ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Function/L2Space.lean
α : Type u_1
𝕜 : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : RCLike 𝕜
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : CompactSpace α
f g : C(α, 𝕜)
hf_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) =ᶠ[ae μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᶠ[ae μ] ⇑g
a✝ : α
hf : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝ = f a✝
hg : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝ = g a✝

⊢ ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝⟫_𝕜 = (starRingEnd 𝕜) (f a✝) * g a✝
-- Mathlib/Algebra/Polynomial/Basic.lean
R : Type u
a b : R
m n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
r : R
n : ℕ

⊢ (C r).coeff (n + 1) = 0
-- Mathlib/RingTheory/Polynomial/Quotient.lean
R : Type u_1
inst✝ : CommRing R
I : Ideal R
a : R
ha : a ∈ I

⊢ ((Quotient.mk (map C I)).comp C) a = 0
-- Mathlib/RingTheory/Polynomial/Quotient.lean
R : Type u_1
inst✝ : CommRing R
I : Ideal R
a : R
ha : a ∈ I

⊢ C a ∈ map C I
-- Mathlib/InformationTheory/Hamming.lean
α : Type u_1
ι : Type u_2
β : ι → Type u_3
inst✝⁴ : Fintype ι
inst✝³ : (i : ι) → DecidableEq (β i)
γ : ι → Type u_4
inst✝² : (i : ι) → DecidableEq (γ i)
inst✝¹ : (i : ι) → Zero (β i)
inst✝ : (i : ι) → Zero (γ i)
f : (i : ι) → γ i → β i
x : (i : ι) → γ i
hf : ∀ (i : ι), f i 0 = 0

⊢ (hammingNorm fun i => f i (x i)) ≤ hammingNorm x
-- Mathlib/Algebra/Group/Subgroup/Basic.lean
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁴ : Group G
inst✝³ : Group G'
inst✝² : Group G''
A : Type u_4
inst✝¹ : AddGroup A
N : Type u_5
inst✝ : Group N
f✝ f : G →* N
hf : Injective ⇑f
H K : Subgroup G

⊢ map f H ≤ map f K ↔ H ≤ K
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A

⊢ ((constantCommuteCompose J U).app (F.val.obj (op t))).hom ≫ (constantSheafAdj J B ht).counit.app ((sheafCompose J U).obj F) = (sheafCompose J U).map ((constantSheafAdj J A ht).counit.app F)
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A

⊢ (presheafToSheaf J B).map ((constantPresheafAdj B ht).counit.app (F.val ⋙ U)) ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app F.val ≫ (sheafCompose J U).map ((sheafificationAdjunction J A).counit.app F) = (presheafToSheaf J B).map ((compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t))) ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app ((const Cᵒᵖ).obj (F.val.obj (op t))) ≫ (sheafCompose J U).map ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val)) ≫ (sheafCompose J U).map ((sheafificationAdjunction J A).counit.app F)
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A

⊢ ((presheafToSheaf J B).map ((constantPresheafAdj B ht).counit.app (F.val ⋙ U)) ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app F.val) ≫ (sheafCompose J U).map ((sheafificationAdjunction J A).counit.app F) = (((presheafToSheaf J B).map ((compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t))) ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app ((const Cᵒᵖ).obj (F.val.obj (op t)))) ≫ (sheafCompose J U).map ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val))) ≫ (sheafCompose J U).map ((sheafificationAdjunction J A).counit.app F)
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A

⊢ (presheafToSheaf J B).map ((constantPresheafAdj B ht).counit.app (F.val ⋙ U)) ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app F.val = ((presheafToSheaf J B).map ((compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t))) ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app ((const Cᵒᵖ).obj (F.val.obj (op t)))) ≫ (sheafCompose J U).map ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val))
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A
this : (compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t)) ≫ { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } = (constantPresheafAdj B ht).counit.app (F.val ⋙ U)

⊢ (presheafToSheaf J B).map ((constantPresheafAdj B ht).counit.app (F.val ⋙ U)) ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app F.val = ((presheafToSheaf J B).map ((compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t))) ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app ((const Cᵒᵖ).obj (F.val.obj (op t)))) ≫ (sheafCompose J U).map ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val))
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A
this : (compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t)) ≫ { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } = (constantPresheafAdj B ht).counit.app (F.val ⋙ U)

⊢ (presheafToSheaf J B).map ((compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t))) ≫ (presheafToSheaf J B).map { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app F.val = (presheafToSheaf J B).map ((compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t))) ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app ((const Cᵒᵖ).obj (F.val.obj (op t))) ≫ (sheafCompose J U).map ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val))
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A
this : (compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t)) ≫ { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } = (constantPresheafAdj B ht).counit.app (F.val ⋙ U)

⊢ (presheafToSheaf J B).map { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app F.val = (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app ((const Cᵒᵖ).obj (F.val.obj (op t))) ≫ (sheafCompose J U).map ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val))
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A
this : (compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t)) ≫ { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } = (constantPresheafAdj B ht).counit.app (F.val ⋙ U)

⊢ ((presheafToSheaf J B).map { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app F.val).val = ((sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app ((const Cᵒᵖ).obj (F.val.obj (op t))) ≫ (sheafCompose J U).map ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val))).val
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A
this : (compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t)) ≫ { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } = (constantPresheafAdj B ht).counit.app (F.val ⋙ U)

⊢ toSheafify J ((const Cᵒᵖ ⋙ (whiskeringRight Cᵒᵖ A B).obj U).obj (F.val.obj (op t))) ≫ ((presheafToSheaf J B).map { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } ≫ (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app F.val).val = toSheafify J ((const Cᵒᵖ ⋙ (whiskeringRight Cᵒᵖ A B).obj U).obj (F.val.obj (op t))) ≫ ((sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app ((const Cᵒᵖ).obj (F.val.obj (op t))) ≫ (sheafCompose J U).map ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val))).val
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A
this : (compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t)) ≫ { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } = (constantPresheafAdj B ht).counit.app (F.val ⋙ U)

⊢ toSheafify J ((const Cᵒᵖ).obj (F.val.obj (op t)) ⋙ U) ≫ ((presheafToSheaf J B).map { app := fun Y => U.map (F.val.map (ht.from (unop Y)).op), naturality := ⋯ }).val ≫ ((sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app F.val).val = toSheafify J ((const Cᵒᵖ).obj (F.val.obj (op t)) ⋙ U) ≫ ((sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).hom.app ((const Cᵒᵖ).obj (F.val.obj (op t)))).val ≫ whiskerRight ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val)).val U
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A
this : (compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t)) ≫ { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } = (constantPresheafAdj B ht).counit.app (F.val ⋙ U)

⊢ { app := fun Y => U.map (F.val.map (ht.from (unop Y)).op), naturality := ⋯ } ≫ whiskerRight (toSheafify J F.val) U = whiskerRight (toSheafify J ((const Cᵒᵖ).obj (F.val.obj (op t)))) U ≫ whiskerRight ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val)).val U
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A
this : (compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t)) ≫ { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } = (constantPresheafAdj B ht).counit.app (F.val ⋙ U)
x✝ : Cᵒᵖ

⊢ ({ app := fun Y => U.map (F.val.map (ht.from (unop Y)).op), naturality := ⋯ } ≫ whiskerRight (toSheafify J F.val) U).app x✝ = (whiskerRight (toSheafify J ((const Cᵒᵖ).obj (F.val.obj (op t)))) U ≫ whiskerRight ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val)).val U).app x✝
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A
this : (compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t)) ≫ { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } = (constantPresheafAdj B ht).counit.app (F.val ⋙ U)
x✝ : Cᵒᵖ

⊢ U.map (F.val.map (ht.from (unop x✝)).op ≫ (toSheafify J F.val).app x✝) = U.map ((toSheafify J ((const Cᵒᵖ).obj (F.val.obj (op t)))).app x✝ ≫ ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val)).val.app x✝)
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A
this : (compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t)) ≫ { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } = (constantPresheafAdj B ht).counit.app (F.val ⋙ U)
x✝ : Cᵒᵖ

⊢ F.val.map (ht.from (unop x✝)).op ≫ (toSheafify J F.val).app x✝ = (toSheafify J ((const Cᵒᵖ).obj (F.val.obj (op t)))).app x✝ ≫ ((presheafToSheaf J A).map ((constantPresheafAdj A ht).counit.app F.val)).val.app x✝
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A
this : (compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t)) ≫ { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } = (constantPresheafAdj B ht).counit.app (F.val ⋙ U)
x✝ : Cᵒᵖ

⊢ (toSheafify J F.val).app (op t) ≫ (sheafify J F.val).map (ht.from (unop x✝)).op = (toSheafify J ((const Cᵒᵖ).obj (F.val.obj (op t)))).app x✝ ≫ ((presheafToSheaf J A).map { app := fun x => F.val.map (ht.from (unop x)).op, naturality := ⋯ }).val.app x✝
-- Mathlib/CategoryTheory/Sites/Discrete.lean
C : Type u_1
inst✝¹³ : Category.{u_4, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹² : Category.{u_5, u_2} A
inst✝¹¹ : HasWeakSheafify J A
inst✝¹⁰ : (constantSheaf J A).Faithful
inst✝⁹ : (constantSheaf J A).Full
t : C
ht : IsTerminal t
B : Type u_3
inst✝⁸ : Category.{u_6, u_3} B
U : A ⥤ B
inst✝⁷ : HasWeakSheafify J A
inst✝⁶ : HasWeakSheafify J B
inst✝⁵ : (constantSheaf J A).Faithful
inst✝⁴ : (constantSheaf J A).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F✝ F : Sheaf J A
this : (compConstIso Cᵒᵖ U).hom.app (F.val.obj (op t)) ≫ { app := fun Y => (F.val ⋙ U).map (ht.from (unop Y)).op, naturality := ⋯ } = (constantPresheafAdj B ht).counit.app (F.val ⋙ U)
x✝ : Cᵒᵖ

⊢ (toSheafify J F.val).app (op t) ≫ (sheafify J F.val).map (ht.from (unop x✝)).op = ({ app := fun x => F.val.map (ht.from (unop x)).op, naturality := ⋯ } ≫ toSheafify J F.val).app x✝
-- Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν

⊢ μ.rnDeriv (μ.withDensity (ν.rnDeriv μ)) =ᶠ[ae μ] μ.rnDeriv ν
-- Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν

⊢ μ.rnDeriv (μ.withDensity (ν.rnDeriv μ)) =ᶠ[ae μ] μ.rnDeriv (μ.withDensity (ν.rnDeriv μ) + ν.singularPart μ)
-- Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν

⊢ μ.rnDeriv (μ.withDensity (ν.rnDeriv μ)) =ᶠ[ae (μ.withDensity (ν.rnDeriv μ))] μ.rnDeriv (μ.withDensity (ν.rnDeriv μ) + ν.singularPart μ)
-- Mathlib/Data/Nat/Cast/Defs.lean
R : Type u_1
inst✝ : AddMonoidWithOne R

⊢ 2 + 1 = 3
-- Mathlib/Data/Nat/Cast/Defs.lean
R : Type u_1
inst✝ : AddMonoidWithOne R

⊢ ↑(1 + 1 + 1) = 3
-- Mathlib/Data/Nat/Cast/Defs.lean
R : Type u_1
inst✝ : AddMonoidWithOne R

⊢ 1 + 1 + 1 = 3
-- Mathlib/Analysis/SpecialFunctions/Integrals.lean
a b : ℝ
n : ℕ

⊢ ∫ (x : ℝ) in a..b, sin x * cos x = (sin b ^ 2 - sin a ^ 2) / 2
-- Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
w : σ → M
n : M
φ ψ : MvPolynomial σ R
inst✝ : DecidableEq M
x : ⨁ (i : M), ↥(weightedHomogeneousSubmodule R w i)

⊢ (coeLinearMap fun i => weightedHomogeneousSubmodule R w i) x = ∑ᶠ (m : M), ↑(x m)
-- Mathlib/Algebra/DirectSum/Basic.lean
ι : Type v
β : ι → Type w
inst✝ : (i : ι) → AddCommMonoid (β i)
κ : Type u_1
h : ι ≃ κ
f : ⨁ (i : ι), β i
k : κ

⊢ ((equivCongrLeft h) f) k = f (h.symm k)
-- Mathlib/LinearAlgebra/Reflection.lean
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
x✝ : M
f✝ : Dual R M
y : M
inst✝¹ : CharZero R
inst✝ : NoZeroSMulDivisors R M
x : M
hx : x ≠ 0
Φ : Set M
hΦ₁ : Φ.Finite
hΦ₂ : span R Φ = ⊤
f g : Dual R M
hf₁ : f x = 2
hf₂ : MapsTo (⇑(preReflection x f)) Φ Φ
hg₁ : g x = 2
hg₂ : MapsTo (⇑(preReflection x g)) Φ Φ
u : M ≃ₗ[R] M := reflection hg₁ * reflection hf₁

⊢ f = g
-- Mathlib/LinearAlgebra/Reflection.lean
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
x✝ : M
f✝ : Dual R M
y : M
inst✝¹ : CharZero R
inst✝ : NoZeroSMulDivisors R M
x : M
hx : x ≠ 0
Φ : Set M
hΦ₁ : Φ.Finite
hΦ₂ : span R Φ = ⊤
f g : Dual R M
hf₁ : f x = 2
hf₂ : MapsTo (⇑(preReflection x f)) Φ Φ
hg₁ : g x = 2
hg₂ : MapsTo (⇑(preReflection x g)) Φ Φ
u : M ≃ₗ[R] M := reflection hg₁ * reflection hf₁
hu : ↑u = LinearMap.id + LinearMap.smulRight (f - g) x

⊢ f = g
-- Mathlib/LinearAlgebra/Reflection.lean
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
x✝ : M
f✝ : Dual R M
y : M
inst✝¹ : CharZero R
inst✝ : NoZeroSMulDivisors R M
x : M
hx : x ≠ 0
Φ : Set M
hΦ₁ : Φ.Finite
hΦ₂ : span R Φ = ⊤
f g : Dual R M
hf₁ : f x = 2
hf₂ : MapsTo (⇑(preReflection x f)) Φ Φ
hg₁ : g x = 2
hg₂ : MapsTo (⇑(preReflection x g)) Φ Φ
u : M ≃ₗ[R] M := reflection hg₁ * reflection hf₁
hu : ∀ (n : ℕ), ↑(u ^ n) = LinearMap.id + ↑n • LinearMap.smulRight (f - g) x

⊢ f = g
-- Mathlib/LinearAlgebra/Reflection.lean
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
x✝ : M
f✝ : Dual R M
y : M
inst✝¹ : CharZero R
inst✝ : NoZeroSMulDivisors R M
x : M
hx : x ≠ 0
Φ : Set M
hΦ₁ : Φ.Finite
hΦ₂ : span R Φ = ⊤
f g : Dual R M
hf₁ : f x = 2
hf₂ : MapsTo (⇑(preReflection x f)) Φ Φ
hg₁ : g x = 2
hg₂ : MapsTo (⇑(preReflection x g)) Φ Φ
u : M ≃ₗ[R] M := reflection hg₁ * reflection hf₁
hu : ∀ (n : ℕ), ↑(u ^ n) = LinearMap.id + ↑n • LinearMap.smulRight (f - g) x

⊢ IsOfFinOrder u