name
stringlengths 3
112
| file
stringlengths 21
116
| statement
stringlengths 17
8.64k
| state
stringlengths 7
205k
| tactic
stringlengths 3
4.55k
| result
stringlengths 7
205k
| id
stringlengths 16
16
|
|---|---|---|---|---|---|---|
ContinuousMap.ker_evalStarAlgHom_inter_adjoin_id
|
Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean
|
lemma ker_evalStarAlgHom_inter_adjoin_id (s : Set 𝕜) (h0 : 0 ∈ s) :
(StarAlgebra.adjoin 𝕜 {restrict s (.id 𝕜)} : Set C(s, 𝕜)) ∩
RingHom.ker (evalStarAlgHom 𝕜 𝕜 (⟨0, h0⟩ : s)) = adjoin 𝕜 {restrict s (.id 𝕜)}
|
case h.mp.intro
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f : C(↑s, 𝕜)
hf₁ : f ∈ ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)})
hf₂ : f ∈ RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)
⊢ f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}
|
simp_rw [adjoin_id_eq_span_one_add, Set.mem_add, SetLike.mem_coe, mem_span_singleton] at hf₁
|
case h.mp.intro
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f : C(↑s, 𝕜)
hf₂ : f ∈ RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)
hf₁ : ∃ x, (∃ a, a • 1 = x) ∧ ∃ y ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}, x + y = f
⊢ f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}
|
c6f4688dc57cb558
|
AlgebraicGeometry.Scheme.IdealSheafData.ideal_iSup
|
Mathlib/AlgebraicGeometry/IdealSheaf.lean
|
@[simp]
lemma ideal_iSup {ι : Type*} {I : ι → IdealSheafData X} : (iSup I).ideal = ⨆ i, (I i).ideal
|
X : Scheme
ι : Type u_1
I : ι → X.IdealSheafData
⊢ (iSup I).ideal = ⨆ i, (I i).ideal
|
rw [← sSup_range, ← sSup_range, ideal_sSup, ← Set.range_comp, Function.comp_def]
|
no goals
|
7459fbe32e481a66
|
Encodable.decode₂_ne_none_iff
|
Mathlib/Logic/Encodable/Basic.lean
|
theorem decode₂_ne_none_iff [Encodable α] {n : ℕ} :
decode₂ α n ≠ none ↔ n ∈ Set.range (encode : α → ℕ)
|
α : Type u_1
inst✝ : Encodable α
n : ℕ
⊢ decode₂ α n ≠ none ↔ n ∈ Set.range encode
|
simp_rw [Set.range, Set.mem_setOf_eq, Ne, Option.eq_none_iff_forall_not_mem,
Encodable.mem_decode₂, not_forall, not_not]
|
no goals
|
41dd8e696b7b01f7
|
MeasureTheory.AEStronglyMeasurable.integral_prod_right'
|
Mathlib/MeasureTheory/Integral/Prod.lean
|
theorem MeasureTheory.AEStronglyMeasurable.integral_prod_right' [SFinite ν] [NormedSpace ℝ E]
⦃f : α × β → E⦄ (hf : AEStronglyMeasurable f (μ.prod ν)) :
AEStronglyMeasurable (fun x => ∫ y, f (x, y) ∂ν) μ :=
⟨fun x => ∫ y, hf.mk f (x, y) ∂ν, hf.stronglyMeasurable_mk.integral_prod_right', by
filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with _ hx using integral_congr_ae hx⟩
|
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝² : NormedAddCommGroup E
inst✝¹ : SFinite ν
inst✝ : NormedSpace ℝ E
f : α × β → E
hf : AEStronglyMeasurable f (μ.prod ν)
⊢ (fun x => ∫ (y : β), f (x, y) ∂ν) =ᶠ[ae μ] fun x => ∫ (y : β), AEStronglyMeasurable.mk f hf (x, y) ∂ν
|
filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with _ hx using integral_congr_ae hx
|
no goals
|
fe24fedfafefecf5
|
LinearIndependent.pair_iff'
|
Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean
|
theorem LinearIndependent.pair_iff' {x y : V} (hx : x ≠ 0) :
LinearIndependent K ![x, y] ↔ ∀ a : K, a • x ≠ y
|
case neg
K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x y : V
hx : x ≠ 0
H : ∀ (a : K), a • x ≠ y
s t : K
ht : ¬t = 0
hst : t⁻¹ • (s • x + t • y) = t⁻¹ • 0
⊢ s = 0 ∧ t = 0
|
simp only [smul_add, smul_smul, inv_mul_cancel₀ ht] at hst
|
case neg
K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x y : V
hx : x ≠ 0
H : ∀ (a : K), a • x ≠ y
s t : K
ht : ¬t = 0
hst : (t⁻¹ * s) • x + 1 • y = t⁻¹ • 0
⊢ s = 0 ∧ t = 0
|
19c2c5ebd4d42798
|
lp.inner_single_left
|
Mathlib/Analysis/InnerProductSpace/l2Space.lean
|
theorem inner_single_left [DecidableEq ι] (i : ι) (a : G i) (f : lp G 2) :
⟪lp.single 2 i a, f⟫ = ⟪a, f i⟫
|
case pos
ι : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
G : ι → Type u_4
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝ : DecidableEq ι
f : ↥(lp G 2)
j : ι
a : G j
⊢ inner (Pi.single j a j) (↑f j) = inner a (↑f j)
|
rw [Pi.single_eq_same]
|
no goals
|
5ec876fe1b1e7ef5
|
CategoryTheory.preserves_fin_of_preserves_binary_and_initial
|
Mathlib/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.lean
|
/-- If `F` preserves the initial object and binary coproducts, then it preserves products indexed by
`Fin n` for any `n`.
-/
lemma preserves_fin_of_preserves_binary_and_initial :
∀ (n : ℕ) (f : Fin n → C), PreservesColimit (Discrete.functor f) F
| 0 => fun f => by
letI : PreservesColimitsOfShape (Discrete (Fin 0)) F :=
preservesColimitsOfShape_of_equiv.{0, 0} (Discrete.equivalence finZeroEquiv'.symm) _
infer_instance
| n + 1 => by
haveI := preserves_fin_of_preserves_binary_and_initial n
intro f
apply
preservesColimit_of_preserves_colimit_cocone
(extendCofanIsColimit f (colimit.isColimit _) (colimit.isColimit _)) _
apply (isColimitMapCoconeCofanMkEquiv _ _ _).symm _
let this :=
extendCofanIsColimit (fun i => F.obj (f i))
(isColimitOfHasCoproductOfPreservesColimit F _)
(isColimitOfHasBinaryCoproductOfPreservesColimit F _ _)
refine IsColimit.ofIsoColimit this ?_
apply Cocones.ext _ _
· apply Iso.refl _
rintro ⟨j⟩
refine Fin.inductionOn j ?_ ?_
· apply Category.comp_id
· rintro i _
dsimp [extendCofan_ι_app, Iso.refl_hom, Cofan.mk_ι_app]
rw [comp_id, ← F.map_comp]
|
C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
F : C ⥤ D
inst✝² : PreservesColimitsOfShape (Discrete WalkingPair) F
inst✝¹ : PreservesColimitsOfShape (Discrete PEmpty.{1}) F
inst✝ : HasFiniteCoproducts C
n : ℕ
this✝ : ∀ (f : Fin n → C), PreservesColimit (Discrete.functor f) F
f : Fin (n + 1) → C
this : IsColimit
(extendCofan (Cofan.mk (F.obj (∐ fun i => f i.succ)) fun j => F.map (Sigma.ι (fun i => f i.succ) j))
(BinaryCofan.mk (F.map coprod.inl) (F.map coprod.inr))) :=
extendCofanIsColimit (fun i => F.obj (f i)) (isColimitOfHasCoproductOfPreservesColimit F fun i => f i.succ)
(isColimitOfHasBinaryCoproductOfPreservesColimit F (f 0) (∐ fun i => f i.succ))
⊢ (extendCofan (Cofan.mk (F.obj (∐ fun i => f i.succ)) fun j => F.map (Sigma.ι (fun i => f i.succ) j))
(BinaryCofan.mk (F.map coprod.inl) (F.map coprod.inr))).pt ≅
(Cofan.mk (F.obj (colimit.cocone (pair (f 0) (colimit.cocone (Discrete.functor fun i => f i.succ)).pt)).pt) fun j =>
F.map
(Fin.cases
(BinaryCofan.inl (colimit.cocone (pair (f 0) (colimit.cocone (Discrete.functor fun i => f i.succ)).pt)))
(fun i =>
(colimit.cocone (Discrete.functor fun i => f i.succ)).ι.app { as := i } ≫
BinaryCofan.inr (colimit.cocone (pair (f 0) (colimit.cocone (Discrete.functor fun i => f i.succ)).pt)))
j)).pt
|
apply Iso.refl _
|
no goals
|
974807fb6f70ed28
|
CStarAlgebra.isClosed_nonneg
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean
|
/-- The set of nonnegative elements in a C⋆-algebra is closed. -/
lemma isClosed_nonneg : IsClosed {a : A | 0 ≤ a}
|
case h.e'_3.h
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
this : IsClosed {a | 0 ≤ a}
a : Unitization ℂ A
⊢ (∃ x, 0 ≤ x ∧ ↑x = a) ↔ ∃ x, 0 ≤ a ∧ ↑x = a
|
congr! 2 with x
|
case h.e'_3.h.a.h.e'_2.h.a
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
this : IsClosed {a | 0 ≤ a}
a : Unitization ℂ A
x : A
⊢ 0 ≤ x ∧ ↑x = a ↔ 0 ≤ a ∧ ↑x = a
|
330fbc0341ffcf7d
|
szemeredi_regularity
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Lemma.lean
|
theorem szemeredi_regularity (hε : 0 < ε) (hl : l ≤ card α) :
∃ P : Finpartition univ,
P.IsEquipartition ∧ l ≤ #P.parts ∧ #P.parts ≤ bound ε l ∧ P.IsUniform G ε
|
case pos
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : bound ε l ≤ Fintype.card α
t : ℕ := initialBound ε l
htα : t ≤ #univ
dum : Finpartition univ
hdum₁ : dum.IsEquipartition
hdum₂ : #dum.parts = initialBound ε l
hε₁ : ε ≤ 1
this : Nonempty α
i : ℕ
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : t ≤ #P.parts
hP₃ : #P.parts ≤ stepBound^[i] t
hP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)
huniform : P.IsUniform G ε
⊢ ∃ P,
P.IsEquipartition ∧
t ≤ #P.parts ∧ #P.parts ≤ stepBound^[i + 1] t ∧ (P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑(i + 1) ≤ ↑(P.energy G))
|
refine ⟨P, hP₁, hP₂, ?_, Or.inl huniform⟩
|
case pos
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : ℝ
l : ℕ
hε : 0 < ε
hl : l ≤ Fintype.card α
hα : bound ε l ≤ Fintype.card α
t : ℕ := initialBound ε l
htα : t ≤ #univ
dum : Finpartition univ
hdum₁ : dum.IsEquipartition
hdum₂ : #dum.parts = initialBound ε l
hε₁ : ε ≤ 1
this : Nonempty α
i : ℕ
P : Finpartition univ
hP₁ : P.IsEquipartition
hP₂ : t ≤ #P.parts
hP₃ : #P.parts ≤ stepBound^[i] t
hP₄ : P.IsUniform G ε ∨ ε ^ 5 / 4 * ↑i ≤ ↑(P.energy G)
huniform : P.IsUniform G ε
⊢ #P.parts ≤ stepBound^[i + 1] t
|
be596e961a4dc7ee
|
Int.lt_add_succ
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean
|
theorem lt_add_succ (a : Int) (n : Nat) : a < a + Nat.succ n :=
le.intro n <| by rw [Int.add_comm, Int.add_left_comm]; rfl
|
a : Int
n : Nat
⊢ a + 1 + ↑n = a + ↑n.succ
|
rw [Int.add_comm, Int.add_left_comm]
|
a : Int
n : Nat
⊢ a + (↑n + 1) = a + ↑n.succ
|
a407616c1a6ec4df
|
ContinuousLinearMap.norm_smulRight_apply
|
Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean
|
theorem norm_smulRight_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRight c f‖ = ‖c‖ * ‖f‖
|
𝕜 : Type u_1
E : Type u_4
Fₗ : Type u_7
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : SeminormedAddCommGroup Fₗ
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 Fₗ
c : E →L[𝕜] 𝕜
f : Fₗ
x : E
⊢ ‖c‖ * ‖x‖ * ‖f‖ = ‖c‖ * ‖f‖ * ‖x‖
|
ring
|
no goals
|
cc50e5b485f6d483
|
Filter.Eventually.diag_of_prod_right
|
Mathlib/Order/Filter/Prod.lean
|
theorem Eventually.diag_of_prod_right {f : Filter α} {g : Filter γ} {p : α × γ × γ → Prop} :
(∀ᶠ x in f ×ˢ (g ×ˢ g), p x) → ∀ᶠ x : α × γ in f ×ˢ g, p (x.1, x.2, x.2)
|
α : Type u_1
γ : Type u_3
f : Filter α
g : Filter γ
p : α × γ × γ → Prop
h : ∀ᶠ (x : α × γ × γ) in f ×ˢ g ×ˢ g, p x
⊢ ∀ᶠ (x : α × γ) in f ×ˢ g, p (x.1, x.2, x.2)
|
obtain ⟨t, ht, s, hs, hst⟩ := eventually_prod_iff.1 h
|
case intro.intro.intro.intro
α : Type u_1
γ : Type u_3
f : Filter α
g : Filter γ
p : α × γ × γ → Prop
h : ∀ᶠ (x : α × γ × γ) in f ×ˢ g ×ˢ g, p x
t : α → Prop
ht : ∀ᶠ (x : α) in f, t x
s : γ × γ → Prop
hs : ∀ᶠ (y : γ × γ) in g ×ˢ g, s y
hst : ∀ {x : α}, t x → ∀ {y : γ × γ}, s y → p (x, y)
⊢ ∀ᶠ (x : α × γ) in f ×ˢ g, p (x.1, x.2, x.2)
|
e2fbd788adeae135
|
Quotient.finLiftOn_mk
|
Mathlib/Data/Fintype/Quotient.lean
|
@[simp]
lemma finLiftOn_mk (a : ∀ i, α i) :
finLiftOn (S := S) (β := β) (⟦a ·⟧) = fun f _ ↦ f a
|
case h.h
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
α : ι → Sort u_2
S : (i : ι) → Setoid (α i)
β : Sort u_3
a : (i : ι) → α i
f : ((i : ι) → α i) → β
h : ∀ (a b : (i : ι) → α i), (∀ (i : ι), a i ≈ b i) → f a = f b
⊢ finLiftOn (fun x => ⟦a x⟧) f h = f a
|
dsimp [finLiftOn]
|
case h.h
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
α : ι → Sort u_2
S : (i : ι) → Setoid (α i)
β : Sort u_3
a : (i : ι) → α i
f : ((i : ι) → α i) → β
h : ∀ (a b : (i : ι) → α i), (∀ (i : ι), a i ≈ b i) → f a = f b
⊢ (finChoice fun x => ⟦a x⟧).liftOn f h = f a
|
ce66dfdf18f63aec
|
Fin.cons_le_cons
|
Mathlib/Data/Fin/Tuple/Basic.lean
|
theorem cons_le_cons [∀ i, Preorder (α i)] {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ} :
cons x₀ x ≤ cons y₀ y ↔ x₀ ≤ y₀ ∧ x ≤ y :=
forall_fin_succ.trans <| and_congr_right' <| by simp only [cons_succ, Pi.le_def]
|
n : ℕ
α : Fin (n + 1) → Type u_1
inst✝ : (i : Fin (n + 1)) → Preorder (α i)
x₀ y₀ : α 0
x y : (i : Fin n) → α i.succ
⊢ (∀ (i : Fin n), cons x₀ x i.succ ≤ cons y₀ y i.succ) ↔ x ≤ y
|
simp only [cons_succ, Pi.le_def]
|
no goals
|
739901c897d22006
|
Profinite.NobelingProof.Products.prop_of_isGood_of_contained
|
Mathlib/Topology/Category/Profinite/Nobeling.lean
|
theorem Products.prop_of_isGood_of_contained {l : Products I} (o : Ordinal) (h : l.isGood C)
(hsC : contained C o) (i : I) (hi : i ∈ l.val) : ord I i < o
|
case h
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
l : Products I
o : Ordinal.{u}
h : isGood C l
hsC : contained C o
i : I
hi : i ∈ ↑l
x : ↑C
h' : (∀ i ∈ ↑l, ↑x i = true) ∧ ¬False
⊢ ord I i < o
|
exact hsC x.val x.prop i (h'.1 i hi)
|
no goals
|
6b65398bdbf67509
|
Monotone.le_leftLim
|
Mathlib/Topology/Order/LeftRightLim.lean
|
theorem le_leftLim (h : x < y) : f x ≤ leftLim f y
|
case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[<] y ≠ ⊥
⊢ f x ≤ sSup (f '' Iio y)
|
refine le_csSup ⟨f y, ?_⟩ (mem_image_of_mem _ h)
|
case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[<] y ≠ ⊥
⊢ f y ∈ upperBounds (f '' Iio y)
|
55e2cef9a9579060
|
MeasureTheory.MemLp.of_bilin
|
Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean
|
theorem MemLp.of_bilin {p q r : ℝ≥0∞} {f : α → E} {g : α → F} (b : E → F → G) (c : ℝ≥0)
(hf : MemLp f p μ) (hg : MemLp g q μ)
(h : AEStronglyMeasurable (fun x ↦ b (f x) (g x)) μ)
(hb : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊)
[hpqr : HolderTriple p q r] :
MemLp (fun x ↦ b (f x) (g x)) r μ
|
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
p q r : ℝ≥0∞
f : α → E
g : α → F
b : E → F → G
c : ℝ≥0
hf : MemLp f p μ
hg : MemLp g q μ
h : AEStronglyMeasurable (fun x => b (f x) (g x)) μ
hb : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr : p.HolderTriple q r
this✝ : eLpNorm f p μ < ⊤
this : eLpNorm g q μ < ⊤
⊢ ↑c * eLpNorm f p μ * eLpNorm g q μ < ⊤
|
finiteness
|
no goals
|
098c91fdc550ca3e
|
List.toFinsupp_append
|
Mathlib/Data/List/ToFinsupp.lean
|
theorem toFinsupp_append {R : Type*} [AddZeroClass R] (l₁ l₂ : List R)
[DecidablePred (getD (l₁ ++ l₂) · 0 ≠ 0)] [DecidablePred (getD l₁ · 0 ≠ 0)]
[DecidablePred (getD l₂ · 0 ≠ 0)] :
toFinsupp (l₁ ++ l₂) =
toFinsupp l₁ + (toFinsupp l₂).embDomain (addLeftEmbedding l₁.length)
|
case h.inr.intro
R : Type u_2
inst✝³ : AddZeroClass R
l₁ l₂ : List R
inst✝² : DecidablePred fun x => (l₁ ++ l₂).getD x 0 ≠ 0
inst✝¹ : DecidablePred fun x => l₁.getD x 0 ≠ 0
inst✝ : DecidablePred fun x => l₂.getD x 0 ≠ 0
k : ℕ
h : l₁.length ≤ l₁.length + k
⊢ l₂.getD k 0 = (Finsupp.embDomain (addLeftEmbedding l₁.length) l₂.toFinsupp) (l₁.length + k)
|
exact Eq.symm (Finsupp.embDomain_apply _ _ _)
|
no goals
|
827e4cf140b9a24e
|
ValuationRing.iff_isInteger_or_isInteger
|
Mathlib/RingTheory/Valuation/ValuationRing.lean
|
theorem iff_isInteger_or_isInteger :
ValuationRing R ↔ ∀ x : K, IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹
|
case mp
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
H : ValuationRing R
x : K
⊢ IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹
|
obtain ⟨x : R, y, hy, rfl⟩ := IsFractionRing.div_surjective (A := R) x
|
case mp.intro.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
H : ValuationRing R
x y : R
hy : y ∈ nonZeroDivisors R
⊢ IsLocalization.IsInteger R ((algebraMap R K) x / (algebraMap R K) y) ∨
IsLocalization.IsInteger R ((algebraMap R K) x / (algebraMap R K) y)⁻¹
|
7ab24d9ebd501080
|
Equiv.Perm.mem_cycleType_iff
|
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
theorem mem_cycleType_iff {n : ℕ} {σ : Perm α} :
n ∈ cycleType σ ↔ ∃ c τ, σ = c * τ ∧ Disjoint c τ ∧ IsCycle c ∧ c.support.card = n
|
case mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
σ : Perm α
⊢ (∃ c τ, σ = c * τ ∧ c.Disjoint τ ∧ c.IsCycle ∧ c.support.card = n) → n ∈ σ.cycleType
|
rintro ⟨c, t, rfl, hd, hc, rfl⟩
|
case mpr.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
c t : Perm α
hd : c.Disjoint t
hc : c.IsCycle
⊢ c.support.card ∈ (c * t).cycleType
|
d7502a60859fe51e
|
CategoryTheory.regularTopology.mem_sieves_iff_hasEffectiveEpi
|
Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean
|
theorem mem_sieves_iff_hasEffectiveEpi (S : Sieve X) :
(S ∈ (regularTopology C) X) ↔
∃ (Y : C) (π : Y ⟶ X), EffectiveEpi π ∧ (S.arrows π)
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preregular C
X : C
S : Sieve X
⊢ S ∈ (regularTopology C) X ↔ ∃ Y π, EffectiveEpi π ∧ S.arrows π
|
constructor
|
case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preregular C
X : C
S : Sieve X
⊢ S ∈ (regularTopology C) X → ∃ Y π, EffectiveEpi π ∧ S.arrows π
case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preregular C
X : C
S : Sieve X
⊢ (∃ Y π, EffectiveEpi π ∧ S.arrows π) → S ∈ (regularTopology C) X
|
174547292021b135
|
CategoryTheory.Limits.preservesLimit_of_preservesEqualizers_and_product
|
Mathlib/CategoryTheory/Limits/Constructions/LimitsOfProductsAndEqualizers.lean
|
/-- If a functor preserves equalizers and the appropriate products, it preserves limits. -/
lemma preservesLimit_of_preservesEqualizers_and_product :
PreservesLimitsOfShape J G where
preservesLimit {K}
|
C : Type u
inst✝⁸ : Category.{v, u} C
J : Type w
inst✝⁷ : SmallCategory J
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasLimitsOfShape (Discrete J) C
inst✝⁴ : HasLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) C
inst✝³ : HasEqualizers C
G : C ⥤ D
inst✝² : PreservesLimitsOfShape WalkingParallelPair G
inst✝¹ : PreservesLimitsOfShape (Discrete J) G
inst✝ : PreservesLimitsOfShape (Discrete ((p : J × J) × (p.1 ⟶ p.2))) G
K : J ⥤ C
P : C := ∏ᶜ K.obj
Q : C := ∏ᶜ fun f => K.obj f.fst.2
s : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.1 } ≫ K.map f.snd
t : P ⟶ Q := Pi.lift fun f => limit.π (Discrete.functor K.obj) { as := f.fst.2 }
I : C := equalizer s t
i : I ⟶ P := equalizer.ι s t
⊢ (Fan.mk (G.obj (∏ᶜ K.obj)) fun j => G.map (Pi.π K.obj j)).pt ⟶
(Fan.mk (G.obj Q) fun f => G.map (Pi.π (fun f => K.obj f.fst.2) f)).pt
|
apply G.map s
|
no goals
|
1664359590cff5a6
|
blimsup_cthickening_ae_le_of_eventually_mul_le
|
Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
|
theorem blimsup_cthickening_ae_le_of_eventually_mul_le (p : ℕ → Prop) {s : ℕ → Set α} {M : ℝ}
(hM : 0 < M) {r₁ r₂ : ℕ → ℝ} (hr : Tendsto r₁ atTop (𝓝[>] 0))
(hMr : ∀ᶠ i in atTop, M * r₁ i ≤ r₂ i) :
(blimsup (fun i => cthickening (r₁ i) (s i)) atTop p : Set α) ≤ᵐ[μ]
(blimsup (fun i => cthickening (r₂ i) (s i)) atTop p : Set α)
|
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => 0 ⊔ r₁ i
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
|
let R₂ i := max 0 (r₂ i)
|
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => 0 ⊔ r₁ i
R₂ : ℕ → ℝ := fun i => 0 ⊔ r₂ i
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
|
33861a164c277922
|
ae_restrict_of_ae_restrict_inter_Ioo
|
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
theorem ae_restrict_of_ae_restrict_inter_Ioo {μ : Measure ℝ} [NoAtoms μ] {s : Set ℝ} {p : ℝ → Prop}
(h : ∀ a b, a ∈ s → b ∈ s → a < b → ∀ᵐ x ∂μ.restrict (s ∩ Ioo a b), p x) :
∀ᵐ x ∂μ.restrict s, p x
|
case intro.intro.right
μ : Measure ℝ
inst✝ : NoAtoms μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x
T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
this : s ⊆ s \ u ∪ ⋃ p ∈ A, s ∩ T p
⊢ ∀ i ∈ A, ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ T i), p x
|
rintro ⟨⟨a, as⟩, ⟨b, bs⟩⟩ -
|
case intro.intro.right.mk.mk.mk
μ : Measure ℝ
inst✝ : NoAtoms μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x
T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
this : s ⊆ s \ u ∪ ⋃ p ∈ A, s ∩ T p
a : ℝ
as : a ∈ s
b : ℝ
bs : b ∈ s
⊢ ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ T (⟨a, as⟩, ⟨b, bs⟩)), p x
|
40ad57d747c092a9
|
FirstOrder.Language.Substructure.subset_closure_withConstants
|
Mathlib/ModelTheory/Substructures.lean
|
theorem subset_closure_withConstants : A ⊆ closure (L[[A]]) s
|
L : Language
M : Type w
inst✝ : L.Structure M
A s : Set M
a : M
ha : a ∈ A
⊢ a ∈ ↑((closure (L[[↑A]])).toFun s)
|
simp only [SetLike.mem_coe]
|
L : Language
M : Type w
inst✝ : L.Structure M
A s : Set M
a : M
ha : a ∈ A
⊢ a ∈ (closure (L[[↑A]])).toFun s
|
0a5a93be9f4bcd75
|
Real.volume_ball
|
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
theorem volume_ball (a r : ℝ) : volume (Metric.ball a r) = ofReal (2 * r)
|
a r : ℝ
⊢ volume (Metric.ball a r) = ofReal (2 * r)
|
rw [ball_eq_Ioo, volume_Ioo, ← sub_add, add_sub_cancel_left, two_mul]
|
no goals
|
1f6bccc88203c80e
|
Orientation.oangle_eq_of_angle_eq_of_sign_eq
|
Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean
|
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
(h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
(hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z
|
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : (o.oangle w x).sign = (o.oangle y z).sign
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : (o.oangle w x).sign = 0
hsyz : (o.oangle y z).sign = 0
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
⊢ π / 2 ≠ π
|
intro hpi
|
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : (o.oangle w x).sign = (o.oangle y z).sign
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : (o.oangle w x).sign = 0
hsyz : (o.oangle y z).sign = 0
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
hpi : π / 2 = π
⊢ False
|
ec88cb80dbeb7f16
|
MaximalSpectrum.finite_of_toPiLocalization_pi_surjective
|
Mathlib/RingTheory/Spectrum/Maximal/Localization.lean
|
theorem finite_of_toPiLocalization_pi_surjective
(h : Function.Surjective (toPiLocalization (Π i, R i))) :
Finite ι
|
ι : Type u_5
R : ι → Type u_4
inst✝¹ : (i : ι) → CommSemiring (R i)
inst✝ : ∀ (i : ι), Nontrivial (R i)
h : ¬Finite ι
⊢ ¬Function.Surjective ⇑(toPiLocalization ((i : ι) → R i))
|
rw [not_finite_iff_infinite] at h
|
ι : Type u_5
R : ι → Type u_4
inst✝¹ : (i : ι) → CommSemiring (R i)
inst✝ : ∀ (i : ι), Nontrivial (R i)
h : Infinite ι
⊢ ¬Function.Surjective ⇑(toPiLocalization ((i : ι) → R i))
|
ba6a09a5dacaa2e4
|
blimsup_cthickening_mul_ae_eq
|
Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
|
theorem blimsup_cthickening_mul_ae_eq (p : ℕ → Prop) (s : ℕ → Set α) {M : ℝ} (hM : 0 < M)
(r : ℕ → ℝ) (hr : Tendsto r atTop (𝓝 0)) :
(blimsup (fun i => cthickening (M * r i) (s i)) atTop p : Set α) =ᵐ[μ]
(blimsup (fun i => cthickening (r i) (s i)) atTop p : Set α)
|
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
∀ (p : ℕ → Prop) {r : ℕ → ℝ},
Tendsto r atTop (𝓝[>] 0) →
blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[>] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ 0 < r i) ⊔
blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ r i ≤ 0) =ᶠ[ae μ]
(blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ 0 < r i) ⊔
blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0
|
simp only [sup_eq_union]
|
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
∀ (p : ℕ → Prop) {r : ℕ → ℝ},
Tendsto r atTop (𝓝[>] 0) →
blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[>] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ 0 < r i) ∪
blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ r i ≤ 0) =ᶠ[ae μ]
(blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ 0 < r i) ∪
blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0
|
8749d72cf710712d
|
IsCompact.finite_of_discrete
|
Mathlib/Topology/Compactness/Compact.lean
|
theorem IsCompact.finite_of_discrete [DiscreteTopology X] (hs : IsCompact s) : s.Finite
|
case intro.intro
X : Type u
inst✝¹ : TopologicalSpace X
s : Set X
inst✝ : DiscreteTopology X
hs : IsCompact s
this : ∀ (x : X), {x} ∈ 𝓝 x
t : Finset X
left✝ : ∀ x ∈ t, x ∈ s
hst : s ⊆ ↑t
⊢ s.Finite
|
exact t.finite_toSet.subset hst
|
no goals
|
9c90eb234e0887ea
|
EuclideanGeometry.exists_circumradius_eq_of_cospherical
|
Mathlib/Geometry/Euclidean/Circumcenter.lean
|
theorem exists_circumradius_eq_of_cospherical {ps : Set P} {n : ℕ} [FiniteDimensional ℝ V]
(hd : finrank ℝ V = n) (hc : Cospherical ps) :
∃ r : ℝ, ∀ sx : Simplex ℝ P n, Set.range sx.points ⊆ ps → sx.circumradius = r
|
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
ps : Set P
n : ℕ
inst✝ : FiniteDimensional ℝ V
hd : finrank ℝ ↥⊤.direction = n
hc : Cospherical ps
this : Nonempty ↥⊤
⊢ ps ⊆ ↑⊤
|
exact Set.subset_univ _
|
no goals
|
28a82da64f0b935b
|
Submodule.LinearDisjoint.linearIndependent_mul_of_flat_left
|
Mathlib/LinearAlgebra/LinearDisjoint.lean
|
theorem linearIndependent_mul_of_flat_left (H : M.LinearDisjoint N) [Module.Flat R M]
{κ ι : Type*} {m : κ → M} {n : ι → N} (hm : LinearIndependent R m)
(hn : LinearIndependent R n) : LinearIndependent R fun (i : κ × ι) ↦ (m i.1).1 * (n i.2).1
|
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : Ring S
inst✝¹ : Algebra R S
M N : Submodule R S
H : M.LinearDisjoint N
inst✝ : Module.Flat R ↥M
κ : Type u_1
ι : Type u_2
m : κ → ↥M
n : ι → ↥N
hm : Function.Injective ⇑(Finsupp.linearCombination R m)
hn : Function.Injective ⇑(Finsupp.linearCombination R n)
i0 : (κ × ι →₀ R) ≃ₗ[R] (κ →₀ R) ⊗[R] (ι →₀ R) := (finsuppTensorFinsupp' R κ ι).symm
i1 : (κ →₀ R) ⊗[R] (ι →₀ R) →ₗ[R] ↥M ⊗[R] (ι →₀ R) := LinearMap.rTensor (ι →₀ R) (Finsupp.linearCombination R m)
i2 : ↥M ⊗[R] (ι →₀ R) →ₗ[R] ↥M ⊗[R] ↥N := LinearMap.lTensor (↥M) (Finsupp.linearCombination R n)
i : (κ × ι →₀ R) →ₗ[R] S := M.mulMap N ∘ₗ i2 ∘ₗ i1 ∘ₗ ↑i0
h1 : Function.Injective ⇑i1
h2 : Function.Injective ⇑i2
h : Function.Injective ⇑i
⊢ i = Finsupp.linearCombination R fun i => ↑(m i.1) * ↑(n i.2)
|
ext x
|
case h.h
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : Ring S
inst✝¹ : Algebra R S
M N : Submodule R S
H : M.LinearDisjoint N
inst✝ : Module.Flat R ↥M
κ : Type u_1
ι : Type u_2
m : κ → ↥M
n : ι → ↥N
hm : Function.Injective ⇑(Finsupp.linearCombination R m)
hn : Function.Injective ⇑(Finsupp.linearCombination R n)
i0 : (κ × ι →₀ R) ≃ₗ[R] (κ →₀ R) ⊗[R] (ι →₀ R) := (finsuppTensorFinsupp' R κ ι).symm
i1 : (κ →₀ R) ⊗[R] (ι →₀ R) →ₗ[R] ↥M ⊗[R] (ι →₀ R) := LinearMap.rTensor (ι →₀ R) (Finsupp.linearCombination R m)
i2 : ↥M ⊗[R] (ι →₀ R) →ₗ[R] ↥M ⊗[R] ↥N := LinearMap.lTensor (↥M) (Finsupp.linearCombination R n)
i : (κ × ι →₀ R) →ₗ[R] S := M.mulMap N ∘ₗ i2 ∘ₗ i1 ∘ₗ ↑i0
h1 : Function.Injective ⇑i1
h2 : Function.Injective ⇑i2
h : Function.Injective ⇑i
x : κ × ι
⊢ (i ∘ₗ Finsupp.lsingle x) 1 = ((Finsupp.linearCombination R fun i => ↑(m i.1) * ↑(n i.2)) ∘ₗ Finsupp.lsingle x) 1
|
f1b7c2beac5cf02c
|
Array.flatMap_toArray
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem flatMap_toArray {β} (f : α → Array β) (as : List α) :
as.toArray.flatMap f = (as.flatMap (fun a => (f a).toList)).toArray
|
α : Type u_1
β : Type u_2
f : α → Array β
as : List α
⊢ flatMap f as.toArray = (List.flatMap (fun a => (f a).toList) as).toArray
|
induction as with
| nil => simp
| cons a as ih =>
apply ext'
simp [ih, flatMap_toArray_cons]
|
no goals
|
e3ac00879d1ac7cf
|
AkraBazziRecurrence.GrowsPolynomially.add_isLittleO
|
Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
|
lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
(hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x
|
case h
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≤ 0
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : 0 < c₂
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖f x‖
x : ℝ
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg' : ∀ (y : ℝ), b * id x ≤ y → ‖g y‖ ≤ 1 / 2 * ‖f y‖
hf₂ : ∀ (y : ℝ), b * id x ≤ y → f y ≤ 0
hx_nonneg : 0 ≤ x
hbx : b * x ≤ x
hfg₂ : ‖g x‖ ≤ -1 / 2 * f x
hx_ub : f x + g x ≤ 1 / 2 * f x
hx_lb : 3 / 2 * f x ≤ f x + g x
⊢ ∀ u ∈ Set.Icc (b * x) x, f u + g u ∈ Set.Icc (3 * c₁ * (f x + g x)) (c₂ / 3 * (f x + g x))
|
intro u ⟨hu_lb, hu_ub⟩
|
case h
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, f x ≤ 0
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : 0 < c₂
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖f x‖
x : ℝ
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg' : ∀ (y : ℝ), b * id x ≤ y → ‖g y‖ ≤ 1 / 2 * ‖f y‖
hf₂ : ∀ (y : ℝ), b * id x ≤ y → f y ≤ 0
hx_nonneg : 0 ≤ x
hbx : b * x ≤ x
hfg₂ : ‖g x‖ ≤ -1 / 2 * f x
hx_ub : f x + g x ≤ 1 / 2 * f x
hx_lb : 3 / 2 * f x ≤ f x + g x
u : ℝ
hu_lb : b * x ≤ u
hu_ub : u ≤ x
⊢ f u + g u ∈ Set.Icc (3 * c₁ * (f x + g x)) (c₂ / 3 * (f x + g x))
|
dd533a6f58fe3819
|
PrimeSpectrum.existsUnique_idempotent_basicOpen_eq_of_isClopen
|
Mathlib/RingTheory/Spectrum/Prime/Topology.lean
|
@[stacks 00EE]
lemma existsUnique_idempotent_basicOpen_eq_of_isClopen {s : Set (PrimeSpectrum R)}
(hs : IsClopen s) : ∃! e : R, IsIdempotentElem e ∧ s = basicOpen e
|
case refine_1.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_3
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
e : x + y = 1
⊢ ↑(basicOpen x) ⊆ s
|
rw [PrimeSpectrum.basicOpen_eq_zeroLocus_compl, Set.compl_subset_comm, ← hI']
|
case refine_1.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_3
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
e : x + y = 1
⊢ zeroLocus ↑I ⊆ zeroLocus {x}
|
f72a236e53847d1c
|
HomologicalComplex₂.D₂_totalShift₂XIso_hom
|
Mathlib/Algebra/Homology/TotalComplexShift.lean
|
@[reassoc]
lemma D₂_totalShift₂XIso_hom (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + y = n₀') (h₁ : n₁ + y = n₁') :
((shiftFunctor₂ C y).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
y.negOnePow • ((K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')
|
case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : (up ℤ).Rel n₀ n₁
