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
|
|---|---|---|---|---|---|---|
Submodule.closure_induction
|
Mathlib/LinearAlgebra/Span/Defs.lean
|
theorem closure_induction {p : (x : M) → x ∈ span R s → Prop}
(zero : p 0 (Submodule.zero_mem _))
(add : ∀ x y hx hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›))
(smul_mem : ∀ (r x) (h : x ∈ s), p (r • x) (Submodule.smul_mem _ _ <| subset_span h)) {x}
(hx : x ∈ span R s) : p x hx
|
R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
p : (x : M) → x ∈ span R s → Prop
zero : p 0 ⋯
add : ∀ (x y : M) (hx : x ∈ span R s) (hy : y ∈ span R s), p x hx → p y hy → p (x + y) ⋯
smul_mem : ∀ (r : R) (x : M) (h : x ∈ s), p (r • x) ⋯
x : M
hx : x ∈ span R s
key : ∀ {v : M}, v ∈ span R s ↔ v ∈ closure (univ • s)
⊢ ∀ (x : M) (h : x ∈ univ • s), (fun x hx => p x ⋯) x ⋯
|
rintro - ⟨r, -, x, hx, rfl⟩
|
case intro.intro.intro.intro
R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
p : (x : M) → x ∈ span R s → Prop
zero : p 0 ⋯
add : ∀ (x y : M) (hx : x ∈ span R s) (hy : y ∈ span R s), p x hx → p y hy → p (x + y) ⋯
smul_mem : ∀ (r : R) (x : M) (h : x ∈ s), p (r • x) ⋯
x✝ : M
hx✝ : x✝ ∈ span R s
key : ∀ {v : M}, v ∈ span R s ↔ v ∈ closure (univ • s)
r : R
left✝ : r ∈ univ
x : M
hx : x ∈ s
⊢ p ((fun x1 x2 => x1 • x2) r x) ⋯
|
3007157e1dc12f6a
|
Std.DHashMap.Internal.Raw₀.wfImp_alterₘ
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean
|
theorem wfImp_alterₘ [BEq α] [Hashable α] [LawfulBEq α] {m : Raw₀ α β} (h : Raw.WFImp m.1) {a : α}
{f : Option (β a) → Option (β a)} : Raw.WFImp (m.alterₘ a f).1 where
buckets_hash_self := isHashSelf_alterₘ m h a f
distinct := DistinctKeys.perm (toListModel_alterₘ h) h.distinct.alterKey
size_eq
|
α : Type u
β : α → Type v
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : LawfulBEq α
m : Raw₀ α β
h : Raw.WFImp m.val
a : α
f : Option (β a) → Option (β a)
h₁ : ¬containsKey a (toListModel m.val.buckets) = true
⊢ (match f none with
| none => m
| some b => (m.consₘ a b).expandIfNecessary).val.size =
(alterKey a f (toListModel m.val.buckets)).length
|
rw [alterKey]
|
α : Type u
β : α → Type v
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : LawfulBEq α
m : Raw₀ α β
h : Raw.WFImp m.val
a : α
f : Option (β a) → Option (β a)
h₁ : ¬containsKey a (toListModel m.val.buckets) = true
⊢ (match f none with
| none => m
| some b => (m.consₘ a b).expandIfNecessary).val.size =
(match f (getValueCast? a (toListModel m.val.buckets)) with
| none => eraseKey a (toListModel m.val.buckets)
| some v => insertEntry a v (toListModel m.val.buckets)).length
|
c03239080d29fbda
|
Polynomial.support_integralNormalization
|
Mathlib/RingTheory/Polynomial/IntegralNormalization.lean
|
theorem support_integralNormalization {f : R[X]} :
(integralNormalization f).support = f.support
|
R : Type u
inst✝¹ : Semiring R
inst✝ : IsCancelMulZero R
f : R[X]
a✝ : Nontrivial R
⊢ f.integralNormalization.support = f.support
|
have : IsDomain R := {}
|
R : Type u
inst✝¹ : Semiring R
inst✝ : IsCancelMulZero R
f : R[X]
a✝ : Nontrivial R
this : IsDomain R
⊢ f.integralNormalization.support = f.support
|
9178f9ebeb3af80a
|
LinearPMap.supSpanSingleton_apply_mk
|
Mathlib/LinearAlgebra/LinearPMap.lean
|
theorem supSpanSingleton_apply_mk (f : E →ₗ.[K] F) (x : E) (y : F) (hx : x ∉ f.domain) (x' : E)
(hx' : x' ∈ f.domain) (c : K) :
f.supSpanSingleton x y hx
⟨x' + c • x, mem_sup.2 ⟨x', hx', _, mem_span_singleton.2 ⟨c, rfl⟩, rfl⟩⟩ =
f ⟨x', hx'⟩ + c • y
|
E : Type u_2
inst✝⁴ : AddCommGroup E
F : Type u_3
inst✝³ : AddCommGroup F
K : Type u_5
inst✝² : DivisionRing K
inst✝¹ : Module K E
inst✝ : Module K F
f : E →ₗ.[K] F
x : E
y : F
hx : x ∉ f.domain
x' : E
hx' : x' ∈ f.domain
c : K
⊢ ↑(f.supSpanSingleton x y hx) ⟨x' + c • x, ⋯⟩ = ↑f ⟨x', hx'⟩ + c • y
|
erw [sup_apply _ ⟨x', hx'⟩ ⟨c • x, _⟩, mkSpanSingleton'_apply]
|
case h
E : Type u_2
inst✝⁴ : AddCommGroup E
F : Type u_3
inst✝³ : AddCommGroup F
K : Type u_5
inst✝² : DivisionRing K
inst✝¹ : Module K E
inst✝ : Module K F
f : E →ₗ.[K] F
x : E
y : F
hx : x ∉ f.domain
x' : E
hx' : x' ∈ f.domain
c : K
⊢ c • x ∈ (mkSpanSingleton' x y ⋯).domain
case hz
E : Type u_2
inst✝⁴ : AddCommGroup E
F : Type u_3
inst✝³ : AddCommGroup F
K : Type u_5
inst✝² : DivisionRing K
inst✝¹ : Module K E
inst✝ : Module K F
f : E →ₗ.[K] F
x : E
y : F
hx : x ∉ f.domain
x' : E
hx' : x' ∈ f.domain
c : K
⊢ ↑⟨x', hx'⟩ + ↑⟨c • x, ?h⟩ = ↑⟨x' + c • x, ⋯⟩
|
80d977329e541b68
|
Polynomial.ofFinsupp_pow
|
Mathlib/Algebra/Polynomial/Basic.lean
|
theorem ofFinsupp_pow (a) (n : ℕ) : (⟨a ^ n⟩ : R[X]) = ⟨a⟩ ^ n
|
R : Type u
inst✝ : Semiring R
a : R[ℕ]
n : ℕ
⊢ { toFinsupp := a ^ n } = { toFinsupp := a } ^ n
|
change _ = npowRec n _
|
R : Type u
inst✝ : Semiring R
a : R[ℕ]
n : ℕ
⊢ { toFinsupp := a ^ n } = npowRec n { toFinsupp := a }
|
ffb2239c238f94f0
|
IsGenericPoint.disjoint_iff
|
Mathlib/Topology/Sober.lean
|
theorem disjoint_iff (h : IsGenericPoint x S) (hU : IsOpen U) : Disjoint S U ↔ x ∉ U
|
α : Type u_1
inst✝ : TopologicalSpace α
x : α
S U : Set α
h : IsGenericPoint x S
hU : IsOpen U
⊢ Disjoint S U ↔ x ∉ U
|
rw [h.mem_open_set_iff hU, ← not_disjoint_iff_nonempty_inter, Classical.not_not]
|
no goals
|
94f69857791b5b7c
|
CategoryTheory.Limits.biprod.conePointUniqueUpToIso_inv
|
Mathlib/CategoryTheory/Limits/Shapes/BinaryBiproducts.lean
|
theorem biprod.conePointUniqueUpToIso_inv (X Y : C) [HasBinaryBiproduct X Y] {b : BinaryBicone X Y}
(hb : b.IsBilimit) :
(hb.isLimit.conePointUniqueUpToIso (BinaryBiproduct.isLimit _ _)).inv =
biprod.desc b.inl b.inr
|
case refine_2.mk.right
C : Type uC
inst✝² : Category.{uC', uC} C
inst✝¹ : HasZeroMorphisms C
X Y : C
inst✝ : HasBinaryBiproduct X Y
b : BinaryBicone X Y
hb : b.IsBilimit
⊢ inr ≫ (BinaryBiproduct.bicone X Y).toCone.π.app { as := WalkingPair.right } =
inr ≫ desc b.inl b.inr ≫ b.toCone.π.app { as := WalkingPair.right }
|
simp
|
no goals
|
c06efc3447119b63
|
CompactIccSpace.mk'
|
Mathlib/Topology/Order/Compact.lean
|
lemma CompactIccSpace.mk' [TopologicalSpace α] [Preorder α]
(h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)) : CompactIccSpace α where
isCompact_Icc {a b}
|
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : Preorder α
h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)
a b : α
hab : ¬a ≤ b
⊢ IsCompact (Icc a b)
|
rw [Icc_eq_empty hab]
|
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : Preorder α
h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)
a b : α
hab : ¬a ≤ b
⊢ IsCompact ∅
|
a8529e4e18c5748e
|
Subalgebra.rank_eq_one_iff
|
Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean
|
theorem rank_eq_one_iff [Nontrivial E] [Module.Free F S] : Module.rank F S = 1 ↔ S = ⊥
|
case intro.mk
F : Type u_1
E : Type u_2
inst✝⁵ : CommRing F
inst✝⁴ : StrongRankCondition F
inst✝³ : Ring E
inst✝² : Algebra F E
inst✝¹ : Nontrivial E
inst✝ : Free F ↥⊥
κ : Type u_2
b : Basis κ F ↥⊥
⊢ Module.rank F ↥⊥ = 1
|
refine le_antisymm ?_ ?_
|
case intro.mk.refine_1
F : Type u_1
E : Type u_2
inst✝⁵ : CommRing F
inst✝⁴ : StrongRankCondition F
inst✝³ : Ring E
inst✝² : Algebra F E
inst✝¹ : Nontrivial E
inst✝ : Free F ↥⊥
κ : Type u_2
b : Basis κ F ↥⊥
⊢ Module.rank F ↥⊥ ≤ 1
case intro.mk.refine_2
F : Type u_1
E : Type u_2
inst✝⁵ : CommRing F
inst✝⁴ : StrongRankCondition F
inst✝³ : Ring E
inst✝² : Algebra F E
inst✝¹ : Nontrivial E
inst✝ : Free F ↥⊥
κ : Type u_2
b : Basis κ F ↥⊥
⊢ 1 ≤ Module.rank F ↥⊥
|
eb0ef629fef399e8
|
CategoryTheory.AB5StarOfSize_of_univLE
|
Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Basic.lean
|
lemma AB5StarOfSize_of_univLE [HasCofilteredLimitsOfSize.{w₂, w₂'} C] [UnivLE.{w, w₂}]
[UnivLE.{w', w₂'}] [AB5StarOfSize.{w₂, w₂'} C] :
haveI : HasCofilteredLimitsOfSize.{w, w'} C := hasCofilteredLimitsOfSize_of_univLE.{w}
AB5StarOfSize.{w, w'} C
|
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasCofilteredLimitsOfSize.{w₂, w₂', v, u} C
inst✝² : UnivLE.{w, w₂}
inst✝¹ : UnivLE.{w', w₂'}
inst✝ : AB5StarOfSize.{w₂, w₂', v, u} C
this : HasCofilteredLimitsOfSize.{w, w', v, u} C
⊢ AB5StarOfSize.{w, w', v, u} C
|
constructor
|
case ofShape
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasCofilteredLimitsOfSize.{w₂, w₂', v, u} C
inst✝² : UnivLE.{w, w₂}
inst✝¹ : UnivLE.{w', w₂'}
inst✝ : AB5StarOfSize.{w₂, w₂', v, u} C
this : HasCofilteredLimitsOfSize.{w, w', v, u} C
⊢ ∀ (J : Type w') [inst : Category.{w, w'} J] [inst_1 : IsCofiltered J], HasExactLimitsOfShape J C
|
f48846e21d253650
|
intervalIntegral.integral_pos_iff_support_of_nonneg_ae'
|
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
theorem integral_pos_iff_support_of_nonneg_ae' (hf : 0 ≤ᵐ[μ.restrict (Ι a b)] f)
(hfi : IntervalIntegrable f μ a b) :
(0 < ∫ x in a..b, f x ∂μ) ↔ a < b ∧ 0 < μ (support f ∩ Ioc a b)
|
case inr
f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hf : 0 ≤ᶠ[ae (μ.restrict (Ι a b))] f
hfi : IntervalIntegrable f μ a b
hba : b ≤ a
⊢ 0 ≤ ∫ (x : ℝ) in Ioc b a, f x ∂μ
|
rw [uIoc_comm, uIoc_of_le hba] at hf
|
case inr
f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hf : 0 ≤ᶠ[ae (μ.restrict (Ioc b a))] f
hfi : IntervalIntegrable f μ a b
hba : b ≤ a
⊢ 0 ≤ ∫ (x : ℝ) in Ioc b a, f x ∂μ
|
98c9d1e3c30988ab
|
MeasureTheory.Measure.haar.haarContent_self
|
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
theorem haarContent_self {K₀ : PositiveCompacts G} : haarContent K₀ K₀.toCompacts = 1
|
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
⊢ (haarContent K₀) K₀.toCompacts = 1
|
simp_rw [← ENNReal.coe_one, haarContent_apply, ENNReal.coe_inj, chaar_self]
|
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
⊢ ⟨1, ⋯⟩ = 1
|
813491877e1017b7
|
denseRange_discrete
|
Mathlib/Topology/Order.lean
|
theorem denseRange_discrete {ι : Type*} {f : ι → α} : DenseRange f ↔ Surjective f
|
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
ι : Type u_3
f : ι → α
⊢ DenseRange f ↔ Surjective f
|
rw [DenseRange, dense_discrete, range_eq_univ]
|
no goals
|
a720371fe21bcd06
|
TensorProduct.smul_tmul_smul
|
Mathlib/LinearAlgebra/TensorProduct/Basic.lean
|
theorem smul_tmul_smul (r s : R) (m : M) (n : N) : (r • m) ⊗ₜ[R] (s • n) = (r * s) • m ⊗ₜ[R] n
|
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_5
N : Type u_6
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
r s : R
m : M
n : N
⊢ (r • m) ⊗ₜ[R] (s • n) = (r * s) • m ⊗ₜ[R] n
|
simp_rw [smul_tmul, tmul_smul, mul_smul]
|
no goals
|
74fdfcacd095fb85
|
Finset.diag_mem_sym2_mem_iff
|
Mathlib/Data/Finset/Sym.lean
|
theorem diag_mem_sym2_mem_iff : (∀ b, b ∈ Sym2.diag a → b ∈ s) ↔ a ∈ s
|
α : Type u_1
s : Finset α
a : α
⊢ Sym2.diag a ∈ s.sym2 ↔ a ∈ s
|
exact mk_mem_sym2_iff.trans <| and_self_iff
|
no goals
|
cbd60f602a44e478
|
Ring.eq_mul_inverse_iff_mul_eq
|
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
theorem eq_mul_inverse_iff_mul_eq (x y z : M₀) (h : IsUnit z) : x = y * inverse z ↔ x * z = y :=
⟨fun h1 => by rw [h1, inverse_mul_cancel_right _ _ h],
fun h1 => by rw [← h1, mul_inverse_cancel_right _ _ h]⟩
|
M₀ : Type u_2
inst✝ : MonoidWithZero M₀
x y z : M₀
h : IsUnit z
h1 : x = y * inverse z
⊢ x * z = y
|
rw [h1, inverse_mul_cancel_right _ _ h]
|
no goals
|
db768333e2c6e800
|
Ideal.subset_union_prime'
|
Mathlib/RingTheory/Ideal/Operations.lean
|
theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
|
ι : Type u_1
R : Type u
inst✝ : CommRing R
s : Finset ι
f : ι → Ideal R
a b : ι
hp : ∀ i ∈ s, (f i).IsPrime
I : Ideal R
n : ℕ
hn : s.card = n
⊢ ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
|
intro h
|
ι : Type u_1
R : Type u
inst✝ : CommRing R
s : Finset ι
f : ι → Ideal R
a b : ι
hp : ∀ i ∈ s, (f i).IsPrime
I : Ideal R
n : ℕ
hn : s.card = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i)
⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
|
fa41770a7d2d9fd9
|
LieModule.lowerCentralSeriesLast_le_of_not_isTrivial
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
theorem lowerCentralSeriesLast_le_of_not_isTrivial [IsNilpotent L M] (h : ¬ IsTrivial L M) :
lowerCentralSeriesLast R L M ≤ lowerCentralSeries R L M 1
|
case zero
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : IsNilpotent L M
h : 1 < 0
hk : nilpotencyLength L M = 0
⊢ (match 0 with
| 0 => ⊥
| k.succ => lowerCentralSeries R L M k) ≤
lowerCentralSeries R L M 1
|
contradiction
|
no goals
|
9fbbbf8b7fa9fbd1
|
NonUnitalSubring.closure_induction'
|
Mathlib/RingTheory/NonUnitalSubring/Basic.lean
|
theorem closure_induction' {s : Set R} {p : closure s → Prop} (a : closure s)
(mem : ∀ (x) (hx : x ∈ s), p ⟨x, subset_closure hx⟩) (zero : p 0)
(add : ∀ x y, p x → p y → p (x + y)) (neg : ∀ x, p x → p (-x))
(mul : ∀ x y, p x → p y → p (x * y)) : p a :=
Subtype.recOn a fun b hb => by
induction hb using closure_induction with
| mem x hx => exact mem x hx
| zero => exact zero
| add x y hx hy h₁ h₂ => exact add _ _ h₁ h₂
| neg x hx h => exact neg _ h
| mul x y hx hy h₁ h₂ => exact mul _ _ h₁ h₂
|
case neg
R : Type u
inst✝ : NonUnitalNonAssocRing R
s : Set R
p : ↥(closure s) → Prop
a : ↥(closure s)
mem : ∀ (x : R) (hx : x ∈ s), p ⟨x, ⋯⟩
zero : p 0
add : ∀ (x y : ↥(closure s)), p x → p y → p (x + y)
neg : ∀ (x : ↥(closure s)), p x → p (-x)
mul : ∀ (x y : ↥(closure s)), p x → p y → p (x * y)
b x : R
hx : x ∈ closure s
h : p ⟨x, hx⟩
⊢ p ⟨-x, ⋯⟩
|
exact neg _ h
|
no goals
|
8be1f3e87edfc4cb
|
Function.Embedding.schroeder_bernstein
|
Mathlib/SetTheory/Cardinal/SchroederBernstein.lean
|
theorem schroeder_bernstein {f : α → β} {g : β → α} (hf : Function.Injective f)
(hg : Function.Injective g) : ∃ h : α → β, Bijective h
|
case inr
α : Type u
β : Type v
f : α → β
g : β → α
hf : Injective f
hg : Injective g
hβ : Nonempty β
F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := ⋯ }
s : Set α := OrderHom.lfp F
hs : (g '' (f '' s)ᶜ)ᶜ = s
hns : g '' (f '' s)ᶜ = sᶜ
g' : α → β := invFun g
g'g : LeftInverse g' g
⊢ ∃ h, Bijective h
|
have hg'ns : g' '' sᶜ = (f '' s)ᶜ := by rw [← hns, g'g.image_image]
|
case inr
α : Type u
β : Type v
f : α → β
g : β → α
hf : Injective f
hg : Injective g
hβ : Nonempty β
F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := ⋯ }
s : Set α := OrderHom.lfp F
hs : (g '' (f '' s)ᶜ)ᶜ = s
hns : g '' (f '' s)ᶜ = sᶜ
g' : α → β := invFun g
g'g : LeftInverse g' g
hg'ns : g' '' sᶜ = (f '' s)ᶜ
⊢ ∃ h, Bijective h
|
95bf487be3670949
|
Ordinal.exists_blsub_cof
|
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
theorem exists_blsub_cof (o : Ordinal) :
∃ f : ∀ a < (cof o).ord, Ordinal, blsub.{u, u} _ f = o
|
case intro.intro.intro
o : Ordinal.{u}
ι : Type u
f : ι → Ordinal.{u}
hf : lsub f = o
hι : #ι = o.cof
⊢ ∃ f, o.cof.ord.blsub f = o
|
rcases Cardinal.ord_eq ι with ⟨r, hr, hι'⟩
|
case intro.intro.intro.intro.intro
o : Ordinal.{u}
ι : Type u
f : ι → Ordinal.{u}
hf : lsub f = o
hι : #ι = o.cof
r : ι → ι → Prop
hr : IsWellOrder ι r
hι' : (#ι).ord = type r
⊢ ∃ f, o.cof.ord.blsub f = o
|
068b935e7bf4fed0
|
WeierstrassCurve.Jacobian.toAffine_addX_of_eq
|
Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
|
private lemma toAffine_addX_of_eq {P : Fin 3 → F} (hPz : P z ≠ 0) {n d : F} (hd : d ≠ 0) :
W.toAffine.addX (P x / P z ^ 2) (P x / P z ^ 2) (-n / (P z * d)) =
(n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * d ^ 2) / (P z * d) ^ 2
|
F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
n d : F
hd : d ≠ 0
⊢ (toAffine W).addX (P x / P z ^ 2) (P x / P z ^ 2) (-n / (P z * d)) =
(n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * d ^ 2) / (P z * d) ^ 2
|
field_simp [mul_ne_zero hPz hd]
|
F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
n d : F
hd : d ≠ 0
⊢ (((n ^ 2 * (P z * d) + -((toAffine W).a₁ * n * (P z * d) ^ 2) - (P z * d) ^ 2 * (P z * d) * (toAffine W).a₂) *
P z ^ 2 -
(P z * d) ^ 2 * (P z * d) * P x) *
P z ^ 2 -
(P z * d) ^ 2 * (P z * d) * P z ^ 2 * P x) *
(P z * d) ^ 2 =
(n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * d ^ 2) *
((P z * d) ^ 2 * (P z * d) * P z ^ 2 * P z ^ 2)
|
66255a2fa05782fd
|
Valued.integer.totallyBounded_iff_finite_residueField
|
Mathlib/Topology/Algebra/Valued/LocallyCompact.lean
|
lemma totallyBounded_iff_finite_residueField [IsDiscreteValuationRing 𝒪[K]] :
TotallyBounded (Set.univ (α := 𝒪[K])) ↔ Finite 𝓀[K]
|
case mpr
K : Type u_2
inst✝² : NontriviallyNormedField K
inst✝¹ : IsUltrametricDist K
inst✝ : IsDiscreteValuationRing ↥𝒪[K]
H : Finite 𝓀[K]
ε : ℝ
εpos : ε > 0
⊢ ∃ t, t.Finite ∧ Set.univ ⊆ ⋃ y ∈ t, Metric.ball y ε
|
obtain ⟨p, hp⟩ := IsDiscreteValuationRing.exists_irreducible 𝒪[K]
|
case mpr.intro
K : Type u_2
inst✝² : NontriviallyNormedField K
inst✝¹ : IsUltrametricDist K
inst✝ : IsDiscreteValuationRing ↥𝒪[K]
H : Finite 𝓀[K]
ε : ℝ
εpos : ε > 0
p : ↥𝒪[K]
hp : Irreducible p
⊢ ∃ t, t.Finite ∧ Set.univ ⊆ ⋃ y ∈ t, Metric.ball y ε
|
368b1da03339c547
|
Lean.Omega.IntList.gcd_dvd
|
Mathlib/.lake/packages/lean4/src/lean/Init/Omega/IntList.lean
|
theorem gcd_dvd (xs : IntList) {a : Int} (m : a ∈ xs) : (xs.gcd : Int) ∣ a
|
case tail
xs : IntList
a b : Int
as✝ : List Int
m : List.Mem a as✝
ih : gcd as✝ ∣ a.natAbs
⊢ b.natAbs.gcd (gcd as✝) ∣ a.natAbs
|
exact Nat.dvd_trans (Nat.gcd_dvd_right _ _) ih
|
no goals
|
a2ce29a1d3b80c5c
|
Nat.exists_coprime
|
Mathlib/.lake/packages/batteries/Batteries/Data/Nat/Gcd.lean
|
theorem exists_coprime (m n : Nat) :
∃ m' n', Coprime m' n' ∧ m = m' * gcd m n ∧ n = n' * gcd m n
|
case inr
m n : Nat
hpos : m.gcd n > 0
⊢ ∃ m' n', m'.Coprime n' ∧ m = m' * m.gcd n ∧ n = n' * m.gcd n
|
exact ⟨_, _, coprime_div_gcd_div_gcd hpos,
(Nat.div_mul_cancel (gcd_dvd_left m n)).symm,
(Nat.div_mul_cancel (gcd_dvd_right m n)).symm⟩
|
no goals
|
86c6a2ec263f3a53
|
PFunctor.M.mk_dest
|
Mathlib/Data/PFunctor/Univariate/M.lean
|
theorem mk_dest (x : M F) : M.mk (dest x) = x
|
case H.succ.intro.e_a.h
F : PFunctor.{u}
x : F.M
n : ℕ
a✝ : Approx.sMk x.dest n = x.approx n
hd : F.A
h' : hd = head' (x.approx 1)
ch : F.B (head' (x.approx 1)) → CofixA F n
h : x.approx n.succ = CofixA.intro (head' (x.approx 1)) ch
a : F.B (head' (x.approx 1))
⊢ children' (x.approx n.succ) (cast ⋯ a) = ch a
|
generalize hh : cast _ a = a''
|
case H.succ.intro.e_a.h
F : PFunctor.{u}
x : F.M
n : ℕ
a✝ : Approx.sMk x.dest n = x.approx n
hd : F.A
h' : hd = head' (x.approx 1)
ch : F.B (head' (x.approx 1)) → CofixA F n
h : x.approx n.succ = CofixA.intro (head' (x.approx 1)) ch
a : F.B (head' (x.approx 1))
a'' : F.B (head' (x.approx n.succ))
hh : cast ⋯ a = a''
⊢ children' (x.approx n.succ) a'' = ch a
|
475d78b79340343f
|
map_mul_left_nhds_one
|
Mathlib/Topology/Algebra/Group/Basic.lean
|
theorem map_mul_left_nhds_one (x : G) : map (x * ·) (𝓝 1) = 𝓝 x
|
G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
x : G
⊢ map (fun x_1 => x * x_1) (𝓝 1) = 𝓝 x
|
simp
|
no goals
|
af53bf0f875f4353
|
WittVector.ghostComponent_teichmullerFun
|
Mathlib/RingTheory/WittVector/Teichmuller.lean
|
theorem ghostComponent_teichmullerFun (r : R) (n : ℕ) :
ghostComponent n (teichmullerFun p r) = r ^ p ^ n
|
case h₀
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
r : R
n : ℕ
⊢ ∀ b ∈ Finset.range (n + 1), b ≠ 0 → ↑p ^ b * (teichmullerFun p r).coeff b ^ p ^ (n - b) = 0
|
intro i _ h0
|
case h₀
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
r : R
n i : ℕ
a✝ : i ∈ Finset.range (n + 1)
h0 : i ≠ 0
⊢ ↑p ^ i * (teichmullerFun p r).coeff i ^ p ^ (n - i) = 0
|
64564ca6964107de
|
MeasureTheory.LevyProkhorov.continuous_equiv_probabilityMeasure
|
Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
|
/-- The identity map `LevyProkhorov (ProbabilityMeasure Ω) → ProbabilityMeasure Ω` is continuous. -/
lemma LevyProkhorov.continuous_equiv_probabilityMeasure :
Continuous (LevyProkhorov.equiv (α := ProbabilityMeasure Ω))
|
case h.refine_4
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := (equiv (ProbabilityMeasure Ω)) ν
Ps : ℕ → ProbabilityMeasure Ω := fun n => (equiv (ProbabilityMeasure Ω)) (μs n)
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
εs : ℕ → ℝ
left✝ : StrictAnti εs
εs_lim : Tendsto εs atTop (𝓝 0)
ε_of_room : Tendsto (fun x => dist (μs x) ν + εs x) atTop (𝓝 0)
ε_of_room' : Tendsto (fun n => dist (μs n) ν + εs n) atTop (𝓝[>] 0)
key :
Tendsto
((fun ε => ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P (thickening ε {a | t ≤ f a}))).toReal) ∘ fun n => dist (μs n) ν + εs n)
atTop (𝓝 (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P {a | t ≤ f a})).toReal))
aux : ∀ (z : ℝ), Iio (z + δ / 2) ∈ 𝓝 z
n : ℕ
hn :
((fun ε => ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P (thickening ε {a | t ≤ f a}))).toReal) ∘ fun n => dist (μs n) ν + εs n) n <
(∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P {a | t ≤ f a})).toReal) + δ / 2
hn' : dist (μs n) ν + εs n < ‖f‖⁻¹ * δ / 2
εs_pos : 0 < εs n
bound :
∫ (ω : Ω), f ω ∂↑(Ps n) ≤
(∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P (thickening (dist (μs n) ν + εs n) {a | t ≤ f a})).toReal) +
(dist (μs n) ν + εs n) * ‖f‖
⊢ (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P {a | t ≤ f a})).toReal) + δ / 2 + (dist (μs n) ν + εs n) * ‖f‖ ≤ ∫ (x : Ω), f x ∂↑P + δ
|
rw [BoundedContinuousFunction.integral_eq_integral_meas_le]
|
case h.refine_4
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := (equiv (ProbabilityMeasure Ω)) ν
Ps : ℕ → ProbabilityMeasure Ω := fun n => (equiv (ProbabilityMeasure Ω)) (μs n)
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
εs : ℕ → ℝ
left✝ : StrictAnti εs
εs_lim : Tendsto εs atTop (𝓝 0)
ε_of_room : Tendsto (fun x => dist (μs x) ν + εs x) atTop (𝓝 0)
ε_of_room' : Tendsto (fun n => dist (μs n) ν + εs n) atTop (𝓝[>] 0)
key :
Tendsto
((fun ε => ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P (thickening ε {a | t ≤ f a}))).toReal) ∘ fun n => dist (μs n) ν + εs n)
atTop (𝓝 (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P {a | t ≤ f a})).toReal))
aux : ∀ (z : ℝ), Iio (z + δ / 2) ∈ 𝓝 z
n : ℕ
hn :
((fun ε => ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P (thickening ε {a | t ≤ f a}))).toReal) ∘ fun n => dist (μs n) ν + εs n) n <
(∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P {a | t ≤ f a})).toReal) + δ / 2
hn' : dist (μs n) ν + εs n < ‖f‖⁻¹ * δ / 2
εs_pos : 0 < εs n
bound :
∫ (ω : Ω), f ω ∂↑(Ps n) ≤
(∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P (thickening (dist (μs n) ν + εs n) {a | t ≤ f a})).toReal) +
(dist (μs n) ν + εs n) * ‖f‖
⊢ (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P {a | t ≤ f a})).toReal) + δ / 2 + (dist (μs n) ν + εs n) * ‖f‖ ≤
(∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P {a | t ≤ f a}).toReal) + δ
case h.refine_4.f_nn
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := (equiv (ProbabilityMeasure Ω)) ν
Ps : ℕ → ProbabilityMeasure Ω := fun n => (equiv (ProbabilityMeasure Ω)) (μs n)
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
εs : ℕ → ℝ
left✝ : StrictAnti εs
εs_lim : Tendsto εs atTop (𝓝 0)
ε_of_room : Tendsto (fun x => dist (μs x) ν + εs x) atTop (𝓝 0)
ε_of_room' : Tendsto (fun n => dist (μs n) ν + εs n) atTop (𝓝[>] 0)
key :
Tendsto
((fun ε => ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P (thickening ε {a | t ≤ f a}))).toReal) ∘ fun n => dist (μs n) ν + εs n)
atTop (𝓝 (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P {a | t ≤ f a})).toReal))
aux : ∀ (z : ℝ), Iio (z + δ / 2) ∈ 𝓝 z
n : ℕ
hn :
((fun ε => ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P (thickening ε {a | t ≤ f a}))).toReal) ∘ fun n => dist (μs n) ν + εs n) n <
(∫ (t : ℝ) in Ioc 0 ‖f‖, (↑(P {a | t ≤ f a})).toReal) + δ / 2
hn' : dist (μs n) ν + εs n < ‖f‖⁻¹ * δ / 2
εs_pos : 0 < εs n
bound :
∫ (ω : Ω), f ω ∂↑(Ps n) ≤
(∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P (thickening (dist (μs n) ν + εs n) {a | t ≤ f a})).toReal) +
(dist (μs n) ν + εs n) * ‖f‖
⊢ 0 ≤ᶠ[ae ↑P] ⇑f
|
e3f8df53302ac07c
|
CategoryTheory.Limits.biproduct.map_lift_mapBiprod
|
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
|
theorem biproduct.map_lift_mapBiprod (g : ∀ j, W ⟶ f j) :
-- Porting note: twice we need haveI to tell Lean about hasBiproduct_of_preserves F f
haveI : HasBiproduct fun j => F.obj (f j) := hasBiproduct_of_preserves F f
F.map (biproduct.lift g) ≫ (F.mapBiproduct f).hom = biproduct.lift fun j => F.map (g j)
|
case w
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
J : Type w₁
f : J → C
inst✝¹ : HasBiproduct f
inst✝ : PreservesBiproduct f F
W : C
g : (j : J) → W ⟶ f j
j : J
⊢ (F.map (lift g) ≫ (F.mapBiproduct f).hom) ≫ π (fun x => F.obj (f x)) j =
(lift fun j => F.map (g j)) ≫ π (fun x => F.obj (f x)) j
|
haveI : HasBiproduct fun j => F.obj (f j) := hasBiproduct_of_preserves F f
|
case w
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
J : Type w₁
f : J → C
inst✝¹ : HasBiproduct f
inst✝ : PreservesBiproduct f F
W : C
g : (j : J) → W ⟶ f j
j : J
this : HasBiproduct fun j => F.obj (f j)
⊢ (F.map (lift g) ≫ (F.mapBiproduct f).hom) ≫ π (fun x => F.obj (f x)) j =
(lift fun j => F.map (g j)) ≫ π (fun x => F.obj (f x)) j
|
b5befb27d4fb8b99
|
Std.DHashMap.Internal.List.eraseKey_of_perm
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean
|
theorem eraseKey_of_perm [BEq α] [EquivBEq α] {l l' : List ((a : α) × β a)} {k : α}
(hl : DistinctKeys l) (h : Perm l l') : Perm (eraseKey k l) (eraseKey k l')
|
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : EquivBEq α
l l' : List ((a : α) × β a)
k : α
hl : DistinctKeys l
h : l.Perm l'
⊢ ∀ (a : α), getEntry? a (eraseKey k l) = getEntry? a (eraseKey k l')
|
simp [getEntry?_eraseKey hl, getEntry?_eraseKey (hl.perm h.symm), getEntry?_of_perm hl h]
|
no goals
|
fe56f8c60e5a647f
|
ConvexOn.lipschitzOnWith_of_abs_le
|
Mathlib/Analysis/Convex/Continuous.lean
|
lemma ConvexOn.lipschitzOnWith_of_abs_le (hf : ConvexOn ℝ (ball x₀ r) f) (hε : 0 < ε)
(hM : ∀ a, dist a x₀ < r → |f a| ≤ M) :
LipschitzOnWith (2 * M / ε).toNNReal f (ball x₀ (r - ε))
|
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConvexOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
K : ℝ := 2 * M / ε
hK : K = 2 * M / ε
x y : E
hx : x ∈ ball x₀ (r - ε)
hy : y ∈ ball x₀ (r - ε)
hx₀r : ball x₀ (r - ε) ⊆ ball x₀ r
hx' : x ∈ ball x₀ r
hy' : y ∈ ball x₀ r
z : E := x + (ε / ‖x - y‖) • (x - y)
hxy : 0 < ‖x - y‖
hz : z ∈ ball x₀ r
⊢ f x - f y ≤ K * ‖x - y‖
|
let a := ε / (ε + ‖x - y‖)
|
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConvexOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
K : ℝ := 2 * M / ε
hK : K = 2 * M / ε
x y : E
hx : x ∈ ball x₀ (r - ε)
hy : y ∈ ball x₀ (r - ε)
hx₀r : ball x₀ (r - ε) ⊆ ball x₀ r
hx' : x ∈ ball x₀ r
hy' : y ∈ ball x₀ r
z : E := x + (ε / ‖x - y‖) • (x - y)
hxy : 0 < ‖x - y‖
hz : z ∈ ball x₀ r
a : ℝ := ε / (ε + ‖x - y‖)
⊢ f x - f y ≤ K * ‖x - y‖
|
f794e80453a6298e
|
Submodule.le_traceDual
|
Mathlib/RingTheory/DedekindDomain/Different.lean
|
lemma le_traceDual {I J : Submodule B L} :
I ≤ Jᵛ ↔ I * J ≤ 1ᵛ
|
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁰ : CommRing A
inst✝⁹ : Field K
inst✝⁸ : CommRing B
inst✝⁷ : Field L
inst✝⁶ : Algebra A K
inst✝⁵ : Algebra B L
inst✝⁴ : Algebra A B
inst✝³ : Algebra K L
inst✝² : Algebra A L
inst✝¹ : IsScalarTower A K L
inst✝ : IsScalarTower A B L
I J : Submodule B L
⊢ I ≤ Jᵛ ↔ I * J ≤ 1ᵛ
|
rw [← le_traceDual_mul_iff, mul_one]
|
no goals
|
c9d13a23548775cc
|
Submodule.spanRank_finite_iff_fg
|
Mathlib/Algebra/Module/SpanRank.lean
|
/-- A submodule's `spanRank` is finite if and only if it is finitely generated. -/
@[simp]
lemma spanRank_finite_iff_fg {p : Submodule R M} : p.spanRank < aleph0 ↔ p.FG
|
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ ⨅ s, #↑↑s < ℵ₀ ↔ ∃ S, S.Finite ∧ span R S = p
|
constructor
|
case mp
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ ⨅ s, #↑↑s < ℵ₀ → ∃ S, S.Finite ∧ span R S = p
case mpr
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ (∃ S, S.Finite ∧ span R S = p) → ⨅ s, #↑↑s < ℵ₀
|
57fd9ea29407f8a5
|
List.mem_eraseIdx_iff_getElem?
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Basic.lean
|
theorem mem_eraseIdx_iff_getElem? {x : α} {l} {k} : x ∈ eraseIdx l k ↔ ∃ i ≠ k, l[i]? = some x
|
case mp
α : Type u_1
x : α
l : List α
k i : Nat
⊢ (∃ x_1, l[i]? = some x) → l[i]? = some x
|
rintro ⟨_, h⟩
|
case mp.intro
α : Type u_1
x : α
l : List α
k i : Nat
w✝ : i < l.length
h : l[i]? = some x
⊢ l[i]? = some x
|
b696caa5872e7aba
|
LSeries.tendsto_cpow_mul_atTop
|
Mathlib/NumberTheory/LSeries/Injectivity.lean
|
/-- If the coefficients `f m` of an L-series are zero for `m ≤ n` and the L-series converges
at some point, then `f (n+1)` is the limit of `(n+1)^x * LSeries f x` as `x → ∞`. -/
lemma LSeries.tendsto_cpow_mul_atTop {f : ℕ → ℂ} {n : ℕ} (h : ∀ m ≤ n, f m = 0)
(ha : abscissaOfAbsConv f < ⊤):
Tendsto (fun x : ℝ ↦ (n + 1) ^ (x : ℂ) * LSeries f x) atTop (nhds (f (n + 1)))
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
|
conv => enter [3, 1]; rw [← add_zero (f _)]
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1) + 0))
|
0f9b595cc43bb900
|
Matroid.mem_closure_iff_exists_isCircuit
|
Mathlib/Data/Matroid/Circuit.lean
|
lemma mem_closure_iff_exists_isCircuit (he : e ∉ X) :
e ∈ M.closure X ↔ ∃ C ⊆ insert e X, M.IsCircuit C ∧ e ∈ C :=
⟨fun h ↦ exists_isCircuit_of_mem_closure h he, fun ⟨C, hCX, hC, heC⟩ ↦ mem_of_mem_of_subset
(hC.mem_closure_diff_singleton_of_mem heC) (M.closure_subset_closure (by simpa))⟩
|
α : Type u_1
M : Matroid α
X : Set α
e : α
he : e ∉ X
x✝ : ∃ C ⊆ insert e X, M.IsCircuit C ∧ e ∈ C
C : Set α
hCX : C ⊆ insert e X
hC : M.IsCircuit C
heC : e ∈ C
⊢ C \ {e} ⊆ X
|
simpa
|
no goals
|
8f3c13c96dde755f
|
Set.iUnion_range_eq_iUnion
|
Mathlib/Data/Set/Lattice.lean
|
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x
|
case h
α : Type u_1
β : Type u_2
ι : Sort u_5
C : ι → Set α
f : (x : ι) → β → ↑(C x)
hf : ∀ (x : ι), Surjective (f x)
x : α
⊢ (x ∈ ⋃ y, range fun x => ↑(f x y)) ↔ x ∈ ⋃ x, C x
|
rw [mem_iUnion, mem_iUnion]
|
case h
α : Type u_1
β : Type u_2
ι : Sort u_5
C : ι → Set α
f : (x : ι) → β → ↑(C x)
hf : ∀ (x : ι), Surjective (f x)
x : α
⊢ (∃ i, x ∈ range fun x => ↑(f x i)) ↔ ∃ i, x ∈ C i
|
9d4f3f5bc914a4d2
|
Option.forall_mem_map
|
Mathlib/Data/Option/Basic.lean
|
theorem forall_mem_map {f : α → β} {o : Option α} {p : β → Prop} :
(∀ y ∈ o.map f, p y) ↔ ∀ x ∈ o, p (f x)
|
α : Type u_1
β : Type u_2
f : α → β
o : Option α
p : β → Prop
⊢ (∀ (y : β), y ∈ Option.map f o → p y) ↔ ∀ (x : α), x ∈ o → p (f x)
|
simp
|
no goals
|
e4f68a2bbadcba76
|
MvPowerSeries.X_pow_dvd_iff
|
Mathlib/RingTheory/MvPowerSeries/Basic.lean
|
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
(X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0
|
case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m j : σ →₀ ℕ
hij : (single s n, j).1 + (single s n, j).2 = m
⊢ n ≤ m s
|
rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
|
case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m j : σ →₀ ℕ
hij : (single s n, j).1 + (single s n, j).2 = m
⊢ n ≤ n + (single s n, j).2 s
|
c235f2aaea0f4970
|
SuccOrder.forall_ne_bot_iff
|
Mathlib/Order/SuccPred/Archimedean.lean
|
lemma SuccOrder.forall_ne_bot_iff
[Nontrivial α] [PartialOrder α] [OrderBot α] [SuccOrder α] [IsSuccArchimedean α]
(P : α → Prop) :
(∀ i, i ≠ ⊥ → P i) ↔ (∀ i, P (SuccOrder.succ i))
|
case intro
α : Type u_1
inst✝⁴ : Nontrivial α
inst✝³ : PartialOrder α
inst✝² : OrderBot α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
P : α → Prop
h : ∀ (i : α), P (succ i)
j : ℕ
hi : Order.succ^[j] ⊥ ≠ ⊥
hj : 0 < j
⊢ P (Order.succ^[j.pred.succ] ⊥)
|
simp only [Function.iterate_succ', Function.comp_apply]
|
case intro
α : Type u_1
inst✝⁴ : Nontrivial α
inst✝³ : PartialOrder α
inst✝² : OrderBot α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
P : α → Prop
h : ∀ (i : α), P (succ i)
j : ℕ
hi : Order.succ^[j] ⊥ ≠ ⊥
hj : 0 < j
⊢ P (Order.succ (Order.succ^[j.pred] ⊥))
|
de964cadf712f800
|
List.map₂Left_eq_zipWith
|
Mathlib/Data/List/Map2.lean
|
theorem map₂Left_eq_zipWith :
∀ as bs, length as ≤ length bs → map₂Left f as bs = zipWith (fun a b => f a (some b)) as bs
| [], [], _ => by simp
| [], _ :: _, _ => by simp
| a :: as, [], h => by
simp at h
| a :: as, b :: bs, h => by
simp only [length_cons, succ_le_succ_iff] at h
simp [h, map₂Left_eq_zipWith]
|
α : Type u
β : Type v
γ : Type w
f : α → Option β → γ
head✝ : β
tail✝ : List β
x✝ : [].length ≤ (head✝ :: tail✝).length
⊢ map₂Left f [] (head✝ :: tail✝) = zipWith (fun a b => f a (some b)) [] (head✝ :: tail✝)
|
simp
|
no goals
|
d79f18ce8e9359bc
|
ENNReal.toNNReal_iSup
|
Mathlib/Data/ENNReal/Real.lean
|
theorem toNNReal_iSup (hf : ∀ i, f i ≠ ∞) : (iSup f).toNNReal = ⨆ i, (f i).toNNReal
|
case intro
ι : Sort u_1
f : ι → ℝ≥0
⊢ (⨆ i, ↑(f i)).toNNReal = ⨆ i, ((fun i => ↑(f i)) i).toNNReal
|
simp_rw [toNNReal_coe]
|
case intro
ι : Sort u_1
f : ι → ℝ≥0
⊢ (⨆ i, ↑(f i)).toNNReal = ⨆ i, f i
|
d664c6a853b7cbe6
|
Stream'.Seq.eq_of_bisim
|
Mathlib/Data/Seq/Seq.lean
|
theorem eq_of_bisim (bisim : IsBisimulation R) {s₁ s₂} (r : s₁ ~ s₂) : s₁ = s₂
|
case cons.cons.intro
α : Type u
R : Seq α → Seq α → Prop
bisim : IsBisimulation R
s₁ s₂ : Seq α
r : R s₁ s₂
t₁ t₂ : Stream' (Option α)
e : ∃ s s', ↑s = t₁ ∧ ↑s' = t₂ ∧ R s s'
x✝¹ : α
s✝¹ : Seq α
x✝ : α
s✝ : Seq α
r✝ : R (cons x✝¹ s✝¹) (cons x✝ s✝)
h1 : x✝¹ = x✝
h2 : R s✝¹ s✝
⊢ some x✝¹ = some x✝ ∧ R s✝¹ s✝
|
constructor
|
case cons.cons.intro.left
α : Type u
R : Seq α → Seq α → Prop
bisim : IsBisimulation R
s₁ s₂ : Seq α
r : R s₁ s₂
t₁ t₂ : Stream' (Option α)
e : ∃ s s', ↑s = t₁ ∧ ↑s' = t₂ ∧ R s s'
x✝¹ : α
s✝¹ : Seq α
x✝ : α
s✝ : Seq α
r✝ : R (cons x✝¹ s✝¹) (cons x✝ s✝)
h1 : x✝¹ = x✝
h2 : R s✝¹ s✝
⊢ some x✝¹ = some x✝
case cons.cons.intro.right
α : Type u
R : Seq α → Seq α → Prop
bisim : IsBisimulation R
s₁ s₂ : Seq α
r : R s₁ s₂
t₁ t₂ : Stream' (Option α)
e : ∃ s s', ↑s = t₁ ∧ ↑s' = t₂ ∧ R s s'
x✝¹ : α
s✝¹ : Seq α
x✝ : α
s✝ : Seq α
r✝ : R (cons x✝¹ s✝¹) (cons x✝ s✝)
h1 : x✝¹ = x✝
h2 : R s✝¹ s✝
⊢ R s✝¹ s✝
|
1f640110e2fa7129
|
Polynomial.dickson_of_two_le
|
Mathlib/RingTheory/Polynomial/Dickson.lean
|
theorem dickson_of_two_le {n : ℕ} (h : 2 ≤ n) :
dickson k a n = X * dickson k a (n - 1) - C a * dickson k a (n - 2)
|
case intro
R : Type u_1
inst✝ : CommRing R
k : ℕ
a : R
n : ℕ
h : 2 ≤ 2 + n
⊢ dickson k a (n + 2) = X * dickson k a (n + 2 - 1) - C a * dickson k a (n + 2 - 2)
|
exact dickson_add_two k a n
|
no goals
|
4bd239701f69d797
|
Algebra.FinitePresentation.ker_fg_of_mvPolynomial
|
Mathlib/RingTheory/FinitePresentation.lean
|
theorem ker_fg_of_mvPolynomial {n : ℕ} (f : MvPolynomial (Fin n) R →ₐ[R] A)
(hf : Function.Surjective f) [FinitePresentation R A] : f.toRingHom.ker.FG
|
R : Type w₁
A : Type w₂
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
inst✝ : FinitePresentation R A
⊢ (RingHom.ker f.toRingHom).FG
|
obtain ⟨m, f', hf', s, hs⟩ := FinitePresentation.out (R := R) (A := A)
|
case intro.intro.intro.intro
R : Type w₁
A : Type w₂
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
inst✝ : FinitePresentation R A
m : ℕ
f' : MvPolynomial (Fin m) R →ₐ[R] A
hf' : Surjective ⇑f'
s : Finset (MvPolynomial (Fin m) R)
hs : Ideal.span ↑s = RingHom.ker f'.toRingHom
⊢ (RingHom.ker f.toRingHom).FG
|
37c27acde7e67205
|
MeasureTheory.Integrable.norm_toL1
|
Mathlib/MeasureTheory/Function/L1Space/AEEqFun.lean
|
theorem norm_toL1 (f : α → β) (hf : Integrable f μ) :
‖hf.toL1 f‖ = (∫⁻ a, edist (f a) 0 ∂μ).toReal
|
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : Integrable f μ
⊢ ‖toL1 f hf‖ = (∫⁻ (a : α), edist (f a) 0 ∂μ).toReal
|
simp [toL1, Lp.norm_toLp, eLpNorm, eLpNorm'_eq_lintegral_enorm]
|
no goals
|
48ed02ba7f9a526f
|
MeasureTheory.measurableSet_generateFrom_singleton_iff
|
Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
|
theorem measurableSet_generateFrom_singleton_iff {s t : Set α} :
MeasurableSet[MeasurableSpace.generateFrom {s}] t ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ
|
case h
α : Type u_1
s : Set α
x : Set Prop
hT : True ∈ x
hF : False ∉ x
p : Prop
hp : p ∈ x
hpneg : ¬p = True
⊢ False
|
rw [eq_iff_iff, iff_true, ← false_iff] at hpneg
|
case h
α : Type u_1
s : Set α
x : Set Prop
hT : True ∈ x
hF : False ∉ x
p : Prop
hp : p ∈ x
hpneg : False ↔ p
⊢ False
|
39127bdfcef76ebf
|
Algebra.Presentation.differentialsSolution_isPresentation
|
Mathlib/Algebra/Module/Presentation/Differentials.lean
|
lemma differentialsSolution_isPresentation :
pres.differentialsSolution.IsPresentation
|
case left
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
pres : Presentation R S
⊢ Submodule.span S (Set.range pres.differentialsSolution.var) = ⊤
|
rw [← Module.Relations.Solution.surjective_π_iff_span_eq_top, ← comm₂₃]
|
case left
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
pres : Presentation R S
⊢ Function.Surjective ⇑(pres.toExtension.toKaehler ∘ₗ ↑pres.cotangentSpaceBasis.repr.symm)
|
d3c9201e4431d782
|
CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.hom_ext
|
Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean
|
lemma IsPointwiseLeftKanExtension.hom_ext
{G : LeftExtension L F} {f₁ f₂ : E ⟶ G} : f₁ = f₂
|
case h.w.h
C : Type u_1
D : Type u_2
H : Type u_3
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_6, u_2} D
inst✝ : Category.{u_5, u_3} H
L : C ⥤ D
F : C ⥤ H
E : L.LeftExtension F
h : E.IsPointwiseLeftKanExtension
G : L.LeftExtension F
f₁ f₂ : E ⟶ G
Y : D
X : CostructuredArrow L Y
⊢ (E.coconeAt Y).ι.app X ≫ f₁.right.app Y = (E.coconeAt Y).ι.app X ≫ f₂.right.app Y
|
have eq₁ := congr_app (StructuredArrow.w f₁) X.left
|
case h.w.h
C : Type u_1
D : Type u_2
H : Type u_3
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_6, u_2} D
inst✝ : Category.{u_5, u_3} H
L : C ⥤ D
F : C ⥤ H
E : L.LeftExtension F
h : E.IsPointwiseLeftKanExtension
G : L.LeftExtension F
f₁ f₂ : E ⟶ G
Y : D
X : CostructuredArrow L Y
eq₁ : (E.hom ≫ ((whiskeringLeft C D H).obj L).map f₁.right).app X.left = G.hom.app X.left
⊢ (E.coconeAt Y).ι.app X ≫ f₁.right.app Y = (E.coconeAt Y).ι.app X ≫ f₂.right.app Y
|
9c52946883f48623
|
Computation.LiftRelAux.ret_left
|
Mathlib/Data/Seq/Computation.lean
|
theorem LiftRelAux.ret_left (R : α → β → Prop) (C : Computation α → Computation β → Prop) (a cb) :
LiftRelAux R C (Sum.inl a) (destruct cb) ↔ ∃ b, b ∈ cb ∧ R a b
|
α : Type u
β : Type v
R : α → β → Prop
C : Computation α → Computation β → Prop
a : α
cb : Computation β
⊢ ∀ (cb : Computation β), LiftRelAux R C (Sum.inl a) cb.think.destruct ↔ ∃ b, b ∈ cb.think ∧ R a b
|
intro
|
α : Type u
β : Type v
R : α → β → Prop
C : Computation α → Computation β → Prop
a : α
cb cb✝ : Computation β
⊢ LiftRelAux R C (Sum.inl a) cb✝.think.destruct ↔ ∃ b, b ∈ cb✝.think ∧ R a b
|
3b346dca3baf4786
|
CategoryTheory.Sheaf.isLocallyBijective_iff_isIso
|
Mathlib/CategoryTheory/Sites/LocallyBijective.lean
|
lemma isLocallyBijective_iff_isIso :
IsLocallyInjective f ∧ IsLocallySurjective f ↔ IsIso f
|
case mp
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F G : Sheaf J A
f : F ⟶ G
inst✝¹ : (forget A).ReflectsIsomorphisms
inst✝ : J.HasSheafCompose (forget A)
⊢ IsLocallyInjective f ∧ IsLocallySurjective f → IsIso f
|
rintro ⟨_, _⟩
|
case mp.intro
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F G : Sheaf J A
f : F ⟶ G
inst✝¹ : (forget A).ReflectsIsomorphisms
inst✝ : J.HasSheafCompose (forget A)
left✝ : IsLocallyInjective f
right✝ : IsLocallySurjective f
⊢ IsIso f
|
c23e5f228ca366c1
|
Nat.ofDigits_digits
|
Mathlib/Data/Nat/Digits.lean
|
theorem ofDigits_digits (b n : ℕ) : ofDigits b (digits b n) = n
|
case succ
n b : ℕ
⊢ ofDigits (b + 1) ((b + 1).digits n) = n
|
rcases b with - | b
|
case succ.zero
n : ℕ
⊢ ofDigits (0 + 1) ((0 + 1).digits n) = n
case succ.succ
n b : ℕ
⊢ ofDigits (b + 1 + 1) ((b + 1 + 1).digits n) = n
|
ccfc3fb23072b183
|
EulerProduct.summable_and_hasSum_factoredNumbers_prod_filter_prime_geometric
|
Mathlib/NumberTheory/EulerProduct/Basic.lean
|
/-- Given a (completely) multiplicative function `f : ℕ → F`, where `F` is a normed field,
such that `‖f p‖ < 1` for all primes `p`, we can express the sum of `f n` over all `s`-factored
positive integers `n` as a product of `(1 - f p)⁻¹` over the primes `p ∈ s`. At the same time,
we show that the sum involved converges absolutely. -/
lemma summable_and_hasSum_factoredNumbers_prod_filter_prime_geometric {f : ℕ →* F}
(h : ∀ {p : ℕ}, p.Prime → ‖f p‖ < 1) (s : Finset ℕ) :
Summable (fun m : factoredNumbers s ↦ ‖f m‖) ∧
HasSum (fun m : factoredNumbers s ↦ f m) (∏ p ∈ s with p.Prime, (1 - f p)⁻¹)
|
F : Type u_1
inst✝¹ : NormedField F
inst✝ : CompleteSpace F
f : ℕ →* F
h : ∀ {p : ℕ}, Nat.Prime p → ‖f p‖ < 1
s : Finset ℕ
hmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n
H₁ : ∏ p ∈ filter (fun p => Nat.Prime p) s, ∑' (n : ℕ), f (p ^ n) = ∏ p ∈ filter (fun p => Nat.Prime p) s, (1 - f p)⁻¹
⊢ ∀ {p : ℕ}, Nat.Prime p → Summable fun n => ‖f (p ^ n)‖
|
intro p hp
|
F : Type u_1
inst✝¹ : NormedField F
inst✝ : CompleteSpace F
f : ℕ →* F
h : ∀ {p : ℕ}, Nat.Prime p → ‖f p‖ < 1
s : Finset ℕ
hmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n
H₁ : ∏ p ∈ filter (fun p => Nat.Prime p) s, ∑' (n : ℕ), f (p ^ n) = ∏ p ∈ filter (fun p => Nat.Prime p) s, (1 - f p)⁻¹
p : ℕ
hp : Nat.Prime p
⊢ Summable fun n => ‖f (p ^ n)‖
|
b953f848801a28f5
|
IsAdjoinRootMonic.coeff_algebraMap
|
Mathlib/RingTheory/IsAdjoinRoot.lean
|
theorem coeff_algebraMap [Nontrivial S] (h : IsAdjoinRootMonic S f) (x : R) :
h.coeff (algebraMap R S x) = Pi.single 0 x
|
case h
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : Ring S
f : R[X]
inst✝¹ : Algebra R S
inst✝ : Nontrivial S
h : IsAdjoinRootMonic S f
x : R
i : ℕ
⊢ h.coeff ((algebraMap R S) x) i = Pi.single 0 x i
|
rw [Algebra.algebraMap_eq_smul_one, map_smul, coeff_one, Pi.smul_apply, smul_eq_mul]
|
case h
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : Ring S
f : R[X]
inst✝¹ : Algebra R S
inst✝ : Nontrivial S
h : IsAdjoinRootMonic S f
x : R
i : ℕ
⊢ x * Pi.single 0 1 i = Pi.single 0 x i
|
272d1168eee9a4a9
|
zpow_pos
|
Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean
|
lemma zpow_pos [PosMulStrictMono G₀] (ha : 0 < a) : ∀ n : ℤ, 0 < a ^ n
| (n : ℕ) => by rw [zpow_natCast]; exact pow_pos ha _
|-(n + 1 : ℕ) => by rw [zpow_neg, inv_pos, zpow_natCast]; exact pow_pos ha _
|
G₀ : Type u_2
inst✝⁴ : GroupWithZero G₀
inst✝³ : PartialOrder G₀
inst✝² : ZeroLEOneClass G₀
inst✝¹ : PosMulReflectLT G₀
a : G₀
inst✝ : PosMulStrictMono G₀
ha : 0 < a
n : ℕ
⊢ 0 < a ^ (-↑(n + 1))
|
rw [zpow_neg, inv_pos, zpow_natCast]
|
G₀ : Type u_2
inst✝⁴ : GroupWithZero G₀
inst✝³ : PartialOrder G₀
inst✝² : ZeroLEOneClass G₀
inst✝¹ : PosMulReflectLT G₀
a : G₀
inst✝ : PosMulStrictMono G₀
ha : 0 < a
n : ℕ
⊢ 0 < a ^ (n + 1)
|
e7eef4564eadb7cb
|
WeierstrassCurve.VariableChange.comp_left_inv
|
Mathlib/AlgebraicGeometry/EllipticCurve/VariableChange.lean
|
lemma comp_left_inv (C : VariableChange R) : comp (inv C) C = id
|
R : Type u
inst✝ : CommRing R
C : VariableChange R
⊢ C.inv.comp C = id
|
rw [comp, id, inv]
|
R : Type u
inst✝ : CommRing R
C : VariableChange R
⊢ { u := { u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.u * C.u,
r :=
{ u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.r * ↑C.u ^ 2 +
C.r,
s :=
↑C.u * { u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.s + C.s,
t :=
{ u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.t * ↑C.u ^ 3 +
{ u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.r * C.s *
↑C.u ^ 2 +
C.t } =
{ u := 1, r := 0, s := 0, t := 0 }
|
3094bd89e944291c
|
sum_div_nat_floor_pow_sq_le_div_sq
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
(∑ i ∈ range N with j < ⌊c ^ i⌋₊, (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2) ≤
c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2
|
case hab
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun i => j < c ^ i) (range N)
⊢ c ^ i ≤ (1 - c⁻¹)⁻¹ * ↑⌊c ^ i⌋₊
|
rw [← div_eq_inv_mul, le_div_iff₀ A, mul_comm]
|
case hab
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun i => j < c ^ i) (range N)
⊢ (1 - c⁻¹) * c ^ i ≤ ↑⌊c ^ i⌋₊
|
a23322a47b698b23
|
orthogonalProjection_orthogonalProjection_of_le
|
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
theorem orthogonalProjection_orthogonalProjection_of_le {U V : Submodule 𝕜 E}
[HasOrthogonalProjection U] [HasOrthogonalProjection V] (h : U ≤ V) (x : E) :
orthogonalProjection U (orthogonalProjection V x) = orthogonalProjection U x :=
Eq.symm <| by
simpa only [sub_eq_zero, map_sub] using
orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero
(Submodule.orthogonal_le h (sub_orthogonalProjection_mem_orthogonal x))
|
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
U V : Submodule 𝕜 E
inst✝¹ : HasOrthogonalProjection U
inst✝ : HasOrthogonalProjection V
h : U ≤ V
x : E
⊢ (orthogonalProjection U) x = (orthogonalProjection U) ↑((orthogonalProjection V) x)
|
simpa only [sub_eq_zero, map_sub] using
orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero
(Submodule.orthogonal_le h (sub_orthogonalProjection_mem_orthogonal x))
|
no goals
|
12f7e1d041eb0827
|
Polynomial.integralNormalization_coeff_degree_ne
|
Mathlib/RingTheory/Polynomial/IntegralNormalization.lean
|
theorem integralNormalization_coeff_degree_ne {i : ℕ} (hi : p.degree ≠ i) :
coeff (integralNormalization p) i = coeff p i * p.leadingCoeff ^ (p.natDegree - 1 - i)
|
R : Type u
inst✝ : Semiring R
p : R[X]
i : ℕ
hi : p.degree ≠ ↑i
⊢ p.integralNormalization.coeff i = p.coeff i * p.leadingCoeff ^ (p.natDegree - 1 - i)
|
rw [integralNormalization_coeff, if_neg hi]
|
no goals
|
209d6cef1bed6b4b
|
LieAlgebra.IsSemisimple.finitelyAtomistic
|
Mathlib/Algebra/Lie/Semisimple/Basic.lean
|
/--
In a semisimple Lie algebra,
Lie ideals that are contained in the supremum of a finite collection of atoms
are themselves the supremum of a finite subcollection of those atoms.
By a compactness argument, this statement can be extended to arbitrary sets of atoms.
See `atomistic`.
The proof is by induction on the finite set of atoms.
-/
private
lemma finitelyAtomistic : ∀ s : Finset (LieIdeal R L), ↑s ⊆ {I : LieIdeal R L | IsAtom I} →
∀ I : LieIdeal R L, I ≤ s.sup id → ∃ t ⊆ s, I = t.sup id
|
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
s : Finset (LieIdeal R L)
hs : ↑s ⊆ {I | IsAtom I}
I : LieIdeal R L
hI✝ : I ≤ s.sup id
S : Set (LieIdeal R L) := {I | IsAtom I}
hI : I < s.sup id
J : LieIdeal R L
hJs : J ∈ s
hJI : ¬J ≤ I
s' : Finset (LieIdeal R L) := s.erase J
hs' : s' ⊂ s
hs'S : ↑s' ⊆ S
K : LieIdeal R L := s'.sup id
y : L
hy : y ∈ id J
z : L
hz : z ∈ K
hx : y + z ∈ I
j : ↥J
this : J ⊓ I = ⊥
⊢ ⁅↑j, y + z⁆ ∈ ⊥
|
apply this.le
|
case a
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
s : Finset (LieIdeal R L)
hs : ↑s ⊆ {I | IsAtom I}
I : LieIdeal R L
hI✝ : I ≤ s.sup id
S : Set (LieIdeal R L) := {I | IsAtom I}
hI : I < s.sup id
J : LieIdeal R L
hJs : J ∈ s
hJI : ¬J ≤ I
s' : Finset (LieIdeal R L) := s.erase J
hs' : s' ⊂ s
hs'S : ↑s' ⊆ S
K : LieIdeal R L := s'.sup id
y : L
hy : y ∈ id J
z : L
hz : z ∈ K
hx : y + z ∈ I
j : ↥J
this : J ⊓ I = ⊥
⊢ ⁅↑j, y + z⁆ ∈ J ⊓ I
|
9feaa310a91f7482
|
HasDerivAt.lhopital_zero_left_on_Ioo
|
Mathlib/Analysis/Calculus/LHopital.lean
|
theorem lhopital_zero_left_on_Ioo (hab : a < b) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
(hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
(hfb : Tendsto f (𝓝[<] b) (𝓝 0)) (hgb : Tendsto g (𝓝[<] b) (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l) :
Tendsto (fun x => f x / g x) (𝓝[<] b) l
|
a b : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hab : a < b
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : Tendsto f (𝓝[<] b) (𝓝 0)
hgb : Tendsto g (𝓝[<] b) (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l
hdnf : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
this✝ : Tendsto (fun x => (f ∘ Neg.neg) x / (g ∘ Neg.neg) x) (𝓝[>] (-b)) l
this : Tendsto (fun x => (fun x => f (-x) / g (-x)) (-x)) (𝓝[<] b) l
⊢ Tendsto (fun x => f x / g x) (𝓝[<] b) l
|
simpa only [neg_neg]
|
no goals
|
8222c911c0270cee
|
exists_lub_Iio
|
Mathlib/Order/Bounds/Basic.lean
|
theorem exists_lub_Iio (i : γ) : ∃ j, IsLUB (Iio i) j
|
case neg
γ : Type v
inst✝ : LinearOrder γ
i : γ
h_exists_lt : ¬∃ j ∈ upperBounds (Iio i), j < i
h : ¬∀ x ∈ upperBounds (Iio i), i ≤ x
⊢ False
|
refine h_exists_lt ?_
|
case neg
γ : Type v
inst✝ : LinearOrder γ
i : γ
h_exists_lt : ¬∃ j ∈ upperBounds (Iio i), j < i
h : ¬∀ x ∈ upperBounds (Iio i), i ≤ x
⊢ ∃ j ∈ upperBounds (Iio i), j < i
|
726072efd7470ae2
|
List.erase_eq_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Erase.lean
|
theorem erase_eq_iff [LawfulBEq α] {a : α} {l : List α} :
l.erase a = l' ↔
(a ∉ l ∧ l = l') ∨
∃ l₁ l₂, a ∉ l₁ ∧ l = l₁ ++ a :: l₂ ∧ l' = l₁ ++ l₂
|
case mp.inr.intro.intro.intro.intro.intro.intro.h
α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a' : α
l' : List α
h : ¬a' ∈ l'
x : List α
⊢ ∃ l₁, ¬a' ∈ l₁ ∧ ∃ x_1, l' ++ a' :: x = l₁ ++ a' :: x_1 ∧ l' ++ x = l₁ ++ x_1
|
refine ⟨l', h, x, by simp⟩
|
no goals
|
f35279b2c0c6abba
|
compactlyGeneratedSpace_of_isClosed_of_t2
|
Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean
|
theorem compactlyGeneratedSpace_of_isClosed_of_t2
(h : ∀ s, (∀ (K : Set X), IsCompact K → IsClosed (s ∩ K)) → IsClosed s) :
CompactlyGeneratedSpace X
|
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : T2Space X
h : ∀ (s : Set X), (∀ (K : Set X), IsCompact K → IsClosed (s ∩ K)) → IsClosed s
s : Set X
hs :
∀ (K : Type u) [inst : TopologicalSpace K] [inst_1 : CompactSpace K] [inst_2 : T2Space K] (f : K → X),
Continuous f → IsClosed (f ⁻¹' s)
K : Set X
hK : IsCompact K
⊢ IsClosed (s ∩ K)
|
rw [Set.inter_comm, ← Subtype.image_preimage_coe]
|
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : T2Space X
h : ∀ (s : Set X), (∀ (K : Set X), IsCompact K → IsClosed (s ∩ K)) → IsClosed s
s : Set X
hs :
∀ (K : Type u) [inst : TopologicalSpace K] [inst_1 : CompactSpace K] [inst_2 : T2Space K] (f : K → X),
Continuous f → IsClosed (f ⁻¹' s)
K : Set X
hK : IsCompact K
⊢ IsClosed (Subtype.val '' (Subtype.val ⁻¹' s))
|
3a8ef05dc35c2a4e
|
round_le
|
Mathlib/Algebra/Order/Round.lean
|
theorem round_le (x : α) (z : ℤ) : |x - round x| ≤ |x - z|
|
case inr.h
α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
x : α
z : ℤ
hx : x < ↑z
⊢ 1 - fract x ≤ |x - ↑z|
|
rw [abs_eq_neg_self.mpr (sub_neg.mpr hx).le]
|
case inr.h
α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
x : α
z : ℤ
hx : x < ↑z
⊢ 1 - fract x ≤ -(x - ↑z)
|
c274f3c721c0243d
|
Finset.le_mulEnergy
|
Mathlib/Combinatorics/Additive/Energy.lean
|
@[to_additive] lemma le_mulEnergy : s.card * t.card ≤ Eₘ[s, t]
|
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Mul α
s t : Finset α
a : α × α
x✝¹ : a ∈ ↑(s ×ˢ t)
b : α × α
x✝ : b ∈ ↑(s ×ˢ t)
⊢ (fun x => ((x.1, x.1), x.2, x.2)) a = (fun x => ((x.1, x.1), x.2, x.2)) b → a = b
|
simp only [Prod.mk.inj_iff, and_self_iff, and_imp]
|
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Mul α
s t : Finset α
a : α × α
x✝¹ : a ∈ ↑(s ×ˢ t)
b : α × α
x✝ : b ∈ ↑(s ×ˢ t)
⊢ a.1 = b.1 → a.2 = b.2 → a = b
|
9ead259eb013602b
|
convexOn_zpow
|
Mathlib/Analysis/Convex/Mul.lean
|
/-- `x^m`, `m : ℤ` is convex on `(0, +∞)` for all `m`. -/
lemma convexOn_zpow : ∀ n : ℤ, ConvexOn 𝕜 (Ioi 0) fun x : 𝕜 ↦ x ^ n
| (n : ℕ) => by
simp_rw [zpow_natCast]
exact (convexOn_pow n).subset Ioi_subset_Ici_self (convex_Ioi _)
| -[n+1] => by
simp_rw [zpow_negSucc, ← inv_pow]
refine (convexOn_iff_forall_pos.2 ⟨convex_Ioi _, ?_⟩).pow (fun x (hx : 0 < x) ↦ by positivity) _
rintro x (hx : 0 < x) y (hy : 0 < y) a b ha hb hab
field_simp
rw [div_le_div_iff₀, ← sub_nonneg]
· calc
0 ≤ a * b * (x - y) ^ 2
|
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
n : ℕ
⊢ ConvexOn 𝕜 (Ioi 0) fun x => x⁻¹ ^ (n + 1)
|
refine (convexOn_iff_forall_pos.2 ⟨convex_Ioi _, ?_⟩).pow (fun x (hx : 0 < x) ↦ by positivity) _
|
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
n : ℕ
⊢ ∀ ⦃x : 𝕜⦄,
x ∈ Ioi 0 → ∀ ⦃y : 𝕜⦄, y ∈ Ioi 0 → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → (a • x + b • y)⁻¹ ≤ a • x⁻¹ + b • y⁻¹
|
b154fb3321707078
|
Basis.coe_toOrthonormalBasis
|
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
theorem _root_.Basis.coe_toOrthonormalBasis (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 v) :
(v.toOrthonormalBasis hv : ι → E) = (v : ι → E) :=
calc
(v.toOrthonormalBasis hv : ι → E) = ((v.toOrthonormalBasis hv).toBasis : ι → E)
|
ι : Type u_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype ι
v : Basis ι 𝕜 E
hv : Orthonormal 𝕜 ⇑v
⊢ ⇑(v.toOrthonormalBasis hv).toBasis = ⇑v
|
simp
|
no goals
|
c770b3de998236b1
|
Prime.exists_mem_multiset_dvd
|
Mathlib/Algebra/BigOperators/Associated.lean
|
theorem exists_mem_multiset_dvd (hp : Prime p) {s : Multiset α} : p ∣ s.prod → ∃ a ∈ s, p ∣ a :=
Multiset.induction_on s (fun h => (hp.not_dvd_one h).elim) fun a s ih h =>
have : p ∣ a * s.prod
|
α : Type u_1
inst✝ : CommMonoidWithZero α
p : α
hp : Prime p
s✝ : Multiset α
a : α
s : Multiset α
ih : p ∣ s.prod → ∃ a ∈ s, p ∣ a
h : p ∣ (a ::ₘ s).prod
⊢ p ∣ a * s.prod
|
simpa using h
|
no goals
|
1d0eae0f00e03dec
|
Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle
|
Mathlib/Algebra/Group/Pi/Lemmas.lean
|
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
{k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
(mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n
|
case refine_1.inr.inr
I : Type u
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m n : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v
hk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1
hl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1
hm : ((if True then u else 1) * if m = n then v else 1) = (if m = k then u else 1) * if m = l then v else 1
hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1
hkm : k ≠ m
hmn : m ≠ n
⊢ k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n
|
rw [if_neg hkm.symm, if_neg hmn, one_mul, mul_one] at hm
|
case refine_1.inr.inr
I : Type u
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m n : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v
hk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1
hl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1
hm : (if True then u else 1) = if m = l then v else 1
hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1
hkm : k ≠ m
hmn : m ≠ n
⊢ k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n
|
e106ea91b86a42a8
|
top_himp
|
Mathlib/Order/Heyting/Basic.lean
|
theorem top_himp : ⊤ ⇨ a = a :=
eq_of_forall_le_iff fun b => by rw [le_himp_iff, inf_top_eq]
|
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
⊢ b ≤ ⊤ ⇨ a ↔ b ≤ a
|
rw [le_himp_iff, inf_top_eq]
|
no goals
|
e7fb1b8b9764dc3f
|
PowerSeries.coeff_mul_X_pow
|
Mathlib/RingTheory/PowerSeries/Basic.lean
|
theorem coeff_mul_X_pow (p : R⟦X⟧) (n d : ℕ) :
coeff R (d + n) (p * X ^ n) = coeff R d p
|
case h₀.mk.hnc
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
d i j : ℕ
h1 : (i, j) ∈ antidiagonal (d + (i, j).2)
h2 : (i, j) ≠ (d, (i, j).2)
⊢ False
|
apply h2
|
case h₀.mk.hnc
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
d i j : ℕ
h1 : (i, j) ∈ antidiagonal (d + (i, j).2)
h2 : (i, j) ≠ (d, (i, j).2)
⊢ (i, j) = (d, (i, j).2)
|
e458456f8b304f1d
|
Basis.finTwoProd_one
|
Mathlib/LinearAlgebra/Basis/Basic.lean
|
theorem finTwoProd_one (R : Type*) [Semiring R] : Basis.finTwoProd R 1 = (0, 1)
|
R : Type u_7
inst✝ : Semiring R
⊢ (Basis.finTwoProd R) 1 = (0, 1)
|
simp [Basis.finTwoProd, LinearEquiv.finTwoArrow]
|
no goals
|
98c21ee8ddca98c0
|
ConjClasses.card_carrier
|
Mathlib/GroupTheory/GroupAction/Quotient.lean
|
theorem ConjClasses.card_carrier {G : Type*} [Group G] [Fintype G] (g : G)
[Fintype (ConjClasses.mk g).carrier] [Fintype <| MulAction.stabilizer (ConjAct G) g] :
Fintype.card (ConjClasses.mk g).carrier =
Fintype.card G / Fintype.card (MulAction.stabilizer (ConjAct G) g)
|
G : Type u_1
inst✝³ : Group G
inst✝² : Fintype G
g : G
inst✝¹ : Fintype ↑(ConjClasses.mk g).carrier
inst✝ : Fintype ↥(MulAction.stabilizer (ConjAct G) g)
⊢ Fintype.card ↑(ConjClasses.mk g).carrier =
Fintype.card (ConjAct G) / Fintype.card ↥(MulAction.stabilizer (ConjAct G) g)
|
rw [← MulAction.card_orbit_mul_card_stabilizer_eq_card_group (ConjAct G) g, Nat.mul_div_cancel]
|
G : Type u_1
inst✝³ : Group G
inst✝² : Fintype G
g : G
inst✝¹ : Fintype ↑(ConjClasses.mk g).carrier
inst✝ : Fintype ↥(MulAction.stabilizer (ConjAct G) g)
⊢ Fintype.card ↑(ConjClasses.mk g).carrier = Fintype.card ↑(MulAction.orbit (ConjAct G) g)
case H
G : Type u_1
inst✝³ : Group G
inst✝² : Fintype G
g : G
inst✝¹ : Fintype ↑(ConjClasses.mk g).carrier
inst✝ : Fintype ↥(MulAction.stabilizer (ConjAct G) g)
⊢ 0 < Fintype.card ↥(MulAction.stabilizer (ConjAct G) g)
|
2414b4aad73a0546
|
TrivSqZeroExt.hasSum_snd_expSeries_of_smul_comm
|
Mathlib/Analysis/Normed/Algebra/TrivSqZeroExt.lean
|
theorem hasSum_snd_expSeries_of_smul_comm (x : tsze R M)
(hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) {e : R}
(h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
HasSum (fun n => snd (expSeries 𝕜 (tsze R M) n fun _ => x)) (e • x.snd)
|
𝕜 : Type u_1
R : Type u_3
M : Type u_4
inst✝¹⁶ : Field 𝕜
inst✝¹⁵ : CharZero 𝕜
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Algebra 𝕜 R
inst✝¹¹ : Module 𝕜 M
inst✝¹⁰ : Module R M
inst✝⁹ : Module Rᵐᵒᵖ M
inst✝⁸ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁷ : IsScalarTower 𝕜 R M
inst✝⁶ : IsScalarTower 𝕜 Rᵐᵒᵖ M
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : IsTopologicalRing R
inst✝² : IsTopologicalAddGroup M
inst✝¹ : ContinuousSMul R M
inst✝ : ContinuousSMul Rᵐᵒᵖ M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
e : R
h : HasSum (fun n => (↑n.factorial)⁻¹ • x.fst ^ n) e
⊢ HasSum (fun n => ((↑n.factorial)⁻¹ • x.fst ^ n) • x.snd) (e • x.snd)
|
exact h.smul_const _
|
no goals
|
80324e42321d0c0b
|
Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux
|
Mathlib/MeasureTheory/Covering/Besicovitch.lean
|
theorem exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux (μ : Measure α)
[IsFiniteMeasure μ] (f : α → Set ℝ) (s : Set α)
(hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) :
∃ t : Set (α × ℝ), t.Countable ∧ (∀ p ∈ t, p.1 ∈ s) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧
μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2) = 0 ∧
t.PairwiseDisjoint fun p => closedBall p.1 p.2
|
case intro.intro.intro.refine_3
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
fun t => ((↑t).PairwiseDisjoint fun p => closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : Finset (α × ℝ) → Finset (α × ℝ)
hF :
∀ (t : Finset (α × ℝ)),
P t →
t ⊆ F t ∧ P (F t) ∧ μ (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * μ (s \ ⋃ p ∈ t, closedBall p.1 p.2)
u : ℕ → Finset (α × ℝ) := fun n => F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
Pu : ∀ (n : ℕ), P (u n)
⊢ μ (s \ ⋃ p ∈ ⋃ n, ↑(u n), closedBall p.1 p.2) = 0
|
have A :
∀ n,
μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ ⋃ n : ℕ, (u n : Set (α × ℝ))), closedBall p.fst p.snd) ≤
μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ u n), closedBall p.fst p.snd) := by
intro n
gcongr μ (s \ ?_)
exact biUnion_subset_biUnion_left (subset_iUnion (fun i => (u i : Set (α × ℝ))) n)
|
case intro.intro.intro.refine_3
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
fun t => ((↑t).PairwiseDisjoint fun p => closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : Finset (α × ℝ) → Finset (α × ℝ)
hF :
∀ (t : Finset (α × ℝ)),
P t →
t ⊆ F t ∧ P (F t) ∧ μ (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * μ (s \ ⋃ p ∈ t, closedBall p.1 p.2)
u : ℕ → Finset (α × ℝ) := fun n => F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
Pu : ∀ (n : ℕ), P (u n)
A : ∀ (n : ℕ), μ (s \ ⋃ p ∈ ⋃ n, ↑(u n), closedBall p.1 p.2) ≤ μ (s \ ⋃ p ∈ u n, closedBall p.1 p.2)
⊢ μ (s \ ⋃ p ∈ ⋃ n, ↑(u n), closedBall p.1 p.2) = 0
|
fb6f6078be13e9db
|
SupClosed.biSup_mem
|
Mathlib/Order/SupClosed.lean
|
lemma SupClosed.biSup_mem {ι : Type*} {t : Set ι} {f : ι → α} (hs : SupClosed s)
(ht : t.Finite) (hbot : ⊥ ∈ s) (hf : ∀ i ∈ t, f i ∈ s) : ⨆ i ∈ t, f i ∈ s
|
α : Type u_3
inst✝ : CompleteLattice α
s : Set α
ι : Type u_5
t : Set ι
f : ι → α
hs : SupClosed s
ht : t.Finite
hbot : ⊥ ∈ s
hf : ∀ i ∈ t, f i ∈ s
⊢ ⨆ i ∈ t, f i ∈ s
|
rw [← sSup_image]
|
α : Type u_3
inst✝ : CompleteLattice α
s : Set α
ι : Type u_5
t : Set ι
f : ι → α
hs : SupClosed s
ht : t.Finite
hbot : ⊥ ∈ s
hf : ∀ i ∈ t, f i ∈ s
⊢ sSup (f '' t) ∈ s
|
a418015b4f12e2f7
|
AddCircle.norm_eq
|
Mathlib/Analysis/Normed/Group/AddCircle.lean
|
theorem norm_eq {x : ℝ} : ‖(x : AddCircle p)‖ = |x - round (p⁻¹ * x) * p|
|
case a
x : ℝ
h₁ : BddBelow (abs '' {m | ↑m = ↑x})
h₂ : (abs '' {m | ↑m = ↑x}).Nonempty
⊢ fract x ⊓ (1 - fract x) ≤ sInf (norm '' {m | ↑m = ↑x})
|
simp only [QuotientAddGroup.mk'_apply, Real.norm_eq_abs, le_csInf_iff h₁ h₂]
|
case a
x : ℝ
h₁ : BddBelow (abs '' {m | ↑m = ↑x})
h₂ : (abs '' {m | ↑m = ↑x}).Nonempty
⊢ ∀ b ∈ abs '' {m | ↑m = ↑x}, fract x ⊓ (1 - fract x) ≤ b
|
e14e944eccb5c30b
|
PowerSeries.X_prime
|
Mathlib/RingTheory/PowerSeries/Basic.lean
|
theorem X_prime : Prime (X : R⟦X⟧)
|
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
⊢ Prime X
|
rw [← Ideal.span_singleton_prime]
|
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
⊢ (Ideal.span {X}).IsPrime
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
⊢ X ≠ 0
|
b6b9c2687b01e6ad
|
MeasureTheory.measurableSet_generateFrom_singleton_iff
|
Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
|
theorem measurableSet_generateFrom_singleton_iff {s t : Set α} :
MeasurableSet[MeasurableSpace.generateFrom {s}] t ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ
|
case neg
α : Type u_1
s : Set α
x : Set Prop
hT : True ∉ x
⊢ (fun x => x ∈ s) ⁻¹' x = ∅ ∨ (fun x => x ∈ s) ⁻¹' x = s ∨ (fun x => x ∈ s) ⁻¹' x = sᶜ ∨ (fun x => x ∈ s) ⁻¹' x = univ
|
by_cases hF : False ∈ x
|
case pos
α : Type u_1
s : Set α
x : Set Prop
hT : True ∉ x
hF : False ∈ x
⊢ (fun x => x ∈ s) ⁻¹' x = ∅ ∨ (fun x => x ∈ s) ⁻¹' x = s ∨ (fun x => x ∈ s) ⁻¹' x = sᶜ ∨ (fun x => x ∈ s) ⁻¹' x = univ
case neg
α : Type u_1
s : Set α
x : Set Prop
hT : True ∉ x
hF : False ∉ x
⊢ (fun x => x ∈ s) ⁻¹' x = ∅ ∨ (fun x => x ∈ s) ⁻¹' x = s ∨ (fun x => x ∈ s) ⁻¹' x = sᶜ ∨ (fun x => x ∈ s) ⁻¹' x = univ
|
39127bdfcef76ebf
|
Polynomial.flt_catalan
|
Mathlib/NumberTheory/FLT/Polynomial.lean
|
theorem Polynomial.flt_catalan
{p q r : ℕ} (hp : 0 < p) (hq : 0 < q) (hr : 0 < r)
(hineq : q * r + r * p + p * q ≤ p * q * r)
(chp : (p : k) ≠ 0) (chq : (q : k) ≠ 0) (chr : (r : k) ≠ 0)
{a b c : k[X]} (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) (hab : IsCoprime a b)
{u v w : k} (hu : u ≠ 0) (hv : v ≠ 0) (hw : w ≠ 0)
(heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0) :
a.natDegree = 0 ∧ b.natDegree = 0 ∧ c.natDegree = 0
|
case refine_2
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : 0 < p
hq : 0 < q
hr : 0 < r
hineq : r * p + p * q + q * r ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hbc : IsCoprime b c
heq' : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hca : IsCoprime c a
⊢ b.natDegree = 0
|
rw [mul_rotate] at hineq
|
case refine_2
k : Type u_1
inst✝ : Field k
p q r : ℕ
hp : 0 < p
hq : 0 < q
hr : 0 < r
hineq : r * p + p * q + q * r ≤ q * r * p
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hbc : IsCoprime b c
heq' : C v * b ^ q + C w * c ^ r + C u * a ^ p = 0
hca : IsCoprime c a
⊢ b.natDegree = 0
|
0ef13b75af03c673
|
Matroid.IsBasis.iUnion_isBasis_iUnion
|
Mathlib/Data/Matroid/Basic.lean
|
theorem IsBasis.iUnion_isBasis_iUnion {ι : Type _} (X I : ι → Set α)
(hI : ∀ i, M.IsBasis (I i) (X i)) (h_ind : M.Indep (⋃ i, I i)) :
M.IsBasis (⋃ i, I i) (⋃ i, X i)
|
case intro.intro.intro.intro.refl
α : Type u_1
M : Matroid α
ι : Type u_2
X I : ι → Set α
hI : ∀ (i : ι), M.IsBasis (I i) (X i)
h_ind : M.Indep (⋃ i, I i)
e : α
he' : ∀ (x : ι), e ∉ I x
i : ι
hes : e ∈ X i
⊢ e ∈ M.E
|
exact (hI i).subset_ground hes
|
no goals
|
178721ff1f281c37
|
PreTilt.isDomain
|
Mathlib/RingTheory/Perfection.lean
|
theorem isDomain : IsDomain (PreTilt O p)
|
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
hp : Nat.Prime p
this✝ : Nontrivial (PreTilt O p)
this : NoZeroDivisors (PreTilt O p)
⊢ IsDomain (PreTilt O p)
|
exact NoZeroDivisors.to_isDomain _
|
no goals
|
2dffd41f28def1f7
|
TopCat.Presheaf.locally_surjective_iff_surjective_on_stalks
|
Mathlib/Topology/Sheaves/LocallySurjective.lean
|
theorem locally_surjective_iff_surjective_on_stalks (T : ℱ ⟶ 𝒢) :
IsLocallySurjective T ↔ ∀ x : X, Function.Surjective ((stalkFunctor C x).map T)
|
case h
C : Type u
inst✝⁴ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
X : TopCat
ℱ 𝒢 : Presheaf C X
inst✝¹ : Limits.HasColimits C
inst✝ : Limits.PreservesFilteredColimits (forget C)
T : ℱ ⟶ 𝒢
hT : IsLocallySurjective T
x : ↑X
U : Opens ↑X
hxU : x ∈ U
t : ToType (𝒢.obj (op U))
V : Opens ↑X
ι : V ⟶ U
hxV : x ∈ V
s : ToType (ℱ.obj (op V))
h_eq : (ConcreteCategory.hom (T.app (op V))) s = (ConcreteCategory.hom (𝒢.map ι.op)) t
⊢ (ConcreteCategory.hom ((stalkFunctor C x).map T)) ((ConcreteCategory.hom (ℱ.germ V x hxV)) s) =
(ConcreteCategory.hom (𝒢.germ U x hxU)) t
|
simp [h_eq, germ_res_apply]
|
no goals
|
564dc8774c703ba6
|
Turing.TM1to1.tr_supports
|
Mathlib/Computability/PostTuringMachine.lean
|
theorem tr_supports [Inhabited Λ] {S : Finset Λ} (ss : Supports M S) :
Supports (tr enc dec M) (trSupp M S) :=
⟨Finset.mem_biUnion.2 ⟨_, ss.1, Finset.mem_insert_self _ _⟩, fun q h ↦ by
suffices ∀ q, SupportsStmt S q → (∀ q' ∈ writes q, q' ∈ trSupp M S) →
SupportsStmt (trSupp M S) (trNormal dec q) ∧
∀ q' ∈ writes q, SupportsStmt (trSupp M S) (tr enc dec M q') by
rcases Finset.mem_biUnion.1 h with ⟨l, hl, h⟩
have :=
this _ (ss.2 _ hl) fun q' hq ↦ Finset.mem_biUnion.2 ⟨_, hl, Finset.mem_insert_of_mem hq⟩
rcases Finset.mem_insert.1 h with (rfl | h)
exacts [this.1, this.2 _ h]
intro q hs hw
induction q with
| move d q IH =>
unfold writes at hw ⊢
replace IH := IH hs hw; refine ⟨?_, IH.2⟩
cases d <;> simp only [trNormal, iterate, supportsStmt_move, IH]
| write f q IH =>
unfold writes at hw ⊢
simp only [Finset.mem_image, Finset.mem_union, Finset.mem_univ, exists_prop, true_and]
at hw ⊢
replace IH := IH hs fun q hq ↦ hw q (Or.inr hq)
refine ⟨supportsStmt_read _ fun a _ s ↦ hw _ (Or.inl ⟨_, rfl⟩), fun q' hq ↦ ?_⟩
rcases hq with (⟨a, q₂, rfl⟩ | hq)
· simp only [tr, supportsStmt_write, supportsStmt_move, IH.1]
· exact IH.2 _ hq
| load a q IH =>
unfold writes at hw ⊢
replace IH := IH hs hw
exact ⟨supportsStmt_read _ fun _ ↦ IH.1, IH.2⟩
| branch p q₁ q₂ IH₁ IH₂ =>
unfold writes at hw ⊢
simp only [Finset.mem_union] at hw ⊢
replace IH₁ := IH₁ hs.1 fun q hq ↦ hw q (Or.inl hq)
replace IH₂ := IH₂ hs.2 fun q hq ↦ hw q (Or.inr hq)
exact ⟨supportsStmt_read _ fun _ ↦ ⟨IH₁.1, IH₂.1⟩, fun q ↦ Or.rec (IH₁.2 _) (IH₂.2 _)⟩
| goto l =>
simp only [writes, Finset.not_mem_empty]; refine ⟨?_, fun _ ↦ False.elim⟩
refine supportsStmt_read _ fun a _ s ↦ ?_
exact Finset.mem_biUnion.2 ⟨_, hs _ _, Finset.mem_insert_self _ _⟩
| halt =>
simp only [writes, Finset.not_mem_empty]; refine ⟨?_, fun _ ↦ False.elim⟩
simp only [SupportsStmt, supportsStmt_move, trNormal]⟩
|
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
enc : Γ → List.Vector Bool n
dec : List.Vector Bool n → Γ
M : Λ → Stmt Γ Λ σ
inst✝¹ : Fintype Γ
inst✝ : Inhabited Λ
S : Finset Λ
ss : Supports M S
q : Λ' Γ Λ σ
h : q ∈ trSupp M S
⊢ ∀ (q : Stmt Γ Λ σ),
SupportsStmt S q →
(∀ q' ∈ writes q, q' ∈ trSupp M S) →
SupportsStmt (trSupp M S) (trNormal dec q) ∧ ∀ q' ∈ writes q, SupportsStmt (trSupp M S) (tr enc dec M q')
|
intro q hs hw
|
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
enc : Γ → List.Vector Bool n
dec : List.Vector Bool n → Γ
M : Λ → Stmt Γ Λ σ
inst✝¹ : Fintype Γ
inst✝ : Inhabited Λ
S : Finset Λ
ss : Supports M S
q✝ : Λ' Γ Λ σ
h : q✝ ∈ trSupp M S
q : Stmt Γ Λ σ
hs : SupportsStmt S q
hw : ∀ q' ∈ writes q, q' ∈ trSupp M S
⊢ SupportsStmt (trSupp M S) (trNormal dec q) ∧ ∀ q' ∈ writes q, SupportsStmt (trSupp M S) (tr enc dec M q')
|
bea148d15df2302d
|
SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source_of_isClosed
|
Mathlib/Geometry/Manifold/PartitionOfUnity.lean
|
theorem exists_isSubordinate_chartAt_source_of_isClosed {s : Set M} (hs : IsClosed s) :
∃ f : SmoothPartitionOfUnity s I M s,
f.IsSubordinate (fun x ↦ (chartAt H (x : M)).source)
|
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : SigmaCompactSpace M
s : Set M
hs : IsClosed s
x : M
hx : x ∈ s
⊢ x ∈ ⋃ i, (chartAt H ↑i).source
|
exact mem_iUnion_of_mem ⟨x, hx⟩ (mem_chart_source H x)
|
no goals
|
0a676c026e8923b0
|
Num.le_to_nat
|
Mathlib/Data/Num/Lemmas.lean
|
theorem le_to_nat {m n : Num} : (m : ℕ) ≤ n ↔ m ≤ n
|
m n : Num
⊢ ¬↑n < ↑m ↔ m ≤ n
|
exact not_congr lt_to_nat
|
no goals
|
030e1471e406d779
|
Lean.Order.admissible_or
|
Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Basic.lean
|
theorem admissible_or (P Q : α → Prop)
(hadm₁ : admissible P) (hadm₂ : admissible Q) : admissible (fun x => P x ∨ Q x)
|
case inl.h
α : Sort u
inst✝ : CCPO α
P Q : α → Prop
hadm₁ : admissible P
hadm₂ : admissible Q
c : α → Prop
hchain : chain c
h : ∀ (x : α), c x → (fun x => P x ∨ Q x) x
hP : ∀ (x : α), c x → ∃ y, c y ∧ x ⊑ y ∧ P y
⊢ P (csup fun x => c x ∧ P x)
|
apply hadm₁ _ (chain_conj _ _ hchain)
|
case inl.h
α : Sort u
inst✝ : CCPO α
P Q : α → Prop
hadm₁ : admissible P
hadm₂ : admissible Q
c : α → Prop
hchain : chain c
h : ∀ (x : α), c x → (fun x => P x ∨ Q x) x
hP : ∀ (x : α), c x → ∃ y, c y ∧ x ⊑ y ∧ P y
⊢ ∀ (x : α), c x ∧ P x → P x
|
d9092059bf3f532d
|
SimpleGraph.Walk.IsCycle.snd_ne_penultimate
|
Mathlib/Combinatorics/SimpleGraph/Path.lean
|
lemma IsCycle.snd_ne_penultimate {p : G.Walk u u} (hp : p.IsCycle) : p.snd ≠ p.penultimate
|
V : Type u
G : SimpleGraph V
u : V
p : G.Walk u u
hp : p.IsCycle
h : p.snd = p.penultimate
this : 3 ≤ p.length
⊢ 1 ≤ p.length
|
omega
|
no goals
|
0a5276b29b15aedb
|
ByteArray.data_append
|
Mathlib/.lake/packages/batteries/Batteries/Data/ByteArray.lean
|
theorem data_append (a b : ByteArray) : (a ++ b).data = a.data ++ b.data
|
a b : ByteArray
⊢ a.data ++ (b.data ++ a.data.extract (a.data.size + b.data.size)) = a.data ++ b.data
|
rw [Array.extract_empty_of_stop_le_start (h:=Nat.le_add_right ..), Array.append_empty]
|
no goals
|
8e1c51ba23736459
|
Std.DHashMap.Internal.Raw.insertMany_val
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/Raw.lean
|
theorem insertMany_val [BEq α][Hashable α] {m : Raw₀ α β} {ρ : Type w} [ForIn Id ρ ((a : α) × β a)] {l : ρ} :
m.val.insertMany l = m.insertMany l
|
α : Type u
β : α → Type v
inst✝² : BEq α
inst✝¹ : Hashable α
m : Raw₀ α β
ρ : Type w
inst✝ : ForIn Id ρ ((a : α) × β a)
l : ρ
⊢ m.val.insertMany l = (m.insertMany l).val.val
|
simp [Raw.insertMany, m.2]
|
no goals
|
bc16c55568456d90
|
LinearMap.isClosedEmbedding_of_injective
|
Mathlib/Topology/Algebra/Module/FiniteDimension.lean
|
theorem LinearMap.isClosedEmbedding_of_injective [T2Space E] [FiniteDimensional 𝕜 E] {f : E →ₗ[𝕜] F}
(hf : LinearMap.ker f = ⊥) : IsClosedEmbedding f :=
let g := LinearEquiv.ofInjective f (LinearMap.ker_eq_bot.mp hf)
{ IsEmbedding.subtypeVal.comp g.toContinuousLinearEquiv.toHomeomorph.isEmbedding with
isClosed_range
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : CompleteSpace 𝕜
inst✝¹² : AddCommGroup E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : IsTopologicalAddGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : ContinuousSMul 𝕜 E
inst✝⁷ : AddCommGroup F
inst✝⁶ : TopologicalSpace F
inst✝⁵ : T2Space F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : Module 𝕜 F
inst✝² : ContinuousSMul 𝕜 F
inst✝¹ : T2Space E
inst✝ : FiniteDimensional 𝕜 E
f : E →ₗ[𝕜] F
hf : ker f = ⊥
g : E ≃ₗ[𝕜] ↥(range f) := LinearEquiv.ofInjective f ⋯
⊢ IsClosed (Set.range ⇑f)
|
haveI := f.finiteDimensional_range
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : CompleteSpace 𝕜
inst✝¹² : AddCommGroup E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : IsTopologicalAddGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : ContinuousSMul 𝕜 E
inst✝⁷ : AddCommGroup F
inst✝⁶ : TopologicalSpace F
inst✝⁵ : T2Space F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : Module 𝕜 F
inst✝² : ContinuousSMul 𝕜 F
inst✝¹ : T2Space E
inst✝ : FiniteDimensional 𝕜 E
f : E →ₗ[𝕜] F
hf : ker f = ⊥
g : E ≃ₗ[𝕜] ↥(range f) := LinearEquiv.ofInjective f ⋯
this : FiniteDimensional 𝕜 ↥(range f)
⊢ IsClosed (Set.range ⇑f)
|
0572f7ac8190d223
|
CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le
|
Mathlib/Order/CompactlyGenerated/Basic.lean
|
theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
IsCompactElement k ↔
∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x
|
case mpr.intro.intro
α : Type u_2
inst✝ : CompleteLattice α
k : α
hk : ∀ (s : Set α), s.Nonempty → DirectedOn (fun x1 x2 => x1 ≤ x2) s → k ≤ sSup s → ∃ x ∈ s, k ≤ x
s : Set α
hsup : k ≤ sSup s
S : Set α := {x | ∃ t, ↑t ⊆ s ∧ x = t.sup id}
dir_US : DirectedOn (fun x1 x2 => x1 ≤ x2) S
sup_S : sSup s ≤ sSup S
Sne : S.Nonempty
j : α
hjS : j ∈ S
hjk : k ≤ j
⊢ ∃ t, ↑t ⊆ s ∧ k ≤ t.sup id
|
obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS
|
case mpr.intro.intro.intro.intro
α : Type u_2
inst✝ : CompleteLattice α
k : α
hk : ∀ (s : Set α), s.Nonempty → DirectedOn (fun x1 x2 => x1 ≤ x2) s → k ≤ sSup s → ∃ x ∈ s, k ≤ x
s : Set α
hsup : k ≤ sSup s
S : Set α := {x | ∃ t, ↑t ⊆ s ∧ x = t.sup id}
dir_US : DirectedOn (fun x1 x2 => x1 ≤ x2) S
sup_S : sSup s ≤ sSup S
Sne : S.Nonempty
j : α
hjk : k ≤ j
t : Finset α
htS : ↑t ⊆ s
htsup : j = t.sup id
⊢ ∃ t, ↑t ⊆ s ∧ k ≤ t.sup id
|
445bf3d7fb93c3e6
|
WeierstrassCurve.Affine.equation_negAdd
|
Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
|
/-- The negated addition of two affine points in `W` on a sloped line lies in `W`. -/
lemma equation_negAdd {x₁ x₂ y₁ y₂ : F} (h₁ : W.Equation x₁ y₁) (h₂ : W.Equation x₂ y₂)
(hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)) : W.Equation
(W.addX x₁ x₂ <| W.slope x₁ x₂ y₁ y₂) (W.negAddY x₁ x₂ y₁ <| W.slope x₁ x₂ y₁ y₂)
|
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
⊢ -((W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₁) * (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₂) *
(W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))) =
0
|
rw [neg_eq_zero, sub_self, mul_zero]
|
no goals
|
62648cb1aa459396
|
isExtreme_iInter
|
Mathlib/Analysis/Convex/Extreme.lean
|
theorem isExtreme_iInter {ι : Sort*} [Nonempty ι] {F : ι → Set E}
(hAF : ∀ i : ι, IsExtreme 𝕜 A (F i)) : IsExtreme 𝕜 A (⋂ i : ι, F i)
|
𝕜 : Type u_1
E : Type u_2
inst✝³ : OrderedSemiring 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : SMul 𝕜 E
A : Set E
ι : Sort u_6
inst✝ : Nonempty ι
F : ι → Set E
hAF : ∀ (i : ι), IsExtreme 𝕜 A (F i)
i : ι
⊢ IsExtreme 𝕜 A (⋂ i, F i)
|
refine ⟨iInter_subset_of_subset i (hAF i).1, fun x₁ hx₁A x₂ hx₂A x hxF hx ↦ ?_⟩
|
𝕜 : Type u_1
E : Type u_2
inst✝³ : OrderedSemiring 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : SMul 𝕜 E
A : Set E
ι : Sort u_6
inst✝ : Nonempty ι
F : ι → Set E
hAF : ∀ (i : ι), IsExtreme 𝕜 A (F i)
i : ι
x₁ : E
hx₁A : x₁ ∈ A
x₂ : E
hx₂A : x₂ ∈ A
x : E
hxF : x ∈ ⋂ i, F i
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₁ ∈ ⋂ i, F i ∧ x₂ ∈ ⋂ i, F i
|
bc168c036470e611
|
Finset.sum_Ioc_by_parts
|
Mathlib/Algebra/BigOperators/Module.lean
|
theorem sum_Ioc_by_parts (hmn : m < n) :
∑ i ∈ Ioc m n, f i • g i =
f n • G (n + 1) - f (m + 1) • G (m + 1)
- ∑ i ∈ Ioc m (n - 1), (f (i + 1) - f i) • G (i + 1)
|
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ℕ → R
g : ℕ → M
m n : ℕ
hmn : m < n
⊢ ∑ i ∈ Ioc m n, f i • g i =
f n • ∑ i ∈ range (n + 1), g i - f (m + 1) • ∑ i ∈ range (m + 1), g i -
∑ i ∈ Ioc m (n - 1), (f (i + 1) - f i) • ∑ i ∈ range (i + 1), g i
|
simpa only [← Nat.Ico_succ_succ, Nat.succ_eq_add_one, Nat.sub_add_cancel (Nat.one_le_of_lt hmn),
add_tsub_cancel_right] using sum_Ico_by_parts f g (Nat.succ_lt_succ hmn)
|
no goals
|
30ed679af84a904e
|
Std.Sat.AIG.RefVec.denote_map
|
Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Map.lean
|
theorem denote_map {aig : AIG α} (target : MapTarget aig len) :
∀ (idx : Nat) (hidx : idx < len),
⟦(map aig target).aig, (map aig target).vec.get idx hidx, assign⟧
=
⟦target.func aig (target.vec.get idx hidx), assign⟧
|
case a
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
target : MapTarget aig len
idx : Nat
hidx : idx < len
⊢ 0 ≤ idx
|
omega
|
no goals
|
964751d60bf234e3
|
memℓp_gen
|
Mathlib/Analysis/Normed/Lp/lpSpace.lean
|
theorem memℓp_gen {f : ∀ i, E i} (hf : Summable fun i => ‖f i‖ ^ p.toReal) : Memℓp f p
|
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hf : Summable fun i => ‖f i‖ ^ ⊤.toReal
⊢ Summable fun x => 1
|
simpa using hf
|
no goals
|
a63b48f332040979
|
Complex.arg_eq_pi_iff
|
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0
|
z : ℂ
⊢ z.arg = π ↔ z.re < 0 ∧ z.im = 0
|
by_cases h₀ : z = 0
|
case pos
z : ℂ
h₀ : z = 0
⊢ z.arg = π ↔ z.re < 0 ∧ z.im = 0
case neg
z : ℂ
h₀ : ¬z = 0
⊢ z.arg = π ↔ z.re < 0 ∧ z.im = 0
|
1ed9b6f4a418a885
|
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