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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E F : PreZeroHypercover S
e : E ≅ F
i : F.I₀
⊢ inv (e.inv.h₀ i) ≫ F.f i = E.f (e.inv.s₀ i)

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S T : C
E✝¹ : PreZeroHypercover S
F✝¹ : PreZeroHypercover S
E✝ : PreZeroHypercover S
F✝ : PreZeroHypercover S
G : PreZeroHypercover S
E F : PreZeroHypercover S
e : E ≅ F
x✝ : E.I₀
⊢ e.inv.s₀ (e.hom.s₀ x✝) = x✝

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S T : C
E✝¹ : PreZeroHypercover S
F✝¹ : PreZeroHypercover S
E✝ : PreZeroHypercover S
F✝ : PreZeroHypercover S
G : PreZeroHypercover S
E F : PreZeroHypercover S
e : E ≅ F
x✝ : F.I₀
⊢ e.hom.s₀ (e.inv.s₀ x✝) = x✝

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
S T : C
E : PreZeroHypercover S
F✝ F : PreZeroHypercover S
G : PreZeroHypercover S
inst✝ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)
f : G.Hom E
g : G.Hom F
i : G.I₀
⊢ f.h₀ i ≫ E.f ((Equiv.sigmaEquivProd E.I₀ F.1).symm (f.s₀ i, g.s₀ i)).fst =
    g.h₀ i ≫ F.f ((Equiv.sigmaEquivProd E.I₀ F.1).symm (f.s₀ i, g.s₀ i)).snd

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
R : Presieve S
⊢ R = R.preZeroHypercover.presieve₀

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : C
S T : Presieve X
h : S ≤ T
Y : C
g : Y ⟶ X
hg : S g
⊢ 𝟙 Y ≫ g = g

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J K : Precoverage C
inst✝ : J.Small
of_le : ∀ ⦃X : C⦄ ⦃R S : Presieve X⦄, R ≤ S → S ∈ K.coverings X → R ∈ K.coverings X
S : C
E : (J ⊓ K).ZeroHypercover S
⊢ (E.restrictIndex (ZeroHypercover.Small.restrictFun (E.weaken ⋯))).presieve₀ ≤ E.presieve₀

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : Precoverage C
inst✝ : J.IsStableUnderBaseChange
S : C
E F : PreZeroHypercover S
e : E ≅ F
h : E.presieve₀ ∈ J.coverings S
i : E.I₀
⊢ e.inv.s₀ (e.hom.s₀ i) = i

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : Precoverage C
inst✝ : J.IsStableUnderBaseChange
S : C
E F : PreZeroHypercover S
e : E ≅ F
h : E.presieve₀ ∈ J.coverings S
i : F.I₀
⊢ CommSq (F.f i) (e.inv.h₀ i) (𝟙 S) (E.f (e.inv.s₀ i))

Tactic:
by simp

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

Goal state before:
pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
J : Pretopology C
T : C
S : Sieve T
x✝ : ∃ R ∈ J.coverings T, R ≤ S.arrows
R : Presieve T
hR : R ∈ J.coverings T
hle : R ≤ S.arrows
⊢ J.toCoverage.Saturate T S

Tactic:
refine Coverage.saturate_of_superset _ ?_ (.of _ _ hR)

Goal state after:
case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasPullbacks C
J : Pretopology C
T : C
S : Sieve T
x✝ : ∃ R ∈ J.coverings T, R ≤ S.arrows
R : Presieve T
hR : R ∈ J.coverings T
hle : R ≤ S.arrows
⊢ Sieve.generate R ≤ S
Module: Mathlib.CategoryTheory.Sites.Coverage

Goal state before:
pp: C : Type u_2
inst✝ : Category.{v_1, u_2} C
K : Coverage C
P : Cᵒᵖ ⥤ Type u_1
H : ∀ {X : C}, ∀ R ∈ K.coverings X, IsSheafFor P R
X Y : C
f : Y ⟶ X
S : Presieve X
hS : S ∈ K.coverings X
T : Presieve Y
hT1 : T ∈ K.coverings Y
hT2 : T.FactorsThruAlong S f
Z : C
g : Z ⟶ Y
hg : T g
W : C
i : Z ⟶ W
e : W ⟶ X
h1 : S e
h2 : i ≫ e = g ≫ f
⊢ 𝟙 Z ≫ g = g

