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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
G : C ⥤ D
A : Type w
inst✝³ : Category.{w', w} A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝² : G.IsCocontinuous J K
inst✝¹ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F
inst✝ : G.IsContinuous J K
F : Sheaf K A
H : Sheaf J A
f : (G.sheafPushforwardContinuous A J K).obj F ⟶ H
⊢ (G.op.ranAdjunction A).unit.app F.obj ≫
      𝟙 (G.op.ran.obj (G.op ⋙ F.obj)) ≫ G.op.ran.map f.hom ≫ 𝟙 ((G.sheafPushforwardCocontinuous A J K).obj H).obj =
    ((G.op.ranAdjunction A).homEquiv F.obj H.obj) f.hom

Tactic:
Category.comp_id,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
G : C ⥤ D
A : Type w
inst✝³ : Category.{w', w} A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝² : G.IsCocontinuous J K
inst✝¹ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F
inst✝ : G.IsContinuous J K
F : Sheaf K A
H : Sheaf J A
f : (G.sheafPushforwardContinuous A J K).obj F ⟶ H
⊢ (G.op.ranAdjunction A).unit.app F.obj ≫ 𝟙 (G.op.ran.obj (G.op ⋙ F.obj)) ≫ G.op.ran.map f.hom =
    ((G.op.ranAdjunction A).homEquiv F.obj H.obj) f.hom
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
D : Type u_2
inst✝¹ : Category.{v_2, u_2} D
K : GrothendieckTopology D
G : C ⥤ D
ℱ : Dᵒᵖ ⥤ Type v
ℱ' : Sheaf K (Type v)
α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj
inst✝ : G.IsLocallyFull K
X Y : C
i : G.obj X ⟶ G.obj Y
x : (fun X ↦ X) ((G.op ⋙ ℱ).obj (op Y))
⊢ (ConcreteCategory.hom (ℱ'.obj.map i.op)) ((ConcreteCategory.hom (α.app (op Y))) x) =
    (ConcreteCategory.hom (α.app (op X))) ((ConcreteCategory.hom (ℱ.map i.op)) x)

Tactic:
by
  have {X Y} (i : X ⟶ Y) (x) :
      ℱ'.1.map (G.map i).op (α.app _ x) = α.app _ (ℱ.map (G.map i).op x) := by
    exact ConcreteCategory.congr_hom (α.naturality i.op).symm x
  refine IsLocallyFull.ext G _ i fun V iVX iVY e ↦ ?_
  simp only [← Functor.map_comp_apply, ← op_comp, ← e, this]

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
Y : C
f : Y ⟶ W
⊢ ((E.inter F).sieve₁ p₁ p₂).arrows f ↔
    (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)))).arrows
      f

Tactic:
let p : W ⟶ pullback ((inter E F).f i) ((inter E F).f j) :=
    pullback.lift p₁ p₂ w

Goal state after:
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
⊢ ((E.inter F).sieve₁ p₁ p₂).arrows f ↔
    (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)))).arrows
      f
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 : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj
X : Cᵒᵖ
⊢ yoneda.map ((G.op.whiskerLeft (sheafHom α)).app X) = yoneda.map (α.app X)

Tactic:
ext U

Goal state after:
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 : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj
X : Cᵒᵖ
U : Aᵒᵖ
x✝ : (yoneda.obj ((G.op ⋙ ℱ).obj X)).obj U
⊢ (ConcreteCategory.hom ((yoneda.map ((G.op.whiskerLeft (sheafHom α)).app X)).app U)).toFun x✝ =
    (ConcreteCategory.hom ((yoneda.map (α.app X)).app U)).toFun x✝
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
X : Dᵒᵖ
⊢ yoneda.map ((sheafHom (G.op.whiskerLeft α)).app X) = yoneda.map (α.app X)

Tactic:
ext U

Goal state after:
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
X : Dᵒᵖ
U : Aᵒᵖ
x✝ : (yoneda.obj (ℱ.obj X)).obj U
⊢ (ConcreteCategory.hom ((yoneda.map ((sheafHom (G.op.whiskerLeft α)).app X)).app U)).toFun x✝ =
    (ConcreteCategory.hom ((yoneda.map (α.app X)).app U)).toFun x✝
Module: Mathlib.CategoryTheory.Sites.InducedTopology

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
F : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : Sheaf K (Type (max u₁ v₁ u₂ v₂))
⊢ Presieve.IsSheaf (F.inducedTopology K) (F.op ⋙ G.obj)

Tactic:
apply Sheaf.sheaf_for_finestTopology

Goal state after:
C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
F : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : Sheaf K (Type (max u₁ v₁ u₂ v₂))
⊢ F.op ⋙ G.obj ∈ Set.range fun G ↦ F.op ⋙ G.obj
Module: Mathlib.CategoryTheory.Sites.InducedTopology

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
K : GrothendieckTopology D
inst✝ : LocallySmall.{max u₁ v₁ u₂ v₂, v₁, u₁} C
X : C
S : Sieve X
G : (Cᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)) ⥤ Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
adj : G ⊣ (whiskeringLeft Cᵒᵖ Dᵒᵖ (Type (max u₁ v₁ u₂ v₂))).obj F.op
⊢ S ∈ (F.inducedTopology K) X ↔
    ∀ ⦃Y : C⦄ (f : Y ⟶ X), K.W (G.map (Sieve.shrinkFunctor.{max u₁ v₁ u₂ v₂, v₁, u₁} (Sieve.pullback f S)).ι)

Tactic:
by
  refine ⟨?_, ?_⟩
  · intro hS Y f
    apply Functor.W_map_of_adjunction_of_isContinuous (F.inducedTopology K) K _ G adj
    refine Sieve.W_shrinkFunctor_ι_of_mem (F.inducedTopology K) (Sieve.pullback f S) ?_
    exact GrothendieckTopology.pullback_stable (F.inducedTopology K) f hS
  · intro H
    apply Sheaf.mem_finestTopology_of_forall_isSheafFor
    rintro - ⟨P, rfl⟩ Y f
    dsimp
    rw [Presieve.isSheafFor_iff_bijective_shrinkFunctor_ι_comp]
    exact (adj.map_comp_bijective_iff _ _).mp (H f _ P.property)

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

Goal state before:
pp: case pullback
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
G : C ⥤ D
K : GrothendieckTopology D
inst✝² : G.LocallyCoverDense K
inst✝¹ : G.IsLocallyFull K
inst✝ : G.IsLocallyFaithful K
U✝ : C
S✝ : Sieve U✝
X : C
S : Sieve X
a✝ : (Precoverage.comap G K.toPrecoverage).Saturate X S
Y : C
f : Y ⟶ X
ih : Sieve.functorPushforward G S ∈ K (G.obj X)
U : C
iUY : U ⟶ Y
W : C
iWX : W ⟶ X
iUW : G.obj U ⟶ G.obj W
hiWX : S.arrows iWX
e₁ : G.map iUY ≫ G.map f = iUW ≫ G.map iWX
U₁ : C
iU₁U : U₁ ⟶ U
iU₁W : U₁ ⟶ W
e₂ : G.map iU₁W = G.map iU₁U ≫ iUW
⊢ Sieve.pullback (G.map iU₁U) (Sieve.pullback (G.map iUY) (Sieve.functorPushforward G (Sieve.pullback f S))) ∈
    K (G.obj U₁)

Tactic:
apply K.superset_covering ?_ (G.functorPushforward_equalizer_mem _
        (iU₁U ≫ iUY ≫ f) (iU₁W ≫ iWX) (by simp [e₁, e₂]))

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
G : C ⥤ D
K : GrothendieckTopology D
inst✝² : G.LocallyCoverDense K
inst✝¹ : G.IsLocallyFull K
inst✝ : G.IsLocallyFaithful K
U✝ : C
S✝ : Sieve U✝
X : C
S : Sieve X
a✝ : (Precoverage.comap G K.toPrecoverage).Saturate X S
Y : C
f : Y ⟶ X
ih : Sieve.functorPushforward G S ∈ K (G.obj X)
U : C
iUY : U ⟶ Y
W : C
iWX : W ⟶ X
iUW : G.obj U ⟶ G.obj W
hiWX : S.arrows iWX
e₁ : G.map iUY ≫ G.map f = iUW ≫ G.map iWX
U₁ : C
iU₁U : U₁ ⟶ U
iU₁W : U₁ ⟶ W
e₂ : G.map iU₁W = G.map iU₁U ≫ iUW
⊢ Sieve.functorPushforward G (Sieve.equalizer (iU₁U ≫ iUY ≫ f) (iU₁W ≫ iWX)) ≤
    Sieve.pullback (G.map iU₁U) (Sieve.pullback (G.map iUY) (Sieve.functorPushforward G (Sieve.pullback f S)))
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 F : PreOneHypercover S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
h₁ : ⦃i j : E.I₀⦄ → (k : E.I₁ i j) → E.Y k ≅ F.Y (s₁ k)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f (s₀ i) = E.f i
w₁₁ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₁ (s₁ k) = E.p₁ k ≫ (h₀ i).hom
w₁₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₂ (s₁ k) = E.p₂ k ≫ (h₀ j).hom
i j : E.I₀
k : E.I₁ i j
⊢ (congrIndexOneOfEqIso ⋯ ⋯ k).inv ≫ E.p₁ ((congrIndexOneOfEq ⋯ ⋯) k) ≫ 𝟙 (E.X (s₀.symm (s₀ i))) = E.p₁ k ≫ eqToHom ⋯

pp: case w
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 F : PreOneHypercover S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
h₁ : ⦃i j : E.I₀⦄ → (k : E.I₁ i j) → E.Y k ≅ F.Y (s₁ k)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f (s₀ i) = E.f i
w₁₁ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₁ (s₁ k) = E.p₁ k ≫ (h₀ i).hom
w₁₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₂ (s₁ k) = E.p₂ k ≫ (h₀ j).hom
i j : E.I₀
k : E.I₁ i j
⊢ (congrIndexOneOfEq ⋯ ⋯) k = s₁.symm ((congrIndexOneOfEq ⋯ ⋯) (s₁ k))

Tactic:
Category.comp_id,

Goal state after:
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 F : PreOneHypercover S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
h₁ : ⦃i j : E.I₀⦄ → (k : E.I₁ i j) → E.Y k ≅ F.Y (s₁ k)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f (s₀ i) = E.f i
w₁₁ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₁ (s₁ k) = E.p₁ k ≫ (h₀ i).hom
w₁₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₂ (s₁ k) = E.p₂ k ≫ (h₀ j).hom
i j : E.I₀
k : E.I₁ i j
⊢ (congrIndexOneOfEqIso ⋯ ⋯ k).inv ≫ E.p₁ ((congrIndexOneOfEq ⋯ ⋯) k) = E.p₁ k ≫ eqToHom ⋯

case w
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 F : PreOneHypercover S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
h₁ : ⦃i j : E.I₀⦄ → (k : E.I₁ i j) → E.Y k ≅ F.Y (s₁ k)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f (s₀ i) = E.f i
w₁₁ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₁ (s₁ k) = E.p₁ k ≫ (h₀ i).hom
w₁₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₂ (s₁ k) = E.p₂ k ≫ (h₀ j).hom
i j : E.I₀
k : E.I₁ i j
⊢ (congrIndexOneOfEq ⋯ ⋯) k = s₁.symm ((congrIndexOneOfEq ⋯ ⋯) (s₁ k))
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 F : PreOneHypercover S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
h₁ : ⦃i j : E.I₀⦄ → (k : E.I₁ i j) → E.Y k ≅ F.Y (s₁ k)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f (s₀ i) = E.f i
w₁₁ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₁ (s₁ k) = E.p₁ k ≫ (h₀ i).hom
w₁₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₂ (s₁ k) = E.p₂ k ≫ (h₀ j).hom
i j : E.I₀
k : E.I₁ i j
⊢ (congrIndexOneOfEqIso ⋯ ⋯ k).inv ≫ E.p₂ ((congrIndexOneOfEq ⋯ ⋯) k) ≫ 𝟙 (E.X (s₀.symm (s₀ j))) = E.p₂ k ≫ eqToHom ⋯

pp: case w
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 F : PreOneHypercover S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
h₁ : ⦃i j : E.I₀⦄ → (k : E.I₁ i j) → E.Y k ≅ F.Y (s₁ k)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f (s₀ i) = E.f i
w₁₁ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₁ (s₁ k) = E.p₁ k ≫ (h₀ i).hom
w₁₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₂ (s₁ k) = E.p₂ k ≫ (h₀ j).hom
i j : E.I₀
k : E.I₁ i j
⊢ (congrIndexOneOfEq ⋯ ⋯) k = s₁.symm ((congrIndexOneOfEq ⋯ ⋯) (s₁ k))

Tactic:
Category.comp_id,

Goal state after:
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 F : PreOneHypercover S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
h₁ : ⦃i j : E.I₀⦄ → (k : E.I₁ i j) → E.Y k ≅ F.Y (s₁ k)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f (s₀ i) = E.f i
w₁₁ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₁ (s₁ k) = E.p₁ k ≫ (h₀ i).hom
w₁₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₂ (s₁ k) = E.p₂ k ≫ (h₀ j).hom
i j : E.I₀
k : E.I₁ i j
⊢ (congrIndexOneOfEqIso ⋯ ⋯ k).inv ≫ E.p₂ ((congrIndexOneOfEq ⋯ ⋯) k) = E.p₂ k ≫ eqToHom ⋯

case w
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 F : PreOneHypercover S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
h₁ : ⦃i j : E.I₀⦄ → (k : E.I₁ i j) → E.Y k ≅ F.Y (s₁ k)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f (s₀ i) = E.f i
w₁₁ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₁ (s₁ k) = E.p₁ k ≫ (h₀ i).hom
w₁₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), (h₁ k).hom ≫ F.p₂ (s₁ k) = E.p₂ k ≫ (h₀ j).hom
i j : E.I₀
k : E.I₁ i j
⊢ (congrIndexOneOfEq ⋯ ⋯) k = s₁.symm ((congrIndexOneOfEq ⋯ ⋯) (s₁ k))
Module: Mathlib.CategoryTheory.Limits.Shapes.Connected

Goal state before:
pp: J : Type u_1
⊢ ∀ {α : Type u_1} (F : WidePullbackShape J → α),
    (∀ {j₁ j₂ : WidePullbackShape J} (x : j₁ ⟶ j₂), F j₁ = F j₂) → ∀ (j j' : WidePullbackShape J), F j = F j'

Tactic:
intro α F H

Goal state after:
J α : Type u_1
F : WidePullbackShape J → α
H : ∀ {j₁ j₂ : WidePullbackShape J} (x : j₁ ⟶ j₂), F j₁ = F j₂
⊢ ∀ (j j' : WidePullbackShape J), F j = F j'
Module: Mathlib.CategoryTheory.Limits.Shapes.Connected

Goal state before:
pp: J : Type u_1
⊢ ∀ {α : Type u_1} (F : WidePushoutShape J → α),
    (∀ {j₁ j₂ : WidePushoutShape J} (x : j₁ ⟶ j₂), F j₁ = F j₂) → ∀ (j j' : WidePushoutShape J), F j = F j'

Tactic:
intro α F H

Goal state after:
J α : Type u_1
F : WidePushoutShape J → α
H : ∀ {j₁ j₂ : WidePushoutShape J} (x : j₁ ⟶ j₂), F j₁ = F j₂
⊢ ∀ (j j' : WidePushoutShape J), F j = F j'
Module: Mathlib.CategoryTheory.Sites.Equivalence

Goal state before:
pp: C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁷ : Category.{v₃, u₃} A
B : Type u₄
inst✝⁶ : Category.{v₄, u₄} B
F : A ⥤ B
inst✝⁵ : G.IsCoverDense J
inst✝⁴ : G.Full
inst✝³ : G.IsContinuous K J
inst✝² : (G.sheafPushforwardContinuous B K J).EssSurj
inst✝¹ : (G.sheafPushforwardContinuous A K J).EssSurj
inst✝ : K.PreservesSheafification F
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
hf : K.W (G.op.whiskerLeft f)
this✝ : K.W ((G.op.associator P F).hom ≫ G.op.whiskerLeft (whiskerRight f F) ≫ (G.op.associator Q F).inv)
this : K.W (G.op.whiskerLeft (whiskerRight f F))
⊢ J.W.inverseImage ((whiskeringRight Cᵒᵖ A B).obj F) f

Tactic:
K.W_whiskerLeft_iff (G := G) (J := J) (f := whiskerRight f F)

Goal state after:
C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁷ : Category.{v₃, u₃} A
B : Type u₄
inst✝⁶ : Category.{v₄, u₄} B
F : A ⥤ B
inst✝⁵ : G.IsCoverDense J
inst✝⁴ : G.Full
inst✝³ : G.IsContinuous K J
inst✝² : (G.sheafPushforwardContinuous B K J).EssSurj
inst✝¹ : (G.sheafPushforwardContinuous A K J).EssSurj
inst✝ : K.PreservesSheafification F
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
hf : K.W (G.op.whiskerLeft f)
this✝ : K.W ((G.op.associator P F).hom ≫ G.op.whiskerLeft (whiskerRight f F) ≫ (G.op.associator Q F).inv)
this : J.W (whiskerRight f F)
⊢ J.W.inverseImage ((whiskeringRight Cᵒᵖ A B).obj F) f
Module: Mathlib.CategoryTheory.Sites.CoversTop.Basic

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
I : Type u_1
Y : I → C
hY : J.CoversTop Y
F : Sheaf J (Type u_2)
x y : ↑F.obj.sections
h : ∀ (i : I), ↑x (Opposite.op (Y i)) = ↑y (Opposite.op (Y i))
⊢ x = y

Tactic:
ext W

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
I : Type u_1
Y : I → C
hY : J.CoversTop Y
F : Sheaf J (Type u_2)
x y : ↑F.obj.sections
h : ∀ (i : I), ↑x (Opposite.op (Y i)) = ↑y (Opposite.op (Y i))
W : Cᵒᵖ
⊢ ↑x W = ↑y W
Module: Mathlib.CategoryTheory.Sites.Over

Goal state before:
pp: case mpr
C : Type u
inst✝ : Category.{v, u} C
X : C
Y₁ Y₂ : Over X
f : Y₁ ⟶ Y₂
S : Sieve Y₂
Z : C
g : Z ⟶ Y₁.left
W : Over X
a : W ⟶ Y₂
b : Z ⟶ W.left
h : S.arrows a
w : g ≫ Over.Hom.left f = b ≫ Over.Hom.left a
T : Over X := Over.mk (b ≫ W.hom)
c : T ⟶ Y₁ := Over.homMk g ⋯
d : T ⟶ W := Over.homMk b ⋯
⊢ S.arrows (c ≫ f)

Tactic:
rw [show c ≫ f = d ≫ a by ext; exact w]

Goal state after:
case mpr
C : Type u
inst✝ : Category.{v, u} C
X : C
Y₁ Y₂ : Over X
f : Y₁ ⟶ Y₂
S : Sieve Y₂
Z : C
g : Z ⟶ Y₁.left
W : Over X
a : W ⟶ Y₂
b : Z ⟶ W.left
h : S.arrows a
w : g ≫ Over.Hom.left f = b ≫ Over.Hom.left a
T : Over X := Over.mk (b ≫ W.hom)
c : T ⟶ Y₁ := Over.homMk g ⋯
d : T ⟶ W := Over.homMk b ⋯
⊢ S.arrows (d ≫ a)
Module: Mathlib.CategoryTheory.Sites.Over

Goal state before:
pp: case mpr
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f : X ⟶ Y
Z : Over X
S : Sieve Z.left
W : Over Y
g : W ⟶ (Over.map f).obj Z
⊢ ((overEquiv ((Over.map f).obj Z)).symm S).arrows g → (functorPushforward (Over.map f) ((overEquiv Z).symm S)).arrows g

Tactic:
· intro hg
    exact ⟨Over.mk (g.left ≫ Z.hom), Over.homMk g.left,
      Over.homMk (𝟙 _) (by simpa using Over.w g), hg, by cat_disch⟩

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf.Generators

Goal state before:
pp: C : Type u'
inst✝⁴ : Category.{v', u'} C
J : GrothendieckTopology C
R : Sheaf J RingCat
inst✝³ : HasWeakSheafify J AddCommGrpCat
inst✝² : J.WEqualsLocallyBijective AddCommGrpCat
M✝ N P : SheafOfModules R
inst✝¹ : ∀ (X : C), HasWeakSheafify (J.over X) AddCommGrpCat
inst✝ : ∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrpCat
M : SheafOfModules R
q : M.LocalGeneratorsData
X : C
⊢ Sieve.ofObjects (fun i ↦ q.X (Exists.choose ⋯)) X ∈ J X

Tactic:
refine J.superset_covering (fun Y hY H ↦ ?_) (q.coversTop X)

Goal state after:
C : Type u'
inst✝⁴ : Category.{v', u'} C
J : GrothendieckTopology C
R : Sheaf J RingCat
inst✝³ : HasWeakSheafify J AddCommGrpCat
inst✝² : J.WEqualsLocallyBijective AddCommGrpCat
M✝ N P : SheafOfModules R
inst✝¹ : ∀ (X : C), HasWeakSheafify (J.over X) AddCommGrpCat
inst✝ : ∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrpCat
M : SheafOfModules R
q : M.LocalGeneratorsData
X Y : C
hY : Y ⟶ X
H : (Sieve.ofObjects q.X X).arrows hY
⊢ (Sieve.ofObjects (fun i ↦ q.X (Exists.choose ⋯)) X).arrows hY
Module: Mathlib.CategoryTheory.Sites.Over

Goal state before:
pp: case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u_1
inst✝ : Category.{v_1, u_1} D
F : C ⥤ D
X : C
U : Over X
S : Sieve ((Over.post F).obj U)
⊢ functorPullback F ((overEquiv ((Over.post F).obj U)) S) ≤ (overEquiv U) (functorPullback (Over.post F) S)

Tactic:
intro Z g hg
    rw [Sieve.overEquiv_iff]
    dsimp [Presieve.functorPullback]
    convert! (Sieve.overEquiv_iff _ _).mp hg
    simp

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Over

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
K : Precoverage C
inst✝¹ : K.HasPullbacks
inst✝ : K.IsStableUnderBaseChange
X : C
⊢ (Precoverage.comap (Over.forget X) K).toGrothendieck ≤ K.toGrothendieck.over X

Tactic:
rw [Precoverage.toGrothendieck_le_iff_le_toPrecoverage]
    intro Y R hR
    rw [Precoverage.mem_comap_iff] at hR
    rw [GrothendieckTopology.mem_toPrecoverage_iff, GrothendieckTopology.mem_over_iff,
      Sieve.overEquiv, RelIso.coe_fn_mk, Equiv.coe_fn_mk,
      ← Sieve.generate_map_eq_functorPushforward]
    exact Precoverage.Saturate.of _ _ hR

Goal state after:
Module: Mathlib.CategoryTheory.Simple

Goal state before:
pp: case mpr.inr
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
X : C
inst✝ : IsSimpleOrder (Subobject X)
Y : C
f : Y ⟶ X
hf : Mono f
i : f ≠ 0
h : mk f = ⊤
⊢ IsIso f

Tactic:
exact (Subobject.isIso_iff_mk_eq_top _).mpr h

Goal state after:
Module: Mathlib.Topology.Category.TopCat.Opens

Goal state before:
pp: X Y : TopCat
f : X ⟶ Y
hf : IsOpenEmbedding ⇑(ConcreteCategory.hom f)
ι : Type u_1
inst✝¹ : Nonempty ι
inst✝ : Finite ι
g : ι → Opens ↑X
⊢ (fun a ↦ (ConcreteCategory.hom f) a) '' ⋂ i, ↑(g i) = ⋂ i, (fun a ↦ (ConcreteCategory.hom f) a) '' ↑(g i)

Tactic:
Set.InjOn.image_iInter_eq

Goal state after:
X Y : TopCat
f : X ⟶ Y
hf : IsOpenEmbedding ⇑(ConcreteCategory.hom f)
ι : Type u_1
inst✝¹ : Nonempty ι
inst✝ : Finite ι
g : ι → Opens ↑X
⊢ ⋂ i, (fun a ↦ (ConcreteCategory.hom f) a) '' ↑(g i) = ⋂ i, (fun a ↦ (ConcreteCategory.hom f) a) '' ↑(g i)

X Y : TopCat
f : X ⟶ Y
hf : IsOpenEmbedding ⇑(ConcreteCategory.hom f)
ι : Type u_1
inst✝¹ : Nonempty ι
inst✝ : Finite ι
g : ι → Opens ↑X
⊢ Set.InjOn (fun a ↦ (ConcreteCategory.hom f) a) (⋃ i, ↑(g i))
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : TopCat
P Q : Presheaf C X
f g : P ⟶ Q
w : ∀ (U : Opens ↑X), f.app (op U) = g.app (op U)
⊢ f.app = g.app

Tactic:
ext U

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
P Q : Presheaf C X
f g : P ⟶ Q
w : ∀ (U : Opens ↑X), f.app (op U) = g.app (op U)
U : (Opens ↑X)ᵒᵖ
⊢ f.app U = g.app U
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X : TopCat
P Q : Presheaf C X
f g : P ⟶ Q
w : ∀ (U : Opens ↑X), f.app (op U) = g.app (op U)
U : (Opens ↑X)ᵒᵖ
⊢ f.app U = g.app U

Tactic:
induction U with | _ U => ?_

Goal state after:
case op
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
P Q : Presheaf C X
f g : P ⟶ Q
w : ∀ (U : Opens ↑X), f.app (op U) = g.app (op U)
U : Opens ↑X
⊢ f.app (op U) = g.app (op U)
Module: Mathlib.Topology.Sheaves.Presheaf

Goal state before:
pp: C✝ : Type u
inst✝⁴ : Category.{v, u} C✝
X✝ : TopCat
C : Type u
inst✝³ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type u_2
inst✝² : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝¹ : ConcreteCategory C FC
inst✝ : HasColimits C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U : Opens ↑X
H : IsOpen (⇑(ConcreteCategory.hom f) '' ↑U)
x : CostructuredArrow (Opens.map f).op (op U) := CostructuredArrow.mk (homOfLE ⋯).op
hx : IsTerminal x
⊢ ((pullback C f).obj ℱ).obj (op U) ≅ ℱ.obj (op { carrier := ⇑(ConcreteCategory.hom f) '' ↑U, is_open' := H })

Tactic:
exact IsColimit.coconePointUniqueUpToIso
    ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op U))
    (colimitOfDiagramTerminal hx _)

Goal state after: