟨_, sup_of_le_left⟩ := sup_eq_left
#align sup_of_le_left sup_of_le_left
alias ⟨le_of_sup_eq, sup_of_le_right⟩ := sup_eq_right
#align sup_of_le_right sup_of_le_right
#align le_of_sup_eq le_of_sup_eq
attribute [simp] sup_of_le_left sup_of_le_right
@[simp]
theorem left_lt_sup : a < a ⊔ b ↔ ¬b ≤ a :=
le_sup_left.lt_iff_ne.trans $ not_congr left_eq_sup
#align left_lt_sup left_lt_sup
@[simp]
theorem right_lt_sup : b < a ⊔ b ↔ ¬a ≤ b :=
le_sup_right.lt_iff_ne.trans $ not_congr right_eq_sup
#align right_lt_sup right_lt_sup
theorem left_or_right_lt_sup (h : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
h.not_le_or_not_le.symm.imp left_lt_sup.2 right_lt_sup.2
#align left_or_right_lt_sup left_or_right_lt_sup
theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
#align le_iff_exists_sup le_iff_exists_sup
@[gcongr]
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
#align sup_le_sup sup_le_sup
@[gcongr]
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
#align sup_le_sup_left sup_le_sup_left
@[gcongr]
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
#align sup_le_sup_right sup_le_sup_right
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_idem : a ⊔ a = a := by simp
#align sup_idem sup_idem
instance : IsIdempotent α (· ⊔ ·) :=
⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a := by apply le_antisymm <;> simp
#align sup_comm sup_comm
instance : IsCommutative α (· ⊔ ·) :=
⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
eq_of_forall_ge_iff $ fun x => by simp only [sup_le_iff]; rw [and_assoc]
#align sup_assoc sup_assoc
instance : IsAssociative α (· ⊔ ·) :=
⟨@sup_assoc _ _⟩
theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
rw [sup_comm, @sup_comm _ _ a, sup_assoc]
#align sup_left_right_swap sup_left_right_swap
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b := by simp
#align sup_left_idem sup_left_idem
-- Porting note: was @[simp], but now proved by simp so not needed.
theorem sup_right_idem : a ⊔ b ⊔ b = a ⊔ b := by simp
#align sup_right_idem sup_right_idem
theorem sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) := by
rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
#align sup_left_comm sup_left_comm
theorem sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b := by
rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
#align sup_right_comm sup_right_comm
theorem sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) := by
rw [sup_assoc, sup_left_comm b, ← sup_assoc]
#align sup_sup_sup_comm sup_sup_sup_comm
theorem sup_sup_distrib_left (a b c : α) : a ⊔ (b ⊔ c) = a ⊔ b ⊔ (a ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_left sup_sup_distrib_left
theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by
rw [sup_sup_sup_comm, sup_idem]
#align sup_sup_distrib_right sup_sup_distrib_right
theorem sup_congr_left (hb : b ≤ a ⊔ c) (hc : c ≤ a ⊔ b) : a ⊔ b = a ⊔ c :=
(sup_le le_sup_left hb).antisymm $ sup_le le_sup_left hc
#align sup_congr_left sup_congr_left
theorem sup_congr_right (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ⊔ c = b ⊔ c :=
(sup_le ha le_sup_right).antisymm $ sup_le hb le_sup_right
#align sup_congr_right sup_congr_right
theorem sup_eq_sup_iff_left : a ⊔ b = a ⊔ c ↔ b ≤ a ⊔ c ∧ c ≤ a ⊔ b :=
⟨fun h => ⟨h ▸ le_sup_right, h.symm ▸ le_sup_right⟩, fun h => sup_congr_left h.1 h.2⟩
#align sup_eq_sup_iff_left sup_eq_sup_iff_left
theorem sup_eq_sup_iff_right : a ⊔ c = b ⊔ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
⟨fun h => ⟨h ▸ le_sup_left, h.symm ▸ le_sup_left⟩, fun h => sup_congr_right h.1 h.2⟩
#align sup_eq_sup_iff_right sup_eq_sup_iff_right
theorem Ne.lt_sup_or_lt_sup (hab : a ≠ b) : a < a ⊔ b ∨ b < a ⊔ b :=
hab.symm.not_le_or_not_le.imp left_lt_sup.2 right_lt_sup.2
#align ne.lt_sup_or_lt_sup Ne.lt_sup_or_lt_sup
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem Monotone.forall_le_of_antitone {β : Type*} [Preorder β] {f g : α → β} (hf : Monotone f)
(hg : Antitone g) (h : f ≤ g) (m n : α) : f m ≤ g n :=
calc
f m ≤ f (m ⊔ n) := hf le_sup_left
_ ≤ g (m ⊔ n) := h _
_ ≤ g n := hg le_sup_right
#align monotone.forall_le_of_antitone Monotone.forall_le_of_antitone
theorem SemilatticeSup.ext_sup {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊔ y) = x ⊔ y :=
eq_of_forall_ge_iff $ fun c => by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
#align semilattice_sup.ext_sup SemilatticeSup.ext_sup
theorem SemilatticeSup.ext {α} {A B : SemilatticeSup α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_sup.ext SemilatticeSup.ext
theorem ite_le_sup (s s' : α) (P : Prop) [Decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
#align ite_le_sup ite_le_sup
end SemilatticeSup
/-!
### Meet-semilattices
-/
/-- A `SemilatticeInf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
/-- The infimum is a lower bound on the first argument -/
protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
/-- The infimum is a lower bound on the second argument -/
protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
/-- The infimum is the *greatest* lower bound -/
protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
#align semilattice_inf SemilatticeInf
instance OrderDual.semilatticeSup (α) [SemilatticeInf α] : SemilatticeSup αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Sup αᵒᵈ)
le_sup_left := @SemilatticeInf.inf_le_left α _
le_sup_right := @SemilatticeInf.inf_le_right α _
sup_le := fun _ _ _ hca hcb => @SemilatticeInf.le_inf α _ _ _ _ hca hcb
instance OrderDual.semilatticeInf (α) [SemilatticeSup α] : SemilatticeInf αᵒᵈ where
__ := inferInstanceAs (PartialOrder αᵒᵈ)
__ := inferInstanceAs (Inf αᵒᵈ)
inf_le_left := @le_sup_left α _
inf_le_right := @le_sup_right α _
le_inf := fun _ _ _ hca hcb => @sup_le α _ _ _ _ hca hcb
theorem SemilatticeSup.dual_dual (α : Type*) [H : SemilatticeSup α] :
OrderDual.semilatticeSup αᵒᵈ = H :=
SemilatticeSup.ext $ fun _ _ => Iff.rfl
#align semilattice_sup.dual_dual SemilatticeSup.dual_dual
section SemilatticeInf
variable [SemilatticeInf α] {a b c d : α}
@[simp]
theorem inf_le_left : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left inf_le_left
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_left' : a ⊓ b ≤ a :=
SemilatticeInf.inf_le_left a b
#align inf_le_left' inf_le_left'
@[simp]
theorem inf_le_right : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right inf_le_right
-- Porting note: no ematch attribute
--@[ematch]
theorem inf_le_right' : a ⊓ b ≤ b :=
SemilatticeInf.inf_le_right a b
#align inf_le_right' inf_le_right'
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
SemilatticeInf.le_inf a b c
#align le_inf le_inf
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
#align inf_le_of_left_le inf_le_of_left_le
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
#align inf_le_of_right_le inf_le_of_right_le
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
#align inf_lt_of_left_lt inf_lt_of_left_lt
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
#align inf_lt_of_right_lt inf_lt_of_right_lt
@[simp]
theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff αᵒᵈ _ _ _ _
#align le_inf_iff le_inf_iff
@[simp]
theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_left inf_eq_left
@[simp]
theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_rfl]
#align inf_eq_right inf_eq_right
@[simp]
theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
#align left_eq_inf left_eq_inf
@[simp]
theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
#align right_eq_inf right_eq_inf
alias ⟨le_of_inf_eq, inf_of_le_left⟩ := inf_eq_left
#align inf_of_le_left inf_of_le_left
