mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
inst✝ : HasPullbacks C
i j : E.I₀ × F.I₀
W : C
p₁ : W ⟶ pullback (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
Y : C
f : Y ⟶ W
p : W ⟶ pullback ((E.inter F).f i) ((E.inter F).f j) := pullback.lift p₁ p₂ w
x✝ : ((E.inter F).sieve₁ p₁ p₂).arrows f
k : (E.inter F).I₁ i j
a : Y ⟶ (E.inter F).Y k
h₁ : f ≫ p₁ = a ≫ (E.inter F).p₁ k
h₂ : f ≫ p₂ = a ≫ (E.inter F).p₂ k
⊢ (pullback.lift f a ⋯ ≫ pullback.fst p ((E.inter F).toPullback k)) ≫ p₂ ≫ pullback.snd (E.f j.1) (F.f j.2) =
    (a ≫ pullback.snd (E.p₁ k.1 ≫ E.f i.1) (F.p₁ k.2 ≫ F.f i.2)) ≫ F.p₂ k.2

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
K : GrothendieckTopology D
A : Type u_4
inst✝² : Category.{v_4, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
α : ℱ ⟶ ℱ'.obj
⊢ sheafHom (G.op.whiskerLeft α) = α

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
K : GrothendieckTopology D
A : Type u_4
inst✝² : Category.{v_4, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
f : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj
𝒢' : Sheaf K A
g : ℱ' ⟶ 𝒢'
⊢ restrictHomEquivHom (f ≫ G.op.whiskerLeft g.hom) = restrictHomEquivHom f ≫ g.hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
K : GrothendieckTopology D
A : Type u_4
inst✝² : Category.{v_4, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
𝒢 : Dᵒᵖ ⥤ A
f : 𝒢 ⟶ ℱ
g : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj
⊢ restrictHomEquivHom (G.op.whiskerLeft f ≫ g) = f ≫ restrictHomEquivHom g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit

Goal state before:
pp: J : Type u₁
K : Type u₂
inst✝⁴ : SmallCategory J
inst✝³ : Category.{v₂, u₂} K
inst✝² : Small.{v, u₂} K
inst✝¹ : FinCategory J
F : J × K ⥤ Type v
inst✝ : IsFiltered K
⊢ Function.Surjective ⇑(ConcreteCategory.hom (colimitLimitToLimitColimit F))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
K : GrothendieckTopology D
A : Type u_4
inst✝² : Category.{v_4, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ ℱ' : Sheaf K A
α : ℱ ⟶ ℱ'
i : IsIso (G.op.whiskerLeft α.hom)
⊢ IsIso α

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝² : G.IsCoverDense K
inst✝¹ : G.IsLocallyFull K
inst✝ : G.IsLocallyFaithful K
ℱ : Sheaf K (Type u_5)
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.obj) T
hx : x.Compatible
Y₁ Y₂ : C
X : D
f₁ : X ⟶ G.obj Y₁
f₂ : X ⟶ G.obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
eq : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂
W : C
i : G.obj W ⟶ X
V₁ : C
iVW : V₁ ⟶ W
iV₁Y₁ : V₁ ⟶ Y₁
e₁ : G.map iV₁Y₁ = G.map iVW ≫ i ≫ f₁
V₂ : C
iV₂V₁ : V₂ ⟶ V₁
iV₂Y₂ : V₂ ⟶ Y₂
e₂ : G.map iV₂Y₂ = G.map iV₂V₁ ≫ G.map iVW ≫ i ≫ f₂
⊢ G.map (iV₂V₁ ≫ iV₁Y₁ ≫ g₁) = G.map (iV₂Y₂ ≫ g₂)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝² : G.IsCoverDense K
inst✝¹ : G.IsLocallyFull K
inst✝ : G.IsLocallyFaithful K
⊢ CompatiblePreserving K G

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁵ : Category.{v_2, u_2} D
E : Type u_3
inst✝⁴ : Category.{v_3, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
A : Type u_4
inst✝³ : Category.{v_4, u_4} A
G : C ⥤ D
inst✝² : G.IsCoverDense K
inst✝¹ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
inst✝ : G.IsContinuous J K
⊢ ∀ {X Y : Sheaf K A}, Function.Injective (G.sheafPushforwardContinuous A J K).map