p q : ℤ
hpq : (up ℤ).π (up ℤ) (up ℤ) (p, q) = n₀
⊢ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
((shiftFunctor₂ C y).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
(y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')
|
dsimp at h hpq
|
case pos.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : n₀ + 1 = n₁
p q : ℤ
hpq : p + q = n₀
⊢ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
((shiftFunctor₂ C y).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p q n₀ hpq ≫
(y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁')
|
f1ece82a737eae57
|
Complex.hasDerivAt_Gammaℂ_one
|
Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
|
lemma hasDerivAt_Gammaℂ_one : HasDerivAt Gammaℂ (-(γ + log (2 * π)) / π) 1
|
⊢ HasDerivAt Gammaℂ (-(↑γ + log (2 * ↑π)) / ↑π) 1
|
let f (s : ℂ) : ℂ := 2 * (2 * π) ^ (-s)
|
f : ℂ → ℂ := fun s => 2 * (2 * ↑π) ^ (-s)
⊢ HasDerivAt Gammaℂ (-(↑γ + log (2 * ↑π)) / ↑π) 1
|
154450c4585fc111
|
List.permutations_perm_permutations'
|
Mathlib/Data/List/Permutation.lean
|
theorem permutations_perm_permutations' (ts : List α) : ts.permutations ~ ts.permutations'
|
case intro.succ.refine_2
α : Type u_1
n : ℕ
IH : ∀ (ts : List α), ts.length < n → ts.permutations ~ ts.permutations'
ts✝ : List α
h✝ : ts✝.length < n + 1
ts : List α
t : α
x✝ : ts.length < n + 1 → ts.permutations ~ ts.permutations'
h : ts.length < n
IH₂ : ts.reverse.permutations ~ ts.permutations'
⊢ map (fun x => x ++ [t]) ts.permutations ++
flatMap (fun y => (permutationsAux2 t [] [] y id).2) ts.reverse.permutations ~
(ts ++ [t]).permutations'
|
refine
(perm_append_comm.trans ((IH₂.flatMap_right _).append ((IH _ h).map _))).trans
(Perm.trans ?_ perm_append_comm.permutations')
|
case intro.succ.refine_2
α : Type u_1
n : ℕ
IH : ∀ (ts : List α), ts.length < n → ts.permutations ~ ts.permutations'
ts✝ : List α
h✝ : ts✝.length < n + 1
ts : List α
t : α
x✝ : ts.length < n + 1 → ts.permutations ~ ts.permutations'
h : ts.length < n
IH₂ : ts.reverse.permutations ~ ts.permutations'
⊢ flatMap (fun y => (permutationsAux2 t [] [] y id).2) ts.permutations' ++ map (fun x => x ++ [t]) ts.permutations' ~
([t] ++ ts).permutations'
|
c620c610f6f713f1
|
LieAlgebra.InvariantForm.restrict_nondegenerate
|
Mathlib/Algebra/Lie/InvariantForm.lean
|
lemma restrict_nondegenerate (I : LieIdeal K L) (hI : IsAtom I) :
(Φ.restrict I).Nondegenerate
|
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
Φ : LinearMap.BilinForm K L
hΦ_nondeg : Φ.Nondegenerate
hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ
hΦ_refl : Φ.IsRefl
hL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥I
I : LieIdeal K L
hI : IsAtom I
⊢ IsCompl (LieIdeal.toLieSubalgebra K L I).toSubmodule (Φ.orthogonal (LieIdeal.toLieSubalgebra K L I).toSubmodule)
|
exact orthogonal_isCompl_toSubmodule Φ hΦ_nondeg hΦ_inv hΦ_refl hL I hI
|
no goals
|
7eb68de4008f4a5e
|
HomogeneousLocalization.mk_eq_zero_of_num
|
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
|
lemma mk_eq_zero_of_num (f : NumDenSameDeg 𝒜 x) (h : f.num = 0) : mk f = 0
|
case a
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
𝒜 : ι → Submodule R A
x : Submonoid A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
inst✝ : GradedAlgebra 𝒜
f : NumDenSameDeg 𝒜 x
h : f.num = 0
⊢ (mk f).val = val 0
|
simp only [val_mk, val_zero, h, ZeroMemClass.coe_zero, Localization.mk_zero]
|
no goals
|
d979ef3d9145bed7
|
SSet.StrictSegal.isPointwiseRightKanExtensionAt.fac_aux₂
|
Mathlib/AlgebraicTopology/SimplicialSet/Coskeletal.lean
|
lemma fac_aux₂ {n : ℕ}
(s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X))
(x : s.pt) (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n) :
X.map (mkOfLe ⟨i, by omega⟩ ⟨j, by omega⟩ hij).op (lift s x) =
s.π.app (strArrowMk₂ (mkOfLe ⟨i, by omega⟩ ⟨j, by omega⟩ hij)) x
|
case a.h.h.h
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x✝ : s.pt
k : ℕ
hk :
∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),
i + k = j → X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op (lift s x✝) = s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) ⋯) x✝
i j : ℕ
hij : i ≤ j
hj : j ≤ n
hik : i + (k + 1) = j
α : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLeComp ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯ ⋯) ⋯
α₀ : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLe ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯) ⋯
α₁ : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ ⋯) ⋯
α₂ : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⋯) ⋯
β₀ : α ⟶ α₀ := homMk (mkOfSucc 1).op ⋯
x : Fin ((unop ((Truncated.inclusion 2).op.obj α₁.right)).len + 1)
⊢ ↑((Hom.toOrderHom (α.hom ≫ (Truncated.inclusion 2).op.map (δ 1).op).unop) x) = ↑((Hom.toOrderHom α₁.hom.unop) x)
|
fin_cases x <;> rfl
|
no goals
|
764b0c7acb432811
|
Ideal.quotientMap_injective'
|
Mathlib/RingTheory/Ideal/Quotient/Operations.lean
|
theorem quotientMap_injective' {J : Ideal R} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided]
{f : R →+* S} {H : J ≤ I.comap f} (h : I.comap f ≤ J) :
Function.Injective (quotientMap I f H)
|
R : Type u
inst✝³ : Ring R
S : Type v
inst✝² : Ring S
J : Ideal R
I : Ideal S
inst✝¹ : I.IsTwoSided
inst✝ : J.IsTwoSided
f : R →+* S
H : J ≤ comap f I
h : comap f I ≤ J
⊢ Injective ⇑(quotientMap I f H)
|
refine (injective_iff_map_eq_zero (quotientMap I f H)).2 fun a ha => ?_
|
R : Type u
inst✝³ : Ring R
S : Type v
inst✝² : Ring S
J : Ideal R
I : Ideal S
inst✝¹ : I.IsTwoSided
inst✝ : J.IsTwoSided
f : R →+* S
H : J ≤ comap f I
h : comap f I ≤ J
a : R ⧸ J
ha : (quotientMap I f H) a = 0
⊢ a = 0
|
e89d6b071a709d49
|
Real.two_le_pi
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
theorem two_le_pi : (2 : ℝ) ≤ π :=
(div_le_div_iff_of_pos_right (show (0 : ℝ) < 2 by norm_num)).1
(by rw [div_self (two_ne_zero' ℝ)]; exact one_le_pi_div_two)
|
⊢ 1 ≤ π / 2
|
exact one_le_pi_div_two
|
no goals
|
d399a33e85a2e9c9
|
Submodule.exists_smith_normal_form_of_le
|
Mathlib/LinearAlgebra/FreeModule/PID.lean
|
theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M) (N O : Submodule R M)
(N_le_O : N ≤ O) :
∃ (n o : ℕ) (hno : n ≤ o) (bO : Basis (Fin o) R O) (bN : Basis (Fin n) R N) (a : Fin n → R),
∀ i, (bN i : M) = a i • bO (Fin.castLE hno i)
|
case neg
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : Finite ι
b : Basis ι R M
val✝ : Fintype ι
M0 : Submodule R M
ih :
∀ N' ≤ M0,
∀ x ∈ M0,
(∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) →
∀ N ≤ N', ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ↑(bO (Fin.castLE hno i))
N : Submodule R M
N_le_O : N ≤ M0
m : ℕ
b'M : Basis (Fin m) R ↥M0
N_bot : ¬N = ⊥
⊢ ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ↑(bO (Fin.castLE hno i))
|
obtain ⟨y, hy, a, _, M', M'_le_M, N', _, N'_le_M', y_ortho, _, h⟩ :=
Submodule.basis_of_pid_aux M0 N b'M N_bot N_le_O
|
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : Finite ι
b : Basis ι R M
val✝ : Fintype ι
M0 : Submodule R M
ih :
∀ N' ≤ M0,
∀ x ∈ M0,
(∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) →
∀ N ≤ N', ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ↑(bO (Fin.castLE hno i))
N : Submodule R M
N_le_O : N ≤ M0
m : ℕ
b'M : Basis (Fin m) R ↥M0
N_bot : ¬N = ⊥
y : M
hy : y ∈ M0
a : R
left✝² : a • y ∈ N
M' : Submodule R M
M'_le_M : M' ≤ M0
N' : Submodule R M
left✝¹ : N' ≤ N
N'_le_M' : N' ≤ M'
y_ortho : ∀ (c : R), ∀ z ∈ M', c • y + z = 0 → c = 0
left✝ : ∀ (c : R), ∀ z ∈ N', c • a • y + z = 0 → c = 0
h :
∀ (n' : ℕ) (bN' : Basis (Fin n') R ↥N'),
∃ bN,
∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'),
∃ (hnm : n' + 1 ≤ m' + 1),
∃ bM,
∀ (as : Fin n' → R),
(∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) →
∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))
⊢ ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ↑(bO (Fin.castLE hno i))
|
ffa033f1b6ed3faa
|
AntitoneOn.integral_le_sum
|
Mathlib/Analysis/SumIntegralComparisons.lean
|
theorem AntitoneOn.integral_le_sum (hf : AntitoneOn f (Icc x₀ (x₀ + a))) :
(∫ x in x₀..x₀ + a, f x) ≤ ∑ i ∈ Finset.range a, f (x₀ + i)
|
x₀ : ℝ
a : ℕ
f : ℝ → ℝ
hf : AntitoneOn f (Icc x₀ (x₀ + ↑a))
k : ℕ
hk : k < a
⊢ uIcc (x₀ + ↑k) (x₀ + ↑(k + 1)) ⊆ Icc x₀ (x₀ + ↑a)
|
rw [uIcc_of_le]
|
x₀ : ℝ
a : ℕ
f : ℝ → ℝ
hf : AntitoneOn f (Icc x₀ (x₀ + ↑a))
k : ℕ
hk : k < a
⊢ Icc (x₀ + ↑k) (x₀ + ↑(k + 1)) ⊆ Icc x₀ (x₀ + ↑a)
x₀ : ℝ
a : ℕ
f : ℝ → ℝ
hf : AntitoneOn f (Icc x₀ (x₀ + ↑a))
k : ℕ
hk : k < a
⊢ x₀ + ↑k ≤ x₀ + ↑(k + 1)
|
709ac30a4a15e5af
|
Std.DHashMap.Raw.mem_of_mem_insertMany_list
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean
|
theorem mem_of_mem_insertMany_list [EquivBEq α] [LawfulHashable α] (h : m.WF)
{l : List ((a : α) × β a)} {k : α} :
k ∈ (m.insertMany l) → (l.map Sigma.fst).contains k = false → k ∈ m
|
α : Type u
β : α → Type v
m : Raw α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.WF
l : List ((a : α) × β a)
k : α
⊢ k ∈ m.insertMany l → (List.map Sigma.fst l).contains k = false → k ∈ m
|
simp only [mem_iff_contains]
|
α : Type u
β : α → Type v
m : Raw α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.WF
l : List ((a : α) × β a)
k : α
⊢ (m.insertMany l).contains k = true → (List.map Sigma.fst l).contains k = false → m.contains k = true
|
74b6f31dbf66a121
|
UniqueFactorizationMonoid.normalizedFactors_one
|
Mathlib/RingTheory/UniqueFactorizationDomain/NormalizedFactors.lean
|
theorem normalizedFactors_one : normalizedFactors (1 : α) = 0
|
case inr.hg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
h : Nontrivial α
x : α
hx : x ∈ 0
⊢ Irreducible x
|
exfalso
|
case inr.hg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
h : Nontrivial α
x : α
hx : x ∈ 0
⊢ False
|