#align le_of_inf_eq le_of_inf_eq
alias ⟨_, inf_of_le_right⟩ := inf_eq_right
#align inf_of_le_right inf_of_le_right
attribute [simp] inf_of_le_left inf_of_le_right
@[simp]
theorem inf_lt_left : a ⊓ b < a ↔ ¬a ≤ b :=
@left_lt_sup αᵒᵈ _ _ _
#align inf_lt_left inf_lt_left
@[simp]
theorem inf_lt_right : a ⊓ b < b ↔ ¬b ≤ a :=
@right_lt_sup αᵒᵈ _ _ _
#align inf_lt_right inf_lt_right
theorem inf_lt_left_or_right (h : a ≠ b) : a ⊓ b < a ∨ a ⊓ b < b :=
@left_or_right_lt_sup αᵒᵈ _ _ _ h
#align inf_lt_left_or_right inf_lt_left_or_right
@[gcongr]
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
@sup_le_sup αᵒᵈ _ _ _ _ _ h₁ h₂
#align inf_le_inf inf_le_inf
@[gcongr]
theorem inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
#align inf_le_inf_right inf_le_inf_right
@[gcongr]
theorem inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
#align inf_le_inf_left inf_le_inf_left
-- Porting note: was @[simp]
theorem inf_idem : a ⊓ a = a :=
@sup_idem αᵒᵈ _ _
#align inf_idem inf_idem
instance : IsIdempotent α (· ⊓ ·) :=
⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm αᵒᵈ _ _ _
#align inf_comm inf_comm
instance : IsCommutative α (· ⊓ ·) :=
⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc αᵒᵈ _ a b c
#align inf_assoc inf_assoc
instance : IsAssociative α (· ⊓ ·) :=
⟨@inf_assoc _ _⟩
theorem inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
@sup_left_right_swap αᵒᵈ _ _ _ _
#align inf_left_right_swap inf_left_right_swap
-- Porting note: was @[simp]
theorem inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem αᵒᵈ _ a b
#align inf_left_idem inf_left_idem
-- Porting note: was @[simp]
theorem inf_right_idem : a ⊓ b ⊓ b = a ⊓ b :=
@sup_right_idem αᵒᵈ _ a b
#align inf_right_idem inf_right_idem
theorem inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm αᵒᵈ _ a b c
#align inf_left_comm inf_left_comm
theorem inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm αᵒᵈ _ a b c
#align inf_right_comm inf_right_comm
theorem inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm αᵒᵈ _ _ _ _ _
#align inf_inf_inf_comm inf_inf_inf_comm
theorem inf_inf_distrib_left (a b c : α) : a ⊓ (b ⊓ c) = a ⊓ b ⊓ (a ⊓ c) :=
@sup_sup_distrib_left αᵒᵈ _ _ _ _
#align inf_inf_distrib_left inf_inf_distrib_left
theorem inf_inf_distrib_right (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ (b ⊓ c) :=
@sup_sup_distrib_right αᵒᵈ _ _ _ _
#align inf_inf_distrib_right inf_inf_distrib_right
theorem inf_congr_left (hb : a ⊓ c ≤ b) (hc : a ⊓ b ≤ c) : a ⊓ b = a ⊓ c :=
@sup_congr_left αᵒᵈ _ _ _ _ hb hc
#align inf_congr_left inf_congr_left
theorem inf_congr_right (h1 : b ⊓ c ≤ a) (h2 : a ⊓ c ≤ b) : a ⊓ c = b ⊓ c :=
@sup_congr_right αᵒᵈ _ _ _ _ h1 h2
#align inf_congr_right inf_congr_right
theorem inf_eq_inf_iff_left : a ⊓ b = a ⊓ c ↔ a ⊓ c ≤ b ∧ a ⊓ b ≤ c :=
@sup_eq_sup_iff_left αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_left inf_eq_inf_iff_left
theorem inf_eq_inf_iff_right : a ⊓ c = b ⊓ c ↔ b ⊓ c ≤ a ∧ a ⊓ c ≤ b :=
@sup_eq_sup_iff_right αᵒᵈ _ _ _ _
#align inf_eq_inf_iff_right inf_eq_inf_iff_right
theorem Ne.inf_lt_or_inf_lt : a ≠ b → a ⊓ b < a ∨ a ⊓ b < b :=
@Ne.lt_sup_or_lt_sup αᵒᵈ _ _ _
#align ne.inf_lt_or_inf_lt Ne.inf_lt_or_inf_lt
theorem SemilatticeInf.ext_inf {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y)
(x y : α) :
(haveI := A; x ⊓ y) = x ⊓ y :=
eq_of_forall_le_iff $ fun c => by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
#align semilattice_inf.ext_inf SemilatticeInf.ext_inf
theorem SemilatticeInf.ext {α} {A B : SemilatticeInf α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
have ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H
cases A
cases B
cases PartialOrder.ext H
congr
#align semilattice_inf.ext SemilatticeInf.ext
theorem SemilatticeInf.dual_dual (α : Type*) [H : SemilatticeInf α] :
OrderDual.semilatticeInf αᵒᵈ = H :=
SemilatticeInf.ext $ fun _ _ => Iff.rfl
#align semilattice_inf.dual_dual SemilatticeInf.dual_dual
theorem inf_le_ite (s s' : α) (P : Prop) [Decidable P] : s ⊓ s' ≤ ite P s s' :=
@ite_le_sup αᵒᵈ _ _ _ _ _
#align inf_le_ite inf_le_ite
end SemilatticeInf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def SemilatticeInf.mk' {α : Type*} [Inf α] (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a) :
SemilatticeInf α := by
haveI : SemilatticeSup αᵒᵈ := SemilatticeSup.mk' inf_comm inf_assoc inf_idem
haveI i := OrderDual.semilatticeInf αᵒᵈ
exact i
#align semilattice_inf.mk' SemilatticeInf.mk'
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
#align lattice Lattice
instance OrderDual.lattice (α) [Lattice α] : Lattice αᵒᵈ :=
{ OrderDual.semilatticeSup α, OrderDual.semilatticeInf α with }
/-- The partial orders from `SemilatticeSup_mk'` and `SemilatticeInf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
theorem semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
{α : Type*} [Sup α] [Inf α]
(sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a) (sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ a : α, a ⊔ a = a) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (inf_idem : ∀ a : α, a ⊓ a = a)
(sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a) (inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) :
@SemilatticeSup.toPartialOrder _ (SemilatticeSup.mk' sup_comm sup_assoc sup_idem) =
@SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
PartialOrder.ext $ fun a b =>
show a ⊔ b = b ↔ b ⊓ a = a from
⟨fun h => by rw [← h, inf_comm, inf_sup_self], fun h => by rw [← h, sup_comm, sup_inf_self]⟩
#align semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder
/-- A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b = b ⊔ a)
(sup_assoc : ∀ a b c : α, a ⊔ b ⊔ c = a ⊔ (b ⊔ c)) (inf_comm : ∀ a b : α, a ⊓ b = b ⊓ a)
(inf_assoc : ∀ a b c : α, a ⊓ b ⊓ c = a ⊓ (b ⊓ c)) (sup_inf_self : ∀ a b : α, a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ a b : α, a ⊓ (a ⊔ b) = a) : Lattice α :=
have sup_idem : ∀ b : α, b ⊔ b = b := fun b =>
calc
b ⊔ b = b ⊔ b ⊓ (b ⊔ b) := by rw [inf_sup_self]
_ = b := by rw [sup_inf_self]
have inf_idem : ∀ b : α, b ⊓ b = b := fun b =>
calc
b ⊓ b = b ⊓ (b ⊔ b ⊓ b) := by rw [sup_inf_self]
_ = b := by rw [inf_sup_self]
let semilatt_inf_inst := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
let semilatt_sup_inst := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
have partial_order_eq : @SemilatticeSup.toPartialOrder _ semilatt_sup_inst =
@SemilatticeInf.toPartialOrder _ semilatt_inf_inst :=
semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder _ _ _ _ _ _
sup_inf_self inf_sup_self
{ semilatt_sup_inst, semilatt_inf_inst with
inf_le_left := fun a b => by
rw [partial_order_eq]
apply inf_le_left,
inf_le_right := fun a b => by
rw [partial_order_eq]
apply inf_le_right,
le_inf := fun a b c => by
rw [partial_order_eq]
apply le_inf }
#align lattice.mk' Lattice.mk'
section Lattice
variable [Lattice α] {a b c d : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
#align inf_le_sup inf_le_sup
-- Porting note: was @[simp]
theorem sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
#align sup_le_inf sup_le_inf
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
#align inf_eq_sup inf_eq_sup
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
#align sup_eq_inf sup_eq_inf
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]
#align inf_lt_sup inf_lt_sup
lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine' ⟨fun h ↦ _, _⟩
{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h }
{ rintro ⟨rfl, rfl⟩
exact ⟨inf_idem, sup_idem⟩ }
#align inf_eq_and_sup_eq_iff inf_eq_and_sup_eq_iff
/-!