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type w
inst✝² : Category.{w', w} A
inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst✝ : IsDenseSubsite J K G
Y : Sheaf J A
U : C
X : A
f : (yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (G.imageSieve g).arrows f → (X ⟶ Z)
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
W₁ : C
g₁ : G.obj W₁ ⟶ G.obj U
W₂ : C
i : W₂ ⟶ W₁
hi : (StructuredArrow.mk (op g₁)).hom ≫ G.op.map (op i) = (StructuredArrow.mk (op (G.map i ≫ g₁))).hom
I : GrothendieckTopology.Cover.Arrow ⟨G.imageSieve (G.map i ≫ g₁), ⋯⟩
⊢ G.map (l I.Y W₂ U I.f (G.map i ≫ g₁) ⋯) = G.map (I.f ≫ i) ≫ g₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : C ⥤ D
A : Type u_4
inst✝¹ : Category.{v_4, u_4} A
inst✝ : IsDenseSubsite J K G
F : Sheaf J A
X Y : C
f₁ f₂ : X ⟶ Y
h : G.map f₁ = G.map f₂
⊢ ∀ (I : GrothendieckTopology.Cover.Arrow ⟨Sieve.equalizer f₁ f₂, ⋯⟩),
    F.obj.map f₁.op ≫ F.obj.map I.f.op = F.obj.map f₂.op ≫ F.obj.map I.f.op

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type w
inst✝² : Category.{w', w} A
inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst✝ : IsDenseSubsite J K G
Y : Sheaf J A
U : C
X : A
f : (yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (G.imageSieve g).arrows f → (X ⟶ Z)
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
W₁ : C
g₁ : G.obj W₁ ⟶ G.obj U
W₂ : C
i : W₂ ⟶ W₁
hi : (StructuredArrow.mk (op g₁)).hom ≫ G.op.map (op i) = (StructuredArrow.mk (op (G.map i ≫ g₁))).hom
I : GrothendieckTopology.Cover.Arrow ⟨G.imageSieve (G.map i ≫ g₁), ⋯⟩
I' : GrothendieckTopology.Cover.Arrow ⟨G.imageSieve g₁, ⋯⟩ := { Y := I.Y, f := I.f ≫ i, hf := ⋯ }
⊢ G.map (l I.Y W₂ U I.f (G.map i ≫ g₁) ⋯) = G.map (l I.Y W₁ U I'.f g₁ ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
G : C ⥤ D
A : Type u_4
inst✝¹ : Category.{v_4, u_4} A
inst✝ : IsDenseSubsite J K G
F : Sheaf J A
X Y : C
f : G.obj X ⟶ G.obj Y
⊢ ∀ ⦃I₁ I₂ : GrothendieckTopology.Cover.Arrow ⟨G.imageSieve f, ⋯⟩⦄ (r : I₁.Relation I₂),
    (match I₁ with
        | { Y := W₀, f := a, hf := ha } => F.obj.map (Exists.choose ha).op) ≫
        F.obj.map r.g₁.op =
      (match I₂ with
        | { Y := W₀, f := a, hf := ha } => F.obj.map (Exists.choose ha).op) ≫
        F.obj.map r.g₂.op

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type w
inst✝² : Category.{w', w} A
inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst✝ : IsDenseSubsite J K G
Y : Sheaf J A
U : C
X : A
f : (yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (G.imageSieve g).arrows f → (X ⟶ Z)
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
⊢ Limits.Cone (StructuredArrow.proj (op (G.obj U)) G.op ⋙ Y.obj)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : C ⥤ D
A : Type u_4
inst✝¹ : Category.{v_4, u_4} A
inst✝ : IsDenseSubsite J K G
F : Sheaf J A
X Y Z : C
f : G.obj X ⟶ G.obj Y
p : Z ⟶ X
g : Z ⟶ Y
fac : G.map p ≫ f = G.map g
⊢ ∀ (I : GrothendieckTopology.Cover.Arrow ⟨Sieve.pullback p (G.imageSieve f), ⋯⟩),
    (mapPreimage K G F f ≫ F.obj.map p.op) ≫ F.obj.map I.f.op = F.obj.map g.op ≫ F.obj.map I.f.op

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : C ⥤ D
A : Type u_4
inst✝¹ : Category.{v_4, u_4} A
inst✝ : IsDenseSubsite J K G
F : Sheaf J A
X Y : C
f : G.obj X ⟶ G.obj Y
g : X ⟶ Y
h : G.map g = f
⊢ G.map (𝟙 X) ≫ f = G.map g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : C ⥤ D
A : Type u_4
inst✝¹ : Category.{v_4, u_4} A
inst✝ : IsDenseSubsite J K G
F : Sheaf J A
X Y : C
f : G.obj X ⟶ G.obj Y
g : X ⟶ Y
h : G.map g = f
⊢ mapPreimage K G F f = F.obj.map g.op

