ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))

⊢ ∫⁻ (x : α), ∫⁻ (y : β), ↑‖f (x, y)‖₊ ∂ν ∂μ < ⊤ ↔ (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ↑‖f (x, a)‖₊ ∂ν < ⊤) ∧ ∫⁻ (a : α), ENNReal.ofReal (∫⁻ (a_1 : β), ↑‖f (a, a_1)‖₊ ∂ν).toReal ∂μ < ⊤
-- Mathlib/Data/Sum/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : LT α
inst✝ : LT β
x✝ : NoMinOrder (α ⊕ β)
a : α

⊢ ∃ b, b < a
-- Mathlib/Data/Sum/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : LT α
inst✝ : LT β
x✝ : NoMinOrder (α ⊕ β)
a b : α
h : inl b < inl a

⊢ ∃ b, b < a
-- Mathlib/Data/Sum/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : LT α
inst✝ : LT β
x✝ : NoMinOrder (α ⊕ β)
a : α
b : β
h : inr b < inl a

⊢ ∃ b, b < a
-- Mathlib/Data/Sum/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : LT α
inst✝ : LT β
x✝ : NoMinOrder (α ⊕ β)
a : β

⊢ ∃ b, b < a
-- Mathlib/Data/Sum/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : LT α
inst✝ : LT β
x✝ : NoMinOrder (α ⊕ β)
a : β
b : α
h : inl b < inr a

⊢ ∃ b, b < a
-- Mathlib/Data/Sum/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : LT α
inst✝ : LT β
x✝ : NoMinOrder (α ⊕ β)
a b : β
h : inr b < inr a

⊢ ∃ b, b < a
-- Mathlib/CategoryTheory/Limits/HasLimits.lean
J : Type u₁
inst✝⁴ : Category.{v₁, u₁} J
K : Type u₂
inst✝³ : Category.{v₂, u₂} K
C : Type u
inst✝² : Category.{v, u} C
F✝ F G : J ⥤ C
inst✝¹ : HasLimit F
inst✝ : HasLimit G
c : Cone F
α : F ⟶ G
j✝ : J

⊢ (lift F c ≫ limMap α) ≫ π G j✝ = lift G ((Cones.postcompose α).obj c) ≫ π G j✝
-- Mathlib/CategoryTheory/Limits/HasLimits.lean
J : Type u₁
inst✝⁴ : Category.{v₁, u₁} J
K : Type u₂
inst✝³ : Category.{v₂, u₂} K
C : Type u
inst✝² : Category.{v, u} C
F✝ F G : J ⥤ C
inst✝¹ : HasLimit F
inst✝ : HasLimit G
c : Cone F
α : F ⟶ G
j✝ : J

⊢ c.π.app j✝ ≫ α.app j✝ = ((Cones.postcompose α).obj c).π.app j✝
-- Mathlib/Analysis/BoxIntegral/Basic.lean
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I J : Box ι
π : TaggedPrepartition I
inst✝ : Fintype ι
l : IntegrationParams
f g : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y y' : F
h : ¬Integrable I l f vol

⊢ integral I l (-f) vol = -integral I l f vol
-- Mathlib/MeasureTheory/Covering/VitaliFamily.lean
α : Type u_1
inst✝ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
ι : Sort u_2
p : ι → Prop
s : ι → Set α
x : α
h : (𝓝 x).HasBasis p s

⊢ (v.filterAt x).HasBasis p fun i => {t | t ∈ v.setsAt x ∧ t ⊆ s i}
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X : C
P : ProjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive

⊢ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) 0).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((projectiveResolutions C).obj X).as) ≫ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((projectiveResolutions C).obj X).as).homologyι 0 ≫ (projectiveResolution X).fromLeftDerivedZero' F = ((HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) 0).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) 0).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) ≫ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex).homologyι 0 ≫ P.fromLeftDerivedZero' F
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X : C
P : ProjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ : HomologicalComplex.opcyclesMap ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map (lift (𝟙 X) P (projectiveResolution X))) 0 ≫ (projectiveResolution X).fromLeftDerivedZero' F = P.fromLeftDerivedZero' F ≫ F.map (𝟙 X)

⊢ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) 0).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((projectiveResolutions C).obj X).as) ≫ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((projectiveResolutions C).obj X).as).homologyι 0 ≫ (projectiveResolution X).fromLeftDerivedZero' F = ((HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) 0).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) 0).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) ≫ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex).homologyι 0 ≫ P.fromLeftDerivedZero' F
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X : C
P : ProjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ : HomologicalComplex.opcyclesMap ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map (lift (𝟙 X) P (projectiveResolution X))) 0 ≫ (projectiveResolution X).fromLeftDerivedZero' F = P.fromLeftDerivedZero' F ≫ F.map (𝟙 X)
h₂ : (P.isoLeftDerivedToHomotopyCategoryObj F).inv = (F.mapHomologicalComplex (ComplexShape.down ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.down ℕ)).map (lift (𝟙 X) P (projectiveResolution X))

⊢ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) 0).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((projectiveResolutions C).obj X).as) ≫ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((projectiveResolutions C).obj X).as).homologyι 0 ≫ (projectiveResolution X).fromLeftDerivedZero' F = ((HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) 0).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) 0).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) ≫ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex).homologyι 0 ≫ P.fromLeftDerivedZero' F
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X : C
P : ProjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₂ : (P.isoLeftDerivedToHomotopyCategoryObj F).inv = (F.mapHomologicalComplex (ComplexShape.down ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.down ℕ)).map (lift (𝟙 X) P (projectiveResolution X))
h₁ : HomologicalComplex.opcyclesMap ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map (lift (𝟙 X) P (projectiveResolution X))) 0 ≫ (projectiveResolution X).fromLeftDerivedZero' F = P.fromLeftDerivedZero' F

⊢ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) 0).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((projectiveResolutions C).obj X).as) ≫ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((projectiveResolutions C).obj X).as).homologyι 0 ≫ (projectiveResolution X).fromLeftDerivedZero' F = ((HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) 0).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) 0).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) ≫ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex).homologyι 0 ≫ P.fromLeftDerivedZero' F
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X : C
P : ProjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₂ : (P.isoLeftDerivedToHomotopyCategoryObj F).inv = (F.mapHomologicalComplex (ComplexShape.down ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.down ℕ)).map (lift (𝟙 X) P (projectiveResolution X))
h₁ : HomologicalComplex.opcyclesMap ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map (lift (𝟙 X) P (projectiveResolution X))) 0 ≫ (projectiveResolution X).fromLeftDerivedZero' F = P.fromLeftDerivedZero' F

⊢ (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) 0).map ((F.mapHomologicalComplex (ComplexShape.down ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.down ℕ)).map (lift (𝟙 X) P (projectiveResolution X))) ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) 0).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((projectiveResolutions C).obj X).as) ≫ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj ((projectiveResolutions C).obj X).as).homologyι 0 ≫ (projectiveResolution X).fromLeftDerivedZero' F = (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) 0).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex) ≫ HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map (lift (𝟙 X) P (projectiveResolution X))) 0 ≫ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj (projectiveResolution X).complex).homologyι 0 ≫ (projectiveResolution X).fromLeftDerivedZero' F
-- Mathlib/MeasureTheory/Measure/AEDisjoint.lean
ι : Type u_1
α : Type u_2
m : MeasurableSpace α
μ : Measure α
s t u v : Set α

⊢ AEDisjoint μ (s ∪ t) u ↔ AEDisjoint μ s u ∧ AEDisjoint μ t u
-- Mathlib/Algebra/GroupWithZero/Units/Basic.lean
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝ : MonoidWithZero M₀
a✝ : Nontrivial M₀

⊢ inverse 0 = 0
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), 0 < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧ ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε

⊢ levyProkhorovEDist μ ν ≤ δ
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), 0 < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧ ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε

⊢ ∀ (ε : ℝ≥0), 0 < ε → δ < ⊤ → levyProkhorovEDist μ ν ≤ δ + ↑ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), 0 < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧ ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε
ε : ℝ≥0
hε : 0 < ε
a✝ : δ < ⊤
ε_top : ¬↑ε = ⊤

⊢ levyProkhorovEDist μ ν ≤ δ + ↑ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), 0 < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧ ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε
ε : ℝ≥0
hε : 0 < ε
a✝ : δ < ⊤
ε_top : ¬↑ε = ⊤

⊢ δ + ↑ε ∈ {ε | ∀ (B : Set Ω), MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε ∧ ν B ≤ μ (thickening ε.toReal B) + ε}
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), 0 < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧ ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε
ε : ℝ≥0
hε : 0 < ε
a✝ : δ < ⊤
ε_top : ¬↑ε = ⊤
B : Set Ω
B_mble : MeasurableSet B

⊢ μ B ≤ ν (thickening (δ + ↑ε).toReal B) + (δ + ↑ε) ∧ ν B ≤ μ (thickening (δ + ↑ε).toReal B) + (δ + ↑ε)
-- Mathlib/Data/Int/Defs.lean
a b c d m n✝ : ℤ
n : ℕ

⊢ 2 * n = n + n
-- Mathlib/Data/Int/Defs.lean
a b c d m n✝ : ℤ
n : ℕ

⊢ ↑2 * -[n+1] = -[n+1] + -[n+1]
-- Mathlib/Data/Int/Defs.lean
a b c d m n✝ : ℤ
n : ℕ

⊢ -↑n.succ + -↑n.succ = -[n+1] + -[n+1]
-- Mathlib/Algebra/BigOperators/Group/Multiset.lean
F : Type u_1
ι : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
inst✝¹ : CommMonoid α
inst✝ : CommMonoid β
s✝ t : Multiset α
a : α
m : Multiset ι
f g : ι → α
s : Multiset α
x y z : α

⊢ (fun x x_1 => x * x_1) x ((fun x x_1 => x * x_1) y z) = (fun x x_1 => x * x_1) y ((fun x x_1 => x * x_1) x z)
-- Mathlib/Combinatorics/SimpleGraph/Basic.lean
ι : Sort u_1
V : Type u
G : SimpleGraph V
a b c u v✝ w✝ : V
e : Sym2 V
s✝ s t : Set (Sym2 V)
h : s ⊆ t
v w : V

⊢ (fromEdgeSet s).Adj v w → (fromEdgeSet t).Adj v w
-- Mathlib/Combinatorics/SimpleGraph/Basic.lean
ι : Sort u_1
V : Type u
G : SimpleGraph V
a b c u v✝ w✝ : V
e : Sym2 V
s✝ s t : Set (Sym2 V)
h : s ⊆ t
v w : V

⊢ s(v, w) ∈ s → ¬v = w → s(v, w) ∈ t
-- Mathlib/Order/Filter/Basic.lean
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
l f✝ f₁ f₂ : Filter α
g g₁ g₂ : Filter β
m✝ : α → β
m' : β → γ
s✝ : Set α
t : Set β
f : Filter α
m : α → β
h : Injective m
s : Set α
hs : s ∈ f

⊢ m ⁻¹' (m '' s) ⊆ s
-- Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
inst✝¹¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
inst✝⁶ : ChartedSpace H' M'
inst✝⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n✝ n : ℕ∞
U V : Opens M
h : U ≤ V
x : M
hx : x ∈ U

⊢ ContMDiffAt I I n (inclusion h) ⟨x, hx⟩
-- Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
inst✝¹¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
inst✝⁶ : ChartedSpace H' M'
inst✝⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n✝ n : ℕ∞
U V : Opens M
h : U ≤ V
x : M
hx : x ∈ U

⊢ ∀ (y : H), ContDiffWithinAtProp I I n id univ y
-- Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
inst✝¹¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
inst✝⁶ : ChartedSpace H' M'
inst✝⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n✝ n : ℕ∞
U V : Opens M
h : U ≤ V
x : M
hx : x ∈ U
y : H

⊢ ContDiffWithinAtProp I I n id univ y
-- Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
inst✝¹¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
inst✝⁶ : ChartedSpace H' M'
inst✝⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n✝ n : ℕ∞
U V : Opens M
h : U ≤ V
x : M
hx : x ∈ U
y : H

⊢ ContDiffWithinAt 𝕜 n (↑I ∘ ↑I.symm) (univ ∩ range ↑I) (↑I y)
-- Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
𝕜 : Type u_1
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace H M
inst✝¹¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁷ : TopologicalSpace M'
inst✝⁶ : ChartedSpace H' M'
inst✝⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n✝ n : ℕ∞
U V : Opens M
h : U ≤ V
x : M
hx : x ∈ U
y : H

⊢ ContDiffWithinAt 𝕜 n (↑I ∘ ↑I.symm) (range ↑I) (↑I y)
-- Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
R : Type u
inst✝ : CommRing R
W : Affine R

⊢ evalEval 0 0 W.polynomialY = W.a₃
-- Mathlib/Algebra/CharP/Quotient.lean
R : Type u_1
inst✝¹ : CommRing R
n : ℕ
inst✝ : CharP R n
I : Ideal R

⊢ CharP (R ⧸ I) n ↔ Ideal.comap (Nat.castRingHom R) I ≤ RingHom.ker (Nat.castRingHom R)
-- Mathlib/Algebra/CharP/Quotient.lean
R : Type u_1
inst✝¹ : CommRing R
n : ℕ
inst✝ : CharP R n
I : Ideal R

⊢ (∀ (x : ℕ), ↑x ∈ I → ↑x = 0) ↔ Ideal.comap (Nat.castRingHom R) I ≤ Ideal.comap (Nat.castRingHom R) ⊥
-- Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
X : Type u_5
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : TopologicalSpace X
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
s✝ t u : Set H
x✝ : H
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source

⊢ P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ↔ P f s x
-- Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
X : Type u_5
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : TopologicalSpace X
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
s✝ t u : Set H
x✝ : H
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)

⊢ P f s x
-- Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
X : Type u_5
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : TopologicalSpace X
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
s✝ t u : Set H
x✝ : H
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
this : P ((f ∘ ↑e.symm) ∘ ↑e.symm.symm) (↑e.symm.symm ⁻¹' (↑e.symm ⁻¹' s)) (↑e.symm (↑e x))

⊢ P f s x
-- Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
X : Type u_5
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : TopologicalSpace X
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
s✝ t u : Set H
x✝ : H
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
this : P ((f ∘ ↑e.symm) ∘ ↑e) (↑e ⁻¹' (↑e.symm ⁻¹' s)) x

⊢ P f s x
-- Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
X : Type u_5
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : TopologicalSpace X
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
s✝ t u : Set H
x✝ : H
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
this : P ((f ∘ ↑e.symm) ∘ ↑e) (↑e ⁻¹' (↑e.symm ⁻¹' s)) x

⊢ (f ∘ ↑e.symm) ∘ ↑e =ᶠ[𝓝 x] f
-- Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
H : Type u_1
M : Type u_2
H' : Type u_3
M' : Type u_4
X : Type u_5
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
inst✝ : TopologicalSpace X
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
s✝ t u : Set H
x✝ : H
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
this : P ((f ∘ ↑e.symm) ∘ ↑e) (↑e ⁻¹' (↑e.symm ⁻¹' s)) x

⊢ ↑e ⁻¹' (↑e.symm ⁻¹' s) =ᶠ[𝓝 x] s
-- Mathlib/Data/Complex/Exponential.lean
x y : ℂ

⊢ cexp x = cexp ↑x.re * (cos ↑x.im + sin ↑x.im * I)
-- Mathlib/Analysis/Complex/PhragmenLindelof.lean
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
hBf : ∃ c < a, ∃ B, f =O[l] fun z => expR (B * expR (c * |u z|))
hBg : ∃ c < a, ∃ B, g =O[l] fun z => expR (B * expR (c * |u z|))

⊢ ∃ c < a, ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * |u z|))
-- Mathlib/Analysis/Complex/PhragmenLindelof.lean
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
hBf : ∃ c < a, ∃ B, f =O[l] fun z => expR (B * expR (c * |u z|))
hBg : ∃ c < a, ∃ B, g =O[l] fun z => expR (B * expR (c * |u z|))
this : ∀ {c₁ c₂ B₁ B₂ : ℝ}, c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖

⊢ ∃ c < a, ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * |u z|))
-- Mathlib/Analysis/Complex/PhragmenLindelof.lean
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
this : ∀ {c₁ c₂ B₁ B₂ : ℝ}, c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))
cg : ℝ
hcg : cg < a
Bg : ℝ
hOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))

⊢ ∃ c < a, ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * |u z|))