#### Distributivity laws
-/
-- TODO: better names?
theorem sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
#align sup_inf_le sup_inf_le
theorem le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
#align le_inf_sup le_inf_sup
theorem inf_sup_self : a ⊓ (a ⊔ b) = a := by simp
#align inf_sup_self inf_sup_self
theorem sup_inf_self : a ⊔ a ⊓ b = a := by simp
#align sup_inf_self sup_inf_self
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left]
#align sup_eq_iff_inf_eq sup_eq_iff_inf_eq
theorem Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr
#align lattice.ext Lattice.ext
end Lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
#align distrib_lattice DistribLattice
section DistribLattice
variable [DistribLattice α] {x y z : α}
theorem le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z
#align le_sup_inf le_sup_inf
theorem sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
#align sup_inf_left sup_inf_left
theorem sup_inf_right : y ⊓ z ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) := by
simp only [sup_inf_left, fun y : α => @sup_comm α _ y x, eq_self_iff_true]
#align sup_inf_right sup_inf_right
theorem inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z :=
calc
x ⊓ (y ⊔ z) = x ⊓ (x ⊔ z) ⊓ (y ⊔ z) := by rw [inf_sup_self]
_ = x ⊓ (x ⊓ y ⊔ z) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (x ⊔ x ⊓ y) ⊓ (x ⊓ y ⊔ z) := by rw [sup_inf_self]
_ = (x ⊓ y ⊔ x) ⊓ (x ⊓ y ⊔ z) := by rw [sup_comm]
_ = x ⊓ y ⊔ x ⊓ z := by rw [sup_inf_left]
#align inf_sup_left inf_sup_left
instance OrderDual.distribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
__ := inferInstanceAs (Lattice αᵒᵈ)
le_sup_inf := fun _ _ _ => le_of_eq (@inf_sup_left α).symm
theorem inf_sup_right : (y ⊔ z) ⊓ x = y ⊓ x ⊔ z ⊓ x := by
simp only [inf_sup_left, fun y : α => @inf_comm α _ y x, eq_self_iff_true]
#align inf_sup_right inf_sup_right
theorem le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, @sup_comm _ _ x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := sup_inf_left.symm
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤ _ := sup_le (le_refl y) inf_le_left
#align le_of_inf_le_sup_le le_of_inf_le_sup_le
theorem eq_of_inf_eq_sup_eq {α : Type u} [DistribLattice α] {a b c : α} (h₁ : b ⊓ a = c ⊓ a)
(h₂ : b ⊔ a = c ⊔ a) :
b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
#align eq_of_inf_eq_sup_eq eq_of_inf_eq_sup_eq
end DistribLattice
-- See note [reducible non-instances]
/-- Prove distributivity of an existing lattice from the dual distributive law. -/
@[reducible]
def DistribLattice.ofInfSupLe [Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) :
DistribLattice α :=
{ le_sup_inf := (@OrderDual.distribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf, }
#align distrib_lattice.of_inf_sup_le DistribLattice.ofInfSupLe
/-!
### Lattices derived from linear orders
-/
-- see Note [lower instance priority]
instance (priority := 100) LinearOrder.toLattice {α : Type u} [o : LinearOrder α] : Lattice α :=
{ o with
sup := max,
le_sup_left := le_max_left, le_sup_right := le_max_right, sup_le := fun _ _ _ => max_le,
inf := min,
inf_le_left := min_le_left, inf_le_right := min_le_right, le_inf := fun _ _ _ => le_min }
section LinearOrder
variable [LinearOrder α] {a b c d : α}
theorem sup_eq_max : a ⊔ b = max a b :=
rfl
#align sup_eq_max sup_eq_max
theorem inf_eq_min : a ⊓ b = min a b :=
rfl
#align inf_eq_min inf_eq_min
theorem sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
#align sup_ind sup_ind
@[simp]
theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
#align le_sup_iff le_sup_iff
@[simp]
theorem lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
#align lt_sup_iff lt_sup_iff
-- Porting note: why does sup_ind need an explicit motive?
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => @sup_ind α _ b c (fun x => x < a) h.1 h.2⟩
#align sup_lt_iff sup_lt_iff
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
#align inf_ind inf_ind
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
#align inf_le_iff inf_le_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
#align inf_lt_iff inf_lt_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
#align lt_inf_iff lt_inf_iff
variable (a b c d)
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _
#align max_max_max_comm max_max_max_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _
#align min_min_min_comm min_min_min_comm
end LinearOrder
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left $ (total_of (· ≤ ·) x y).resolve_left h']
#align sup_eq_max_default sup_eq_maxDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
[IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right $ (total_of (· ≤ ·) x y).resolve_left h']
#align inf_eq_min_default inf_eq_minDefault
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible]
def Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableRel ((· ≤ ·) : α → α → Prop)]
[DecidableRel ((· < ·) : α → α → Prop)] [IsTotal α (· ≤ ·)] : LinearOrder α :=
{ ‹Lattice α› with
decidableLE := ‹_›,
decidableEq := ‹_›,
decidableLT := ‹_›,
le_total := total_of (· ≤ ·),
max := (· ⊔ ·),
max_def := by exact congr_fun₂ sup_eq_maxDefault,
min := (· ⊓ ·),
min_def := by exact congr_fun₂ inf_eq_minDefault }
#align lattice.to_linear_order Lattice.toLinearOrder
-- see Note [lower instance priority]
instance (priority := 100) {α : Type u} [LinearOrder α] :
DistribLattice α where
__ := inferInstanceAs (Lattice α)
le_sup_inf _ b c :=
match le_total b c with
| Or.inl h => inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| Or.inr h => inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
instance : DistribLattice ℕ := inferInstance
/-! ### Dual order -/
open OrderDual
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
#align of_dual_inf ofDual_inf
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
#align of_dual_sup ofDual_sup
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
#align to_dual_inf toDual_inf
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl
#align to_dual_sup toDual_sup
section LinearOrder
variable [LinearOrder α]
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
#align of_dual_min ofDual_min
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
#align of_dual_max ofDual_max
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
#align to_dual_min toDual_min
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl
#align to_dual_max toDual_max
end LinearOrder
/-! ### Function lattices -/
namespace Pi
variable {ι : Type*} {α' : ι → Type*}
instance [∀ i, Sup (α' i)] : Sup (∀ i, α' i) :=
⟨fun f g i => f i ⊔ g i⟩
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
#align pi.sup_apply Pi.sup_apply
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
#align pi.sup_def Pi.sup_def
instance [∀ i, Inf (α' i)] : Inf (∀ i, α' i) :=
⟨fun f g i => f i ⊓ g i⟩
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
#align pi.inf_apply Pi.inf_apply
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl
#align pi.inf_def Pi.inf_def
instance semilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i)
instance semilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i)
instance lattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where
__ := inferInstanceAs (SemilatticeSup (∀ i, α' i))
__ := inferInstanceAs (SemilatticeInf (∀ i, α' i))
instance distribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
le_sup_inf _ _ _ _ := le_sup_inf
end Pi
namespace Function
variable {ι : Type*} {π : ι → Type*} [DecidableEq ι]
-- porting note: Dot notation on `Function.update` broke
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_sup Function.update_sup
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_noteq, *]
#align function.update_inf Function.update_inf
end Function
/-!
### Monotone functions and lattices
-/
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align monotone.sup Monotone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align monotone.inf Monotone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg
#align monotone.max Monotone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg
#align monotone.min Monotone.min
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
#align monotone.le_map_sup Monotone.le_map_sup
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
#align monotone.map_inf_le Monotone.map_inf_le
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
fun x y hxy => inf_eq_left.1 $ by rw [← h, inf_eq_left.2 hxy]
#align monotone.of_map_inf Monotone.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
#align monotone.of_map_sup Monotone.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left]
#align monotone.map_sup Monotone.map_sup
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _
#align monotone.map_inf Monotone.map_inf
end Monotone
namespace MonotoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align monotone_on.sup MonotoneOn.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align monotone_on.inf MonotoneOn.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align monotone_on.max MonotoneOn.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align monotone_on.min MonotoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align monotone_on.of_map_inf MonotoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align monotone_on.of_map_sup MonotoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right]
#align monotone_on.map_sup MonotoneOn.map_sup
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy
#align monotone_on.map_inf MonotoneOn.map_inf
end MonotoneOn
namespace Antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
#align antitone.sup Antitone.sup
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h)
#align antitone.inf Antitone.inf
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg
#align antitone.max Antitone.max
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg
#align antitone.min Antitone.min
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
#align antitone.map_sup_le Antitone.map_sup_le
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
#align antitone.le_map_inf Antitone.le_map_inf
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
#align antitone.map_sup Antitone.map_sup
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
#align antitone.map_inf Antitone.map_inf
end Antitone
namespace AntitoneOn
variable {f : α → β} {s : Set α} {x y : α}
/-- Pointwise supremum of two antitone functions is an antitone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h)
#align antitone_on.sup AntitoneOn.sup
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
#align antitone_on.inf AntitoneOn.inf
/-- Pointwise maximum of two antitone functions is an antitone function. -/
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg
#align antitone_on.max AntitoneOn.max
/-- Pointwise minimum of two antitone functions is an antitone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg
#align antitone_on.min AntitoneOn.min
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy]
#align antitone_on.of_map_inf AntitoneOn.of_map_inf
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
#align antitone_on.of_map_sup AntitoneOn.of_map_sup
variable [LinearOrder α]
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right]
#align antitone_on.map_sup AntitoneOn.map_sup
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy
#align antitone_on.map_inf AntitoneOn.map_inf
end AntitoneOn
/-!
### Products of (semi-)lattices
-/
namespace Prod
variable (α β)
instance [Sup α] [Sup β] : Sup (α × β) :=
⟨fun p q => ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [Inf α] [Inf β] : Inf (α × β) :=
⟨fun p q => ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
rfl
#align prod.mk_sup_mk Prod.mk_sup_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
#align prod.mk_inf_mk Prod.mk_inf_mk
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
#align prod.fst_sup Prod.fst_sup
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
#align prod.fst_inf Prod.fst_inf
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
#align prod.snd_sup Prod.snd_sup
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
#align prod.snd_inf Prod.snd_inf
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
#align prod.swap_sup Prod.swap_sup
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl
#align prod.swap_inf Prod.swap_inf
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl
#align prod.sup_def Prod.sup_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
#align prod.inf_def Prod.inf_def
instance semilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Sup (α × β))
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩
instance semilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
__ := inferInstanceAs (PartialOrder (α × β))
__ := inferInstanceAs (Inf (α × β))
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩
instance lattice [Lattice α] [Lattice β] : Lattice (α × β) where
__ := inferInstanceAs (SemilatticeSup (α × β))
__ := inferInstanceAs (SemilatticeInf (α × β))
instance distribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
__ := inferInstanceAs (Lattice (α × β))
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩
end Prod
/-!
### Subtypes of (semi-)lattices
-/
namespace Subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
sup := fun x y => ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := fun _ _ => le_sup_left,
le_sup_right := fun _ _ => le_sup_right,
sup_le := fun _ _ _ h1 h2 => sup_le h1 h2 }
#align subtype.semilattice_sup Subtype.semilatticeSup
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } :=
{ inferInstanceAs (PartialOrder (Subtype P)) with
inf := fun x y => ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := fun _ _ => inf_le_left,
inf_le_right := fun _ _ => inf_le_right,
le_inf := fun _ _ _ h1 h2 => le_inf h1 h2 }
#align subtype.semilattice_inf Subtype.semilatticeInf
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } :=
{ Subtype.semilatticeInf Pinf, Subtype.semilatticeSup Psup with }
#align subtype.lattice Subtype.lattice
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
#align subtype.coe_sup Subtype.coe_sup
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
#align subtype.coe_inf Subtype.coe_inf
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
#align subtype.mk_sup_mk Subtype.mk_sup_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl
#align subtype.mk_inf_mk Subtype.mk_inf_mk
end Subtype
section lift
/-- A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeSup [Sup α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) : SemilatticeSup α :=
{ PartialOrder.lift f hf_inj with
sup := Sup.sup,
le_sup_left := fun a b => by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_left,
le_sup_right := fun a b => by
change f b ≤ f (a ⊔ b)
rw [map_sup]
exact le_sup_right,
sup_le := fun a b c ha hb => by
change f (a ⊔ b) ≤ f c
rw [map_sup]
exact sup_le ha hb }
#align function.injective.semilattice_sup Function.Injective.semilatticeSup
/-- A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.semilatticeInf [Inf α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : SemilatticeInf α :=
{ PartialOrder.lift f hf_inj with
inf := Inf.inf,
inf_le_left := fun a b => by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exact inf_le_left,
inf_le_right := fun a b => by
change f (a ⊓ b) ≤ f b
rw [map_inf]
exact inf_le_right,
le_inf := fun a b c ha hb => by
change f a ≤ f (b ⊓ c)
rw [map_inf]
exact le_inf ha hb }
#align function.injective.semilattice_inf Function.Injective.semilatticeInf
/-- A type endowed with `⊔` and `⊓` is a `Lattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `Lattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.lattice [Sup α] [Inf α] [Lattice β] (f : α → β)
(hf_inj : Function.Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) : Lattice α :=
{ hf_inj.semilatticeSup f map_sup, hf_inj.semilatticeInf f map_inf with }
#align function.injective.lattice Function.Injective.lattice
/-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
DistribLattice α :=
{ hf_inj.lattice f map_sup map_inf with
le_sup_inf := fun a b c => by
change f ((a ⊔ b) ⊓ (a ⊔ c)) ≤ f (a ⊔ b ⊓ c)
rw [map_inf, map_sup, map_sup, map_sup, map_inf]
|
exact le_sup_inf
|