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : C ⥤ D
A : Type u_4
inst✝¹ : Category.{v_4, u_4} A
inst✝ : IsDenseSubsite J K G
F : Sheaf J A
X : C
⊢ mapPreimage K G F (𝟙 (G.obj X)) = 𝟙 (F.obj.obj (op X))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
inst✝ : HasPullbacks C
i j : E.I₀ × F.I₀
W : C
p₁ : W ⟶ pullback (E.f i.1) (F.f i.2)
p₂ : W ⟶ pullback (E.f j.1) (F.f j.2)
w : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1) (F.f j.2) ≫ E.f j.1
⊢ (E.inter F).sieve₁ p₁ p₂ =
    Sieve.bind (E.sieve₁ (p₁ ≫ pullback.fst (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.fst (E.f j.1) (F.f j.2))).arrows
      fun x f x_1 ↦
      Sieve.pullback f (F.sieve₁ (p₁ ≫ pullback.snd (E.f i.1) (F.f i.2)) (p₂ ≫ pullback.snd (E.f j.1) (F.f j.2)))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : C ⥤ D
A : Type u_4
inst✝¹ : Category.{v_4, u_4} A
inst✝ : IsDenseSubsite J K G
F : Sheaf J A
X Y Z : C
f : G.obj X ⟶ G.obj Y
g : G.obj Y ⟶ G.obj Z
⊢ ∀ (I : GrothendieckTopology.Cover.Arrow ⟨G.imageSieve f, ⋯⟩),
    mapPreimage K G F (f ≫ g) ≫ F.obj.map I.f.op = (mapPreimage K G F g ≫ mapPreimage K G F f) ≫ F.obj.map I.f.op

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E✝ E : PreOneHypercover S
F : PreOneHypercover S
G : PreOneHypercover S
f : E.Hom F
g : F.Hom G
⊢ ∀ {i j : E.I₀} (k : E.I₁ i j), (f.h₁ k ≫ g.h₁ (f.s₁ k)) ≫ G.p₁ ((g.s₁ ∘ f.s₁) k) = E.p₁ k ≫ (f.comp g.toHom).h₀ i

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : C ⥤ D
A : Type u_4
inst✝¹ : Category.{v_4, u_4} A
inst✝ : IsDenseSubsite J K G
F : Sheaf J A
X Y Z : C
f : G.obj X ⟶ G.obj Y
g : Z ⟶ X
⊢ mapPreimage K G F f ≫ F.obj.map g.op = mapPreimage K G F (G.map g ≫ f)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E✝ E : PreOneHypercover S
F : PreOneHypercover S
G : PreOneHypercover S
f : E.Hom F
g : F.Hom G
⊢ ∀ {i j : E.I₀} (k : E.I₁ i j), (f.h₁ k ≫ g.h₁ (f.s₁ k)) ≫ G.p₂ ((g.s₁ ∘ f.s₁) k) = E.p₂ k ≫ (f.comp g.toHom).h₀ j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E✝ E : PreOneHypercover S
F : PreOneHypercover S
G : PreOneHypercover S
f : E.Hom F
P : Cᵒᵖ ⥤ A
c : Multifork (E.multicospanIndex P)
hc : IsLimit c
d : Multifork (F.multicospanIndex P)
⊢ ∀ (b : E.multicospanShape.R),
    (d.ι (f.s₀ (E.multicospanShape.fst b)) ≫ P.map (f.h₀ (E.multicospanShape.fst b)).op) ≫
        (E.multicospanIndex P).fst b =
      (d.ι (f.s₀ (E.multicospanShape.snd b)) ≫ P.map (f.h₀ (E.multicospanShape.snd b)).op) ≫
        (E.multicospanIndex P).snd b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E : PreOneHypercover S
F : PreOneHypercover S
f : E.Hom F
P : Cᵒᵖ ⥤ A
c : Multifork (E.multicospanIndex P)
hc : IsLimit c
d : Multifork (F.multicospanIndex P)
a : E.I₀
⊢ f.mapMultiforkOfIsLimit P hc d ≫ c.ι a = d.ι (f.s₀ a) ≫ P.map (f.h₀ a).op