bf5e6a4744791dae
|
Stream'.WSeq.exists_of_liftRel_right
|
Mathlib/Data/Seq/WSeq.lean
|
theorem exists_of_liftRel_right {R : α → β → Prop} {s t} (H : LiftRel R s t) {b} (h : b ∈ t) :
∃ a, a ∈ s ∧ R a b
|
α : Type u
β : Type v
R : α → β → Prop
s : WSeq α
t : WSeq β
H : LiftRel R s t
b : β
h : b ∈ t
⊢ ∃ a, a ∈ s ∧ R a b
|
rw [← LiftRel.swap] at H
|
α : Type u
β : Type v
R : α → β → Prop
s : WSeq α
t : WSeq β
H : swap (LiftRel fun x y => R y x) s t
b : β
h : b ∈ t
⊢ ∃ a, a ∈ s ∧ R a b
|
03359b01f37ea099
|
HomologicalComplex.extend.leftHomologyData.lift_d_comp_eq_zero_iff'
|
Mathlib/Algebra/Homology/Embedding/ExtendHomology.lean
|
/-- Auxiliary lemma for `lift_d_comp_eq_zero_iff`. -/
lemma lift_d_comp_eq_zero_iff' ⦃W : C⦄ (f' : K.X i ⟶ cone.pt)
(hf' : f' ≫ cone.ι = K.d i j)
(f'' : (K.extend e).X i' ⟶ cone.pt)
(hf'' : f'' ≫ cone.ι ≫ (extendXIso K e hj').inv = (K.extend e).d i' j')
(φ : cone.pt ⟶ W) :
f' ≫ φ = 0 ↔ f'' ≫ φ = 0
|
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{u_4, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j k : ι
i' j' : ι'
hj' : e.f j = j'
hi : c.prev j = i
hi' : c'.prev j' = i'
cone : KernelFork (K.d j k)
hcone : IsLimit cone
W : C
f' : K.X i ⟶ cone.pt
hf' : f' ≫ Fork.ι cone = K.d i j
f'' : (K.extend e).X i' ⟶ cone.pt
hf'' : f'' ≫ Fork.ι cone ≫ (K.extendXIso e hj').inv = (K.extend e).d i' j'
φ : cone.pt ⟶ W
hij : ¬c.Rel i j
⊢ f' ≫ Fork.ι cone = 0 ≫ Fork.ι cone
|
simp only [zero_comp, hf', K.shape _ _ hij]
|
no goals
|
47e14c7240329723
|
Finset.ciInf_eq_min'_image
|
Mathlib/Order/ConditionallyCompleteLattice/Finset.lean
|
theorem Finset.ciInf_eq_min'_image {s : Finset ι} (h : ∃ x ∈ s, f x ≤ sInf ∅)
(h' : (s.image f).Nonempty
|
ι : Type u_1
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
f : ι → α
s : Finset ι
h : ∃ x ∈ s, f x ≤ sInf ∅
h' : autoParam (image f s).Nonempty _auto✝
⊢ (image (fun i => OrderDual.toDual (f i)) s).max' ⋯ = (image (⇑OrderDual.toDual ∘ f) s).max' ⋯
|
congr
|
no goals
|
b093b26813c26f92
|
ae_eq_const_or_norm_average_lt_of_norm_le_const
|
Mathlib/Analysis/Convex/Integral.lean
|
theorem ae_eq_const_or_norm_average_lt_of_norm_le_const [StrictConvexSpace ℝ E]
(h_le : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) : f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ‖⨍ x, f x ∂μ‖ < C
|
case inr
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
C : ℝ
inst✝ : StrictConvexSpace ℝ E
h_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
hC0 : 0 < C
⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C
|
by_cases hfi : Integrable f μ
|
case pos
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
C : ℝ
inst✝ : StrictConvexSpace ℝ E
h_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
hC0 : 0 < C
hfi : Integrable f μ
⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C
case neg
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
C : ℝ
inst✝ : StrictConvexSpace ℝ E
h_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
hC0 : 0 < C
hfi : ¬Integrable f μ
⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C
|
ad5ef237f6de5f77
|
LinearMap.ker_eq_bot_of_cancel
|
Mathlib/Algebra/Module/Submodule/Range.lean
|
theorem ker_eq_bot_of_cancel {f : M →ₛₗ[τ₁₂] M₂}
(h : ∀ u v : ker f →ₗ[R] M, f.comp u = f.comp v → u = v) : ker f = ⊥
|
R : Type u_1
R₂ : Type u_2
M : Type u_5
M₂ : Type u_6
inst✝⁵ : Semiring R
inst✝⁴ : Semiring R₂
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R₂ M₂
τ₁₂ : R →+* R₂
f : M →ₛₗ[τ₁₂] M₂
h : ∀ (u v : ↥(ker f) →ₗ[R] M), f.comp u = f.comp v → u = v
h₁ : f.comp 0 = 0
⊢ ker f = ⊥
|
rw [← Submodule.range_subtype (ker f),
← h 0 f.ker.subtype (Eq.trans h₁ (comp_ker_subtype f).symm)]
|
R : Type u_1
R₂ : Type u_2
M : Type u_5
M₂ : Type u_6
inst✝⁵ : Semiring R
inst✝⁴ : Semiring R₂
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R₂ M₂
τ₁₂ : R →+* R₂
f : M →ₛₗ[τ₁₂] M₂
h : ∀ (u v : ↥(ker f) →ₗ[R] M), f.comp u = f.comp v → u = v
h₁ : f.comp 0 = 0
⊢ range 0 = ⊥
|
eee034053a4c80f1
|
isSelfAdjoint_iff_isStarNormal_and_spectrumRestricts
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.lean
|
/-- An element in a C⋆-algebra is selfadjoint if and only if it is normal and its spectrum is
contained in `ℝ`. -/
lemma isSelfAdjoint_iff_isStarNormal_and_spectrumRestricts {a : A} :
IsSelfAdjoint a ↔ IsStarNormal a ∧ SpectrumRestricts a Complex.reCLM
|
case refine_2.intro
A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ IsStarNormal
a : A
ha₁ : IsStarNormal a
ha₂ : SpectrumRestricts a ⇑Complex.reCLM
⊢ star a = cfc id a
|
rw [← cfc_star_id a (R := ℂ)]
|
case refine_2.intro
A : Type u_1
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra ℂ A
inst✝ : ContinuousFunctionalCalculus ℂ IsStarNormal
a : A
ha₁ : IsStarNormal a
ha₂ : SpectrumRestricts a ⇑Complex.reCLM
⊢ cfc (fun x => star x) a = cfc id a
|
43364a93391bdcfb
|
PadicInt.lift_sub_val_mem_span
|
Mathlib/NumberTheory/Padics/RingHoms.lean
|
theorem lift_sub_val_mem_span (r : R) (n : ℕ) :
lift f_compat r - (f n r).val ∈ (Ideal.span {(p : ℤ_[p]) ^ n})
|
R : Type u_1
inst✝ : NonAssocSemiring R
p : ℕ
f : (k : ℕ) → R →+* ZMod (p ^ k)
hp_prime : Fact (Nat.Prime p)
f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1
r : R
n : ℕ
⊢ (lift f_compat) r - ↑((f n) r).val ∈ Ideal.span {↑p ^ n}
|
obtain ⟨k, hk⟩ :=
limNthHom_spec f_compat r _
(show (0 : ℝ) < (p : ℝ) ^ (-n : ℤ) from zpow_pos (mod_cast hp_prime.1.pos) _)
|
case intro
R : Type u_1
inst✝ : NonAssocSemiring R
p : ℕ
f : (k : ℕ) → R →+* ZMod (p ^ k)
hp_prime : Fact (Nat.Prime p)
f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1
r : R
n k : ℕ
hk : ∀ n_1 ≥ k, ‖limNthHom f_compat r - ↑(nthHom f r n_1)‖ < ↑p ^ (-↑n)
⊢ (lift f_compat) r - ↑((f n) r).val ∈ Ideal.span {↑p ^ n}
|
86f8daac951c537a
|
Filter.extraction_forall_of_frequently
|
Mathlib/Order/Filter/AtTopBot/Basic.lean
|
theorem extraction_forall_of_frequently {P : ℕ → ℕ → Prop} (h : ∀ n, ∃ᶠ k in atTop, P n k) :
∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, P n (φ n)
|
P : ℕ → ℕ → Prop
h : ∀ (n : ℕ), ∃ᶠ (k : ℕ) in atTop, P n k
⊢ ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), P n (φ n)
|
simp only [frequently_atTop'] at h
|
P : ℕ → ℕ → Prop
h : ∀ (n a : ℕ), ∃ b > a, P n b
⊢ ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), P n (φ n)
|
5438ca62ebe37b43
|
fermatLastTheoremThree
|
Mathlib/NumberTheory/FLT/Three.lean
|
theorem fermatLastTheoremThree : FermatLastTheoremFor 3
|
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : a ^ 3 + b ^ 3 = ↑u * c ^ 3
S' : FermatLastTheoremForThreeGen.Solution' hζ :=
{ a := a, b := b, c := c, u := u, ha := ha, hb := hb, hc := hc, coprime := coprime, hcdvd := hcdvd, H := H }
⊢ False
|
obtain ⟨S, -⟩ := FermatLastTheoremForThreeGen.exists_Solution_of_Solution' S'
|
case intro
K : Type := CyclotomicField 3 ℚ
hζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) ↑3 := IsCyclotomicExtension.zeta_spec 3 ℚ K
this : NumberField K
a b c : NumberField.RingOfIntegers K
u : (NumberField.RingOfIntegers K)ˣ
hc : c ≠ 0
ha : ¬hζ.toInteger - 1 ∣ a
hb : ¬hζ.toInteger - 1 ∣ b
hcdvd : hζ.toInteger - 1 ∣ c
coprime : IsCoprime a b
H : a ^ 3 + b ^ 3 = ↑u * c ^ 3
S' : FermatLastTheoremForThreeGen.Solution' hζ :=
{ a := a, b := b, c := c, u := u, ha := ha, hb := hb, hc := hc, coprime := coprime, hcdvd := hcdvd, H := H }
S : FermatLastTheoremForThreeGen.Solution hζ
⊢ False
|
53f56643c6c85b50
|
MeasureTheory.Measure.exists_positive_of_not_mutuallySingular
|
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
|
theorem exists_positive_of_not_mutuallySingular (μ ν : Measure α) [IsFiniteMeasure μ]
[IsFiniteMeasure ν] (h : ¬ μ ⟂ₘ ν) :
∃ ε : ℝ≥0, 0 < ε ∧
∃ E : Set α, MeasurableSet E ∧ 0 < ν E
∧ ∀ A, MeasurableSet A → ε * ν (A ∩ E) ≤ μ (A ∩ E)
|
case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA : ℝ≥0
hA₃✝¹ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ν A
νA : ℝ≥0
hA₃✝ hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : 0 < νA
⊢ ∀ (b : ℝ≥0), 0 < b → μA ≤ b
|
intro c hc
|
case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ) ≤ ((1 / (↑n + 1)) • ν) (t ∩ (f n)ᶜ)
A : Set α := ⋂ n, (f n)ᶜ
hAmeas : MeasurableSet A
hA₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ A) ≤ ((1 / (↑n + 1)) • ν) (t ∩ A)
μA : ℝ≥0
hA₃✝¹ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ν A
νA : ℝ≥0
hA₃✝ hA₃ : ∀ (n : ℕ), ↑μA ≤ ↑(1 / (↑n + 1)) * ↑νA
hb : 0 < νA
c : ℝ≥0
hc : 0 < c
⊢ μA ≤ c
|
48437d57402f42cd
|
AList.toFinmap_eq
|
Mathlib/Data/Finmap.lean
|
theorem AList.toFinmap_eq {s₁ s₂ : AList β} :
toFinmap s₁ = toFinmap s₂ ↔ s₁.entries ~ s₂.entries
|
α : Type u
β : α → Type v
s₁ s₂ : AList β
⊢ ⟦s₁⟧ = ⟦s₂⟧ ↔ s₁.entries ~ s₂.entries
|
cases s₁
|
case mk
α : Type u
β : α → Type v
s₂ : AList β
entries✝ : List (Sigma β)
nodupKeys✝ : entries✝.NodupKeys
⊢ ⟦{ entries := entries✝, nodupKeys := nodupKeys✝ }⟧ = ⟦s₂⟧ ↔
{ entries := entries✝, nodupKeys := nodupKeys✝ }.entries ~ s₂.entries
|
5aad0d12e928d483
|
IsLocalization.isInteger_smul
|
Mathlib/RingTheory/Localization/Integer.lean
|
theorem isInteger_smul {a : R} {b : S} (hb : IsInteger R b) : IsInteger R (a • b)
|
case intro
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
a : R
b : S
b' : R
hb : (algebraMap R S) b' = b
⊢ IsInteger R (a • b)
|
use a * b'
|
case h
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
a : R
b : S
b' : R
hb : (algebraMap R S) b' = b
⊢ (algebraMap R S) (a * b') = a • b
|
92eaf71c2574e8d9
|
Std.DHashMap.Internal.List.insertList_perm_of_perm_first
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean
|
theorem insertList_perm_of_perm_first [BEq α] [EquivBEq α] {l1 l2 toInsert : List ((a : α) × β a)}
(h : Perm l1 l2) (distinct : DistinctKeys l1) :
