ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Control/Traversable/Equiv.lean
t t' : Type u → Type u
eqv : (α : Type u) → t α ≃ t' α
inst✝⁵ : Traversable t
inst✝⁴ : LawfulTraversable t
F G : Type u → Type u
inst✝³ : Applicative F
inst✝² : Applicative G
inst✝¹ : LawfulApplicative F
inst✝ : LawfulApplicative G
η : ApplicativeTransformation F G
α β γ : Type u
f : β → F γ
g : α → G β
x : t' α
\u22a2 Equiv.traverse eqv (Comp.mk ∘ map f ∘ g) x = Comp.mk (Equiv.traverse eqv f <$> Equiv.traverse eqv g x)
-- Mathlib/Control/Traversable/Equiv.lean
t t' : Type u → Type u
eqv : (α : Type u) → t α ≃ t' α
inst✝⁵ : Traversable t
inst✝⁴ : LawfulTraversable t
F G : Type u → Type u
inst✝³ : Applicative F
inst✝² : Applicative G
inst✝¹ : LawfulApplicative F
inst✝ : LawfulApplicative G
η : ApplicativeTransformation F G
α β γ : Type u
f : β → F γ
g : α → G β
x : t' α
\u22a2 Comp.mk ((fun x => ⇑(eqv γ) <$> x) <$> traverse f <$> traverse g ((eqv α).symm x)) = Comp.mk (Equiv.traverse eqv f <$> Equiv.traverse eqv g x)
-- Mathlib/Analysis/Normed/Group/Bounded.lean
α : Type u_1
ι : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁴ : SeminormedGroup E
inst✝³ : SeminormedGroup F
inst✝² : SeminormedGroup G
s : Set E
a a₁ a₂ b b₁ b₂ : E
r r₁ r₂ : ℝ
inst✝¹ : Nonempty α
inst✝ : SemilatticeSup α
u : α → E
\u22a2 CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ (m : α), N ≤ m → ∀ (n : α), N ≤ n → ‖u m / u n‖ < ε
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝¹ : Group α
inst✝ : Fintype α
x : α
hx : orderOf x = Fintype.card α
\u22a2 IsCyclic α
-- Mathlib/GroupTheory/Congruence/Basic.lean
M : Type u_1
N : Type u_2
P : Type u_3
inst✝ : Monoid M
c : Con M
f : M → M
hf : ∀ (x : M), c (f x * x) 1
x y : M
h : c x y
\u22a2 c (f x) (f y)
-- Mathlib/GroupTheory/Congruence/Basic.lean
M : Type u_1
N : Type u_2
P : Type u_3
inst✝ : Monoid M
c : Con M
f : M → M
x y : M
hf : ∀ (x : M), ↑(f x) * ↑x = 1
h : ↑x = ↑y
\u22a2 ↑(f x) = ↑(f y)
-- Mathlib/GroupTheory/Congruence/Basic.lean
M : Type u_1
N : Type u_2
P : Type u_3
inst✝ : Monoid M
c : Con M
f : M → M
x y : M
hf : ∀ (x : M), ↑(f x) * ↑x = 1
h : ↑x = ↑y
hf' : ∀ (x : M), ↑x * ↑(f x) = 1
\u22a2 ↑(f x) = ↑(f y)
-- Mathlib/GroupTheory/Congruence/Basic.lean
M : Type u_1
N : Type u_2
P : Type u_3
inst✝ : Monoid M
c : Con M
f : M → M
x y : M
hf : ∀ (x : M), ↑(f x) * ↑x = 1
h : ↑x = ↑y
hf' : ∀ (x : M), ↑x * ↑(f x) = 1
this : { val := ↑x, inv := ↑(f x), val_inv := ⋯, inv_val := ⋯ } = { val := ↑y, inv := ↑(f y), val_inv := ⋯, inv_val := ⋯ }
\u22a2 ↑(f x) = ↑(f y)
-- Mathlib/Data/Set/Equitable.lean
α : Type u_1
β : Type u_2
s : Set α
f : α → ℕ
\u22a2 s.EquitableOn f ↔ ∃ b, ∀ a ∈ s, f a = b ∨ f a = b + 1
-- Mathlib/NumberTheory/Bernoulli.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
h_odd : Odd n
hlt : 1 < n
B : ℚ⟦X⟧ := PowerSeries.mk fun n => bernoulli' n / ↑n !
\u22a2 (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1)
-- Mathlib/NumberTheory/Bernoulli.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
h_odd : Odd n
hlt : 1 < n
B : ℚ⟦X⟧ := PowerSeries.mk fun n => bernoulli' n / ↑n !
h : B * (exp ℚ - 1) = X * exp ℚ
\u22a2 (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1)
-- Mathlib/NumberTheory/Bernoulli.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
h_odd : Odd n
hlt : 1 < n
B : ℚ⟦X⟧ := PowerSeries.mk fun n => bernoulli' n / ↑n !
h : B * (exp ℚ - 1) = X * exp ℚ
\u22a2 evalNegHom B * (1 - exp ℚ) = -(X * 1)
-- Mathlib/NumberTheory/Bernoulli.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
h_odd : Odd n
hlt : 1 < n
B : ℚ⟦X⟧ := PowerSeries.mk fun n => bernoulli' n / ↑n !
h : B * (exp ℚ - 1) = X * exp ℚ
\u22a2 evalNegHom (B * (exp ℚ - 1)) * exp ℚ = evalNegHom (X * exp ℚ) * exp ℚ
-- Mathlib/Order/Interval/Set/Basic.lean
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
a a₁ a₂ b b₁ b₂ c d : α
h : min c d < b
\u22a2 Iio b ∪ Ioo c d = Iio (max b d)
-- Mathlib/Order/Interval/Set/Basic.lean
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
a a₁ a₂ b b₁ b₂ c d : α
hcd : c ≤ d
h : c < b
\u22a2 Iio b ∪ Ioo c d = Iio (max b d)
-- Mathlib/Order/Interval/Set/Basic.lean
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
a a₁ a₂ b b₁ b₂ c d : α
hcd : d ≤ c
h : d < b
\u22a2 Iio b ∪ Ioo c d = Iio (max b d)
-- Mathlib/MeasureTheory/Measure/Hausdorff.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
K : ℝ≥0
f : X → Y
s t : Set X
h : LipschitzOnWith K f s
d : ℝ
hd : 0 ≤ d
\u22a2 μH[d] (f '' s) ≤ ↑K ^ d * μH[d] s
-- Mathlib/Order/CompleteLattice.lean
α : Type u_1
β : Type u_2
β₂ : Type u_3
γ : Type u_4
ι : Sort u_5
ι' : Sort u_6
κ : ι → Sort u_7
κ' : ι' → Sort u_8
inst✝ : CompleteLattice α
f✝ g s t : ι → α
a b : α
f : Bool → α
\u22a2 ⨆ b, f b = f true ⊔ f false
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝² : Category.{v', u'} J
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimitsOfShape J TopCat
F : J ⥤ PresheafedSpace C
j : J
\u22a2 { obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.map (𝟙 j) = 𝟙 ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝² : Category.{v', u'} J
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimitsOfShape J TopCat
F : J ⥤ PresheafedSpace C
j : J
\u22a2 (opEquiv ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j)) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.map (𝟙 j)) = (opEquiv ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j)) (𝟙 ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j))
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝² : Category.{v', u'} J
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimitsOfShape J TopCat
F : J ⥤ PresheafedSpace C
j : J
U : (Opens ↑(colimit (F ⋙ forget C)))ᵒᵖ
\u22a2 ((opEquiv ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j)) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.map (𝟙 j))).app U = ((opEquiv ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j)) (𝟙 ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j))).app U
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝² : Category.{v', u'} J
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimitsOfShape J TopCat
F : J ⥤ PresheafedSpace C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
\u22a2 { obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.map (f ≫ g) = { obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.map f ≫ { obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.map g
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝² : Category.{v', u'} J
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimitsOfShape J TopCat
F : J ⥤ PresheafedSpace C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
\u22a2 (opEquiv ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j₁) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j₃)) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.map (f ≫ g)) = (opEquiv ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j₁) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j₃)) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.map f ≫ { obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.map g)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝² : Category.{v', u'} J
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimitsOfShape J TopCat
F : J ⥤ PresheafedSpace C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
U : (Opens ↑(colimit (F ⋙ forget C)))ᵒᵖ
\u22a2 ((opEquiv ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j₁) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j₃)) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.map (f ≫ g))).app U = ((opEquiv ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j₁) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.obj j₃)) ({ obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.map f ≫ { obj := fun j => op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf), map := fun {j j'} f => ((pushforward C (colimit.ι (F ⋙ forget C) j')).map (F.map f).c ≫ (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j') (F.obj j).presheaf).inv ≫ (pushforwardEq ⋯ (F.obj j).presheaf).hom).op }.map g)).app U
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝² : Category.{v', u'} J
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimitsOfShape J TopCat
F : J ⥤ PresheafedSpace C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
U : (Opens ↑(colimit (F ⋙ forget C)))ᵒᵖ
\u22a2 (F.map (f ≫ g)).c.app (op ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U))) ≫ 𝟙 ((F.obj j₁).presheaf.obj (op ((Opens.map (F.map (f ≫ g)).base).obj ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U))))) ≫ (pushforwardEq ⋯ (F.obj j₁).presheaf).hom.app U = ((F.map g).c.app (op ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U))) ≫ 𝟙 ((F.obj j₂).presheaf.obj (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U))))) ≫ (pushforwardEq ⋯ (F.obj j₂).presheaf).hom.app U) ≫ (F.map f).c.app (op ((Opens.map (colimit.ι (F ⋙ forget C) j₂)).obj (unop U))) ≫ 𝟙 ((F.obj j₁).presheaf.obj (op ((Opens.map (F.map f).base).obj ((Opens.map (colimit.ι (F ⋙ forget C) j₂)).obj (unop U))))) ≫ (pushforwardEq ⋯ (F.obj j₁).presheaf).hom.app U
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝² : Category.{v', u'} J
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimitsOfShape J TopCat
F : J ⥤ PresheafedSpace C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
U : (Opens ↑(colimit (F ⋙ forget C)))ᵒᵖ
this : op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U))) = op ((Opens.map (colimit.ι (F ⋙ forget C) j₂)).obj (unop U))
\u22a2 (F.map (f ≫ g)).c.app (op ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U))) ≫ 𝟙 ((F.obj j₁).presheaf.obj (op ((Opens.map (F.map (f ≫ g)).base).obj ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U))))) ≫ (pushforwardEq ⋯ (F.obj j₁).presheaf).hom.app U = ((F.map g).c.app (op ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U))) ≫ 𝟙 ((F.obj j₂).presheaf.obj (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U))))) ≫ (pushforwardEq ⋯ (F.obj j₂).presheaf).hom.app U) ≫ (F.map f).c.app (op ((Opens.map (colimit.ι (F ⋙ forget C) j₂)).obj (unop U))) ≫ 𝟙 ((F.obj j₁).presheaf.obj (op ((Opens.map (F.map f).base).obj ((Opens.map (colimit.ι (F ⋙ forget C) j₂)).obj (unop U))))) ≫ (pushforwardEq ⋯ (F.obj j₁).presheaf).hom.app U
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
J : Type u'
inst✝² : Category.{v', u'} J
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimitsOfShape J TopCat
F : J ⥤ PresheafedSpace C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
U : (Opens ↑(colimit (F ⋙ forget C)))ᵒᵖ
this : op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U))) = op ((Opens.map (colimit.ι (F ⋙ forget C) j₂)).obj (unop U))
\u22a2 (F.map g).c.app (op ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U))) ≫ (F.map f).c.app (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U)))) ≫ (F.obj j₁).presheaf.map (eqToHom ⋯) = (F.map g).c.app (op ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U))) ≫ (F.map f).c.app (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.ι (F ⋙ forget C) j₃)).obj (unop U)))) ≫ (F.obj j₁).presheaf.map ((Opens.map (F.map f).base).map (eqToHom ⋯)).op ≫ (F.obj j₁).presheaf.map (eqToHom ⋯)
-- 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)
\u22a2 ⟪(ContinuousMap.toLp 2 μ 𝕜) f, (ContinuousMap.toLp 2 μ 𝕜) g⟫_𝕜 = ∫ (x : α), (starRingEnd 𝕜) (f x) * g x ∂μ
-- 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)
\u22a2 (fun a => ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a⟫_𝕜) =ᶠ[ae μ] fun a => (starRingEnd 𝕜) (f a) * g a
-- 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
\u22a2 (fun a => ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a⟫_𝕜) =ᶠ[ae μ] fun a => (starRingEnd 𝕜) (f a) * g a
-- 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
\u22a2 (fun a => ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a⟫_𝕜) =ᶠ[ae μ] fun a => (starRingEnd 𝕜) (f a) * g a
-- 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✝
\u22a2 ⟪↑↑((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
\u22a2 (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
\u22a2 ((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
\u22a2 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
\u22a2 (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
\u22a2 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
\u22a2 ((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)