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E : PreOneHypercover S
P : Cᵒᵖ ⥤ A
c : Multifork (E.multicospanIndex P)
hc : IsLimit c
d : Multifork (E.multicospanIndex P)
⊢ (id E).mapMultiforkOfIsLimit P hc d = Multifork.IsLimit.lift hc d.ι ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
G : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
A : Type w
inst✝² : Category.{w', w} A
inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst✝ : IsDenseSubsite J K G
Y : Sheaf J A
U : C
X : A
f : (yoneda.obj (((𝟭 (Cᵒᵖ ⥤ A)).obj Y.obj).obj (op U))).obj (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (G.imageSieve g).arrows f → (X ⟶ Z)
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
c : Limits.Cone (StructuredArrow.proj (op (G.obj U)) G.op ⋙ Y.obj) :=
  { pt := X,
    π :=
      {
        app := fun g ↦
          ⋯.amalgamate ⟨G.imageSieve g.hom.unop, ⋯⟩ (fun I ↦ f ≫ Y.obj.map (l I.Y (unop g.2) U I.f g.hom.unop ⋯).op) ⋯,
        naturality := ⋯ } }
this :
  (G.op.isPointwiseRightKanExtensionRanCounit Y.obj (op (G.obj U))).lift c ≫ (G.op.ranCounit.app Y.obj).app (op U) =
    c.π.app (StructuredArrow.mk (𝟙 (op (G.obj U))))
I : GrothendieckTopology.Cover.Arrow ⟨G.imageSieve (𝟙 (G.obj U)), ⋯⟩
⊢ G.map (l I.Y U U I.f (𝟙 (G.obj U)) ⋯) = G.map I.f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁴ : Category.{v_2, u_2} D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : C ⥤ D
A : Type u_4
inst✝³ : Category.{v_4, u_4} A
inst✝² : IsDenseSubsite J K G
inst✝¹ : (G.sheafPushforwardContinuous A J K).IsEquivalence
inst✝ : HasWeakSheafify J A
P₁ P₂ : Dᵒᵖ ⥤ A
f : P₁ ⟶ P₂
Q : Sheaf K A
g : (sheafifyOfIsEquivalence J K G A).obj P₂ ⟶ Q
⊢ (sheafifyHomEquivOfIsEquivalence J K G) ((sheafifyOfIsEquivalence J K G A).map f ≫ g) =
    f ≫ (sheafifyHomEquivOfIsEquivalence J K G) g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S : C
E : PreOneHypercover S
F : PreOneHypercover S
G : PreOneHypercover S
f : E.Hom F
P : Cᵒᵖ ⥤ A
c : Multifork (E.multicospanIndex P)
hc : IsLimit c
d : Multifork (F.multicospanIndex P)
g : F.Hom G
t : Multifork (G.multicospanIndex P)
hd : IsLimit d
⊢ (f.comp g).mapMultiforkOfIsLimit P hc t = g.mapMultiforkOfIsLimit P hd t ≫ f.mapMultiforkOfIsLimit P hc d

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreOneHypercover S
i j : E.I₀
⊢ congrIndexOneOfEq ⋯ ⋯ = Equiv.refl (E.I₁ i j)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreOneHypercover S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
i'' j'' : E.I₀
hii'' : i' = i''
hjj'' : j' = j''
k : E.I₁ i j
⊢ (congrIndexOneOfEq hii'' hjj'') ((congrIndexOneOfEq hii' hjj') k) = (congrIndexOneOfEq ⋯ ⋯) k

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
u₀ : E.I₀ → F.I₀
u₁ : ⦃i j : E.I₀⦄ → E.I₁ i j → F.I₁ (u₀ i) (u₀ j)
k : E.I₁ i j
⊢ u₁ ((congrIndexOneOfEq hii' hjj') k) = (congrIndexOneOfEq ⋯ ⋯) (u₁ k)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝¹ E✝ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E : PreOneHypercover S
F : PreOneHypercover S
i✝ i' j✝ j' : E.I₀
hii' : i✝ = i'
hjj' : j✝ = j'
u₀ v₀ : E.I₀ → F.I₀
u₁ : ⦃i j : E.I₀⦄ → E.I₁ i j → F.I₁ (u₀ i) (u₀ j)
v₁ : ⦃i j : E.I₀⦄ → E.I₁ i j → F.I₁ (v₀ i) (v₀ j)
h₀ : u₀ = v₀
i j : E.I₀
k : E.I₁ i j
⊢ v₀ i = u₀ i

