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 |
|---|---|---|---|---|---|---|
Std.DHashMap.Internal.List.DistinctKeys.insertEntryIfNew | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem DistinctKeys.insertEntryIfNew [BEq α] [PartialEquivBEq α] {k : α} {v : β k}
{l : List ((a : α) × β a)} (h: DistinctKeys l):
DistinctKeys (insertEntryIfNew k v l) | case isFalse
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
k : α
v : β k
l : List ((a : α) × β a)
h : DistinctKeys l
h✝ : ¬containsKey k l = true
⊢ DistinctKeys l ∧ containsKey k l = false | rename_i h' | case isFalse
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
k : α
v : β k
l : List ((a : α) × β a)
h : DistinctKeys l
h' : ¬containsKey k l = true
⊢ DistinctKeys l ∧ containsKey k l = false | 6431867d6e396e85 |
MeasureTheory.SignedMeasure.totalVariation_absolutelyContinuous_iff | Mathlib/MeasureTheory/Decomposition/Jordan.lean | theorem totalVariation_absolutelyContinuous_iff (s : SignedMeasure α) (μ : Measure α) :
s.totalVariation ≪ μ ↔
s.toJordanDecomposition.posPart ≪ μ ∧ s.toJordanDecomposition.negPart ≪ μ | case mp.left
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
μ : Measure α
h : s.totalVariation ≪ μ
S : Set α
x✝ : MeasurableSet S
hS₂ : μ S = 0
this : s.toJordanDecomposition.posPart S = 0 ∧ s.toJordanDecomposition.negPart S = 0
⊢ s.toJordanDecomposition.posPart S = 0
case mp.right
α : Type u_1
inst✝ : Mea... | exacts [this.1, this.2] | no goals | 5fbb1685e757f365 |
QuasispectrumRestricts.cfc | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.lean | theorem cfc (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S)) (h0 : p 0)
(h : ∀ a, p a ↔ q a ∧ QuasispectrumRestricts a f) :
NonUnitalContinuousFunctionalCalculus R p where
predicate_zero := h0
compactSpace_quasispectrum a | R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : NonUnita... | refine ⟨cfcₙHom_predicate _ _, ?_⟩ | R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : NonUnita... | 3af5d542f22c6b37 |
MeasureTheory.SimpleFunc.setToSimpleFunc_indicator | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToSimpleFunc_indicator (T : Set α → F →L[ℝ] F') (hT_empty : T ∅ = 0)
{m : MeasurableSpace α} {s : Set α} (hs : MeasurableSet s) (x : F) :
SimpleFunc.setToSimpleFunc T
(SimpleFunc.piecewise s hs (SimpleFunc.const α x) (SimpleFunc.const α 0)) =
T s x | case inl
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
T : Set α → F →L[ℝ] F'
hT_empty : T ∅ = 0
m : MeasurableSpace α
x : F
hs : MeasurableSet ∅
⊢ setToSimpleFunc T (piecewise ∅ hs (const α x) (const α 0)) = (T ∅) ... | simp only [hT_empty, ContinuousLinearMap.zero_apply, piecewise_empty, const_zero,
setToSimpleFunc_zero_apply] | no goals | 8d51b9bdca45c311 |
Mathlib.Tactic.Bicategory.of_normalize_eq | Mathlib/Tactic/CategoryTheory/Bicategory/PureCoherence.lean | theorem of_normalize_eq {f g f' : a ⟶ b} {η θ : f ≅ g} (η_f : 𝟙 a ≫ f ≅ f') (η_g : 𝟙 a ≫ g ≅ f')
(h_η : 𝟙 a ◁ η ≪≫ η_g = η_f)
(h_θ : 𝟙 a ◁ θ ≪≫ η_g = η_f) : η = θ | B : Type u
inst✝ : Bicategory B
a b : B
f g f' : a ⟶ b
η θ : f ≅ g
η_f : 𝟙 a ≫ f ≅ f'
η_g : 𝟙 a ≫ g ≅ f'
h_η : 𝟙 a ◁ η ≪≫ η_g = η_f
h_θ : 𝟙 a ◁ θ ≪≫ η_g = η_f
⊢ (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom = θ.hom | simp [← reassoc_of% (congrArg Iso.hom h_θ)] | no goals | 39c6163815cf1890 |
Matrix.exists_mulVec_eq_zero_iff_aux | Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean | theorem exists_mulVec_eq_zero_iff_aux {K : Type*} [DecidableEq n] [Field K] {M : Matrix n n K} :
(∃ v ≠ 0, M *ᵥ v = 0) ↔ M.det = 0 | case mp
n : Type u_1
inst✝² : Fintype n
K : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Field K
M : Matrix n n K
⊢ (∃ v, v ≠ 0 ∧ M *ᵥ v = 0) → M.det = 0 | rintro ⟨v, hv, mul_eq⟩ | case mp.intro.intro
n : Type u_1
inst✝² : Fintype n
K : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Field K
M : Matrix n n K
v : n → K
hv : v ≠ 0
mul_eq : M *ᵥ v = 0
⊢ M.det = 0 | c2c6cb1bbf6f20d0 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.deleteOne_subset | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem deleteOne_subset (f : DefaultFormula n) (id : Nat) (c : DefaultClause n) :
c ∈ toList (deleteOne f id) → c ∈ toList f | case h_3.inl.h.isTrue
n : Nat
f : DefaultFormula n
id : Nat
c : DefaultClause n
x✝¹ : Option (DefaultClause n)
val✝ : DefaultClause n
x✝ :
∀ (l : Literal (PosFin n)) (nodupkey : ∀ (l_1 : PosFin n), ¬(l_1, true) ∈ [l] ∨ ¬(l_1, false) ∈ [l])
(nodup : [l].Nodup), val✝ = { clause := [l], nodupkey := nodupkey, nodup :... | simp only [Array.toList_set] at h1 | case h_3.inl.h.isTrue
n : Nat
f : DefaultFormula n
id : Nat
c : DefaultClause n
x✝¹ : Option (DefaultClause n)
val✝ : DefaultClause n
x✝ :
∀ (l : Literal (PosFin n)) (nodupkey : ∀ (l_1 : PosFin n), ¬(l_1, true) ∈ [l] ∨ ¬(l_1, false) ∈ [l])
(nodup : [l].Nodup), val✝ = { clause := [l], nodupkey := nodupkey, nodup :... | 548982dcd427efd2 |
hasSum_coe_mul_geometric_of_norm_lt_one' | Mathlib/Analysis/SpecificLimits/Normed.lean | theorem hasSum_coe_mul_geometric_of_norm_lt_one'
{x : R} (h : ‖x‖ < 1) :
HasSum (fun n ↦ n * x ^ n : ℕ → R) (x * (Ring.inverse (1 - x)) ^ 2) | case h.e'_5
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
A : HasSum (fun n => (↑n + 1) * x ^ n) (Ring.inverse (1 - x) ^ 2)
B : HasSum (fun n => x ^ n) (Ring.inverse (1 - x))
⊢ (fun n => ↑n * x ^ n) = fun b => (↑b + 1) * x ^ b - x ^ b | ext n | case h.e'_5.h
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
A : HasSum (fun n => (↑n + 1) * x ^ n) (Ring.inverse (1 - x) ^ 2)
B : HasSum (fun n => x ^ n) (Ring.inverse (1 - x))
n : ℕ
⊢ ↑n * x ^ n = (↑n + 1) * x ^ n - x ^ n | b5e22f39cd33f651 |
Monoid.CoprodI.induction_on | Mathlib/GroupTheory/CoprodI.lean | theorem induction_on {C : CoprodI M → Prop} (m : CoprodI M) (h_one : C 1)
(h_of : ∀ (i) (m : M i), C (of m)) (h_mul : ∀ x y, C x → C y → C (x * y)) : C m | ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
C : CoprodI M → Prop
m : CoprodI M
h_one : C 1
h_of : ∀ (i : ι) (m : M i), C (of m)
h_mul : ∀ (x y : CoprodI M), C x → C y → C (x * y)
⊢ C m | induction m using CoprodI.induction_left with
| one => exact h_one
| mul m x hx => exact h_mul _ _ (h_of _ _) hx | no goals | 78c827cd2083f7e2 |
IsPrimitiveRoot.isPrimitiveRoot_iff' | Mathlib/RingTheory/RootsOfUnity/PrimitiveRoots.lean | theorem isPrimitiveRoot_iff' {k : ℕ} [NeZero k] {ζ ξ : Rˣ} (h : IsPrimitiveRoot ζ k) :
IsPrimitiveRoot ξ k ↔ ∃ i < k, i.Coprime k ∧ ζ ^ i = ξ | case mpr
R : Type u_4
inst✝² : CommRing R
inst✝¹ : IsDomain R
k : ℕ
inst✝ : NeZero k
ζ ξ : Rˣ
h : IsPrimitiveRoot ζ k
⊢ (∃ i < k, i.Coprime k ∧ ζ ^ i = ξ) → IsPrimitiveRoot ξ k | rintro ⟨i, -, hi, rfl⟩ | case mpr.intro.intro.intro
R : Type u_4
inst✝² : CommRing R
inst✝¹ : IsDomain R
k : ℕ
inst✝ : NeZero k
ζ : Rˣ
h : IsPrimitiveRoot ζ k
i : ℕ
hi : i.Coprime k
⊢ IsPrimitiveRoot (ζ ^ i) k | bfc0b4a6b9a92f25 |
ProbabilityTheory.Kernel.ae_compProd_of_ae_ae | Mathlib/Probability/Kernel/Composition/CompProd.lean | lemma ae_compProd_of_ae_ae {κ : Kernel α β} {η : Kernel (α × β) γ}
{p : β × γ → Prop} (hp : MeasurableSet {x | p x})
(h : ∀ᵐ b ∂κ a, ∀ᵐ c ∂η (a, b), p (b, c)) :
∀ᵐ bc ∂(κ ⊗ₖ η) a, p bc | case pos.hs
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
a : α
κ : Kernel α β
η : Kernel (α × β) γ
p : β × γ → Prop
hp : MeasurableSet {x | p x}
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
h : (κ a) {a_1 | ¬(η (a, a_1)) {a | ¬p (a_1, a)} = 0} = 0
⊢ Measur... | exact hp.compl | no goals | 8d3cbf329ae11c89 |
Functor.ext | Mathlib/Control/Functor.lean | theorem Functor.ext {F} :
∀ {F1 : Functor F} {F2 : Functor F} [@LawfulFunctor F F1] [@LawfulFunctor F F2],
(∀ (α β) (f : α → β) (x : F α), @Functor.map _ F1 _ _ f x = @Functor.map _ F2 _ _ f x) →
F1 = F2
| ⟨m, mc⟩, ⟨m', mc'⟩, H1, H2, H => by
cases show @m = @m' by funext α β f x; apply H
congr
... | case refl.e_mapConst.h.h
F : Type u_1 → Type u_2
m : {α β : Type u_1} → (α → β) → F α → F β
mc mc' : {α β : Type u_1} → α → F β → F α
H1 : LawfulFunctor F
H2 : LawfulFunctor F
H : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x
α β : Type u_1
E1 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β
⊢ mc =... | have E2 := @map_const _ ⟨@m, @mc'⟩ H2 | case refl.e_mapConst.h.h
F : Type u_1 → Type u_2
m : {α β : Type u_1} → (α → β) → F α → F β
mc mc' : {α β : Type u_1} → α → F β → F α
H1 : LawfulFunctor F
H2 : LawfulFunctor F
H : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x
α β : Type u_1
E1 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β
E2 : ∀... | 2ea74e0927c19bee |
AlgebraicGeometry.HasRingHomProperty.isLocal_ringHomProperty_of_isLocalAtSource_of_isLocalAtTarget | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | lemma isLocal_ringHomProperty_of_isLocalAtSource_of_isLocalAtTarget
[IsLocalAtTarget P] [IsLocalAtSource P] :
RingHom.PropertyIsLocal fun f ↦ P (Spec.map (CommRingCat.ofHom f)) | case localizationAwayPreserves
P : MorphismProperty Scheme
inst✝⁹ : IsLocalAtTarget P
inst✝⁸ : IsLocalAtSource P
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] f => P (Spec.map (CommRingCat.ofHom f))
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'... | refine (RingHom.RespectsIso.is_localization_away_iff hP ..).mp ?_ | case localizationAwayPreserves
P : MorphismProperty Scheme
inst✝⁹ : IsLocalAtTarget P
inst✝⁸ : IsLocalAtSource P
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] f => P (Spec.map (CommRingCat.ofHom f))
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'... | b9bae3dd3914b2b4 |
List.idxOf_append_of_not_mem | Mathlib/Data/List/Basic.lean | theorem idxOf_append_of_not_mem {a : α} (h : a ∉ l₁) :
idxOf a (l₁ ++ l₂) = l₁.length + idxOf a l₂ | α : Type u
l₁ l₂ : List α
inst✝ : DecidableEq α
a : α
h : ¬a ∈ l₁
⊢ idxOf a (l₁ ++ l₂) = l₁.length + idxOf a l₂ | induction' l₁ with d₁ t₁ ih | case nil
α : Type u
l₁ l₂ : List α
inst✝ : DecidableEq α
a : α
h : ¬a ∈ []
⊢ idxOf a ([] ++ l₂) = [].length + idxOf a l₂
case cons
α : Type u
l₁ l₂ : List α
inst✝ : DecidableEq α
a d₁ : α
t₁ : List α
ih : ¬a ∈ t₁ → idxOf a (t₁ ++ l₂) = t₁.length + idxOf a l₂
h : ¬a ∈ d₁ :: t₁
⊢ idxOf a (d₁ :: t₁ ++ l₂) = (d₁ :: t₁).le... | e3ab4907b14ac870 |
MeasureTheory.Measure.toSignedMeasure_add | Mathlib/MeasureTheory/VectorMeasure/Basic.lean | theorem toSignedMeasure_add (μ ν : Measure α) [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
(μ + ν).toSignedMeasure = μ.toSignedMeasure + ν.toSignedMeasure | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
⊢ (μ + ν).toSignedMeasure = μ.toSignedMeasure + ν.toSignedMeasure | ext i hi | case h
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
i : Set α
hi : MeasurableSet i
⊢ ↑(μ + ν).toSignedMeasure i = ↑(μ.toSignedMeasure + ν.toSignedMeasure) i | eafe7052bf05c363 |
FirstOrder.Language.ElementarySubstructure.theory_model_iff | Mathlib/ModelTheory/ElementarySubstructures.lean | theorem theory_model_iff (S : L.ElementarySubstructure M) (T : L.Theory) : S ⊨ T ↔ M ⊨ T | L : Language
M : Type u_1
inst✝ : L.Structure M
S : L.ElementarySubstructure M
T : L.Theory
⊢ ↥S ⊨ T ↔ M ⊨ T | simp only [Theory.model_iff, realize_sentence] | no goals | 7cc6226aca7eba8f |
ContinuousAlgHom.coe_inj | Mathlib/Topology/Algebra/Algebra.lean | theorem coe_inj {f g : A →A[R] B} : (f : A →ₐ[R] B) = g ↔ f = g := by
cases f; cases g; simp only [mk.injEq]; exact Eq.congr_right rfl
| R : Type u_1
inst✝⁶ : CommSemiring R
A : Type u_2
inst✝⁵ : Semiring A
inst✝⁴ : TopologicalSpace A
B : Type u_3
inst✝³ : Semiring B
inst✝² : TopologicalSpace B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
f g : A →A[R] B
⊢ ↑f = ↑g ↔ f = g | cases f | case mk
R : Type u_1
inst✝⁶ : CommSemiring R
A : Type u_2
inst✝⁵ : Semiring A
inst✝⁴ : TopologicalSpace A
B : Type u_3
inst✝³ : Semiring B
inst✝² : TopologicalSpace B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
g : A →A[R] B
toAlgHom✝ : A →ₐ[R] B
cont✝ : Continuous (↑↑toAlgHom✝.toRingHom).toFun
⊢ ↑{ toAlgHom := toAlgHom✝,... | 7d87ac4ae94bb938 |
List.MergeSort.Internal.mergeSortTR₂_run_eq_mergeSort | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sort/Impl.lean | theorem mergeSortTR₂_run_eq_mergeSort : {n : Nat} → (l : { l : List α // l.length = n }) → mergeSortTR₂.run le l = mergeSort l.1 le
| 0, ⟨[], _⟩
| 1, ⟨[a], _⟩ => by simp [mergeSortTR₂.run, mergeSort]
| n+2, ⟨a :: b :: l, h⟩ => by
cases h
simp only [mergeSortTR₂.run, mergeSort]
rw [splitRevInTwo_fst, s... | case refl
α : Type u_1
le : α → α → Bool
a b : α
l : List α
⊢ mergeTR (mergeSort ?m.68695 le) (⟨(splitInTwo ⟨a :: b :: l, ⋯⟩).snd.val, ⋯⟩.val.mergeSort le) le =
((splitInTwo ⟨a :: b :: l, ⋯⟩).fst.val.mergeSort le).merge ((splitInTwo ⟨a :: b :: l, ⋯⟩).snd.val.mergeSort le) le
case refl.w
α : Type u_1
le : α → α → B... | rw [merge_eq_mergeTR] | case refl.w
α : Type u_1
le : α → α → Bool
a b : α
l : List α
⊢ (splitInTwo ⟨a :: b :: l, ⋯⟩).fst.val = ⟨(splitInTwo ⟨a :: b :: l, ⋯⟩).fst.val.reverse, ⋯⟩.val.reverse | f5d955d0f1f20fce |
hasSum_mellin | Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean | /-- Most basic version of the "Mellin transform = Dirichlet series" argument. -/
lemma hasSum_mellin {a : ι → ℂ} {p : ι → ℝ} {F : ℝ → ℂ} {s : ℂ}
(hp : ∀ i, a i = 0 ∨ 0 < p i) (hs : 0 < s.re)
(hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * rexp (-p i * t)) (F t))
(h_sum : Summable fun i ↦ ‖a i‖ / (p i) ^ s.re) :
... | case convert_3.inl
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), a i = 0 ∨ 0 < p i
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => a i * ↑(rexp (-p i * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / p i ^ s.re
i : ι
hai : a i = 0
⊢ Integrable ((fun i t => ↑t ^ (s - 1) * (a i * ↑(re... | simp [hai] | no goals | 01ba79e74bb3c4b4 |
Std.DHashMap.Internal.Raw₀.toListModel_containsThenInsert | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean | theorem toListModel_containsThenInsert [BEq α] [Hashable α] [EquivBEq α] [LawfulHashable α]
{m : Raw₀ α β} (h : Raw.WFImp m.1) {a : α} {b : β a} :
Perm (toListModel (m.containsThenInsert a b).2.1.2) (insertEntry a b (toListModel m.1.2)) | α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
m : Raw₀ α β
h : Raw.WFImp m.val
a : α
b : β a
⊢ toListModel (m.containsThenInsert a b).snd.val.buckets ~ insertEntry a b (toListModel m.val.buckets) | rw [containsThenInsert_eq_insertₘ] | α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
m : Raw₀ α β
h : Raw.WFImp m.val
a : α
b : β a
⊢ toListModel (m.insertₘ a b).val.buckets ~ insertEntry a b (toListModel m.val.buckets) | e7c33ae0b2dd3169 |
ProbabilityTheory.Kernel.compProd_eq_tsum_compProd | Mathlib/Probability/Kernel/Composition/CompProd.lean | theorem compProd_eq_tsum_compProd (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
[IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) :
(κ ⊗ₖ η) a s = ∑' (n : ℕ) (m : ℕ), (seq κ n ⊗ₖ seq η m) a s | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
hs : MeasurableSet s
⊢ κ.compProdFun η a s = ∑' (n : ℕ) (m : ℕ), (κ.seq n).compProdFun (η.seq m) a s | exact compProdFun_eq_tsum κ η a hs | no goals | 0c44d5eecc5f9201 |
Associates.mem_factorSet_top | Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean | theorem mem_factorSet_top {p : Associates α} {hp : Irreducible p} : p ∈ (⊤ : FactorSet α) | α : Type u_1
inst✝ : CancelCommMonoidWithZero α
p : Associates α
hp : Irreducible p
⊢ BfactorSetMem ⟨p, hp⟩ ⊤ | exact trivial | no goals | 4d3dd99302dd40c9 |
Functor.Comp.id_map | Mathlib/Control/Functor.lean | theorem id_map : ∀ x : Comp F G α, Comp.map id x = x
| Comp.mk x => by simp only [Comp.map, id_map, id_map']; rfl
| F : Type u → Type w
G : Type v → Type u
inst✝³ : Functor F
inst✝² : Functor G
inst✝¹ : LawfulFunctor F
inst✝ : LawfulFunctor G
α : Type v
x : F (G α)
⊢ Comp.map id (mk x) = mk x | simp only [Comp.map, id_map, id_map'] | F : Type u → Type w
G : Type v → Type u
inst✝³ : Functor F
inst✝² : Functor G
inst✝¹ : LawfulFunctor F
inst✝ : LawfulFunctor G
α : Type v
x : F (G α)
⊢ mk (mk x) = mk x | e16fb330f14bd726 |
Array.snd_lt_add_of_mem_zipIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Range.lean | theorem snd_lt_add_of_mem_zipIdx {x : α × Nat} {l : Array α} {k : Nat} (h : x ∈ zipIdx l k) :
x.2 < k + l.size | case intro.intro
α : Type u_1
l : Array α
k i : Nat
h' : i < (l.zipIdx k).size
h : (l.zipIdx k)[i] ∈ l.zipIdx k
⊢ (l.zipIdx k)[i].snd < k + l.size | simpa using h' | no goals | bb1e10cd9a541d17 |
tsum_sq_fourierCoeff | Mathlib/Analysis/Fourier/AddCircle.lean | theorem tsum_sq_fourierCoeff (f : Lp ℂ 2 <| @haarAddCircle T hT) :
∑' i : ℤ, ‖fourierCoeff f i‖ ^ 2 = ∫ t : AddCircle T, ‖f t‖ ^ 2 ∂haarAddCircle | T : ℝ
hT : Fact (0 < T)
f : ↥(Lp ℂ 2 haarAddCircle)
H₁ : ‖fourierBasis.repr f‖ ^ 2 = ∑' (i : ℤ), ‖↑(fourierBasis.repr f) i‖ ^ 2
H₂ : ‖fourierBasis.repr f‖ ^ 2 = ‖f‖ ^ 2
H₃ : RCLike.re (inner f f) = ∫ (x : AddCircle T), RCLike.re (inner (↑↑f x) (↑↑f x)) ∂haarAddCircle
⊢ ∑' (i : ℤ), ‖↑(fourierBasis.repr f) i‖ ^ 2 = ∫ (t ... | simp only [← norm_sq_eq_inner] at H₃ | T : ℝ
hT : Fact (0 < T)
f : ↥(Lp ℂ 2 haarAddCircle)
H₁ : ‖fourierBasis.repr f‖ ^ 2 = ∑' (i : ℤ), ‖↑(fourierBasis.repr f) i‖ ^ 2
H₂ : ‖fourierBasis.repr f‖ ^ 2 = ‖f‖ ^ 2
H₃ : ‖f‖ ^ 2 = ∫ (x : AddCircle T), ‖↑↑f x‖ ^ 2 ∂haarAddCircle
⊢ ∑' (i : ℤ), ‖↑(fourierBasis.repr f) i‖ ^ 2 = ∫ (t : AddCircle T), ‖↑↑f t‖ ^ 2 ∂haarAdd... | 116dddb097891070 |
MonoidHom.exists_nhds_isBounded | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | @[to_additive]
lemma _root_.MonoidHom.exists_nhds_isBounded (f : G →* H) (hf : Measurable f) (x : G) :
∃ s ∈ 𝓝 x, IsBounded (f '' s) | case intro
G : Type u_1
H : Type u_2
inst✝⁸ : MeasurableSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
inst✝² : MeasurableSpace H
inst✝¹ : SeminormedGroup H
inst✝ : OpensMeasurableSpace H
f : G →* H
hf : Measurable ⇑f
x : G
K : Pos... | rw [image_op_smul_distrib, image_div] | case intro
G : Type u_1
H : Type u_2
inst✝⁸ : MeasurableSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
inst✝² : MeasurableSpace H
inst✝¹ : SeminormedGroup H
inst✝ : OpensMeasurableSpace H
f : G →* H
hf : Measurable ⇑f
x : G
K : Pos... | 5d039bded9500100 |
Turing.PartrecToTM2.trNormal_supports | Mathlib/Computability/TMToPartrec.lean | theorem trNormal_supports {S c k} (Hk : codeSupp c k ⊆ S) : (trNormal c k).Supports S | case fix
S : Finset Λ'
f : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Λ'.Supports S (trNormal f k)
k : Cont'
Hk : codeSupp f.fix k ⊆ S
⊢ Λ'.Supports S (trNormal f (Cont'.fix f k)) | apply IHf | case fix.Hk
S : Finset Λ'
f : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Λ'.Supports S (trNormal f k)
k : Cont'
Hk : codeSupp f.fix k ⊆ S
⊢ codeSupp f (Cont'.fix f k) ⊆ S | 279b0f47b2c1d269 |
Basis.orientation_eq_or_eq_neg | Mathlib/LinearAlgebra/Orientation.lean | theorem orientation_eq_or_eq_neg (e : Basis ι R M) (x : Orientation R M ι) :
x = e.orientation ∨ x = -e.orientation | R : Type u_1
inst✝⁴ : LinearOrderedCommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
x : Orientation R M ι
⊢ x = e.orientation ∨ x = -e.orientation | induction x using Module.Ray.ind with | h x hx =>
rw [← x.map_basis_ne_zero_iff e] at hx
rwa [Basis.orientation, ray_eq_iff, neg_rayOfNeZero, ray_eq_iff, x.eq_smul_basis_det e,
sameRay_neg_smul_left_iff_of_ne e.det_ne_zero hx, sameRay_smul_left_iff_of_ne e.det_ne_zero hx,
lt_or_lt_iff_ne, ne_comm] | no goals | 0a3a9013b40d0611 |
Matrix.natCast_kronecker | Mathlib/Data/Matrix/Kronecker.lean | theorem natCast_kronecker [NonAssocSemiring α] [DecidableEq l] (a : ℕ) (B : Matrix m n α) :
(a : Matrix l l α) ⊗ₖ B =
Matrix.reindex (Equiv.prodComm _ _) (Equiv.prodComm _ _) (blockDiagonal fun _ => a • B) :=
diagonal_kronecker _ _ |>.trans <| by
congr! 2
ext
simp [(Nat.cast_commute a _).eq]
| case h.e'_6.h.e'_7
α : Type u_2
l : Type u_8
m : Type u_9
n : Type u_10
inst✝¹ : NonAssocSemiring α
inst✝ : DecidableEq l
a : ℕ
B : Matrix m n α
⊢ (fun i => ↑a •> B) = fun x => a •> B | ext | case h.e'_6.h.e'_7.h.a
α : Type u_2
l : Type u_8
m : Type u_9
n : Type u_10
inst✝¹ : NonAssocSemiring α
inst✝ : DecidableEq l
a : ℕ
B : Matrix m n α
x✝ : l
i✝ : m
j✝ : n
⊢ (↑a •> B) i✝ j✝ = (a •> B) i✝ j✝ | 554189108d790b82 |
ProbabilityTheory.variance_le_expectation_sq | Mathlib/Probability/Variance.lean | theorem variance_le_expectation_sq [IsProbabilityMeasure μ] {X : Ω → ℝ}
(hm : AEStronglyMeasurable X μ) : variance X μ ≤ μ[X ^ 2] | case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hm : AEStronglyMeasurable X μ
hX : ¬MemLp X 2 μ
⊢ ∫ (a : Ω), (X a - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ ≤ ∫ (x : Ω), (X ^ 2) x ∂μ | by_cases hint : Integrable X μ | case pos
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hm : AEStronglyMeasurable X μ
hX : ¬MemLp X 2 μ
hint : Integrable X μ
⊢ ∫ (a : Ω), (X a - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ ≤ ∫ (x : Ω), (X ^ 2) x ∂μ
case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabil... | 2896dc06b5e6e469 |
Std.Tactic.BVDecide.BVExpr.bitblast.denote_blastShiftLeftConst | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftLeft.lean | theorem denote_blastShiftLeftConst (aig : AIG α) (target : ShiftTarget aig w)
(assign : α → Bool) :
∀ (idx : Nat) (hidx : idx < w),
⟦
(blastShiftLeftConst aig target).aig,
(blastShiftLeftConst aig target).vec.get idx hidx,
assign
⟧
=
if hidx : idx ... | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
target : aig.ShiftTarget w
assign : α → Bool
⊢ ∀ (idx : Nat) (hidx : idx < w),
⟦assign,
{ aig := (blastShiftLeftConst aig target).aig, ref := (blastShiftLeftConst aig target).vec.get idx hidx }⟧ =
if hidx : idx < target.distance th... | intros | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
target : aig.ShiftTarget w
assign : α → Bool
idx✝ : Nat
hidx✝ : idx✝ < w
⊢ ⟦assign,
{ aig := (blastShiftLeftConst aig target).aig, ref := (blastShiftLeftConst aig target).vec.get idx✝ hidx✝ }⟧ =
if hidx : idx✝ < target.distance then false
... | 5528b5bd214a74db |
Real.Angle.tan_eq_inv_of_two_nsmul_add_two_nsmul_eq_pi | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | theorem tan_eq_inv_of_two_nsmul_add_two_nsmul_eq_pi {θ ψ : Angle}
(h : (2 : ℕ) • θ + (2 : ℕ) • ψ = π) : tan ψ = (tan θ)⁻¹ | case h
ψ : Angle
x✝ : ℝ
h : 2 • ↑x✝ + 2 • ψ = ↑π
⊢ ψ.tan = (↑x✝).tan⁻¹ | induction ψ using Real.Angle.induction_on | case h.h
x✝¹ x✝ : ℝ
h : 2 • ↑x✝¹ + 2 • ↑x✝ = ↑π
⊢ (↑x✝).tan = (↑x✝¹).tan⁻¹ | 5d1f9ed33a4b52c5 |
Ordinal.relIso_enum' | Mathlib/SetTheory/Ordinal/Basic.lean | theorem relIso_enum' {α β : Type u} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder α r]
[IsWellOrder β s] (f : r ≃r s) (o : Ordinal) :
∀ (hr : o < type r) (hs : o < type s), f (enum r ⟨o, hr⟩) = enum s ⟨o, hs⟩ | case intro.intro
α β : Type u
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
f : r ≃r s
o : Ordinal.{u}
γ : Type u
t : γ → γ → Prop
wo : IsWellOrder γ t
g : t ≺i r
h : t ≺i s
⊢ f ((enum r) ⟨type t, ⋯⟩) = (enum s) ⟨type t, ⋯⟩ | rw [enum_type g, enum_type (g.transRelIso f)] | case intro.intro
α β : Type u
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
f : r ≃r s
o : Ordinal.{u}
γ : Type u
t : γ → γ → Prop
wo : IsWellOrder γ t
g : t ≺i r
h : t ≺i s
⊢ f g.top = (g.transRelIso f).top | f8a3f2c2e909af08 |
Nat.Linear.ExprCnstr.eq_true_of_isValid | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Linear.lean | theorem ExprCnstr.eq_true_of_isValid (ctx : Context) (c : ExprCnstr) (h : c.toNormPoly.isValid) : c.denote ctx = True | ctx : Context
c : ExprCnstr
h : c.toNormPoly.isValid = true
this : PolyCnstr.denote ctx c.toNormPoly = True
⊢ denote ctx c = True | simp [-eq_iff_iff] at this | ctx : Context
c : ExprCnstr
h : c.toNormPoly.isValid = true
this : denote ctx c = True
⊢ denote ctx c = True | 11fe104adc3b1b7d |
ShrinkingLemma.PartialRefinement.exists_gt | Mathlib/Topology/ShrinkingLemma.lean | theorem exists_gt [NormalSpace X] (v : PartialRefinement u s ⊤) (hs : IsClosed s)
(i : ι) (hi : i ∉ v.carrier) :
∃ v' : PartialRefinement u s ⊤, v < v' | case intro.intro.intro.refine_7
ι : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝ : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier
I : s ∩ ⋂ j, ⋂ (_ : j ≠ i), (v.toFun j)ᶜ ⊆ v.toFun i
C : IsClosed (s ∩ ⋂ j, ⋂ (_ : j ≠ i), (v.toFun j)ᶜ)
vi : Set X
ovi... | exact fun hle => hi (hle.1 <| mem_insert _ _) | no goals | f07fb14ff2f0d64d |
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 inl.intro.intro.intro.intro
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hfg : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ c * ‖f x‖
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
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.... | specialize hfg (c := 1/2) (by norm_num) | case inl.intro.intro.intro.intro
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, 0 ≤ f x
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 /... | dd533a6f58fe3819 |
Std.DHashMap.Internal.Raw₀.size_expandIfNecessary | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean | theorem size_expandIfNecessary [BEq α] [Hashable α] {m : Raw₀ α β} :
(expandIfNecessary m).val.size = m.val.size | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
⊢ m.expandIfNecessary.val.size = m.val.size | rw [expandIfNecessary] | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
⊢ (if Std.DHashMap.Internal.numBucketsForCapacity✝ m.1.size ≤ m.1.buckets.size then
⟨{ size := m.1.size, buckets := m.1.buckets }, ⋯⟩
else
match expand ⟨m.1.buckets, ⋯⟩ with
| ⟨buckets', h'⟩ => ⟨{ size := m.1.... | 31a9779352ceb6a9 |
GromovHausdorff.ghDist_le_of_approx_subsets | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | theorem ghDist_le_of_approx_subsets {s : Set X} (Φ : s → Y) {ε₁ ε₂ ε₃ : ℝ}
(hs : ∀ x : X, ∃ y ∈ s, dist x y ≤ ε₁) (hs' : ∀ x : Y, ∃ y : s, dist x (Φ y) ≤ ε₃)
(H : ∀ x y : s, |dist x y - dist (Φ x) (Φ y)| ≤ ε₂) : ghDist X Y ≤ ε₁ + ε₂ / 2 + ε₃ | X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
s : Set X
Φ : ↑s → Y
ε₁ ε₂ ε₃ : ℝ
hs : ∀ (x : X), ∃ y ∈ s, dist x y ≤ ε₁
hs' : ∀ (x : Y), ∃ y, dist x (Φ y) ≤ ε₃
H : ∀ (x y : ↑s), |dist x y - dist (Φ x) (Φ y)| ≤ ε₂... | rcases exists_mem_of_nonempty X with ⟨xX, _⟩ | case intro
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
s : Set X
Φ : ↑s → Y
ε₁ ε₂ ε₃ : ℝ
hs : ∀ (x : X), ∃ y ∈ s, dist x y ≤ ε₁
hs' : ∀ (x : Y), ∃ y, dist x (Φ y) ≤ ε₃
H : ∀ (x y : ↑s), |dist x y - dist (Φ x) ... | ca73aa8d17706ff3 |
Orientation.oangle_eq_angle_of_sign_eq_one | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | theorem oangle_eq_angle_of_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) :
o.oangle x y = InnerProductGeometry.angle x y | case neg
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)
x y : V
h : (o.oangle x y).sign = 1
hx : ¬x = 0
hy : ¬y = 0
hxy : o.oangle x y = -↑(InnerProductGeometry.angle x y)
⊢ False | rw [hxy, Real.Angle.sign_neg, neg_eq_iff_eq_neg, ← SignType.neg_iff, ← not_le] at h | case neg
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)
x y : V
h : ¬0 ≤ (↑(InnerProductGeometry.angle x y)).sign
hx : ¬x = 0
hy : ¬y = 0
hxy : o.oangle x y = -↑(InnerProductGeometry.angle x y)
⊢ False | f5667f81397e71ce |
IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed | Mathlib/Topology/Compactness/Compact.lean | theorem IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed
{ι : Type v} [hι : Nonempty ι] (t : ι → Set X) (htd : Directed (· ⊇ ·) t)
(htn : ∀ i, (t i).Nonempty) (htc : ∀ i, IsCompact (t i)) (htcl : ∀ i, IsClosed (t i)) :
(⋂ i, t i).Nonempty | X : Type u
inst✝ : TopologicalSpace X
ι : Type v
hι : Nonempty ι
t : ι → Set X
htd : Directed (fun x1 x2 => x1 ⊇ x2) t
htc : ∀ (i : ι), IsCompact (t i)
htcl : ∀ (i : ι), IsClosed (t i)
i₀ : ι := hι.some
htn : ∀ (i : ι), t i ≠ ∅
⊢ t i₀ ∩ ⋂ i, t i ≠ ∅ | apply mt ((htc i₀).elim_directed_family_closed t htcl) | X : Type u
inst✝ : TopologicalSpace X
ι : Type v
hι : Nonempty ι
t : ι → Set X
htd : Directed (fun x1 x2 => x1 ⊇ x2) t
htc : ∀ (i : ι), IsCompact (t i)
htcl : ∀ (i : ι), IsClosed (t i)
i₀ : ι := hι.some
htn : ∀ (i : ι), t i ≠ ∅
⊢ ¬(Directed (fun x1 x2 => x1 ⊇ x2) t → ∃ i, t i₀ ∩ t i = ∅) | 6ce44587c65756d8 |
MeasureTheory.AnalyticSet.iInter | Mathlib/MeasureTheory/Constructions/Polish/Basic.lean | theorem AnalyticSet.iInter [hι : Nonempty ι] [Countable ι] [T2Space α] {s : ι → Set α}
(hs : ∀ n, AnalyticSet (s n)) : AnalyticSet (⋂ n, s n) | α : Type u_1
ι : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : Countable ι
inst✝ : T2Space α
s : ι → Set α
hs : ∀ (n : ι), AnalyticSet (s n)
i₀ : ι
β : ι → Type
hβ : (n : ι) → TopologicalSpace (β n)
h'β : ∀ (n : ι), PolishSpace (β n)
f : (n : ι) → β n → α
f_cont : ∀ (n : ι), Continuous (f n)
f_range : ∀ (n : ι), range ... | simp [γ, t, F, hx] | no goals | ca81b48ec4adcb41 |
Std.Sat.AIG.toCNF.Cache.IsExtensionBy_trans_left | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/CNF.lean | theorem Cache.IsExtensionBy_trans_left (cache1 : Cache aig cnf1) (cache2 : Cache aig cnf2)
(cache3 : Cache aig cnf3) (h12 : IsExtensionBy cache1 cache2 new1 hnew1)
(h23 : IsExtensionBy cache2 cache3 new2 hnew2) : IsExtensionBy cache1 cache3 new1 hnew1 | case extension
aig : AIG Nat
cnf1 cnf2 cnf3 : CNF aig.CNFVar
new1 : Nat
hnew1 : new1 < aig.decls.size
new2 : Nat
hnew2 : new2 < aig.decls.size
cache1 : Cache aig cnf1
cache2 : Cache aig cnf2
cache3 : Cache aig cnf3
h12 : cache1.IsExtensionBy cache2 new1 hnew1
h23 : cache2.IsExtensionBy cache3 new2 hnew2
⊢ ∀ (idx : Nat)... | intro idx hidx hmarked | case extension
aig : AIG Nat
cnf1 cnf2 cnf3 : CNF aig.CNFVar
new1 : Nat
hnew1 : new1 < aig.decls.size
new2 : Nat
hnew2 : new2 < aig.decls.size
cache1 : Cache aig cnf1
cache2 : Cache aig cnf2
cache3 : Cache aig cnf3
h12 : cache1.IsExtensionBy cache2 new1 hnew1
h23 : cache2.IsExtensionBy cache3 new2 hnew2
idx : Nat
hidx ... | 935167f3a7b54c92 |
minSmoothness_eq_infty | Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean | @[simp] lemma minSmoothness_eq_infty {n : WithTop ℕ∞} :
minSmoothness 𝕜 n = ∞ ↔ (n = ∞ ∧ IsRCLikeNormedField 𝕜) | 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
⊢ minSmoothness 𝕜 n = ∞ ↔ n = ∞ ∧ IsRCLikeNormedField 𝕜 | simp only [minSmoothness] | 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
⊢ (if IsRCLikeNormedField 𝕜 then n else ω) = ∞ ↔ n = ∞ ∧ IsRCLikeNormedField 𝕜 | a08afc42e6fac63d |
List.exists_lt_of_prod_lt' | Mathlib/Algebra/Order/BigOperators/Group/List.lean | @[to_additive exists_lt_of_sum_lt]
lemma exists_lt_of_prod_lt' [LinearOrder M] [MulRightMono M]
[MulLeftMono M] {l : List ι} (f g : ι → M)
(h : (l.map f).prod < (l.map g).prod) : ∃ i ∈ l, f i < g i | ι : Type u_1
M : Type u_3
inst✝³ : Monoid M
inst✝² : LinearOrder M
inst✝¹ : MulRightMono M
inst✝ : MulLeftMono M
l : List ι
f g : ι → M
h : (map f l).prod < (map g l).prod
⊢ ∃ i, i ∈ l ∧ f i < g i | contrapose! h | ι : Type u_1
M : Type u_3
inst✝³ : Monoid M
inst✝² : LinearOrder M
inst✝¹ : MulRightMono M
inst✝ : MulLeftMono M
l : List ι
f g : ι → M
h : ∀ (i : ι), i ∈ l → g i ≤ f i
⊢ (map g l).prod ≤ (map f l).prod | aa9012f6c8aa3aa9 |
Fermat42.neg_of_minimal | Mathlib/NumberTheory/FLT/Four.lean | theorem neg_of_minimal {a b c : ℤ} : Minimal a b c → Minimal a b (-c) | a b c : ℤ
h2 : ∀ (a1 b1 c1 : ℤ), Fermat42 a1 b1 c1 → c.natAbs ≤ c1.natAbs
ha : a ≠ 0
hb : b ≠ 0
heq : a ^ 4 + b ^ 4 = c ^ 2
⊢ c ^ 2 = (-c) ^ 2 | exact (neg_sq c).symm | no goals | 011e62322015e505 |
Quotient.list_ind | Mathlib/Data/Fintype/Quotient.lean | /-- Choice-free induction principle for quotients indexed by a `List`. -/
@[elab_as_elim]
lemma list_ind {l : List ι} {C : (∀ i ∈ l, Quotient (S i)) → Prop}
(f : ∀ a : ∀ i ∈ l, α i, C (⟦a · ·⟧)) (q : ∀ i ∈ l, Quotient (S i)) : C q :=
match l with
| [] => cast (congr_arg _ (funext₂ nofun)) (f nofun)
| i ::... | case a
ι : Type u_1
inst✝ : DecidableEq ι
α : ι → Sort u_2
S : (i : ι) → Setoid (α i)
l✝ : List ι
i : ι
l : List ι
C : ((i_1 : ι) → i_1 ∈ i :: l → Quotient (S i_1)) → Prop
f : ∀ (a : (i_1 : ι) → i_1 ∈ i :: l → α i_1), C fun x1 x2 => ⟦a x1 x2⟧
q : (i_1 : ι) → i_1 ∈ i :: l → Quotient (S i_1)
a : α i
as : (i : ι) → i ∈ l ... | exact f _ | no goals | b8d0229daaf08bea |
IsSimpleRing.isField_center | Mathlib/RingTheory/SimpleRing/Field.lean | lemma isField_center (A : Type*) [Ring A] [IsSimpleRing A] : IsField (Subring.center A) where
exists_pair_ne := ⟨0, 1, zero_ne_one⟩
mul_comm := mul_comm
mul_inv_cancel | A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun x_1 => x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ y * x * a * y = y * (x * a) * y | rw [mul_assoc y x a] | no goals | 08d555f2a5a9e7b4 |
MeasureTheory.exists_measure_iInter_lt | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | theorem exists_measure_iInter_lt {α ι : Type*} {_ : MeasurableSpace α} {μ : Measure α}
[SemilatticeSup ι] [Countable ι] {f : ι → Set α}
(hm : ∀ i, NullMeasurableSet (f i) μ) {ε : ℝ≥0∞} (hε : 0 < ε) (hfin : ∃ i, μ (f i) ≠ ∞)
(hfem : ⋂ n, f n = ∅) : ∃ m, μ (⋂ n ≤ m, f n) < ε | α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
f : ι → Set α
hm : ∀ (i : ι), NullMeasurableSet (f i) μ
ε : ℝ≥0∞
hε : 0 < ε
hfin : ∃ i, μ (f i) ≠ ⊤
hfem : ⋂ n, f n = ∅
F : ι → ℝ≥0∞ := fun m => μ (⋂ n, ⋂ (_ : n ≤ m), f n)
hFAnti : Antitone F
⊢ μ (⋂ n, f n) = 0 | simp only [hfem, measure_empty] | no goals | 1f737bad22d86841 |
List.toFinset_append | Mathlib/Data/Finset/Lattice/Lemmas.lean | theorem toFinset_append : toFinset (l ++ l') = l.toFinset ∪ l'.toFinset | case nil
α : Type u_1
inst✝ : DecidableEq α
l l' : List α
⊢ ([] ++ l').toFinset = [].toFinset ∪ l'.toFinset | simp | no goals | 20e356d9f0eb7ff8 |
padicValInt.self | Mathlib/NumberTheory/Padics/PadicVal/Basic.lean | theorem self (hp : 1 < p) : padicValInt p p = 1 | p : ℕ
hp : 1 < p
⊢ padicValInt p ↑p = 1 | simp [padicValNat.self hp] | no goals | 233d8be6ebe40eb8 |
Vector.mem_iff_append | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem mem_iff_append {a : α} {l : Vector α n} :
a ∈ l ↔ ∃ (m k : Nat) (w : m + 1 + k = n) (s : Vector α m) (t : Vector α k),
l = (s.push a ++ t).cast w :=
⟨append_of_mem, by rintro ⟨m, k, rfl, s, t, rfl⟩; simp⟩
| case intro.intro.intro.intro.intro
α : Type u_1
a : α
m k : Nat
s : Vector α m
t : Vector α k
⊢ a ∈ Vector.cast ⋯ (s.push a ++ t) | simp | no goals | e2ab699503e51a3a |
Ordinal.mul_eq_right_iff_opow_omega0_dvd | Mathlib/SetTheory/Ordinal/FixedPoint.lean | theorem mul_eq_right_iff_opow_omega0_dvd {a b : Ordinal} : a * b = b ↔ a ^ ω ∣ b | case inr.refine_1.inl
a b : Ordinal.{u_1}
ha : 0 < a
hab✝ : a * (b % a ^ ω) = b % a ^ ω
hab : b % a ^ ω = 0
⊢ b % a ^ ω = 0 | exact hab | no goals | 16da28166121502d |
IsDiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | theorem toUniqueFactorizationMonoid (hR : HasUnitMulPowIrreducibleFactorization R) :
UniqueFactorizationMonoid R :=
let p := Classical.choose hR
let spec := Classical.choose_spec hR
UniqueFactorizationMonoid.of_exists_prime_factors fun x hx => by
use Multiset.replicate (Classical.choose (spec.2 hx)) p
... | case h.right
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hR : HasUnitMulPowIrreducibleFactorization R
p : R := Classical.choose hR
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=
Classical.choose_spec hR
x : R
hx : x ≠ 0
⊢ Associated (Multiset.repli... | rw [Multiset.prod_replicate] | case h.right
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hR : HasUnitMulPowIrreducibleFactorization R
p : R := Classical.choose hR
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=
Classical.choose_spec hR
x : R
hx : x ≠ 0
⊢ Associated (p ^ Classical.... | c98f86af3b032a22 |
MeasureTheory.condExp_mul_of_stronglyMeasurable_left | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | theorem condExp_mul_of_stronglyMeasurable_left {f g : α → ℝ} (hf : StronglyMeasurable[m] f)
(hfg : Integrable (f * g) μ) (hg : Integrable g μ) : μ[f * g|m] =ᵐ[μ] f * μ[g|m] | case pos.intro.intro
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
sets : ℕ → Set α
sets_prop : ∀ (n : ℕ), MeasurableSet (sets n) ∧ μ (sets n) < ⊤ ∧ ∀ x ∈ sets n, ‖f x‖ ≤ ↑n
h_univ :... | simp_rw [forall_and] at sets_prop | case pos.intro.intro
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : m ≤ m0
hμm this : SigmaFinite (μ.trim hm)
sets : ℕ → Set α
h_univ : ⋃ i, sets i = Set.univ
sets_prop : (∀ (x : ℕ), MeasurableSet (sets x)) ∧ (∀ (x : ℕ), μ (s... | baffe7f7f5a39936 |
CategoryTheory.Adjunction.ext | Mathlib/CategoryTheory/Adjunction/Basic.lean | @[ext]
lemma ext {F : C ⥤ D} {G : D ⥤ C} {adj adj' : F ⊣ G}
(h : adj.unit = adj'.unit) : adj = adj' | case w.h.a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj adj' : F ⊣ G
h : adj.unit = adj'.unit
X : D
⊢ (adj.homEquiv (G.obj X) ((𝟭 D).obj X)) (adj.counit.app X) = (adj.homEquiv (G.obj X) ((𝟭 D).obj X)) (adj'.counit.app X) | rw [Adjunction.homEquiv_unit, Adjunction.homEquiv_unit,
Adjunction.right_triangle_components, h, Adjunction.right_triangle_components] | no goals | 45fb15cac4e22f40 |
Nat.Prime.dvd_choose_add | Mathlib/Data/Nat/Choose/Dvd.lean | theorem dvd_choose_add (hp : Prime p) (hap : a < p) (hbp : b < p) (h : p ≤ a + b) :
p ∣ choose (a + b) a | p a b : ℕ
hp : Prime p
hap : a < p
hbp : b < p
h : p ≤ a + b
⊢ p ∣ (a + b).choose a | have h₁ : p ∣ (a + b)! := hp.dvd_factorial.2 h | p a b : ℕ
hp : Prime p
hap : a < p
hbp : b < p
h : p ≤ a + b
h₁ : p ∣ (a + b)!
⊢ p ∣ (a + b).choose a | 91355c9bd65a89ac |
HurwitzZeta.continuousOn_sinKernel | Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean | lemma continuousOn_sinKernel (a : UnitAddCircle) : ContinuousOn (sinKernel a) (Ioi 0) | case H
a : ℝ
⊢ ContinuousOn (fun x => ↑(sinKernel (↑a) x)) (Ioi 0) | simp_rw [sinKernel_def] | case H
a : ℝ
⊢ ContinuousOn (fun x => jacobiTheta₂' (↑a) (I * ↑x) / (-2 * ↑π)) (Ioi 0) | b8f60ffe0d55b4c8 |
WeierstrassCurve.Affine.nonsingular_negAdd_of_eval_derivative_ne_zero | Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | lemma nonsingular_negAdd_of_eval_derivative_ne_zero {x₁ x₂ y₁ ℓ : R}
(hx' : W'.Equation (W'.addX x₁ x₂ ℓ) (W'.negAddY x₁ x₂ y₁ ℓ))
(hx : (W'.addPolynomial x₁ y₁ ℓ).derivative.eval (W'.addX x₁ x₂ ℓ) ≠ 0) :
W'.Nonsingular (W'.addX x₁ x₂ ℓ) (W'.negAddY x₁ x₂ y₁ ℓ) | R : Type r
inst✝ : CommRing R
W' : Affine R
x₁ x₂ y₁ ℓ : R
hx' : W'.Equation (W'.addX x₁ x₂ ℓ) (W'.negAddY x₁ x₂ y₁ ℓ)
hx :
W'.a₁ * (ℓ * (W'.addX x₁ x₂ ℓ - x₁) + y₁) - (3 * W'.addX x₁ x₂ ℓ ^ 2 + 2 * W'.a₂ * W'.addX x₁ x₂ ℓ + W'.a₄) = 0 ∧
2 * (ℓ * (W'.addX x₁ x₂ ℓ - x₁) + y₁) + (W'.a₁ * W'.addX x₁ x₂ ℓ + W'.a₃) = ... | rw [addPolynomial, linePolynomial, polynomial] | R : Type r
inst✝ : CommRing R
W' : Affine R
x₁ x₂ y₁ ℓ : R
hx' : W'.Equation (W'.addX x₁ x₂ ℓ) (W'.negAddY x₁ x₂ y₁ ℓ)
hx :
W'.a₁ * (ℓ * (W'.addX x₁ x₂ ℓ - x₁) + y₁) - (3 * W'.addX x₁ x₂ ℓ ^ 2 + 2 * W'.a₂ * W'.addX x₁ x₂ ℓ + W'.a₄) = 0 ∧
2 * (ℓ * (W'.addX x₁ x₂ ℓ - x₁) + y₁) + (W'.a₁ * W'.addX x₁ x₂ ℓ + W'.a₃) = ... | c67d40c30275d2f6 |
RootPairing.pairing_smul_root_eq_of_not_linInd | Mathlib/LinearAlgebra/RootSystem/Reduced.lean | lemma pairing_smul_root_eq_of_not_linInd [NeZero (2 : R)] [NoZeroSMulDivisors R M]
(h : ¬ LinearIndependent R ![P.root i, P.root j]) :
P.pairing j i • P.root i = (2 : R) • P.root j | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
P : RootPairing ι R M N
i j : ι
inst✝¹ : NeZero 2
inst✝ : NoZeroSMulDivisors R M
s t : R
h₂ : s ≠ 0
h₃ : t ≠ 0
h₁ : s • P.root i = -t • P.root j
h₄ : s * 2 = -(... | rw [h₁] | no goals | 1e9f3d77c0025a81 |
CategoryTheory.HasExactLimitsOfShape.of_codomain_equivalence | Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Basic.lean | lemma HasExactLimitsOfShape.of_codomain_equivalence (J : Type*) [Category J] {D : Type*}
[Category D] (e : C ≌ D) [HasLimitsOfShape J C] [HasExactLimitsOfShape J C] :
haveI : HasLimitsOfShape J D := Adjunction.hasLimitsOfShape_of_equivalence e.inverse
HasExactLimitsOfShape J D | C : Type u
inst✝⁴ : Category.{v, u} C
J : Type u_1
inst✝³ : Category.{u_3, u_1} J
D : Type u_2
inst✝² : Category.{u_4, u_2} D
e : C ≌ D
inst✝¹ : HasLimitsOfShape J C
inst✝ : HasExactLimitsOfShape J C
this : HasLimitsOfShape J D
x✝² : Type
x✝¹ : SmallCategory x✝²
x✝ : FinCategory x✝²
K : x✝² ⥤ J ⥤ D
⊢ e.congrRight.inver... | apply e.symm.congrRight.fullyFaithfulFunctor.preimageIso | C : Type u
inst✝⁴ : Category.{v, u} C
J : Type u_1
inst✝³ : Category.{u_3, u_1} J
D : Type u_2
inst✝² : Category.{u_4, u_2} D
e : C ≌ D
inst✝¹ : HasLimitsOfShape J C
inst✝ : HasExactLimitsOfShape J C
this : HasLimitsOfShape J D
x✝² : Type
x✝¹ : SmallCategory x✝²
x✝ : FinCategory x✝²
K : x✝² ⥤ J ⥤ D
⊢ e.symm.congrRight.... | aee7a1266599c512 |
PFunctor.M.agree_iff_agree' | Mathlib/Data/PFunctor/Univariate/M.lean | theorem agree_iff_agree' {n : ℕ} (x y : M F) :
Agree (x.approx n) (y.approx <| n + 1) ↔ Agree' n x y | case mpr.succ.step.f.f
F : PFunctor.{u}
n✝ : ℕ
n_ih : ∀ (x y : F.M), Agree' n✝ x y → Agree (x.approx n✝) (y.approx (n✝ + 1))
a : F.A
x' y' : F.B a → F.M
a✝² : ∀ (i : F.B a), Agree' n✝ (x' i) (y' i)
x_a : F.A
x_f : F.B x_a → F.M
a✝¹ : M.mk ⟨x_a, x_f⟩ = M.mk ⟨a, x'⟩
y_a : F.A
y_f : F.B y_a → F.M
a✝ : M.mk ⟨y_a, y_f⟩ = M.... | have h_a_1 := mk_inj ‹M.mk ⟨x_a, x_f⟩ = M.mk ⟨a, x'⟩› | case mpr.succ.step.f.f
F : PFunctor.{u}
n✝ : ℕ
n_ih : ∀ (x y : F.M), Agree' n✝ x y → Agree (x.approx n✝) (y.approx (n✝ + 1))
a : F.A
x' y' : F.B a → F.M
a✝² : ∀ (i : F.B a), Agree' n✝ (x' i) (y' i)
x_a : F.A
x_f : F.B x_a → F.M
a✝¹ : M.mk ⟨x_a, x_f⟩ = M.mk ⟨a, x'⟩
y_a : F.A
y_f : F.B y_a → F.M
a✝ : M.mk ⟨y_a, y_f⟩ = M.... | 74e48f06950cd29d |
BitVec.getElem?_setWidth | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getElem?_setWidth (m : Nat) (x : BitVec n) (i : Nat) :
(x.setWidth m)[i]? = if i < m then some (x.getLsbD i) else none | n m : Nat
x : BitVec n
i : Nat
⊢ (setWidth m x)[i]? = if i < m then some (x.getLsbD i) else none | simp [getElem?_eq, getElem_setWidth] | no goals | 407d5fc89df3acf8 |
AddCircle.ergodic_zsmul_add | Mathlib/Dynamics/Ergodic/AddCircle.lean | theorem ergodic_zsmul_add (x : AddCircle T) {n : ℤ} (h : 1 < |n|) : Ergodic fun y => n • y + x | T : ℝ
hT : Fact (0 < T)
x : AddCircle T
n : ℤ
f : AddCircle T → AddCircle T := fun y => n • y + x
e : AddCircle T ≃ᵐ AddCircle T := MeasurableEquiv.addLeft (DivisibleBy.div x (n - 1))
he : MeasurePreserving (⇑e) volume volume
h : n - 1 ≠ 0
hnx : n • DivisibleBy.div x (n - 1) = x + DivisibleBy.div x (n - 1)
⊢ ⇑e ∘ f ∘ ⇑... | ext y | case h
T : ℝ
hT : Fact (0 < T)
x : AddCircle T
n : ℤ
f : AddCircle T → AddCircle T := fun y => n • y + x
e : AddCircle T ≃ᵐ AddCircle T := MeasurableEquiv.addLeft (DivisibleBy.div x (n - 1))
he : MeasurePreserving (⇑e) volume volume
h : n - 1 ≠ 0
hnx : n • DivisibleBy.div x (n - 1) = x + DivisibleBy.div x (n - 1)
y : A... | 05c20060c1761793 |
Rat.den_div_intCast_eq_one_iff | Mathlib/Data/Rat/Lemmas.lean | theorem den_div_intCast_eq_one_iff (m n : ℤ) (hn : n ≠ 0) : ((m : ℚ) / n).den = 1 ↔ n ∣ m | case mp
m n : ℤ
hn : ↑n ≠ 0
⊢ (↑m / ↑n).den = 1 → n ∣ m | rw [Rat.den_eq_one_iff, eq_div_iff hn] | case mp
m n : ℤ
hn : ↑n ≠ 0
⊢ ↑(↑m / ↑n).num * ↑n = ↑m → n ∣ m | ccb4bee3bdc0ed12 |
CategoryTheory.Abelian.image_ι_comp_eq_zero | Mathlib/CategoryTheory/Abelian/Basic.lean | theorem image_ι_comp_eq_zero {R : C} {g : Q ⟶ R} (h : f ≫ g = 0) : Abelian.image.ι f ≫ g = 0 :=
zero_of_epi_comp (Abelian.factorThruImage f) <| by simp [h]
| C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
P Q : C
f : P ⟶ Q
R : C
g : Q ⟶ R
h : f ≫ g = 0
⊢ Abelian.factorThruImage f ≫ image.ι f ≫ g = 0 | simp [h] | no goals | c9dad67dd31231d5 |
LieSubmodule.isNilpotentOfIsNilpotentSpanSupEqTop | Mathlib/Algebra/Lie/Engel.lean | theorem isNilpotentOfIsNilpotentSpanSupEqTop (hnp : IsNilpotent <| toEnd R L M x)
(hIM : IsNilpotent I M) : IsNilpotent L M | R : Type u₁
L : Type u₂
M : Type u₄
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✝ : LieModule R L M
I : LieIdeal R L
x : L
hxI : Submodule.span R {x} ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤
hnp : _root_.IsNilpotent ... | obtain ⟨n, hn⟩ := hnp | case intro
R : Type u₁
L : Type u₂
M : Type u₄
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✝ : LieModule R L M
I : LieIdeal R L
x : L
hxI : Submodule.span R {x} ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤
hIM : LieModul... | b5f692b7de26dcf5 |
top_mem_range_transfiniteIterate | Mathlib/Order/TransfiniteIteration.lean | lemma top_mem_range_transfiniteIterate
(hφ' : ∀ i ≠ (⊤ : I), i < φ i) (φtop : φ ⊤ = ⊤)
(H : ¬ Function.Injective (fun (j : J) ↦ transfiniteIterate φ j i₀)) :
∃ (j : J), transfiniteIterate φ j i₀ = ⊤ | I : Type u
inst✝⁴ : CompleteLattice I
φ : I → I
i₀ : I
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hφ' : ∀ (i : I), i ≠ ⊤ → i < φ i
φtop : φ ⊤ = ⊤
H : ¬∀ ⦃a₁ a₂ : J⦄, transfiniteIterate φ a₁ i₀ = transfiniteIterate φ a₂ i₀ → a₁ = a₂
hφ : ∀ (i : I), i ≤ φ i
⊢ ∃ j₁ j... | simp only [not_forall] at H | I : Type u
inst✝⁴ : CompleteLattice I
φ : I → I
i₀ : I
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hφ' : ∀ (i : I), i ≠ ⊤ → i < φ i
φtop : φ ⊤ = ⊤
hφ : ∀ (i : I), i ≤ φ i
H : ∃ x x_1, ∃ (_ : transfiniteIterate φ x i₀ = transfiniteIterate φ x_1 i₀), ¬x = x_1
⊢ ∃ j₁ ... | c68dde9d50bbb271 |
Std.DHashMap.Internal.Raw₀.wfImp_eraseₘ | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean | theorem wfImp_eraseₘ [BEq α] [Hashable α] [EquivBEq α] [LawfulHashable α] {m : Raw₀ α β} {a : α}
(h : Raw.WFImp m.1) : Raw.WFImp (m.eraseₘ a).1 | case isTrue
α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
m : Raw₀ α β
a : α
h : Raw.WFImp m.val
h✝ : m.containsₘ a = true
⊢ Raw.WFImp (m.eraseₘaux a).val | next h' => exact wfImp_eraseₘaux m a h h' | no goals | 2381d6bcd371186b |
MeasureTheory.absolutelyContinuous_map_mul_right | Mathlib/MeasureTheory/Group/Prod.lean | theorem absolutelyContinuous_map_mul_right (g : G) : μ ≪ map (· * g) μ | G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul₂ G
μ : Measure G
inst✝² : SFinite μ
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulLeftInvariant
g : G
s : Set G
hs : MeasurableSet s
⊢ μ s = 0 → μ s = 0 | exact id | no goals | 77d184ee0769232f |
uniformity_eq_comap_inv_mul_nhds_one | Mathlib/Topology/Algebra/UniformGroup/Defs.lean | theorem uniformity_eq_comap_inv_mul_nhds_one :
𝓤 α = comap (fun x : α × α => x.1⁻¹ * x.2) (𝓝 (1 : α)) | α : Type u_1
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : UniformGroup α
⊢ comap (unop ∘ (fun x => x.2 / x.1) ∘ fun p => (op p.1, op p.2)) (𝓝 1) = comap (fun x => x.1⁻¹ * x.2) (𝓝 1) | simp [Function.comp_def] | no goals | 49eca051e5576b93 |
Ideal.homogeneousCore'_eq_sSup | Mathlib/RingTheory/GradedAlgebra/Homogeneous/Ideal.lean | theorem Ideal.homogeneousCore'_eq_sSup :
I.homogeneousCore' 𝒜 = sSup { J : Ideal A | J.IsHomogeneous 𝒜 ∧ J ≤ I } | case h
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
coe_mono : Monotone toIdeal
⊢ IsGreatest {J | IsHomogeneous 𝒜 J ∧ J ≤ I} (homogeneousCore' 𝒜 I) | convert coe_mono.map_isGreatest (Ideal.homogeneousCore.gc 𝒜).isGreatest_u using 1 | case h.e'_3
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
coe_mono : Monotone toIdeal
⊢ {J | IsHomogeneous 𝒜 J ∧ J ≤ I} = toIdeal '' {a | a.toIdeal ≤ I} | 9432fbd8401d3b7e |
CategoryTheory.Limits.Types.coequalizer_preimage_image_eq_of_preimage_eq | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | theorem coequalizer_preimage_image_eq_of_preimage_eq (π : Y ⟶ Z) (e : f ≫ π = g ≫ π)
(h : IsColimit (Cofork.ofπ π e)) (U : Set Y) (H : f ⁻¹' U = g ⁻¹' U) : π ⁻¹' (π '' U) = U | case intro
X Y Z : Type u
f g : X ⟶ Y
π : Y ⟶ Z
e : f ≫ π = g ≫ π
h : IsColimit (Cofork.ofπ π e)
U : Set Y
H : f ⁻¹' U = g ⁻¹' U
x : X
⊢ f x ∈ U ↔ g x ∈ U | change x ∈ f ⁻¹' U ↔ x ∈ g ⁻¹' U | case intro
X Y Z : Type u
f g : X ⟶ Y
π : Y ⟶ Z
e : f ≫ π = g ≫ π
h : IsColimit (Cofork.ofπ π e)
U : Set Y
H : f ⁻¹' U = g ⁻¹' U
x : X
⊢ x ∈ f ⁻¹' U ↔ x ∈ g ⁻¹' U | 4cf815c00d783414 |
MeasureTheory.lintegral_prod_mul | Mathlib/MeasureTheory/Measure/Prod.lean | theorem lintegral_prod_mul {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (hf : AEMeasurable f μ)
(hg : AEMeasurable g ν) : ∫⁻ z, f z.1 * g z.2 ∂μ.prod ν = (∫⁻ x, f x ∂μ) * ∫⁻ y, g y ∂ν | α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
f : α → ℝ≥0∞
g : β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g ν
⊢ ∫⁻ (z : α × β), f z.1 * g z.2 ∂μ.prod ν = (∫⁻ (x : α), f x ∂μ) * ∫⁻ (y : β), g y ∂ν | simp [lintegral_prod _ (hf.fst.mul hg.snd), lintegral_lintegral_mul hf hg] | no goals | 98abb7e7dbb73a13 |
List.isInfix_map_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean | theorem isInfix_map_iff {β} {f : α → β} {l₁ : List α} {l₂ : List β} :
l₂ <:+: l₁.map f ↔ ∃ l, l <:+: l₁ ∧ l₂ = l.map f | α : Type u_1
β : Type u_2
f : α → β
l₁ : List α
l₂ : List β
⊢ l₂ <:+: map f l₁ ↔ ∃ l, l <:+: l₁ ∧ l₂ = map f l | rw [← filterMap_eq_map, isInfix_filterMap_iff] | no goals | f9e26183debaf75e |
Ordinal.principal_add_mul_of_principal_add | Mathlib/SetTheory/Ordinal/Principal.lean | theorem principal_add_mul_of_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)
(hb : Principal (· + ·) b) : Principal (· + ·) (a * b) | case inr.inl
a : Ordinal.{u}
h✝ : 0 < a
hb₁ : 0 ≠ 1
hb : Principal (fun x1 x2 => x1 + x2) 0
⊢ Principal (fun x1 x2 => x1 + x2) (a * 0) | rw [mul_zero] | case inr.inl
a : Ordinal.{u}
h✝ : 0 < a
hb₁ : 0 ≠ 1
hb : Principal (fun x1 x2 => x1 + x2) 0
⊢ Principal (fun x1 x2 => x1 + x2) 0 | 27974d2b8e448b12 |
NormedSpace.map_exp_of_mem_ball | Mathlib/Analysis/Normed/Algebra/Exponential.lean | theorem map_exp_of_mem_ball {F} [FunLike F 𝔸 𝔹] [RingHomClass F 𝔸 𝔹] (f : F) (hf : Continuous f)
(x : 𝔸) (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
f (exp 𝕂 x) = exp 𝕂 (f x) | 𝕂 : Type u_1
𝔸 : Type u_2
𝔹 : Type u_3
inst✝⁷ : NontriviallyNormedField 𝕂
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedRing 𝔹
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : NormedAlgebra 𝕂 𝔹
inst✝² : CompleteSpace 𝔸
F : Type u_4
inst✝¹ : FunLike F 𝔸 𝔹
inst✝ : RingHomClass F 𝔸 𝔹
f : F
hf : Continuous ⇑f
x : 𝔸
hx : x ∈ EMet... | rw [exp_eq_tsum, exp_eq_tsum] | 𝕂 : Type u_1
𝔸 : Type u_2
𝔹 : Type u_3
inst✝⁷ : NontriviallyNormedField 𝕂
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedRing 𝔹
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : NormedAlgebra 𝕂 𝔹
inst✝² : CompleteSpace 𝔸
F : Type u_4
inst✝¹ : FunLike F 𝔸 𝔹
inst✝ : RingHomClass F 𝔸 𝔹
f : F
hf : Continuous ⇑f
x : 𝔸
hx : x ∈ EMet... | 911856a38ac7fba0 |
MeasureTheory.condExpL1_mono | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | theorem condExpL1_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
condExpL1 hm μ f ≤ᵐ[μ] condExpL1 hm μ g | α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝⁴ : SigmaFinite (μ.trim hm)
E : Type u_7
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᶠ[ae μ] g
⊢ condExpL1 hm μ f ≤ condExp... | have h_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x : E, 0 ≤ x → 0 ≤ condExpInd E hm μ s x :=
fun s hs hμs x hx => condExpInd_nonneg hs hμs.ne x hx | α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝⁴ : SigmaFinite (μ.trim hm)
E : Type u_7
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᶠ[ae μ] g
h_nonneg : ∀ (s : Set α), Me... | 1d41e4e6b14e6d48 |
Finpartition.card_filter_equitabilise_small | Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean | theorem card_filter_equitabilise_small (hm : m ≠ 0) :
#{u ∈ (P.equitabilise h).parts | #u = m} = a | α : Type u_1
inst✝ : DecidableEq α
s : Finset α
m a b : ℕ
P : Finpartition s
h : a * m + b * (m + 1) = #s
hm : m ≠ 0
⊢ (equitabilise h).parts =
filter (fun u => #u = m) (equitabilise h).parts ∪ filter (fun u => #u = m + 1) (equitabilise h).parts | rw [← filter_or, filter_true_of_mem] | α : Type u_1
inst✝ : DecidableEq α
s : Finset α
m a b : ℕ
P : Finpartition s
h : a * m + b * (m + 1) = #s
hm : m ≠ 0
⊢ ∀ x ∈ (equitabilise h).parts, #x = m ∨ #x = m + 1 | 6764f0dc61dac4e1 |
Batteries.UnionFind.parent_rootD | Mathlib/.lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | theorem parent_rootD (self : UnionFind) (x : Nat) :
self.parent (self.rootD x) = self.rootD x | case isFalse
self : UnionFind
x : Nat
h✝ : ¬x < self.size
⊢ self.parent x = x | simp [parentD_of_not_lt, *] | no goals | 2d93c5dff10f27ed |
Stream'.WSeq.join_append | Mathlib/Data/Seq/WSeq.lean | theorem join_append (S T : WSeq (WSeq α)) : join (append S T) ~ʷ append (join S) (join T) | α : Type u
S T : WSeq (WSeq α)
⊢ (S.append T).join = nil.append (S.append T).join | simp | no goals | dcbc952c785d70aa |
ProbabilityTheory.Kernel.compProd_prodMkLeft_eq_comp | Mathlib/Probability/Kernel/Composition/Prod.lean | lemma compProd_prodMkLeft_eq_comp
(κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel β γ) [IsSFiniteKernel η] :
κ ⊗ₖ (prodMkLeft α η) = (Kernel.id ×ₖ η) ∘ₖ κ | case h.h.e_f.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel β γ
inst✝ : IsSFiniteKernel η
a : α
s : Set (β × γ)
hs : MeasurableSet s
b : β
⊢ (η b) {c | (b, c) ∈ s} = ∫⁻ (b_1 : β), (η b) {c | (b_1, c) ∈ s}... | rw [lintegral_dirac'] | case h.h.e_f.h.hf
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel β γ
inst✝ : IsSFiniteKernel η
a : α
s : Set (β × γ)
hs : MeasurableSet s
b : β
⊢ Measurable fun b_1 => (η b) {c | (b_1, c) ∈ s} | 62af1bc45781c411 |
IsNilpotent.exp_add_of_commute | Mathlib/RingTheory/Nilpotent/Exp.lean | theorem exp_add_of_commute {a b : A} (h₁ : Commute a b) (h₂ : IsNilpotent a) (h₃ : IsNilpotent b) :
exp (a + b) = exp a * exp b | case refl
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := n₁ ⊔ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ : ∑... | rfl | no goals | 925becc4d11e37f5 |
UniformContinuousOn.cexp | Mathlib/Analysis/SpecialFunctions/Exp.lean | /-- The complex exponential function is uniformly continuous on left half planes. -/
lemma UniformContinuousOn.cexp (a : ℝ) : UniformContinuousOn exp {x : ℂ | x.re ≤ a} | a ε : ℝ
hε : ε > 0
this✝ : ∀ (a : ℂ) (ε : ℝ), 0 < ε → ∀ᶠ (x : ℂ) in 𝓝 a, dist (cexp x) (cexp a) < ε
ha : 0 < ε / (2 * Real.exp a)
δ : ℝ
hδ : δ > 0 ∧ ∀ ⦃y : ℂ⦄, dist y 0 < δ → dist (cexp y) 1 < ε / (2 * Real.exp a)
x : ℂ
a✝ : x ∈ {x | x.re ≤ a}
y : ℂ
hy : y ∈ {x | x.re ≤ a}
hxy : ‖x - y‖ < δ
h3 : ‖cexp (x - y) - 1‖ < ε... | exact hy | no goals | e2639d45c5a634ab |
LinearMap.split_surjective_of_localization_maximal | Mathlib/RingTheory/LocalProperties/Projective.lean | theorem LinearMap.split_surjective_of_localization_maximal
(f : M →ₗ[R] N) [Module.FinitePresentation R N]
(H : ∀ (I : Ideal R) (_ : I.IsMaximal),
∃ (g : _ →ₗ[Localization.AtPrime I] _),
(LocalizedModule.map I.primeCompl f).comp g = LinearMap.id) :
∃ (g : N →ₗ[R] M), f.comp g = LinearMap.id | case h.e.h.a.mpr.intro.intro.mk.a.a.h.h
R : Type u_1
N : Type u_2
M : Type uM
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
f : M →ₗ[R] N
inst✝ : Module.FinitePresentation R N
H : ∀ (I : Ideal R) (x : I.IsMaximal), ∃ g, (LocalizedModule.map I.primeCompl) f ∘... | simp only [coe_comp, coe_restrictScalars, Function.comp_apply,
LocalizedModule.mkLinearMap_apply, LocalizedModule.map_mk, llcomp_apply] | no goals | d80210efccbecc76 |
Algebra.intNorm_eq_of_isLocalization | Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean | lemma Algebra.intNorm_eq_of_isLocalization (x : B) :
algebraMap A Aₘ (Algebra.intNorm A B x) = Algebra.intNorm Aₘ Bₘ (algebraMap B Bₘ x) | case neg.a
A : Type u_1
B : Type u_4
inst✝²⁶ : CommRing A
inst✝²⁵ : CommRing B
inst✝²⁴ : Algebra A B
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²³ : CommRing Aₘ
inst✝²² : CommRing Bₘ
inst✝²¹ : Algebra Aₘ Bₘ
inst✝²⁰ : Algebra A Aₘ
inst✝¹⁹ : Algebra B Bₘ
inst✝¹⁸ : Algebra A Bₘ
inst✝¹⁷ : IsScalarTower A Aₘ Bₘ
inst✝¹⁶ : IsScalarTowe... | rw [← IsScalarTower.algebraMap_apply, Algebra.algebraMap_intNorm_fractionRing,
Algebra.algebraMap_intNorm (L := L), ← IsScalarTower.algebraMap_apply] | no goals | 9e85e6de2d1bcfc1 |
NumberField.Units.rootsOfUnity_eq_one | Mathlib/NumberTheory/NumberField/Units/Basic.lean | theorem rootsOfUnity_eq_one [NumberField K] {k : ℕ+} (hc : Nat.Coprime k (torsionOrder K))
{ζ : (𝓞 K)ˣ} : ζ ∈ rootsOfUnity k (𝓞 K) ↔ ζ = 1 | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
k : ℕ+
hc : (↑k).Coprime ↑(torsionOrder K)
ζ : (𝓞 K)ˣ
⊢ ζ ∈ rootsOfUnity (↑k) (𝓞 K) ↔ ζ = 1 | rw [mem_rootsOfUnity] | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
k : ℕ+
hc : (↑k).Coprime ↑(torsionOrder K)
ζ : (𝓞 K)ˣ
⊢ ζ ^ ↑k = 1 ↔ ζ = 1 | a7e21166a35e2c56 |
CategoryTheory.Subgroupoid.Map.arrows_iff | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | theorem Map.arrows_iff (hφ : Function.Injective φ.obj) (S : Subgroupoid C) {c d : D} (f : c ⟶ d) :
Map.Arrows φ hφ S c d f ↔
∃ (a b : C) (g : a ⟶ b) (ha : φ.obj a = c) (hb : φ.obj b = d) (_hg : g ∈ S.arrows a b),
f = eqToHom ha.symm ≫ φ.map g ≫ eqToHom hb | case mpr.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Groupoid C
D : Type u_1
inst✝ : Groupoid D
φ : C ⥤ D
hφ : Function.Injective φ.obj
S : Subgroupoid C
a b : C
g : a ⟶ b
hg : g ∈ S.arrows a b
⊢ Arrows φ hφ S (φ.obj a) (φ.obj b) (φ.map g) | constructor | case mpr.intro.intro.intro.intro.intro.intro.hf
C : Type u
inst✝¹ : Groupoid C
D : Type u_1
inst✝ : Groupoid D
φ : C ⥤ D
hφ : Function.Injective φ.obj
S : Subgroupoid C
a b : C
g : a ⟶ b
hg : g ∈ S.arrows a b
⊢ g ∈ S.arrows a b | 534931aaf988bf8e |
MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal | Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : α → ℝ≥0)
(fint : Integrable (fun x => (f x : ℝ)) μ) {ε : ℝ} (εpos : 0 < ε) :
∃ g : α → ℝ≥0∞,
(∀ x, (f x : ℝ≥0∞) < g x) ∧
LowerSemicontinuous g ∧
(∀ᵐ x ∂μ, g x < ⊤) ∧
Integrable (fun x => (g x).toReal) μ ∧ (∫ x,... | α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable (fun x => ↑(f x)) μ
ε : ℝ
εpos : 0 < ε
⊢ AEMeasurable f μ | convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable | case h.e'_5.h
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable (fun x => ↑(f x)) μ
ε : ℝ
εpos : 0 < ε
x✝ : α
⊢ f x✝ = (↑(f x✝)).toNNReal | 3f39b00d374e3aed |
CategoryTheory.Adjunction.homEquiv_naturality_right_square_iff | Mathlib/CategoryTheory/Adjunction/Basic.lean | theorem homEquiv_naturality_right_square_iff (f : X' ⟶ X) (g : X ⟶ G.obj Y')
(h : X' ⟶ G.obj Y) (k : Y ⟶ Y') :
(F.map f ≫ (adj.homEquiv X Y').symm g = (adj.homEquiv X' Y).symm h ≫ k) ↔
(f ≫ g = h ≫ G.map k) :=
⟨fun w ↦ by simpa only [Equiv.apply_symm_apply]
using homEquiv_naturality_left_square ad... | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
X' X : C
Y Y' : D
f : X' ⟶ X
g : X ⟶ G.obj Y'
h : X' ⟶ G.obj Y
k : Y ⟶ Y'
w : F.map f ≫ (adj.homEquiv X Y').symm g = (adj.homEquiv X' Y).symm h ≫ k
⊢ f ≫ g = h ≫ G.map k | simpa only [Equiv.apply_symm_apply]
using homEquiv_naturality_left_square adj _ _ _ _ w | no goals | 06905eefb78b0110 |
hyperoperation_ge_four_zero | Mathlib/Data/Nat/Hyperoperation.lean | theorem hyperoperation_ge_four_zero (n k : ℕ) :
hyperoperation (n + 4) 0 k = if Even k then 1 else 0 | case succ
n kk : ℕ
kih : hyperoperation (n + 4) 0 kk = if Even kk then 1 else 0
⊢ hyperoperation (n + 3) 0 (hyperoperation (n + 3 + 1) 0 kk) = if Even (kk + 1) then 1 else 0 | rw [kih] | case succ
n kk : ℕ
kih : hyperoperation (n + 4) 0 kk = if Even kk then 1 else 0
⊢ hyperoperation (n + 3) 0 (if Even kk then 1 else 0) = if Even (kk + 1) then 1 else 0 | cf7b395d2a373738 |
Complex.hasStrictDerivAt_cos | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | theorem hasStrictDerivAt_cos (x : ℂ) : HasStrictDerivAt cos (-sin x) x | case h.e'_9
x : ℂ
⊢ -(cexp (-x * I) - cexp (x * I)) * I * 2⁻¹ = (cexp (x * I) * (1 * I) + cexp (-x * I) * (-1 * I)) * 2⁻¹ | ring | no goals | 45c6e15abc97e36d |
PowerSeries.eq_zero_or_eq_zero_of_mul_eq_zero | Mathlib/RingTheory/PowerSeries/Basic.lean | theorem eq_zero_or_eq_zero_of_mul_eq_zero [NoZeroDivisors R] (φ ψ : R⟦X⟧) (h : φ * ψ = 0) :
φ = 0 ∨ ψ = 0 | case neg
R : Type u_1
inst✝¹ : Ring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
H : ¬φ = 0
ex : ∃ m, (coeff R m) φ ≠ 0
m : ℕ := Nat.find ex
hm₁ : (coeff R m) φ ≠ 0
hm₂ : ∀ k < m, ¬(coeff R k) φ ≠ 0
n : ℕ
ih : ∀ m < n, (coeff R m) ψ = 0
h : ∑ p ∈ antidiagonal (m + n), (coeff R p.1) φ * (coeff R p.2) ψ = 0
i j : ℕ
hij : (i, j)... | rw [mem_antidiagonal] at hij | case neg
R : Type u_1
inst✝¹ : Ring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
H : ¬φ = 0
ex : ∃ m, (coeff R m) φ ≠ 0
m : ℕ := Nat.find ex
hm₁ : (coeff R m) φ ≠ 0
hm₂ : ∀ k < m, ¬(coeff R k) φ ≠ 0
n : ℕ
ih : ∀ m < n, (coeff R m) ψ = 0
h : ∑ p ∈ antidiagonal (m + n), (coeff R p.1) φ * (coeff R p.2) ψ = 0
i j : ℕ
hij : (i, j)... | d0b45343dbc828b7 |
MeasureTheory.SimpleFunc.map_setToSimpleFunc | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem map_setToSimpleFunc (T : Set α → F →L[ℝ] F') (h_add : FinMeasAdditive μ T) {f : α →ₛ G}
(hf : Integrable f μ) {g : G → F} (hg : g 0 = 0) :
(f.map g).setToSimpleFunc T = ∑ x ∈ f.range, T (f ⁻¹' {x}) (g x) | case neg.hSp
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable... | exact h0 hi.2.symm | no goals | f53dabe35f13a020 |
Orientation.norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | theorem norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.sin (o.oangle x (x + y)) = ‖x + y‖ | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
⊢ ‖y‖ / (o.oangle x (x + y)).sin = ‖x + y‖ | have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1
⊢ ‖y‖ / (o.oangle x (x + y)).sin = ‖x + y‖ | 1ccd7c47eb0a292b |
MeasureTheory.QuotientMeasureEqMeasurePreimage.sigmaFiniteQuotient | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | /-- If a measure `μ` on a quotient satisfies `QuotientMeasureEqMeasurePreimage` with respect to a
sigma-finite measure `ν`, then it is itself `SigmaFinite`. -/
@[to_additive]
lemma QuotientMeasureEqMeasurePreimage.sigmaFiniteQuotient
[i : SigmaFinite ν] [i' : HasFundamentalDomain G α ν]
(μ : Measure (Quotient α... | case mk
G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
ν : Measure α
inst✝⁴ : SMulInvariantMeasure G α ν
inst✝³ : Countable G
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableSMul G α
i : SigmaFinite ν
i' : HasFundamentalDomain G α ν
μ : Measure (Quotient α_mod_G)
inst✝ : Quot... | simp only [mem_setOf_eq] at hA_meas | case mk
G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
ν : Measure α
inst✝⁴ : SMulInvariantMeasure G α ν
inst✝³ : Countable G
inst✝² : MeasurableSpace G
inst✝¹ : MeasurableSMul G α
i : SigmaFinite ν
i' : HasFundamentalDomain G α ν
μ : Measure (Quotient α_mod_G)
inst✝ : Quot... | 78476c9fc7a0c287 |
HasCompactMulSupport.continuous_extend_one | Mathlib/Topology/Algebra/Support.lean | theorem continuous_extend_one [TopologicalSpace β] {U : Set α'} (hU : IsOpen U) {f : U → β}
(cont : Continuous f) (supp : HasCompactMulSupport f) :
Continuous (Subtype.val.extend f 1) :=
continuous_of_mulTSupport fun x h ↦ by
rw [show x = ↑(⟨x, Subtype.coe_image_subset _ _
(supp.mulTSupport_extend_o... | α' : Type u_3
β : Type u_4
inst✝³ : TopologicalSpace α'
inst✝² : One β
inst✝¹ : T2Space α'
inst✝ : TopologicalSpace β
U : Set α'
hU : IsOpen U
f : ↑U → β
cont : Continuous f
supp : HasCompactMulSupport f
x : α'
h : x ∈ mulTSupport (extend Subtype.val f 1)
⊢ ContinuousAt f ⟨x, ⋯⟩ | exact cont.continuousAt | no goals | 1616273ecfe898ea |
NumberField.InfinitePlace.nrComplexPlaces_eq_zero_of_finrank_eq_one | Mathlib/NumberTheory/NumberField/Embeddings.lean | theorem nrComplexPlaces_eq_zero_of_finrank_eq_one (h : finrank ℚ K = 1) :
nrComplexPlaces K = 0 | K : Type u_2
inst✝¹ : Field K
inst✝ : NumberField K
h : finrank ℚ K = 1
⊢ nrComplexPlaces K = 0 | linarith [card_add_two_mul_card_eq_rank K] | no goals | 31e214781001676e |
direction_affineSpan | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Defs.lean | theorem direction_affineSpan (s : Set P) : (affineSpan k s).direction = vectorSpan k s | case a.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₁ p₂ : P
hp₂ : p₂ ∈ s
v₁ : V
hv₁ : v₁ ∈ vectorSpan k s
hp₁ : p₁ = v₁ +ᵥ p₂
p₃ p₄ : P
hp₄ : p₄ ∈ s
v₂ : V
hv... | simp only [SetLike.mem_coe] | case a.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₁ p₂ : P
hp₂ : p₂ ∈ s
v₁ : V
hv₁ : v₁ ∈ vectorSpan k s
hp₁ : p₁ = v₁ +ᵥ p₂
p₃ p₄ : P
hp₄ : p₄ ∈ s
v₂ : V
hv... | 002080c61aac1fc7 |
le_inv_mul_iff₀ | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | /-- See `le_inv_mul_iff₀'` for a version with multiplication on the other side. -/
lemma le_inv_mul_iff₀ (hc : 0 < c) : a ≤ c⁻¹ * b ↔ c * a ≤ b where
mp h | G₀ : Type u_2
inst✝⁴ : GroupWithZero G₀
inst✝³ : PartialOrder G₀
inst✝² : ZeroLEOneClass G₀
inst✝¹ : PosMulReflectLT G₀
a b c : G₀
inst✝ : PosMulMono G₀
hc : 0 < c
h : a ≤ c⁻¹ * b
⊢ c * a ≤ b | simpa [hc.ne'] using mul_le_mul_of_nonneg_left h hc.le | no goals | 45d643f4ccbb4d8e |
frontier_Ici' | Mathlib/Topology/Order/DenselyOrdered.lean | theorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} | α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a : α
ha : (Iio a).Nonempty
⊢ frontier (Ici a) = {a} | simp [frontier, ha] | no goals | 321163099cf5b855 |
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