/-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
DistribLattice α :=
{ hf_inj.lattice f map_sup map_inf with
le_sup_inf := fun a b c => by
change f ((a ⊔ b) ⊓ (a ⊔ c)) ≤ f (a ⊔ b ⊓ c)
rw [map_inf, map_sup, map_sup, map_sup, map_inf]
|
Mathlib.Order.Lattice.1502_0.wE3igZl9MFbJBfv
|
/-- A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. -/
@[reducible]
protected def Function.Injective.distribLattice [Sup α] [Inf α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
DistribLattice α
|
Mathlib_Order_Lattice
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
g : σ → τ
p : MvPolynomial σ R
⊢ (map f) ((rename g) p) = (rename g) ((map f) p)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
|
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
|
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
|
Mathlib.Data.MvPolynomial.Rename.72_0.3NqVCwOs1E93kvK
|
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p)
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
g : σ → τ
p : MvPolynomial σ R
a : R
⊢ (map f) ((rename g) (C a)) = (rename g) ((map f) (C a))
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by
|
simp only [map_C, rename_C]
|
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by
|
Mathlib.Data.MvPolynomial.Rename.72_0.3NqVCwOs1E93kvK
|
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p)
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
g : σ → τ
p✝ p q : MvPolynomial σ R
hp : (map f) ((rename g) p) = (rename g) ((map f) p)
hq : (map f) ((rename g) q) = (rename g) ((map f) q)
⊢ (map f) ((rename g) (p + q)) = (rename g) ((map f) (p + q))
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by
|
simp only [hp, hq, AlgHom.map_add, RingHom.map_add]
|
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by
|
Mathlib.Data.MvPolynomial.Rename.72_0.3NqVCwOs1E93kvK
|
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p)
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
g : σ → τ
p✝ p : MvPolynomial σ R
n : σ
hp : (map f) ((rename g) p) = (rename g) ((map f) p)
⊢ (map f) ((rename g) (p * X n)) = (rename g) ((map f) (p * X n))
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
|
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
|
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
|
Mathlib.Data.MvPolynomial.Rename.72_0.3NqVCwOs1E93kvK
|
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p)
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
g : τ → α
p : MvPolynomial σ R
⊢ (rename g) (eval₂ C (X ∘ f) p) = (rename (g ∘ f)) p
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
|
simp only [rename, aeval_eq_eval₂Hom]
|
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
|
Mathlib.Data.MvPolynomial.Rename.80_0.3NqVCwOs1E93kvK
|
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
g : τ → α
p : MvPolynomial σ R
⊢ (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) (eval₂ C (X ∘ f) p) =
(eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g ∘ f)) p
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
|
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
|
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
|
Mathlib.Data.MvPolynomial.Rename.80_0.3NqVCwOs1E93kvK
|
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
g : τ → α
p : MvPolynomial σ R
⊢ eval₂ (RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C)
(⇑(eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) ∘ X ∘ f) p =
(eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g ∘ f)) p
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
|
simp only [(· ∘ ·), eval₂Hom_X']
|
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
|
Mathlib.Data.MvPolynomial.Rename.80_0.3NqVCwOs1E93kvK
|
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
g : τ → α
p : MvPolynomial σ R
⊢ eval₂ (RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) fun x => X (g x)) C) (fun x => X (g (f x))) p =
(eval₂Hom (algebraMap R (MvPolynomial α R)) fun x => X (g (f x))) p
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
|
refine' eval₂Hom_congr _ rfl rfl
|
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
|
Mathlib.Data.MvPolynomial.Rename.80_0.3NqVCwOs1E93kvK
|
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
g : τ → α
p : MvPolynomial σ R
⊢ RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) fun x => X (g x)) C = algebraMap R (MvPolynomial α R)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
|
ext1
|
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
|
Mathlib.Data.MvPolynomial.Rename.80_0.3NqVCwOs1E93kvK
|
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p
|
Mathlib_Data_MvPolynomial_Rename
|
case a
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
g : τ → α
p : MvPolynomial σ R
x✝ : R
⊢ (RingHom.comp (eval₂Hom (algebraMap R (MvPolynomial α R)) fun x => X (g x)) C) x✝ =
(algebraMap R (MvPolynomial α R)) x✝
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1;
|
simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
|
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1;
|
Mathlib.Data.MvPolynomial.Rename.80_0.3NqVCwOs1E93kvK
|
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
d : σ →₀ ℕ
r : R
⊢ (rename f) ((monomial d) r) = (monomial (Finsupp.mapDomain f d)) r
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
|
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
|
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
|
Mathlib.Data.MvPolynomial.Rename.98_0.3NqVCwOs1E93kvK
|
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
d : σ →₀ ℕ
r : R
⊢ ((algebraMap R (MvPolynomial τ R)) r * Finsupp.prod d fun i k => (X ∘ f) i ^ k) =
C r * Finsupp.prod d fun a m => X (f a) ^ m
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
·
|
rfl
|
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
·
|
Mathlib.Data.MvPolynomial.Rename.98_0.3NqVCwOs1E93kvK
|
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r
|
Mathlib_Data_MvPolynomial_Rename
|
case h_zero
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
d : σ →₀ ℕ
r : R
⊢ ∀ (b : τ), X b ^ 0 = 1
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
·
|
exact fun n => pow_zero _
|
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
·
|
Mathlib.Data.MvPolynomial.Rename.98_0.3NqVCwOs1E93kvK
|
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r
|
Mathlib_Data_MvPolynomial_Rename
|
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
d : σ →₀ ℕ
r : R
⊢ ∀ (b : τ) (m₁ m₂ : ℕ), X b ^ (m₁ + m₂) = X b ^ m₁ * X b ^ m₂
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
·
|
exact fun n i₁ i₂ => pow_add _ _ _
|
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
·
|
Mathlib.Data.MvPolynomial.Rename.98_0.3NqVCwOs1E93kvK
|
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
p : MvPolynomial σ R
⊢ (rename f) p = Finsupp.mapDomain (Finsupp.mapDomain f) p
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
|
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
|
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
|
Mathlib.Data.MvPolynomial.Rename.107_0.3NqVCwOs1E93kvK
|
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
p : MvPolynomial σ R
⊢ (sum p fun s a => (monomial (sum s fun a b => fun₀ | f a => b)) a) =
sum p fun a => Finsupp.single (sum a fun a => Finsupp.single (f a))
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
|
rfl
|
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
|
Mathlib.Data.MvPolynomial.Rename.107_0.3NqVCwOs1E93kvK
|
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
⊢ Injective ⇑(rename f)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
|
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
|
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
|
Mathlib.Data.MvPolynomial.Rename.114_0.3NqVCwOs1E93kvK
|
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R)
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
this : ⇑(rename f) = Finsupp.mapDomain (Finsupp.mapDomain f)
⊢ Injective ⇑(rename f)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
|
rw [this]
|
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
|
Mathlib.Data.MvPolynomial.Rename.114_0.3NqVCwOs1E93kvK
|
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R)
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
this : ⇑(rename f) = Finsupp.mapDomain (Finsupp.mapDomain f)
⊢ Injective (Finsupp.mapDomain (Finsupp.mapDomain f))
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
|
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
|
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
|