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝¹ E✝ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E : PreOneHypercover S
F : PreOneHypercover S
i✝ i' j✝ j' : E.I₀
hii' : i✝ = i'
hjj' : j✝ = j'
u₀ v₀ : E.I₀ → F.I₀
u₁ : ⦃i j : E.I₀⦄ → E.I₁ i j → F.I₁ (u₀ i) (u₀ j)
v₁ : ⦃i j : E.I₀⦄ → E.I₁ i j → F.I₁ (v₀ i) (v₀ j)
h₀ : u₀ = v₀
i j : E.I₀
k : E.I₁ i j
⊢ v₀ j = u₀ j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
u₀ v₀ : E.I₀ → F.I₀
u₁ : ⦃i j : E.I₀⦄ → E.I₁ i j → F.I₁ (u₀ i) (u₀ j)
v₁ : ⦃i j : E.I₀⦄ → E.I₁ i j → F.I₁ (v₀ i) (v₀ j)
h₀ : u₀ = v₀
h₁ : ∀ (i j : E.I₀) (k : E.I₁ i j), u₁ k = (congrIndexOneOfEq ⋯ ⋯) (v₁ k)
i j : E.I₀
k : E.I₁ i j
⊢ (congrIndexOneOfEq ⋯ ⋯) (v₁ k) = u₁ k

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreOneHypercover S
a b : E.I₁'
left : a.fst.1 = b.fst.1
right : a.fst.2 = b.fst.2
h : (congrIndexOneOfEq left right) a.snd = b.snd
⊢ a = b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝² E✝¹ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E✝ : PreOneHypercover S
F : PreOneHypercover S
i✝ i'✝ j✝ j'✝ : E✝.I₀
hii'✝ : i✝ = i'✝
hjj'✝ : j✝ = j'✝
E : PreOneHypercover S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ E.Y ((congrIndexOneOfEq hii' hjj') k) = E.Y k

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreOneHypercover S
i j : E.I₀
k : E.I₁ i j
⊢ congrIndexOneOfEqIso ⋯ ⋯ k = Iso.refl (E.Y ((congrIndexOneOfEq ⋯ ⋯) k))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝¹ E✝ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E : PreOneHypercover S
F : PreOneHypercover S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ E.X i' = E.X i

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreOneHypercover S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ (congrIndexOneOfEqIso hii' hjj' k).hom ≫ E.p₁ k = E.p₁ ((congrIndexOneOfEq hii' hjj') k) ≫ eqToHom ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁴ : Category.{v_2, u_2} D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : C ⥤ D
A : Type u_4
inst✝³ : Category.{v_4, u_4} A
inst✝² : IsDenseSubsite J K G
inst✝¹ : (G.sheafPushforwardContinuous A J K).IsEquivalence
inst✝ : HasWeakSheafify J A
P : Dᵒᵖ ⥤ A
Q₁ Q₂ : Sheaf K A
f : (sheafifyOfIsEquivalence J K G A).obj P ⟶ Q₁
g : Q₁ ⟶ Q₂
⊢ (sheafifyHomEquivOfIsEquivalence J K G) (f ≫ g) = (sheafifyHomEquivOfIsEquivalence J K G) f ≫ g.hom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝¹ E✝ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E : PreOneHypercover S
F : PreOneHypercover S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ E.X i = E.X i'

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreOneHypercover S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ (congrIndexOneOfEqIso hii' hjj' k).inv ≫ E.p₁ ((congrIndexOneOfEq hii' hjj') k) = E.p₁ k ≫ eqToHom ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁵ : Category.{v_2, u_2} D
E : Type u_3
inst✝⁴ : Category.{v_3, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
G : C ⥤ D
A : Type u_4
inst✝³ : Category.{v_4, u_4} A
inst✝² : IsDenseSubsite J K G
F : Sheaf J A
inst✝¹ : (G.sheafPushforwardContinuous A J K).IsEquivalence
inst✝ : HasWeakSheafify J A
P₁ P₂ : Dᵒᵖ ⥤ A
Q : Sheaf K A
f : P₁ ⟶ P₂
g : P₂ ⟶ (sheafToPresheaf K A).obj Q
⊢ (sheafifyHomEquivOfIsEquivalence J K G) ((sheafifyHomEquivOfIsEquivalence J K G).symm (f ≫ g)) =
    (sheafifyHomEquivOfIsEquivalence J K G)
      ((sheafifyOfIsEquivalence J K G A).map f ≫ (sheafifyHomEquivOfIsEquivalence J K G).symm g)

Tactic:
by

Goal state after: