ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : SMulCommClass ℝ 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
H : Type u_7
inst✝ : NormedAddCommGroup H
f : α → H
p : ℝ≥0∞
hp1 : p ≠ 0
hp2 : p ≠ ⊤
hf : Memℒp f p μ

⊢ eLpNorm f p μ = ENNReal.ofReal ((∫ (a : α), ‖f a‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹)
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : SMulCommClass ℝ 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
H : Type u_7
inst✝ : NormedAddCommGroup H
f : α → H
p : ℝ≥0∞
hp1 : p ≠ 0
hp2 : p ≠ ⊤
hf : Memℒp f p μ
A : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ = ∫⁻ (a : α), ↑‖f a‖₊ ^ p.toReal ∂μ

⊢ eLpNorm f p μ = ENNReal.ofReal ((∫ (a : α), ‖f a‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹)
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : SMulCommClass ℝ 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
H : Type u_7
inst✝ : NormedAddCommGroup H
f : α → H
p : ℝ≥0∞
hp1 : p ≠ 0
hp2 : p ≠ ⊤
hf : Memℒp f p μ
A : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ = ∫⁻ (a : α), ↑‖f a‖₊ ^ p.toReal ∂μ

⊢ (∫⁻ (x : α), ↑‖f x‖₊ ^ p.toReal ∂μ) ^ p.toReal⁻¹ = ENNReal.ofReal ((∫ (a : α), ‖f a‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹)
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : SMulCommClass ℝ 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
H : Type u_7
inst✝ : NormedAddCommGroup H
f : α → H
p : ℝ≥0∞
hp1 : p ≠ 0
hp2 : p ≠ ⊤
hf : Memℒp f p μ
A : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ = ∫⁻ (a : α), ↑‖f a‖₊ ^ p.toReal ∂μ

⊢ (∫⁻ (x : α), ↑‖f x‖₊ ^ p.toReal ∂μ) ^ p.toReal⁻¹ = ENNReal.ofReal ((∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ).toReal ^ p.toReal⁻¹)
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : SMulCommClass ℝ 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
H : Type u_7
inst✝ : NormedAddCommGroup H
f : α → H
p : ℝ≥0∞
hp1 : p ≠ 0
hp2 : p ≠ ⊤
hf : Memℒp f p μ
A : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ = ∫⁻ (a : α), ↑‖f a‖₊ ^ p.toReal ∂μ

⊢ 0 ≤ᶠ[ae μ] fun a => ‖f a‖ ^ p.toReal
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : SMulCommClass ℝ 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
H : Type u_7
inst✝ : NormedAddCommGroup H
f : α → H
p : ℝ≥0∞
hp1 : p ≠ 0
hp2 : p ≠ ⊤
hf : Memℒp f p μ
A : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ = ∫⁻ (a : α), ↑‖f a‖₊ ^ p.toReal ∂μ

⊢ AEStronglyMeasurable (fun a => ‖f a‖ ^ p.toReal) μ
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : SMulCommClass ℝ 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
H : Type u_7
inst✝ : NormedAddCommGroup H
f : α → H
p : ℝ≥0∞
hp1 : p ≠ 0
hp2 : p ≠ ⊤
hf : Memℒp f p μ
A : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ = ∫⁻ (a : α), ↑‖f a‖₊ ^ p.toReal ∂μ

⊢ ∫⁻ (a : α), ↑‖f a‖₊ ^ p.toReal ∂μ ≠ ⊤
-- Mathlib/Order/Interval/Finset/Nat.lean
a b c x : ℕ

⊢ x ∈ Ico a b.succ ↔ x ∈ Icc a b
-- Mathlib/FieldTheory/Normal.lean
F : Type u_1
K : Type u_2
inst✝¹⁸ : Field F
inst✝¹⁷ : Field K
inst✝¹⁶ : Algebra F K
K₁ : Type u_3
K₂ : Type u_4
K₃ : Type u_5
inst✝¹⁵ : Field F
inst✝¹⁴ : Field K₁
inst✝¹³ : Field K₂
inst✝¹² : Field K₃
inst✝¹¹ : Algebra F K₁
inst✝¹⁰ : Algebra F K₂
inst✝⁹ : Algebra F K₃
ϕ : K₁ →ₐ[F] K₂
χ : K₁ ≃ₐ[F] K₂
ψ : K₂ →ₐ[F] K₃
ω : K₂ ≃ₐ[F] K₃
E : Type u_6
inst✝⁸ : Field E
inst✝⁷ : Algebra F E
inst✝⁶ : Algebra E K₁
inst✝⁵ : Algebra E K₂
inst✝⁴ : Algebra E K₃
inst✝³ : IsScalarTower F E K₁
inst✝² : IsScalarTower F E K₂
inst✝¹ : IsScalarTower F E K₃
inst✝ : Normal F E
x✝ : E

⊢ (algebraMap E K₃) (((ψ.restrictNormal E).comp (ϕ.restrictNormal E)) x✝) = (algebraMap E K₃) (((ψ.comp ϕ).restrictNormal E) x✝)
-- Mathlib/FieldTheory/Adjoin.lean
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
K : IntermediateField F E
h : K.toSubalgebra = Algebra.adjoin F S

⊢ K = adjoin F S
-- Mathlib/FieldTheory/Adjoin.lean
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
K : IntermediateField F E
h : K.toSubalgebra = Algebra.adjoin F S

⊢ K.toSubalgebra = (adjoin F S).toSubalgebra
-- Mathlib/FieldTheory/Adjoin.lean
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
K : IntermediateField F E
h : K.toSubalgebra = Algebra.adjoin F S

⊢ Algebra.adjoin F S = (adjoin F S).toSubalgebra
-- Mathlib/FieldTheory/Adjoin.lean
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
K : IntermediateField F E
h : K.toSubalgebra = Algebra.adjoin F S

⊢ ∀ x ∈ Algebra.adjoin F S, x⁻¹ ∈ Algebra.adjoin F S
-- Mathlib/FieldTheory/Adjoin.lean
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
K : IntermediateField F E
h : K.toSubalgebra = Algebra.adjoin F S
x : E

⊢ x ∈ Algebra.adjoin F S → x⁻¹ ∈ Algebra.adjoin F S
-- Mathlib/Data/Set/Pointwise/Interval.lean
α : Type u_1
inst✝ : LinearOrderedField α
a✝ a c : α
h : c < 0

⊢ (fun x => c * x) ⁻¹' Iic a = Ici (a / c)
-- Mathlib/Data/List/Intervals.lean
n m : ℕ
h : n ≤ m

⊢ Ico n (m + 1) = Ico n m ++ [m]
-- Mathlib/Data/List/Intervals.lean
n m : ℕ
h : n ≤ m

⊢ m ≤ m + 1
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))

⊢ op (⋯.functor.obj (unop (op ((Opens.map (pullback.fst (D.f i j) (D.f i k)).base).toPrefunctor.1 (unop (op U)))))) = (Opens.map (D.f i k).base).op.obj (op (⋯.functor.obj U))
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))

⊢ op { carrier := ⇑(pullback.snd (D.f i j) (D.f i k)).base '' ↑{ carrier := ⇑(pullback.fst (D.f i j) (D.f i k)).base ⁻¹' ↑U, is_open' := ⋯ }, is_open' := ⋯ } = op { carrier := ⇑(D.f i k).base ⁻¹' ↑{ carrier := ⇑(D.f i j).base '' ↑U, is_open' := ⋯ }, is_open' := ⋯ }
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))

⊢ ⇑(pullback.snd (D.f i j) (D.f i k)).base '' ↑{ carrier := ⇑(pullback.fst (D.f i j) (D.f i k)).base ⁻¹' ↑U, is_open' := ⋯ } = ⇑(D.f i k).base ⁻¹' ↑{ carrier := ⇑(D.f i j).base '' ↑U, is_open' := ⋯ }
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))

⊢ invApp (D.f i j) U ≫ (D.f i k).c.app (op ((opensFunctor (D.f i j)).obj U)) = (pullback.fst (D.f i j) (D.f i k)).c.app (op U) ≫ invApp (pullback.snd (D.f i j) (D.f i k)) (unop (op ((Opens.map (pullback.fst (D.f i j) (D.f i k)).base).toPrefunctor.1 (unop (op U))))) ≫ (D.V (i, k)).presheaf.map (eqToHom ⋯)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))
this : ∀ (U : (Opens ↑↑(D.U i))ᵒᵖ), (pullback.fst (D.f i j) (D.f i k) ≫ D.f i j).c.app U = (pullback.snd (D.f i j) (D.f i k) ≫ D.f i k).c.app U ≫ (pullback (D.f i j) (D.f i k)).presheaf.map (eqToHom ⋯)

⊢ invApp (D.f i j) U ≫ (D.f i k).c.app (op ((opensFunctor (D.f i j)).obj U)) = (pullback.fst (D.f i j) (D.f i k)).c.app (op U) ≫ invApp (pullback.snd (D.f i j) (D.f i k)) (unop (op ((Opens.map (pullback.fst (D.f i j) (D.f i k)).base).toPrefunctor.1 (unop (op U))))) ≫ (D.V (i, k)).presheaf.map (eqToHom ⋯)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))
this : ∀ (U : (Opens ↑↑(D.U i))ᵒᵖ), (D.f i j).c.app U ≫ (pullback.fst (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i j).base).obj (unop U))) = ((D.f i k).c.app U ≫ (pullback.snd (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i k).base).obj (unop U)))) ≫ (pullback (D.f i j) (D.f i k)).presheaf.map (eqToHom ⋯)

⊢ invApp (D.f i j) U ≫ (D.f i k).c.app (op ((opensFunctor (D.f i j)).obj U)) = (pullback.fst (D.f i j) (D.f i k)).c.app (op U) ≫ invApp (pullback.snd (D.f i j) (D.f i k)) (unop (op ((Opens.map (pullback.fst (D.f i j) (D.f i k)).base).toPrefunctor.1 (unop (op U))))) ≫ (D.V (i, k)).presheaf.map (eqToHom ⋯)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))
this : ∀ (U : (Opens ↑↑(D.U i))ᵒᵖ), (D.f i j).c.app U ≫ (pullback.fst (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i j).base).obj (unop U))) = ((D.f i k).c.app U ≫ (pullback.snd (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i k).base).obj (unop U)))) ≫ (pullback (D.f i j) (D.f i k)).presheaf.map (eqToHom ⋯)

⊢ (D.f i k).c.app (op ((opensFunctor (D.f i j)).obj U)) = ((D.f i j).c.app (op ((opensFunctor (D.f i j)).obj U)) ≫ (D.V (i, j)).presheaf.map (eqToHom ⋯)) ≫ (pullback.fst (D.f i j) (D.f i k)).c.app (op U) ≫ invApp (pullback.snd (D.f i j) (D.f i k)) (unop (op ((Opens.map (pullback.fst (D.f i j) (D.f i k)).base).toPrefunctor.1 (unop (op U))))) ≫ (D.V (i, k)).presheaf.map (eqToHom ⋯)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))
this : ∀ (U : (Opens ↑↑(D.U i))ᵒᵖ), (D.f i j).c.app U ≫ (pullback.fst (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i j).base).obj (unop U))) = ((D.f i k).c.app U ≫ (pullback.snd (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i k).base).obj (unop U)))) ≫ (pullback (D.f i j) (D.f i k)).presheaf.map (eqToHom ⋯)

⊢ (D.f i k).c.app (op ((opensFunctor (D.f i j)).obj U)) = (D.f i j).c.app (op ((opensFunctor (D.f i j)).obj U)) ≫ (D.V (i, j)).presheaf.map (eqToHom ⋯) ≫ (pullback.fst (D.f i j) (D.f i k)).c.app (op U) ≫ invApp (pullback.snd (D.f i j) (D.f i k)) ((Opens.map (pullback.fst (D.f i j) (D.f i k)).base).toPrefunctor.1 U) ≫ (D.V (i, k)).presheaf.map (eqToHom ⋯)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))
this : ∀ (U : (Opens ↑↑(D.U i))ᵒᵖ), (D.f i j).c.app U ≫ (pullback.fst (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i j).base).obj (unop U))) = ((D.f i k).c.app U ≫ (pullback.snd (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i k).base).obj (unop U)))) ≫ (pullback (D.f i j) (D.f i k)).presheaf.map (eqToHom ⋯)

⊢ (D.f i k).c.app (op ((opensFunctor (D.f i j)).obj U)) = (D.f i k).c.app (op ((opensFunctor (D.f i j)).obj U)) ≫ (D.V (i, k)).presheaf.map ((homOfLE ⋯).op ≫ (opensFunctor (pullback.snd (D.f i j) (D.f i k))).op.map (eqToHom ⋯ ≫ (Opens.map (pullback.fst (D.f i j) (D.f i k)).base).op.map (eqToHom ⋯)) ≫ eqToHom ⋯)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))
this : ∀ (U : (Opens ↑↑(D.U i))ᵒᵖ), (D.f i j).c.app U ≫ (pullback.fst (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i j).base).obj (unop U))) = ((D.f i k).c.app U ≫ (pullback.snd (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i k).base).obj (unop U)))) ≫ (pullback (D.f i j) (D.f i k)).presheaf.map (eqToHom ⋯)

⊢ (D.V (i, k)).presheaf.map ((homOfLE ⋯).op ≫ (opensFunctor (pullback.snd (D.f i j) (D.f i k))).op.map (eqToHom ⋯ ≫ (Opens.map (pullback.fst (D.f i j) (D.f i k)).base).op.map (eqToHom ⋯)) ≫ eqToHom ⋯) = 𝟙 (((TopCat.Presheaf.pushforward C (D.f i k).base).obj (D.V (i, k)).presheaf).obj (op ((opensFunctor (D.f i j)).obj U)))
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(D.V (i, j))
this : ∀ (U : (Opens ↑↑(D.U i))ᵒᵖ), (D.f i j).c.app U ≫ (pullback.fst (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i j).base).obj (unop U))) = ((D.f i k).c.app U ≫ (pullback.snd (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i k).base).obj (unop U)))) ≫ (pullback (D.f i j) (D.f i k)).presheaf.map (eqToHom ⋯)

⊢ 𝟙 ((D.V (i, k)).presheaf.obj (op ((Opens.map (D.f i k).base).obj (unop (op ((opensFunctor (D.f i j)).obj U)))))) = 𝟙 (((TopCat.Presheaf.pushforward C (D.f i k).base).obj (D.V (i, k)).presheaf).obj (op ((opensFunctor (D.f i j)).obj U)))
-- Mathlib/Analysis/Calculus/ContDiff/Basic.lean
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁹ : NormedAddCommGroup D
inst✝⁸ : NormedSpace 𝕜 D
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g✝ : F → G
x x₀ : E
c : F
b : E × F → G
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
q : E → FormalMultilinearSeries 𝕜 E F
g : E → F
hf : HasFTaylorSeriesUpToOn n f p s
hg : HasFTaylorSeriesUpToOn n g q s

⊢ HasFTaylorSeriesUpToOn n (f + g) (p + q) s
-- Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f✝ : MultilinearMap 𝕜 E G
k n : ℕ
f : MultilinearMap 𝕜 (fun x => G) G'
s : Finset (Fin n)
hk : s.card = k
z : G
C : ℝ
H : ∀ (m : Fin n → G), ‖f m‖ ≤ C * ∏ i : Fin n, ‖m i‖
v : Fin k → G

⊢ ‖(f.restr s hk z) v‖ ≤ C * ‖z‖ ^ (n - k) * ∏ i : Fin k, ‖v i‖
-- Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f✝ : MultilinearMap 𝕜 E G
k n : ℕ
f : MultilinearMap 𝕜 (fun x => G) G'
s : Finset (Fin n)
hk : s.card = k
z : G
C : ℝ
H : ∀ (m : Fin n → G), ‖f m‖ ≤ C * ∏ i : Fin n, ‖m i‖
v : Fin k → G

⊢ ‖(f.restr s hk z) v‖ ≤ C * ((∏ i : Fin k, ‖v i‖) * ‖z‖ ^ (n - k))
-- Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f✝ : MultilinearMap 𝕜 E G
k n : ℕ
f : MultilinearMap 𝕜 (fun x => G) G'
s : Finset (Fin n)
hk : s.card = k
z : G
C : ℝ
H : ∀ (m : Fin n → G), ‖f m‖ ≤ C * ∏ i : Fin n, ‖m i‖
v : Fin k → G

⊢ (∏ i : Fin k, ‖v i‖) * ‖z‖ ^ (n - k) = ∏ i : Fin n, ‖if h : i ∈ s then v ((s.orderIsoOfFin hk).symm ⟨i, h⟩) else z‖
-- Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f✝ : MultilinearMap 𝕜 E G
k n : ℕ
f : MultilinearMap 𝕜 (fun x => G) G'
s : Finset (Fin n)
hk : s.card = k
z : G
C : ℝ
H : ∀ (m : Fin n → G), ‖f m‖ ≤ C * ∏ i : Fin n, ‖m i‖
v : Fin k → G

⊢ (∏ i : { x // x ∈ s }, ‖v ((s.orderIsoOfFin hk).symm i)‖) * ‖z‖ ^ (n - k) = (∏ a : { a // a ∈ s }, ‖v ((s.orderIsoOfFin hk).symm ⟨↑a, ⋯⟩)‖) * ‖z‖ ^ (n - k)
-- Mathlib/Analysis/BoxIntegral/Partition/Additive.lean
ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : AddCommMonoid N
I₀ : WithTop (Box ι)
I✝ J : Box ι
i : ι
f : ι →ᵇᵃ[I₀] M
g : M →+ N
I : Box ι
hI : ↑I ≤ I₀
π : Prepartition I
hπ : π.IsPartition

⊢ ∑ Ji ∈ π.boxes, (⇑g ∘ ⇑f) Ji = (⇑g ∘ ⇑f) I
-- Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean
s : ℂ
hs : 1 < s.re

⊢ Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - ↑p ^ (-s))⁻¹) atTop (𝓝 (riemannZeta s))
-- Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean
s : ℂ
hs : 1 < s.re

⊢ Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - ↑p ^ (-s))⁻¹) atTop (𝓝 (∑' (n : ℕ), (riemannZetaSummandHom ⋯) n))
-- Mathlib/Data/DFinsupp/NeLocus.lean
α : Type u_1
N : α → Type u_2
inst✝² : DecidableEq α
inst✝¹ : (a : α) → DecidableEq (N a)
inst✝ : (a : α) → AddGroup (N a)
f f₁ f₂ g g₁ g₂ : Π₀ (a : α), N a

⊢ (f - g₁).neLocus (f - g₂) = g₁.neLocus g₂
-- Mathlib/Analysis/Convex/Integral.lean
α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ : Measure α
s : Set E
t : Set α
f : α → E
g : E → ℝ
C : ℝ
inst✝ : IsFiniteMeasure μ
hs : Convex ℝ s
h0 : μ t ≠ 0
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
ht : ⨍ (x : α) in t, f x ∂μ ∈ interior s

⊢ ⨍ (x : α), f x ∂μ ∈ interior s
-- Mathlib/Analysis/Convex/Integral.lean
α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ : Measure α
s : Set E
t : Set α
f : α → E
g : E → ℝ
C : ℝ
inst✝ : IsFiniteMeasure μ
hs : Convex ℝ s
h0 : μ (toMeasurable μ t) ≠ 0
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
ht : ⨍ (x : α) in t, f x ∂μ ∈ interior s

⊢ ⨍ (x : α), f x ∂μ ∈ interior s
-- Mathlib/Analysis/Convex/Integral.lean
α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ : Measure α
s : Set E
t : Set α
f : α → E
g : E → ℝ
C : ℝ
inst✝ : IsFiniteMeasure μ
hs : Convex ℝ s
h0 : μ (toMeasurable μ t) ≠ 0
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
ht : ⨍ (x : α) in toMeasurable μ t, f x ∂μ ∈ interior s

⊢ ⨍ (x : α), f x ∂μ ∈ interior s
-- Mathlib/Analysis/Convex/Integral.lean
α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ : Measure α
s : Set E
t : Set α
f : α → E
g : E → ℝ
C : ℝ
inst✝ : IsFiniteMeasure μ
hs : Convex ℝ s
h0 : μ (toMeasurable μ t) ≠ 0
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
ht : ⨍ (x : α) in toMeasurable μ t, f x ∂μ ∈ interior s
h0' : ¬μ (toMeasurable μ t)ᶜ = 0

⊢ ⨍ (x : α), f x ∂μ ∈ interior s
-- Mathlib/LinearAlgebra/PerfectPairing.lean
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : PerfectPairing R M N
x : M
f : Dual R M

⊢ (p.toDualRight.symm.dualMap (p.toDualLeft x)) f = ((Dual.eval R M) x) f
-- Mathlib/Topology/Bases.lean
α : Type u
β : Type u_1
t : TopologicalSpace α
B : Set (Set α)
s : Set α
inst✝ : DiscreteTopology α

⊢ SeparableSpace α ↔ Countable α
-- Mathlib/Data/Finsupp/ToDFinsupp.lean
ι : Type u_1
R : Type u_2
M : Type u_3
η : ι → Type u_4
N : Type u_5
inst✝² : Semiring R
inst✝¹ : DecidableEq ι
inst✝ : Zero N
n : N
i : ι
a : η i
j : ι
b : η j
h : ¬i = j

⊢ ((sigmaFinsuppEquivDFinsupp (single ⟨i, a⟩ n)) j) b = ((DFinsupp.single ⟨i, a⟩.fst (single ⟨i, a⟩.snd n)) j) b
-- Mathlib/Data/Finsupp/ToDFinsupp.lean
ι : Type u_1
R : Type u_2
M : Type u_3
η : ι → Type u_4
N : Type u_5
inst✝² : Semiring R
inst✝¹ : DecidableEq ι
inst✝ : Zero N
n : N
i : ι
a : η i
j : ι
b : η j
h : ¬i = j

⊢ (single ⟨i, a⟩ n) ⟨j, b⟩ = 0
-- Mathlib/Data/Finsupp/ToDFinsupp.lean
ι : Type u_1
R : Type u_2
M : Type u_3
η : ι → Type u_4
N : Type u_5
inst✝² : Semiring R
inst✝¹ : DecidableEq ι
inst✝ : Zero N
n : N
i : ι
a : η i
j : ι
b : η j
h : ¬i = j
H : ⟨i, a⟩ ≠ ⟨j, b⟩

⊢ (single ⟨i, a⟩ n) ⟨j, b⟩ = 0
-- Mathlib/Topology/Order/ScottTopology.lean
α : Type u_1
β : Type u_2
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α
s : Set α
a : α

⊢ IsClosed s ↔ IsLowerSet s ∧ DirSupClosed s
-- Mathlib/Topology/UniformSpace/Cauchy.lean
α : Type u
β : Type v
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
hu : CauchySeq u
f g : ℕ → ℕ
hf : Tendsto f atTop atTop
hg : Tendsto g atTop atTop

⊢ ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), ((u ∘ f ∘ φ) n, (u ∘ g ∘ φ) n) ∈ V n
-- Mathlib/Topology/UniformSpace/Cauchy.lean
α : Type u
β : Type v
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
hu : Tendsto (Prod.map u u) atTop (𝓤 α)
f g : ℕ → ℕ
hf : Tendsto f atTop atTop
hg : Tendsto g atTop atTop

⊢ ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), ((u ∘ f ∘ φ) n, (u ∘ g ∘ φ) n) ∈ V n