Mathlib.Data.MvPolynomial.Rename.114_0.3NqVCwOs1E93kvK
|
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R)
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
i : σ
⊢ (AlgHom.comp (killCompl hf) (rename f)) (X i) = (AlgHom.id R (MvPolynomial σ R)) (X i)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
|
dsimp
|
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
|
Mathlib.Data.MvPolynomial.Rename.136_0.3NqVCwOs1E93kvK
|
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
i : σ
⊢ (killCompl hf) ((rename f) (X i)) = X i
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
|
rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply]
|
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
|
Mathlib.Data.MvPolynomial.Rename.136_0.3NqVCwOs1E93kvK
|
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ ≃ τ
src✝ : MvPolynomial σ R →ₐ[R] MvPolynomial τ R := rename ⇑f
p : MvPolynomial σ R
⊢ (rename ⇑f.symm) ((rename ⇑f) p) = p
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply]
#align mv_polynomial.kill_compl_comp_rename MvPolynomial.killCompl_comp_rename
@[simp]
theorem killCompl_rename_app (p : MvPolynomial σ R) : killCompl hf (rename f p) = p :=
AlgHom.congr_fun (killCompl_comp_rename hf) p
#align mv_polynomial.kill_compl_rename_app MvPolynomial.killCompl_rename_app
end
section
variable (R)
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by
|
rw [rename_rename, f.symm_comp_self, rename_id]
|
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by
|
Mathlib.Data.MvPolynomial.Rename.153_0.3NqVCwOs1E93kvK
|
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ ≃ τ
src✝ : MvPolynomial σ R →ₐ[R] MvPolynomial τ R := rename ⇑f
p : MvPolynomial τ R
⊢ (rename ⇑f) ((rename ⇑f.symm) p) = p
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply]
#align mv_polynomial.kill_compl_comp_rename MvPolynomial.killCompl_comp_rename
@[simp]
theorem killCompl_rename_app (p : MvPolynomial σ R) : killCompl hf (rename f p) = p :=
AlgHom.congr_fun (killCompl_comp_rename hf) p
#align mv_polynomial.kill_compl_rename_app MvPolynomial.killCompl_rename_app
end
section
variable (R)
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by rw [rename_rename, f.symm_comp_self, rename_id]
right_inv := fun p => by
|
rw [rename_rename, f.self_comp_symm, rename_id]
|
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by rw [rename_rename, f.symm_comp_self, rename_id]
right_inv := fun p => by
|
Mathlib.Data.MvPolynomial.Rename.153_0.3NqVCwOs1E93kvK
|
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p : MvPolynomial σ R
⊢ eval₂ f g ((rename k) p) = eval₂ f (g ∘ k) p
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply]
#align mv_polynomial.kill_compl_comp_rename MvPolynomial.killCompl_comp_rename
@[simp]
theorem killCompl_rename_app (p : MvPolynomial σ R) : killCompl hf (rename f p) = p :=
AlgHom.congr_fun (killCompl_comp_rename hf) p
#align mv_polynomial.kill_compl_rename_app MvPolynomial.killCompl_rename_app
end
section
variable (R)
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by rw [rename_rename, f.symm_comp_self, rename_id]
right_inv := fun p => by rw [rename_rename, f.self_comp_symm, rename_id] }
#align mv_polynomial.rename_equiv MvPolynomial.renameEquiv
@[simp]
theorem renameEquiv_refl : renameEquiv R (Equiv.refl σ) = AlgEquiv.refl :=
AlgEquiv.ext rename_id
#align mv_polynomial.rename_equiv_refl MvPolynomial.renameEquiv_refl
@[simp]
theorem renameEquiv_symm (f : σ ≃ τ) : (renameEquiv R f).symm = renameEquiv R f.symm :=
rfl
#align mv_polynomial.rename_equiv_symm MvPolynomial.renameEquiv_symm
@[simp]
theorem renameEquiv_trans (e : σ ≃ τ) (f : τ ≃ α) :
(renameEquiv R e).trans (renameEquiv R f) = renameEquiv R (e.trans f) :=
AlgEquiv.ext (rename_rename e f)
#align mv_polynomial.rename_equiv_trans MvPolynomial.renameEquiv_trans
end
section
variable (f : R →+* S) (k : σ → τ) (g : τ → S) (p : MvPolynomial σ R)
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
|
apply MvPolynomial.induction_on p
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
|
Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k)
|
Mathlib_Data_MvPolynomial_Rename
|
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p : MvPolynomial σ R
⊢ ∀ (a : R), eval₂ f g ((rename k) (C a)) = eval₂ f (g ∘ k) (C a)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply]
#align mv_polynomial.kill_compl_comp_rename MvPolynomial.killCompl_comp_rename
@[simp]
theorem killCompl_rename_app (p : MvPolynomial σ R) : killCompl hf (rename f p) = p :=
AlgHom.congr_fun (killCompl_comp_rename hf) p
#align mv_polynomial.kill_compl_rename_app MvPolynomial.killCompl_rename_app
end
section
variable (R)
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by rw [rename_rename, f.symm_comp_self, rename_id]
right_inv := fun p => by rw [rename_rename, f.self_comp_symm, rename_id] }
#align mv_polynomial.rename_equiv MvPolynomial.renameEquiv
@[simp]
theorem renameEquiv_refl : renameEquiv R (Equiv.refl σ) = AlgEquiv.refl :=
AlgEquiv.ext rename_id
#align mv_polynomial.rename_equiv_refl MvPolynomial.renameEquiv_refl
@[simp]
theorem renameEquiv_symm (f : σ ≃ τ) : (renameEquiv R f).symm = renameEquiv R f.symm :=
rfl
#align mv_polynomial.rename_equiv_symm MvPolynomial.renameEquiv_symm
@[simp]
theorem renameEquiv_trans (e : σ ≃ τ) (f : τ ≃ α) :
(renameEquiv R e).trans (renameEquiv R f) = renameEquiv R (e.trans f) :=
AlgEquiv.ext (rename_rename e f)
#align mv_polynomial.rename_equiv_trans MvPolynomial.renameEquiv_trans
end
section
variable (f : R →+* S) (k : σ → τ) (g : τ → S) (p : MvPolynomial σ R)
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
·
|
intros
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
·
|
Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k)
|
Mathlib_Data_MvPolynomial_Rename
|
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p : MvPolynomial σ R
a✝ : R
⊢ eval₂ f g ((rename k) (C a✝)) = eval₂ f (g ∘ k) (C a✝)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply]
#align mv_polynomial.kill_compl_comp_rename MvPolynomial.killCompl_comp_rename
@[simp]
theorem killCompl_rename_app (p : MvPolynomial σ R) : killCompl hf (rename f p) = p :=
AlgHom.congr_fun (killCompl_comp_rename hf) p
#align mv_polynomial.kill_compl_rename_app MvPolynomial.killCompl_rename_app
end
section
variable (R)
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by rw [rename_rename, f.symm_comp_self, rename_id]
right_inv := fun p => by rw [rename_rename, f.self_comp_symm, rename_id] }
#align mv_polynomial.rename_equiv MvPolynomial.renameEquiv
@[simp]
theorem renameEquiv_refl : renameEquiv R (Equiv.refl σ) = AlgEquiv.refl :=
AlgEquiv.ext rename_id
#align mv_polynomial.rename_equiv_refl MvPolynomial.renameEquiv_refl
@[simp]
theorem renameEquiv_symm (f : σ ≃ τ) : (renameEquiv R f).symm = renameEquiv R f.symm :=
rfl
#align mv_polynomial.rename_equiv_symm MvPolynomial.renameEquiv_symm
@[simp]
theorem renameEquiv_trans (e : σ ≃ τ) (f : τ ≃ α) :
(renameEquiv R e).trans (renameEquiv R f) = renameEquiv R (e.trans f) :=
AlgEquiv.ext (rename_rename e f)
#align mv_polynomial.rename_equiv_trans MvPolynomial.renameEquiv_trans
end
section
variable (f : R →+* S) (k : σ → τ) (g : τ → S) (p : MvPolynomial σ R)
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· intros
|
simp [*]
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· intros
|
Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k)
|
Mathlib_Data_MvPolynomial_Rename
|
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p : MvPolynomial σ R
⊢ ∀ (p q : MvPolynomial σ R),
eval₂ f g ((rename k) p) = eval₂ f (g ∘ k) p →
eval₂ f g ((rename k) q) = eval₂ f (g ∘ k) q → eval₂ f g ((rename k) (p + q)) = eval₂ f (g ∘ k) (p + q)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply]
#align mv_polynomial.kill_compl_comp_rename MvPolynomial.killCompl_comp_rename
@[simp]
theorem killCompl_rename_app (p : MvPolynomial σ R) : killCompl hf (rename f p) = p :=
AlgHom.congr_fun (killCompl_comp_rename hf) p
#align mv_polynomial.kill_compl_rename_app MvPolynomial.killCompl_rename_app
end
section
variable (R)
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by rw [rename_rename, f.symm_comp_self, rename_id]
right_inv := fun p => by rw [rename_rename, f.self_comp_symm, rename_id] }
#align mv_polynomial.rename_equiv MvPolynomial.renameEquiv
@[simp]
theorem renameEquiv_refl : renameEquiv R (Equiv.refl σ) = AlgEquiv.refl :=
AlgEquiv.ext rename_id
#align mv_polynomial.rename_equiv_refl MvPolynomial.renameEquiv_refl
@[simp]
theorem renameEquiv_symm (f : σ ≃ τ) : (renameEquiv R f).symm = renameEquiv R f.symm :=
rfl
#align mv_polynomial.rename_equiv_symm MvPolynomial.renameEquiv_symm
@[simp]
theorem renameEquiv_trans (e : σ ≃ τ) (f : τ ≃ α) :
(renameEquiv R e).trans (renameEquiv R f) = renameEquiv R (e.trans f) :=
AlgEquiv.ext (rename_rename e f)
#align mv_polynomial.rename_equiv_trans MvPolynomial.renameEquiv_trans
end
section
variable (f : R →+* S) (k : σ → τ) (g : τ → S) (p : MvPolynomial σ R)
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
·
|
intros
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
·
|
Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k)
|
Mathlib_Data_MvPolynomial_Rename
|
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p p✝ q✝ : MvPolynomial σ R
a✝¹ : eval₂ f g ((rename k) p✝) = eval₂ f (g ∘ k) p✝
a✝ : eval₂ f g ((rename k) q✝) = eval₂ f (g ∘ k) q✝
⊢ eval₂ f g ((rename k) (p✝ + q✝)) = eval₂ f (g ∘ k) (p✝ + q✝)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply]
#align mv_polynomial.kill_compl_comp_rename MvPolynomial.killCompl_comp_rename
@[simp]
theorem killCompl_rename_app (p : MvPolynomial σ R) : killCompl hf (rename f p) = p :=
AlgHom.congr_fun (killCompl_comp_rename hf) p
#align mv_polynomial.kill_compl_rename_app MvPolynomial.killCompl_rename_app
end
section
variable (R)
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by rw [rename_rename, f.symm_comp_self, rename_id]
right_inv := fun p => by rw [rename_rename, f.self_comp_symm, rename_id] }
#align mv_polynomial.rename_equiv MvPolynomial.renameEquiv
@[simp]
theorem renameEquiv_refl : renameEquiv R (Equiv.refl σ) = AlgEquiv.refl :=
AlgEquiv.ext rename_id
#align mv_polynomial.rename_equiv_refl MvPolynomial.renameEquiv_refl
@[simp]
theorem renameEquiv_symm (f : σ ≃ τ) : (renameEquiv R f).symm = renameEquiv R f.symm :=
rfl
#align mv_polynomial.rename_equiv_symm MvPolynomial.renameEquiv_symm
@[simp]
theorem renameEquiv_trans (e : σ ≃ τ) (f : τ ≃ α) :
(renameEquiv R e).trans (renameEquiv R f) = renameEquiv R (e.trans f) :=
AlgEquiv.ext (rename_rename e f)
#align mv_polynomial.rename_equiv_trans MvPolynomial.renameEquiv_trans
end
section
variable (f : R →+* S) (k : σ → τ) (g : τ → S) (p : MvPolynomial σ R)
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· intros
|
simp [*]
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· intros
|
Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k)
|
Mathlib_Data_MvPolynomial_Rename
|
case h_X
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p : MvPolynomial σ R
⊢ ∀ (p : MvPolynomial σ R) (n : σ),
eval₂ f g ((rename k) p) = eval₂ f (g ∘ k) p → eval₂ f g ((rename k) (p * X n)) = eval₂ f (g ∘ k) (p * X n)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply]
#align mv_polynomial.kill_compl_comp_rename MvPolynomial.killCompl_comp_rename
@[simp]
theorem killCompl_rename_app (p : MvPolynomial σ R) : killCompl hf (rename f p) = p :=
AlgHom.congr_fun (killCompl_comp_rename hf) p
#align mv_polynomial.kill_compl_rename_app MvPolynomial.killCompl_rename_app
end
section
variable (R)
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by rw [rename_rename, f.symm_comp_self, rename_id]
right_inv := fun p => by rw [rename_rename, f.self_comp_symm, rename_id] }
#align mv_polynomial.rename_equiv MvPolynomial.renameEquiv
@[simp]
theorem renameEquiv_refl : renameEquiv R (Equiv.refl σ) = AlgEquiv.refl :=
AlgEquiv.ext rename_id
#align mv_polynomial.rename_equiv_refl MvPolynomial.renameEquiv_refl
@[simp]
theorem renameEquiv_symm (f : σ ≃ τ) : (renameEquiv R f).symm = renameEquiv R f.symm :=
rfl
#align mv_polynomial.rename_equiv_symm MvPolynomial.renameEquiv_symm
@[simp]
theorem renameEquiv_trans (e : σ ≃ τ) (f : τ ≃ α) :
(renameEquiv R e).trans (renameEquiv R f) = renameEquiv R (e.trans f) :=
AlgEquiv.ext (rename_rename e f)
#align mv_polynomial.rename_equiv_trans MvPolynomial.renameEquiv_trans
end
section
variable (f : R →+* S) (k : σ → τ) (g : τ → S) (p : MvPolynomial σ R)
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
·
|
intros
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
·
|
Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k)
|
Mathlib_Data_MvPolynomial_Rename
|
case h_X
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g : τ → S
p p✝ : MvPolynomial σ R
n✝ : σ
a✝ : eval₂ f g ((rename k) p✝) = eval₂ f (g ∘ k) p✝
⊢ eval₂ f g ((rename k) (p✝ * X n✝)) = eval₂ f (g ∘ k) (p✝ * X n✝)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply]
#align mv_polynomial.kill_compl_comp_rename MvPolynomial.killCompl_comp_rename
@[simp]
theorem killCompl_rename_app (p : MvPolynomial σ R) : killCompl hf (rename f p) = p :=
AlgHom.congr_fun (killCompl_comp_rename hf) p
#align mv_polynomial.kill_compl_rename_app MvPolynomial.killCompl_rename_app
end
section
variable (R)
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by rw [rename_rename, f.symm_comp_self, rename_id]
right_inv := fun p => by rw [rename_rename, f.self_comp_symm, rename_id] }
#align mv_polynomial.rename_equiv MvPolynomial.renameEquiv
@[simp]
theorem renameEquiv_refl : renameEquiv R (Equiv.refl σ) = AlgEquiv.refl :=
AlgEquiv.ext rename_id
#align mv_polynomial.rename_equiv_refl MvPolynomial.renameEquiv_refl
@[simp]
theorem renameEquiv_symm (f : σ ≃ τ) : (renameEquiv R f).symm = renameEquiv R f.symm :=
rfl
#align mv_polynomial.rename_equiv_symm MvPolynomial.renameEquiv_symm
@[simp]
theorem renameEquiv_trans (e : σ ≃ τ) (f : τ ≃ α) :
(renameEquiv R e).trans (renameEquiv R f) = renameEquiv R (e.trans f) :=
AlgEquiv.ext (rename_rename e f)
#align mv_polynomial.rename_equiv_trans MvPolynomial.renameEquiv_trans
end
section
variable (f : R →+* S) (k : σ → τ) (g : τ → S) (p : MvPolynomial σ R)
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· intros
|
simp [*]
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· intros
|
Mathlib.Data.MvPolynomial.Rename.185_0.3NqVCwOs1E93kvK
|
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k)
|
Mathlib_Data_MvPolynomial_Rename
|
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
g : τ → MvPolynomial σ R
⊢ (rename k) (eval₂ C (g ∘ k) p) = eval₂ C (⇑(rename k) ∘ g) ((rename k) p)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply]
#align mv_polynomial.kill_compl_comp_rename MvPolynomial.killCompl_comp_rename
@[simp]
theorem killCompl_rename_app (p : MvPolynomial σ R) : killCompl hf (rename f p) = p :=
AlgHom.congr_fun (killCompl_comp_rename hf) p
#align mv_polynomial.kill_compl_rename_app MvPolynomial.killCompl_rename_app
end
section
variable (R)
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by rw [rename_rename, f.symm_comp_self, rename_id]
right_inv := fun p => by rw [rename_rename, f.self_comp_symm, rename_id] }
#align mv_polynomial.rename_equiv MvPolynomial.renameEquiv
@[simp]
theorem renameEquiv_refl : renameEquiv R (Equiv.refl σ) = AlgEquiv.refl :=
AlgEquiv.ext rename_id
#align mv_polynomial.rename_equiv_refl MvPolynomial.renameEquiv_refl
@[simp]
theorem renameEquiv_symm (f : σ ≃ τ) : (renameEquiv R f).symm = renameEquiv R f.symm :=
rfl
#align mv_polynomial.rename_equiv_symm MvPolynomial.renameEquiv_symm
@[simp]
theorem renameEquiv_trans (e : σ ≃ τ) (f : τ ≃ α) :
(renameEquiv R e).trans (renameEquiv R f) = renameEquiv R (e.trans f) :=
AlgEquiv.ext (rename_rename e f)
#align mv_polynomial.rename_equiv_trans MvPolynomial.renameEquiv_trans
end
section
variable (f : R →+* S) (k : σ → τ) (g : τ → S) (p : MvPolynomial σ R)
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· intros
simp [*]
#align mv_polynomial.eval₂_rename MvPolynomial.eval₂_rename
theorem eval₂Hom_rename : eval₂Hom f g (rename k p) = eval₂Hom f (g ∘ k) p :=
eval₂_rename _ _ _ _
#align mv_polynomial.eval₂_hom_rename MvPolynomial.eval₂Hom_rename
theorem aeval_rename [Algebra R S] : aeval g (rename k p) = aeval (g ∘ k) p :=
eval₂Hom_rename _ _ _ _
#align mv_polynomial.aeval_rename MvPolynomial.aeval_rename
theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) := by
|
apply MvPolynomial.induction_on p
|
theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) := by
|
Mathlib.Data.MvPolynomial.Rename.199_0.3NqVCwOs1E93kvK
|
theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g)
|
Mathlib_Data_MvPolynomial_Rename
|
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
g : τ → MvPolynomial σ R
⊢ ∀ (a : R), (rename k) (eval₂ C (g ∘ k) (C a)) = eval₂ C (⇑(rename k) ∘ g) ((rename k) (C a))
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on multivariate polynomials,
which modifies the set of variables.