Perm (insertList l1 toInsert) (insertList l2 toInsert)
|
case cons
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : EquivBEq α
hd : (a : α) × β a
tl : List ((a : α) × β a)
ih : ∀ {l1 l2 : List ((a : α) × β a)}, l1.Perm l2 → DistinctKeys l1 → (insertList l1 tl).Perm (insertList l2 tl)
l1 l2 : List ((a : α) × β a)
h : l1.Perm l2
distinct : DistinctKeys l1
⊢ (insertList l1 (hd :: tl)).Perm (insertList l2 (hd :: tl))
|
simp only [insertList]
|
case cons
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : EquivBEq α
hd : (a : α) × β a
tl : List ((a : α) × β a)
ih : ∀ {l1 l2 : List ((a : α) × β a)}, l1.Perm l2 → DistinctKeys l1 → (insertList l1 tl).Perm (insertList l2 tl)
l1 l2 : List ((a : α) × β a)
h : l1.Perm l2
distinct : DistinctKeys l1
⊢ (insertList (insertEntry hd.fst hd.snd l1) tl).Perm (insertList (insertEntry hd.fst hd.snd l2) tl)
|
4f56e4d2011c180d
|
LieIdeal.ker_incl
|
Mathlib/Algebra/Lie/Submodule.lean
|
theorem ker_incl : I.incl.ker = ⊥
|
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I : LieIdeal R L
⊢ I.incl.ker = ⊥
|
ext
|
case h
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I : LieIdeal R L
m✝ : ↥I
⊢ m✝ ∈ I.incl.ker ↔ m✝ ∈ ⊥
|
7fbddbcb1f875af1
|
FirstOrder.Language.Formula.realize_iExs
|
Mathlib/ModelTheory/Semantics.lean
|
theorem _root_.FirstOrder.Language.Formula.realize_iExs
[Finite γ] {φ : L.Formula (α ⊕ γ)} {v : α → M} : (φ.iExs γ).Realize v ↔
∃ (i : γ → M), φ.Realize (Sum.elim v i)
|
L : Language
M : Type w
inst✝¹ : L.Structure M
α : Type u'
γ : Type u_3
inst✝ : Finite γ
φ : L.Formula (α ⊕ γ)
v : α → M
e : γ ≃ Fin (Classical.choose ⋯) := Classical.choice ⋯
⊢ (∃ xs,
Realize φ (fun x => Sum.elim (fun x => v x) (fun x => xs (Fin.cast ⋯ ((Classical.choice ⋯) x))) x) fun x =>
xs (natAdd (Classical.choose ⋯) x)) ↔
∃ i, φ.Realize (Sum.elim v i)
|
refine Equiv.exists_congr ?_ ?_
|
case refine_1
L : Language
M : Type w
inst✝¹ : L.Structure M
α : Type u'
γ : Type u_3
inst✝ : Finite γ
φ : L.Formula (α ⊕ γ)
v : α → M
e : γ ≃ Fin (Classical.choose ⋯) := Classical.choice ⋯
⊢ (Fin (Classical.choose ⋯) → M) ≃ (γ → M)
case refine_2
L : Language
M : Type w
inst✝¹ : L.Structure M
α : Type u'
γ : Type u_3
inst✝ : Finite γ
φ : L.Formula (α ⊕ γ)
v : α → M
e : γ ≃ Fin (Classical.choose ⋯) := Classical.choice ⋯
⊢ ∀ (a : Fin (Classical.choose ⋯) → M),
(Realize φ (fun x => Sum.elim (fun x => v x) (fun x => a (Fin.cast ⋯ ((Classical.choice ⋯) x))) x) fun x =>
a (natAdd (Classical.choose ⋯) x)) ↔
φ.Realize (Sum.elim v (?refine_1 a))
|
edaad082a0094423
|
BitVec.ofNat_sub_ofNat
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem ofNat_sub_ofNat {n} (x y : Nat) : BitVec.ofNat n x - BitVec.ofNat n y = .ofNat n ((2^n - y % 2^n) + x)
|
case a
n x y : Nat
⊢ (BitVec.ofNat n x - BitVec.ofNat n y).toNat = (BitVec.ofNat n (2 ^ n - y % 2 ^ n + x)).toNat
|
simp [BitVec.ofNat, Fin.ofNat'_sub]
|
no goals
|
b9aac0b64a9a17bd
|
Polynomial.bernoulli_generating_function
|
Mathlib/NumberTheory/BernoulliPolynomials.lean
|
theorem bernoulli_generating_function (t : A) :
(mk fun n => aeval t ((1 / n ! : ℚ) • bernoulli n)) * (exp A - 1) =
PowerSeries.X * rescale t (exp A)
|
case h.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
hnp1 : IsUnit ↑(n + 1)!
⊢ (algebraMap ℚ A) ↑(n + 1)! *
∑ x ∈ range (n + 1),
(PowerSeries.coeff A x) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) *
(PowerSeries.coeff A (n + 1 - x)) (exp A - 1) =
t ^ n * (algebraMap ℚ A) (↑((n + 1) * n !) * (1 / ↑n !))
|
rw [cast_mul, mul_assoc,
mul_one_div_cancel (show (n ! : ℚ) ≠ 0 from cast_ne_zero.2 (factorial_ne_zero n)), mul_one,
mul_comm (t ^ n), ← aeval_monomial, cast_add, cast_one]
|
case h.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
hnp1 : IsUnit ↑(n + 1)!
⊢ (algebraMap ℚ A) ↑(n + 1)! *
∑ x ∈ range (n + 1),
(PowerSeries.coeff A x) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) *
(PowerSeries.coeff A (n + 1 - x)) (exp A - 1) =
(aeval t) ((monomial n) (↑n + 1))
|
5c1f5e37cc9d52d3
|
Complex.approx_Gamma_integral_tendsto_Gamma_integral
|
Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean
|
theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 - x / n) ^ n : ℝ) * (x : ℂ) ^ (s - 1)) atTop
(𝓝 <| Gamma s)
|
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n => (Ioc 0 ↑n).indicator fun x => ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n => f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
⊢ 0 ≤ ↑n
|
positivity
|
no goals
|
e883263ba9a935a5
|
Finset.support_sum_eq
|
Mathlib/Data/Finsupp/BigOperators.lean
|
theorem Finset.support_sum_eq [AddCommMonoid M] (s : Finset (ι →₀ M))
(hs : (s : Set (ι →₀ M)).PairwiseDisjoint Finsupp.support) :
(s.sum id).support = Finset.sup s Finsupp.support
|
case h.e'_2.h.e'_4
ι : Type u_1
M : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : AddCommMonoid M
s : Finset (ι →₀ M)
hs : (↑s).PairwiseDisjoint Finsupp.support
this : Multiset.Pairwise (Disjoint on Finsupp.support) s.val
⊢ s.sum id = s.val.sum
|
exact (Finset.sum_val _).symm
|
no goals
|
67f7aca84a3dab37
|
Equiv.Perm.card_support_eq_two
|
Mathlib/GroupTheory/Perm/Support.lean
|
theorem card_support_eq_two {f : Perm α} : #f.support = 2 ↔ IsSwap f
|
case mp.intro.intro.intro.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
h : #f.support = 2
x y : α
hmem : ¬x = y
hins : {x, y} = f.support
ht : #{y} = 1
a : α
key : ∀ (b : α), f b ≠ b ↔ b = x ∨ b = y
⊢ f a = (swap x y) a
|
by_cases ha : f a = a
|
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
h : #f.support = 2
x y : α
hmem : ¬x = y
hins : {x, y} = f.support
ht : #{y} = 1
a : α
key : ∀ (b : α), f b ≠ b ↔ b = x ∨ b = y
ha : f a = a
⊢ f a = (swap x y) a
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
h : #f.support = 2
x y : α
hmem : ¬x = y
hins : {x, y} = f.support
ht : #{y} = 1
a : α
key : ∀ (b : α), f b ≠ b ↔ b = x ∨ b = y
ha : ¬f a = a
⊢ f a = (swap x y) a
|
36f5f162aba5bd84
|
ApproximatesLinearOn.open_image
|
Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean
|
theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
(hs : IsOpen s) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : IsOpen (f '' s)
|
case inr.intro.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInverse
hc : c < f'symm.nnnorm⁻¹
hs : ∀ x ∈ s, ∃ i, 0 < i ∧ closedBall x i ⊆ s
x : E
hx : x ∈ s
ε : ℝ
ε0 : 0 < ε
hε : closedBall x ε ⊆ s
⊢ ∃ i, 0 < i ∧ closedBall (f x) i ⊆ f '' s
|
refine ⟨(f'symm.nnnorm⁻¹ - c) * ε, mul_pos (sub_pos.2 hc) ε0, ?_⟩
|
case inr.intro.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInverse
hc : c < f'symm.nnnorm⁻¹
hs : ∀ x ∈ s, ∃ i, 0 < i ∧ closedBall x i ⊆ s
x : E
hx : x ∈ s
ε : ℝ
ε0 : 0 < ε
hε : closedBall x ε ⊆ s
⊢ closedBall (f x) ((↑f'symm.nnnorm⁻¹ - ↑c) * ε) ⊆ f '' s
|
dff2560cbebf928f
|
PFun.id_comp
|
Mathlib/Data/PFun.lean
|
theorem id_comp (f : α →. β) : (PFun.id β).comp f = f :=
ext fun _ _ => by simp
|
α : Type u_1
β : Type u_2
f : α →. β
x✝¹ : α
x✝ : β
⊢ x✝ ∈ (PFun.id β).comp f x✝¹ ↔ x✝ ∈ f x✝¹
|
simp
|
no goals
|
206a34a628855fea
|
AlgebraicTopology.AlternatingFaceMapComplex.d_squared
|
Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean
|
theorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0
|
case h
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := Finset.filter (fun ij => ↑ij.2 ≤ ↑ij.1) Finset.univ
φ : (ij : P) → ij ∈ S → P := fun ij hij => (ij.2.castLT ⋯, ij.1.succ)
⊢ ∀ (a : P) (ha : a ∈ S),
((-1) ^ ↑a.2 • X.δ a.2) ≫ ((-1) ^ ↑a.1 • X.δ a.1) =
-((-1) ^ ↑(φ a ha).2 • X.δ (φ a ha).2) ≫ ((-1) ^ ↑(φ a ha).1 • X.δ (φ a ha).1)
|
rintro ⟨i, j⟩ hij
|
case h.mk
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := Finset.filter (fun ij => ↑ij.2 ≤ ↑ij.1) Finset.univ
φ : (ij : P) → ij ∈ S → P := fun ij hij => (ij.2.castLT ⋯, ij.1.succ)
i : Fin (n + 2)
j : Fin (n + 3)
hij : (i, j) ∈ S
⊢ ((-1) ^ ↑(i, j).2 • X.δ (i, j).2) ≫ ((-1) ^ ↑(i, j).1 • X.δ (i, j).1) =
-((-1) ^ ↑(φ (i, j) hij).2 • X.δ (φ (i, j) hij).2) ≫ ((-1) ^ ↑(φ (i, j) hij).1 • X.δ (φ (i, j) hij).1)
|
795ab48b2006c7af
|
Metric.closedBall_subset_cthickening_singleton
|
Mathlib/Topology/MetricSpace/Thickening.lean
|
theorem closedBall_subset_cthickening_singleton {α : Type*} [PseudoMetricSpace α] (x : α) (δ : ℝ) :
closedBall x δ ⊆ cthickening δ ({x} : Set α)
|
α : Type u_2
inst✝ : PseudoMetricSpace α
x : α
δ : ℝ
⊢ closedBall x δ ⊆ cthickening δ {x}
|
rcases lt_or_le δ 0 with (hδ | hδ)
|
case inl
α : Type u_2
inst✝ : PseudoMetricSpace α
x : α
δ : ℝ
hδ : δ < 0
⊢ closedBall x δ ⊆ cthickening δ {x}
case inr
α : Type u_2
inst✝ : PseudoMetricSpace α
x : α
δ : ℝ
hδ : 0 ≤ δ
⊢ closedBall x δ ⊆ cthickening δ {x}
|
dbe71a115118f14a
|
Real.rpow_le_rpow_of_exponent_ge_of_imp
|
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
theorem rpow_le_rpow_of_exponent_ge_of_imp (hx0 : 0 ≤ x) (hx1 : x ≤ 1) (hyz : z ≤ y)
(h : x = 0 → y = 0 → z = 0) :
x ^ y ≤ x ^ z
|
x y z : ℝ
hx0 : 0 ≤ x
hx1 : x ≤ 1
hyz : z ≤ y
h : x = 0 → y = 0 → z = 0
⊢ x ^ y ≤ x ^ z
|
rcases eq_or_lt_of_le hx0 with (rfl | hx0')
|
case inl
y z : ℝ
hyz : z ≤ y
hx0 : 0 ≤ 0
hx1 : 0 ≤ 1
h : 0 = 0 → y = 0 → z = 0
⊢ 0 ^ y ≤ 0 ^ z
case inr
x y z : ℝ
hx0 : 0 ≤ x
hx1 : x ≤ 1
hyz : z ≤ y
h : x = 0 → y = 0 → z = 0
hx0' : 0 < x
⊢ x ^ y ≤ x ^ z
|
148ee6ba8dfad2de
|
cfcₙAux_mem_range_inr
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.lean
|
lemma cfcₙAux_mem_range_inr (f : C(σₙ 𝕜 a, 𝕜)₀) :
cfcₙAux hp₁ a ha f ∈ NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A)
|
case refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p₁
inst✝ : CompleteSpace A
f : C(↑(σₙ 𝕜 a), 𝕜)₀
h₁ : (cfcₙAux ⋯ a ha) f ∈ closure (⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id ⋯}))