## Main declarations
* `MvPolynomial.rename`
* `MvPolynomial.renameEquiv`
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ α : Type*` (indexing the variables)
+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R` elements of the coefficient ring
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ α`
-/
noncomputable section
open BigOperators
open Set Function Finsupp AddMonoidAlgebra
open BigOperators
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial
section Rename
/-- Rename all the variables in a multivariable polynomial. -/
def rename (f : σ → τ) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=
aeval (X ∘ f)
#align mv_polynomial.rename MvPolynomial.rename
@[simp]
theorem rename_C (f : σ → τ) (r : R) : rename f (C r) = C r :=
eval₂_C _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_C MvPolynomial.rename_C
@[simp]
theorem rename_X (f : σ → τ) (i : σ) : rename f (X i : MvPolynomial σ R) = X (f i) :=
eval₂_X _ _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.rename_X MvPolynomial.rename_X
theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
#align mv_polynomial.map_rename MvPolynomial.map_rename
@[simp]
theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
rename g (rename f p) = rename (g ∘ f) p :=
show rename g (eval₂ C (X ∘ f) p) = _ by
simp only [rename, aeval_eq_eval₂Hom]
-- porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
-- Hopefully this is less prone to breaking
rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
simp only [(· ∘ ·), eval₂Hom_X']
refine' eval₂Hom_congr _ rfl rfl
ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
#align mv_polynomial.rename_rename MvPolynomial.rename_rename
@[simp]
theorem rename_id (p : MvPolynomial σ R) : rename id p = p :=
eval₂_eta p
#align mv_polynomial.rename_id MvPolynomial.rename_id
theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial
theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by
simp only [rename, aeval_def, eval₂, Finsupp.mapDomain, algebraMap_eq, comp_apply,
X_pow_eq_monomial, ← monomial_finsupp_sum_index]
rfl
#align mv_polynomial.rename_eq MvPolynomial.rename_eq
theorem rename_injective (f : σ → τ) (hf : Function.Injective f) :
Function.Injective (rename f : MvPolynomial σ R → MvPolynomial τ R) := by
have :
(rename f : MvPolynomial σ R → MvPolynomial τ R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=
funext (rename_eq f)
rw [this]
exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)
#align mv_polynomial.rename_injective MvPolynomial.rename_injective
section
variable {f : σ → τ} (hf : Function.Injective f)
open Classical
/-- Given a function between sets of variables `f : σ → τ` that is injective with proof `hf`,
`MvPolynomial.killCompl hf` is the `AlgHom` from `R[τ]` to `R[σ]` that is left inverse to
`rename f : R[σ] → R[τ]` and sends the variables in the complement of the range of `f` to `0`. -/
def killCompl : MvPolynomial τ R →ₐ[R] MvPolynomial σ R :=
aeval fun i => if h : i ∈ Set.range f then X <| (Equiv.ofInjective f hf).symm ⟨i, h⟩ else 0
#align mv_polynomial.kill_compl MvPolynomial.killCompl
theorem killCompl_comp_rename : (killCompl hf).comp (rename f) = AlgHom.id R _ :=
algHom_ext fun i => by
dsimp
rw [rename, killCompl, aeval_X, comp_apply, aeval_X, dif_pos, Equiv.ofInjective_symm_apply]
#align mv_polynomial.kill_compl_comp_rename MvPolynomial.killCompl_comp_rename
@[simp]
theorem killCompl_rename_app (p : MvPolynomial σ R) : killCompl hf (rename f p) = p :=
AlgHom.congr_fun (killCompl_comp_rename hf) p
#align mv_polynomial.kill_compl_rename_app MvPolynomial.killCompl_rename_app
end
section
variable (R)
/-- `MvPolynomial.rename e` is an equivalence when `e` is. -/
@[simps apply]
def renameEquiv (f : σ ≃ τ) : MvPolynomial σ R ≃ₐ[R] MvPolynomial τ R :=
{ rename f with
toFun := rename f
invFun := rename f.symm
left_inv := fun p => by rw [rename_rename, f.symm_comp_self, rename_id]
right_inv := fun p => by rw [rename_rename, f.self_comp_symm, rename_id] }
#align mv_polynomial.rename_equiv MvPolynomial.renameEquiv
@[simp]
theorem renameEquiv_refl : renameEquiv R (Equiv.refl σ) = AlgEquiv.refl :=
AlgEquiv.ext rename_id
#align mv_polynomial.rename_equiv_refl MvPolynomial.renameEquiv_refl
@[simp]
theorem renameEquiv_symm (f : σ ≃ τ) : (renameEquiv R f).symm = renameEquiv R f.symm :=
rfl
#align mv_polynomial.rename_equiv_symm MvPolynomial.renameEquiv_symm
@[simp]
theorem renameEquiv_trans (e : σ ≃ τ) (f : τ ≃ α) :
(renameEquiv R e).trans (renameEquiv R f) = renameEquiv R (e.trans f) :=
AlgEquiv.ext (rename_rename e f)
#align mv_polynomial.rename_equiv_trans MvPolynomial.renameEquiv_trans
end
section
variable (f : R →+* S) (k : σ → τ) (g : τ → S) (p : MvPolynomial σ R)
theorem eval₂_rename : (rename k p).eval₂ f g = p.eval₂ f (g ∘ k) := by
apply MvPolynomial.induction_on p <;>
· intros
simp [*]
#align mv_polynomial.eval₂_rename MvPolynomial.eval₂_rename
theorem eval₂Hom_rename : eval₂Hom f g (rename k p) = eval₂Hom f (g ∘ k) p :=
eval₂_rename _ _ _ _
#align mv_polynomial.eval₂_hom_rename MvPolynomial.eval₂Hom_rename
theorem aeval_rename [Algebra R S] : aeval g (rename k p) = aeval (g ∘ k) p :=
eval₂Hom_rename _ _ _ _
#align mv_polynomial.aeval_rename MvPolynomial.aeval_rename
theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) := by
apply MvPolynomial.induction_on p <;>
·
|
intros
|
theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) := by
apply MvPolynomial.induction_on p <;>
·
|
Mathlib.Data.MvPolynomial.Rename.199_0.3NqVCwOs1E93kvK
|
theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g)
|
Mathlib_Data_MvPolynomial_Rename
|
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