⊢ IsClosed ↑(NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A))
|
have : Continuous (Unitization.fst (R := 𝕜) (A := A)) :=
Unitization.uniformEquivProd.continuous.fst
|
case refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p₁
inst✝ : CompleteSpace A
f : C(↑(σₙ 𝕜 a), 𝕜)₀
h₁ : (cfcₙAux ⋯ a ha) f ∈ closure (⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id ⋯}))
this : Continuous Unitization.fst
⊢ IsClosed ↑(NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A))
|
694a4ae2d7053273
|
Plausible.InjectiveFunction.applyId_injective
|
Mathlib/Testing/Plausible/Functions.lean
|
theorem applyId_injective [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs) (h₁ : xs ~ ys) :
Injective.{u + 1, u + 1} (List.applyId (xs.zip ys))
|
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
i : ℕ
h : ys[i]? = some (applyId (xs.zip ys) y)
hx : xs[i]? = some x
j : ℕ
hy : xs[j]? = some y
h₂ : xs.length = ys.length
⊢ i < xs.length
|
rw [List.getElem?_eq_some_iff] at hx
|
α : Type u
inst✝ : DecidableEq α
xs ys : List α
h₀ : xs.Nodup
h₁ : xs ~ ys
x y : α
i : ℕ
h : ys[i]? = some (applyId (xs.zip ys) y)
hx : ∃ (h : i < xs.length), xs[i] = x
j : ℕ
hy : xs[j]? = some y
h₂ : xs.length = ys.length
⊢ i < xs.length
|
159dee04a35ae3d2
|
MeasureTheory.tendsto_Lp_of_tendsto_ae_of_meas
|
Mathlib/MeasureTheory/Function/UnifTight.lean
|
theorem tendsto_Lp_of_tendsto_ae_of_meas (hp : 1 ≤ p) (hp' : p ≠ ∞)
{f : ℕ → α → β} {g : α → β} (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
(hg' : MemLp g p μ) (hui : UnifIntegrable f p μ) (hut : UnifTight f p μ)
(hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)
|
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ⊤
hμ : μ = 0
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε
|
rw [hμ]
|
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ⊤
hμ : μ = 0
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p 0 ≤ ε
|
77334843f098ce11
|
Real.cos_nonneg_of_mem_Icc
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
theorem cos_nonneg_of_mem_Icc {x : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) : 0 ≤ cos x :=
sin_add_pi_div_two x ▸ sin_nonneg_of_mem_Icc ⟨by linarith [hx.1], by linarith [hx.2]⟩
|
x : ℝ
hx : x ∈ Icc (-(π / 2)) (π / 2)
⊢ 0 ≤ x + π / 2
|
linarith [hx.1]
|
no goals
|
61947baea5937182
|
ContMDiffWithinAt.comp
|
Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
|
theorem ContMDiffWithinAt.comp {t : Set M'} {g : M' → M''} (x : M)
(hg : ContMDiffWithinAt I' I'' n g t (f x)) (hf : ContMDiffWithinAt I I' n f s x)
(st : MapsTo f s t) : ContMDiffWithinAt I I'' n (g ∘ f) s x
|
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
E'' : Type u_8
inst✝⁶ : NormedAddCommGroup E''
inst✝⁵ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝⁴ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝³ : TopologicalSpace M''
inst✝² : ChartedSpace H M
inst✝¹ : ChartedSpace H' M'
inst✝ : ChartedSpace H'' M''
f : M → M'
s : Set M
n : WithTop ℕ∞
t : Set M'
g : M' → M''
x : M
hg :
ContinuousWithinAt g t (f x) ∧
ContDiffWithinAt 𝕜 n (↑(extChartAt I'' (g (f x))) ∘ g ∘ ↑(extChartAt I' (f x)).symm)
(↑(extChartAt I' (f x)).symm ⁻¹' t ∩ range ↑I') (↑(extChartAt I' (f x)) (f x))
hf :
ContinuousWithinAt f s x ∧
ContDiffWithinAt 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
(↑(extChartAt I x) x)
st : MapsTo f s t
⊢ ContinuousWithinAt (g ∘ f) s x ∧
ContDiffWithinAt 𝕜 n (↑(extChartAt I'' ((g ∘ f) x)) ∘ (g ∘ f) ∘ ↑(extChartAt I x).symm)
(↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)
|
refine ⟨hg.1.comp hf.1 st, ?_⟩
|
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
E'' : Type u_8
inst✝⁶ : NormedAddCommGroup E''
inst✝⁵ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝⁴ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝³ : TopologicalSpace M''
inst✝² : ChartedSpace H M
inst✝¹ : ChartedSpace H' M'
inst✝ : ChartedSpace H'' M''
f : M → M'
s : Set M
n : WithTop ℕ∞
t : Set M'
g : M' → M''
x : M
hg :
ContinuousWithinAt g t (f x) ∧
ContDiffWithinAt 𝕜 n (↑(extChartAt I'' (g (f x))) ∘ g ∘ ↑(extChartAt I' (f x)).symm)
(↑(extChartAt I' (f x)).symm ⁻¹' t ∩ range ↑I') (↑(extChartAt I' (f x)) (f x))
hf :
ContinuousWithinAt f s x ∧
ContDiffWithinAt 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)
(↑(extChartAt I x) x)
st : MapsTo f s t
⊢ ContDiffWithinAt 𝕜 n (↑(extChartAt I'' ((g ∘ f) x)) ∘ (g ∘ f) ∘ ↑(extChartAt I x).symm)
(↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)
|
21a30820f7079ed0
|
IsIntegrallyClosed.pow_dvd_pow_iff
|
Mathlib/RingTheory/IntegralClosure/IntegrallyClosed.lean
|
theorem pow_dvd_pow_iff [IsDomain R] [IsIntegrallyClosed R]
{n : ℕ} (hn : n ≠ 0) {a b : R} : a ^ n ∣ b ^ n ↔ a ∣ b
|
case neg.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
hy : IsIntegral R y
k : R
hk : (algebraMap R K) k = y
⊢ a ∣ b
|
refine ⟨k, IsFractionRing.injective R K ?_⟩
|
case neg.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
hy : IsIntegral R y
k : R
hk : (algebraMap R K) k = y
⊢ (algebraMap R K) b = (algebraMap R K) (a * k)
|
fff7a7a39e81336f
|
MvPolynomial.weightedDegree_eq_zero_iff
|
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
|
theorem weightedDegree_eq_zero_iff [CanonicallyOrderedAdd M]
(hw : NonTorsionWeight w) {m : σ →₀ ℕ} :
weight w m = 0 ↔ ∀ x : σ, m x = 0
|
case h.mpr
M : Type u_2
σ : Type u_3
inst✝¹ : OrderedAddCommMonoid M
w : σ → M
inst✝ : CanonicallyOrderedAdd M
hw : NonTorsionWeight w
m : σ →₀ ℕ
x : σ
hax : m x = 0
a✝ : m x ≠ 0
⊢ m x • w x = 0
|
simp only [hax, zero_smul]
|
no goals
|
2175db668788ba02
|
CategoryTheory.presheafHom_isSheafFor
|
Mathlib/CategoryTheory/Sites/SheafHom.lean
|
lemma presheafHom_isSheafFor :
Presieve.IsSheafFor (presheafHom F G) S.arrows
|
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y₁ Y₂ : Over X
φ : Y₂ ⟶ Y₁
Z : Over Y₂.left
hZ : (Sieve.pullback Y₂.hom S).arrows Z.hom
⊢ S.arrows ((Z.hom ≫ φ.left) ≫ Y₁.hom)
|
simpa using hZ
|
no goals
|
03e610e11f4d0fc5
|
intervalIntegral.integral_comp_smul_deriv'''
|
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
theorem integral_comp_smul_deriv''' {f f' : ℝ → ℝ} {g : ℝ → G} (hf : ContinuousOn f [[a, b]])
(hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x)
(hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))) (hg1 : IntegrableOn g (f '' [[a, b]]))
(hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]) :
(∫ x in a..b, f' x • (g ∘ f) x) = ∫ u in f a..f b, g u
|
case intro.intro
a b : ℝ
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f f' : ℝ → ℝ
g : ℝ → G
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (a ⊓ b) (a ⊔ b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] volume
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (a ⊓ b) (a ⊔ b)
c : ℝ
hc : c ∈ Ioo (a ⊓ b) x
d : ℝ
hd : d ∈ Ioo x (a ⊔ b)
cdsub : [[c, d]] ⊆ Ioo (a ⊓ b) (a ⊔ b)
⊢ HasDerivWithinAt (fun u => ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x
|
replace hg_cont := hg_cont.mono (image_subset f cdsub)
|
case intro.intro
a b : ℝ
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f f' : ℝ → ℝ
g : ℝ → G
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] volume
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
x : ℝ
hx : x ∈ Ioo (a ⊓ b) (a ⊔ b)
c : ℝ
hc : c ∈ Ioo (a ⊓ b) x
d : ℝ
hd : d ∈ Ioo x (a ⊔ b)
cdsub : [[c, d]] ⊆ Ioo (a ⊓ b) (a ⊔ b)
hg_cont : ContinuousOn g (f '' [[c, d]])
⊢ HasDerivWithinAt (fun u => ∫ (t : ℝ) in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x
|
c4728850936608e5
|
Unitization.quasispectrum_eq_spectrum_inr'
|
Mathlib/Algebra/Algebra/Quasispectrum.lean
|
lemma quasispectrum_eq_spectrum_inr' (R S : Type*) {A : Type*} [Semifield R]
[Field S] [NonUnitalRing A] [Algebra R S] [Module S A] [IsScalarTower S A A]
[SMulCommClass S A A] [Module R A] [IsScalarTower R S A] (a : A) :
quasispectrum R a = spectrum R (a : Unitization S A)
|
case h
R : Type u_3
S : Type u_4
A : Type u_5
inst✝⁸ : Semifield R
inst✝⁷ : Field S
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Algebra R S
inst✝⁴ : Module S A
inst✝³ : IsScalarTower S A A
inst✝² : SMulCommClass S A A
inst✝¹ : Module R A
inst✝ : IsScalarTower R S A
a : A
r : R
⊢ r ∈ quasispectrum R a ↔ r ∈ spectrum R ↑a
|
have := Set.singleton_subset_iff.mpr (zero_mem_spectrum_inr R S a)
|
case h
R : Type u_3
S : Type u_4
A : Type u_5
inst✝⁸ : Semifield R
inst✝⁷ : Field S
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Algebra R S
inst✝⁴ : Module S A
inst✝³ : IsScalarTower S A A
inst✝² : SMulCommClass S A A
inst✝¹ : Module R A
inst✝ : IsScalarTower R S A
a : A
r : R
this : {0} ⊆ spectrum R ↑a
⊢ r ∈ quasispectrum R a ↔ r ∈ spectrum R ↑a
|
be0e30794ed1ae31
|
AlgebraicGeometry.Scheme.RationalMap.toRationalMap_toPartialMap
|
Mathlib/AlgebraicGeometry/RationalMap.lean
|
@[simp]
lemma RationalMap.toRationalMap_toPartialMap [IsReduced X] [Y.IsSeparated]
(f : X ⤏ Y) : f.toPartialMap.toRationalMap = f
|
X Y : Scheme
inst✝¹ : IsReduced X
inst✝ : Y.IsSeparated
f : X.PartialMap Y
⊢ f.toRationalMap.toPartialMap.toRationalMap = (f.toRationalMap.toPartialMap.restrict f.domain ⋯ ⋯).toRationalMap
|
simp
|
no goals
|
a4d2e7ea40074dfe
|
Nat.shiftLeft_eq
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Basic.lean
|
theorem shiftLeft_eq (a b : Nat) : a <<< b = a * 2 ^ b :=
match b with
| 0 => (Nat.mul_one _).symm
| b+1 => (shiftLeft_eq _ b).trans <| by
simp [Nat.pow_succ, Nat.mul_assoc, Nat.mul_left_comm, Nat.mul_comm]
|
a b✝ b : Nat
⊢ 2 * a * 2 ^ b = a * 2 ^ (b + 1)
|
simp [Nat.pow_succ, Nat.mul_assoc, Nat.mul_left_comm, Nat.mul_comm]
|
no goals
|
1495ef5ce36bd66c
|
List.mapM_toArray
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem mapM_toArray [Monad m] [LawfulMonad m] (f : α → m β) (l : List α) :
l.toArray.mapM f = List.toArray <$> l.mapM f
|
case cons
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → m β
a : α
l : List α
init : Array β
ih : ∀ (init : Array β), List.foldlM (fun bs a => bs.push <$> f a) init l = (fun x => init ++ x.toArray) <$> mapM' f l
⊢ List.foldlM (fun bs a => bs.push <$> f a) init (a :: l) =
(fun x => init ++ x.toArray) <$> do
let __do_lift ← f a
let __do_lift_1 ← mapM' f l
pure (__do_lift :: __do_lift_1)
|
simp [ih]
|
no goals
|
a3f66bae8e466bf7
|
Nat.Prime.primeFactors
|
Mathlib/Data/Nat/PrimeFin.lean
|
@[simp] protected lemma Prime.primeFactors (hp : p.Prime) : p.primeFactors = {p}
|
p : ℕ
hp : Prime p
⊢ p.primeFactors = {p}
|
simp [Nat.primeFactors, primeFactorsList_prime hp]
|
no goals
|
8bea25087cfd338b
|
MeasureTheory.lintegral_tsum
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
theorem lintegral_tsum [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ i, AEMeasurable (f i) μ) :
∫⁻ a, ∑' i, f i a ∂μ = ∑' i, ∫⁻ a, f i a ∂μ
|
case h.left
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (i : β), AEMeasurable (f i) μ
s t : Finset β
⊢ (fun x1 x2 => x1 ≤ x2) ((fun s a => ∑ i ∈ s, f i a) s) ((fun s a => ∑ i ∈ s, f i a) (s ∪ t))
|
exact fun a => Finset.sum_le_sum_of_subset Finset.subset_union_left
|
no goals
|
a9b7b8322de28144
|
IsPrimitiveRoot.minpoly_eq_pow_coprime
|
Mathlib/RingTheory/RootsOfUnity/Minpoly.lean
|
theorem minpoly_eq_pow_coprime {m : ℕ} (hcop : Nat.Coprime m n) :
minpoly ℤ μ = minpoly ℤ (μ ^ m)
|
case refine_2
K : Type u_1
inst✝² : CommRing K
μ : K
inst✝¹ : IsDomain K
inst✝ : CharZero K
m n✝ u : ℕ
hunit : IsUnit u
h✝ : IsPrimitiveRoot μ n✝
hcop✝ : u.Coprime n✝
⊢ minpoly ℤ μ = minpoly ℤ (μ ^ u)
|
congr
|
case refine_2.e_x
K : Type u_1
inst✝² : CommRing K
μ : K
inst✝¹ : IsDomain K
inst✝ : CharZero K
m n✝ u : ℕ
hunit : IsUnit u
h✝ : IsPrimitiveRoot μ n✝
hcop✝ : u.Coprime n✝
⊢ μ = μ ^ u
|
ba6ce0a8c0aa9e21
|
legendreSym.at_two
|
Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean
|
theorem at_two (hp : p ≠ 2) : legendreSym p 2 = χ₈ p
|
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : p ≠ 2
⊢ legendreSym p 2 = χ₈ ↑p
|
have : (2 : ZMod p) = (2 : ℤ) := by norm_cast
|
p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : p ≠ 2
this : 2 = ↑2
⊢ legendreSym p 2 = χ₈ ↑p
|
bcc80ad3cfdac489
|
Std.DHashMap.Raw.get?_ofList_of_mem
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean
|
theorem get?_ofList_of_mem [LawfulBEq α]
{l : List ((a : α) × β a)} {k k' : α} (k_beq : k == k') {v : β k}
(distinct : l.Pairwise (fun a b => (a.1 == b.1) = false))
(mem : ⟨k, v⟩ ∈ l) :
(ofList l).get? k' = some (cast (by congr; apply LawfulBEq.eq_of_beq k_beq) v)
|
α : Type u
β : α → Type v
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : LawfulBEq α
l : List ((a : α) × β a)
k k' : α
k_beq : (k == k') = true
v : β k
distinct : List.Pairwise (fun a b => (a.fst == b.fst) = false) l
mem : ⟨k, v⟩ ∈ l
⊢ (ofList l).get? k' = some (cast ⋯ v)
|
simp_to_raw using Raw₀.get?_insertMany_empty_list_of_mem
|
no goals
|
576fda0473bbbdeb
|
Nat.primeFactors_eq_empty
|
Mathlib/Data/Nat/PrimeFin.lean
|
@[simp] lemma primeFactors_eq_empty : n.primeFactors = ∅ ↔ n = 0 ∨ n = 1
|
case mp
n : ℕ
⊢ n ≠ 0 ∧ n ≠ 1 → n.primeFactors ≠ ∅
|
rintro hn
|
case mp
n : ℕ
hn : n ≠ 0 ∧ n ≠ 1
⊢ n.primeFactors ≠ ∅
|
9bfb50858a6b3406
|
List.cycleType_formPerm
|
Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
|
theorem cycleType_formPerm (hl : Nodup l) (hn : 2 ≤ l.length) :
cycleType l.attach.formPerm = {l.length}
|
α : Type u_1
inst✝ : DecidableEq α
l : List α
hl : l.attach.Nodup
hn : 2 ≤ l.attach.length
x : { x // x ∈ l }
h : l.attach = [x]
⊢ False
|
simp [h, Nat.succ_le_succ_iff] at hn
|
no goals
|
c111839ef78929d8
|
Fin.ofNat'_val_eq_self
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean
|
theorem ofNat'_val_eq_self [NeZero n] (x : Fin n) : (Fin.ofNat' n x) = x
|
case h
n : Nat
inst✝ : NeZero n
x : Fin n
⊢ ↑(Fin.ofNat' n ↑x) = ↑x
|
rw [val_ofNat', Nat.mod_eq_of_lt]
|
case h
n : Nat
inst✝ : NeZero n
x : Fin n
⊢ ↑x < n
|
bf93afaa12d7610d
|
Polynomial.coeff_mul_degree_add_degree
|
Mathlib/Algebra/Polynomial/Degree/Operations.lean
|
theorem coeff_mul_degree_add_degree (p q : R[X]) :
coeff (p * q) (natDegree p + natDegree q) = leadingCoeff p * leadingCoeff q :=
calc
coeff (p * q) (natDegree p + natDegree q) =
∑ x ∈ antidiagonal (natDegree p + natDegree q), coeff p x.1 * coeff q x.2 :=
coeff_mul _ _ _
_ = coeff p (natDegree p) * coeff q (natDegree q)
|
case refine_1
R : Type u
inst✝ : Semiring R
p q : R[X]
⊢ ∀ b ∈ antidiagonal (p.natDegree + q.natDegree), b ≠ (p.natDegree, q.natDegree) → p.coeff b.1 * q.coeff b.2 = 0
|
rintro ⟨i, j⟩ h₁ h₂
|
case refine_1.mk
R : Type u
inst✝ : Semiring R
p q : R[X]
i j : ℕ
h₁ : (i, j) ∈ antidiagonal (p.natDegree + q.natDegree)
h₂ : (i, j) ≠ (p.natDegree, q.natDegree)
⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0
|
90cfb4e75ea0e0dd
|
Function.funext_iff_of_subsingleton
|
Mathlib/Logic/Function/Basic.lean
|
lemma funext_iff_of_subsingleton [Subsingleton α] {g : α → β} (x y : α) :
f x = g y ↔ f = g
|
case refine_2
α : Sort u_1
β : Sort u_2
f : α → β
inst✝ : Subsingleton α
g : α → β
x y : α
h : f = g
⊢ f x = g y
|
rw [h, Subsingleton.elim x y]
|
no goals
|
ba8d242339c93243
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastUmod.denote_go_eq_divRec_r
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Umod.lean
|
theorem denote_go_eq_divRec_r (aig : AIG α) (assign : α → Bool) (curr : Nat) (lhs rhs rbv qbv : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d q r : AIG.RefVec aig w) (wn wr : Nat)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, d.get idx hidx, assign⟧ = rhs.getLsbD idx)
(hq : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, q.get idx hidx, assign⟧ = qbv.getLsbD idx)
(hr : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, r.get idx hidx, assign⟧ = rbv.getLsbD idx)
(hfalse : ⟦aig, falseRef, assign⟧ = false)
(htrue : ⟦aig, trueRef, assign⟧ = true)
:
∀ (idx : Nat) (hidx : idx < w),
⟦
(go aig curr falseRef trueRef n d wn wr q r).aig,
(go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx,
assign
⟧
=
(BitVec.divRec curr { n := lhs, d := rhs} { wn, wr, q := qbv, r := rbv }).r.getLsbD idx
|
case hr.hfalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx) →
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx) →
⟦assign, { aig := aig, ref := falseRef }⟧ = false →
⟦assign, { aig := aig, ref := trueRef }⟧ = true →
∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (go aig curr falseRef trueRef n d wn wr q r).aig,
ref := (go aig curr falseRef trueRef n d wn wr q r).r.get idx hidx }⟧ =
(BitVec.divRec curr { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).r.getLsbD idx
aig : AIG α
rbv qbv : BitVec w
falseRef trueRef : aig.Ref
n d q r : aig.RefVec w
wn wr : Nat
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hq : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
htrue : ⟦assign, { aig := aig, ref := trueRef }⟧ = true
idx✝ : Nat
hidx✝ : idx✝ < w
hdiscr :
{ wn := wn, wr := wr, q := qbv, r := rbv }.r.shiftConcat
(lhs.getLsbD ({ wn := wn, wr := wr, q := qbv, r := rbv }.wn - 1)) <
rhs
idx : Nat
hidx : idx < w
⊢ ⟦assign, { aig := aig, ref := falseRef }⟧ = false
|
exact hfalse
|
no goals
|
123e2e165ae62131
|
CategoryTheory.Limits.MonoFactorisation.ext
|
Mathlib/CategoryTheory/Limits/Shapes/Images.lean
|
theorem ext {F F' : MonoFactorisation f} (hI : F.I = F'.I)
(hm : F.m = eqToHom hI ≫ F'.m) : F = F'
|
case mk.mk.refl.e_e
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f : X ⟶ Y
I✝ : C
Fm : I✝ ⟶ Y
m_mono✝¹ : Mono Fm
e✝¹ : X ⟶ I✝
Ffac : e✝¹ ≫ Fm = f
Fm' : I✝ ⟶ Y
m_mono✝ : Mono Fm'
e✝ : X ⟶ I✝
Ffac' : e✝ ≫ Fm' = f
hm : Fm = Fm'
⊢ e✝¹ ≫ Fm = e✝ ≫ Fm
|
rw [Ffac, hm, Ffac']
|
no goals
|
806c385ef78a3681
|
Ideal.isPrime_of_prime
|
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
theorem Ideal.isPrime_of_prime {P : Ideal A} (h : Prime P) : IsPrime P
|
case refine_1
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
P : Ideal A
h : Prime P
⊢ P ≠ ⊤
|
rintro rfl
|
case refine_1
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
h : Prime ⊤
⊢ False
|
53a6e4534a408bba
|
Ergodic.ae_empty_or_univ_of_ae_le_preimage'
|
Mathlib/Dynamics/Ergodic/Ergodic.lean
|
theorem ae_empty_or_univ_of_ae_le_preimage' (hf : Ergodic f μ) (hs : NullMeasurableSet s μ)
(hs' : s ≤ᵐ[μ] f ⁻¹' s) (h_fin : μ s ≠ ∞) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ
|
α : Type u_1
m : MeasurableSpace α
s : Set α
f : α → α
μ : Measure α
hf : Ergodic f μ
hs : NullMeasurableSet s μ
hs' : s ≤ᶠ[ae μ] f ⁻¹' s
h_fin : μ (f ⁻¹' s) ≠ ⊤
⊢ s =ᶠ[ae μ] ∅ ∨ s =ᶠ[ae μ] univ
|
refine hf.quasiErgodic.ae_empty_or_univ₀ hs ?_
|
α : Type u_1
m : MeasurableSpace α
s : Set α
f : α → α
μ : Measure α
hf : Ergodic f μ
hs : NullMeasurableSet s μ
hs' : s ≤ᶠ[ae μ] f ⁻¹' s
h_fin : μ (f ⁻¹' s) ≠ ⊤
⊢ f ⁻¹' s =ᶠ[ae μ] s
|
0e46e73dd39f99ee
|
Polynomial.IsUnitTrinomial.irreducible_aux1
|
Mathlib/Algebra/Polynomial/UnitTrinomial.lean
|
theorem irreducible_aux1 {k m n : ℕ} (hkm : k < m) (hmn : m < n) (u v w : Units ℤ)
(hp : p = trinomial k m n (u : ℤ) v w) :
C (v : ℤ) * (C (u : ℤ) * X ^ (m + n) + C (w : ℤ) * X ^ (n - m + k + n)) =
⟨Finsupp.filter (· ∈ Set.Ioo (k + n) (n + n)) (p * p.mirror).toFinsupp⟩
|
case h
p : ℤ[X]
k m n : ℕ
hkm : k < m
hmn : m < n
u v w : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
key : n - m + k < n
⊢ m + n ∈ Set.Ioo (k + n) (n + n)
|
exact ⟨add_lt_add_right hkm n, add_lt_add_right hmn n⟩
|
no goals
|
fb6ff88b1ff3f5f1
|
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