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
X✝ : C
S : Sieve X✝
hS : S ∈ J X✝
⊢ ⊤ (Cover.oneHypercover ⟨S, hS⟩)

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
X✝ : C
S : Sieve X✝
hS : S ∈ J X✝
⊢ (Cover.oneHypercover ⟨S, hS⟩).sieve₀ ≤ S

Tactic:
by simp

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

Goal state before:
pp: case mpr
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
H : ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ J.coverings X, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
X : C
S : Sieve X
hS : S ∈ J.toGrothendieck X
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows

Tactic:
induction hS with
    | of X S hS =>
      exact fun _ _ => H S hS
    | top =>
      simp [Presieve.isSheafFor_top P]
    | pullback X S hS _ f ih =>
      intro Y f
      rw [← S.pullback_comp]
      exact ih (f ≫ _)
    | transitive X R S hS h H1 H2 =>
      intro Y f
      simp only [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor] at *
      choose H1 H1' using H1
      choose H2 H2' using H2
      refine ⟨?_, fun x hx => ?_⟩
      · intro x t₁ t₂ h₁ h₂
        refine (H1 f).ext (fun Z g hg => ?_)
        refine (H2 hg (𝟙 _)).ext (fun ZZ gg hgg => ?_)
        simp only [Sieve.pullback_id, Sieve.pullback_apply] at hgg
        simp only [← comp_apply]
        rw [← P.map_comp, ← op_comp, h₁, h₂]
        simpa only [Sieve.pullback_apply, Category.assoc] using hgg
      let y : ∀ ⦃Z : C⦄ (g : Z ⟶ Y),
        ((S.pullback (g ≫ f)).pullback (𝟙 _)).arrows.FamilyOfElements P :=
        fun Z g ZZ gg hgg => x (gg ≫ g) (by simpa using hgg)
      have hy : ∀ ⦃Z : C⦄ (g : Z ⟶ Y), (y g).Compatible := by
        intro Z g Y₁ Y₂ ZZ g₁ g₂ f₁ f₂ h₁ h₂ h
        rw [hx]
        rw [reassoc_of% h]
      choose z hz using fun ⦃Z : C⦄ ⦃g : Z ⟶ Y⦄ (hg : R.pullback f g) =>
        H2' hg (𝟙 _) (y g) (hy g)
      let q : (R.pullback f).arrows.FamilyOfElements P := fun Z g hg => z hg
      have hq : q.Compatible := by
        intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ h
        apply (H2 h₁ g₁).ext
        intro ZZ gg hgg
        simp only [← comp_apply]
        rw [← P.map_comp, ← P.map_comp, ← op_comp, ← op_comp, hz, hz]
        · dsimp [y]; congr 1; simp only [Category.assoc, h]
        · simpa [reassoc_of% h] using hgg
        · simpa using hgg
      obtain ⟨t, ht⟩ := H1' f q hq
      refine ⟨t, fun Z g hg => ?_⟩
      refine (H1 (g ≫ f)).ext (fun ZZ gg hgg => ?_)
      rw [← comp_apply _ (P.map gg.op), ← P.map_comp, ← op_comp, ht]
      on_goal 2 => simpa using hgg
      refine (H2 hgg (𝟙 _)).ext (fun ZZZ ggg hggg => ?_)
      rw [← comp_apply _ (P.map ggg.op), ← P.map_comp, ← op_comp, hz]
      on_goal 2 => simpa using hggg
      refine (H2 hgg ggg).ext (fun ZZZZ gggg _ => ?_)
      rw [← comp_apply _ (P.map gggg.op), ← P.map_comp, ← op_comp]
      apply hx
      simp

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

Goal state before:
pp: C : Type u_2
inst✝⁴ : Category.{u_1, u_2} C
J : Precoverage C
inst✝³ : J.IsStableUnderComposition
inst✝² : J.IsStableUnderBaseChange
inst✝¹ : J.HasPullbacks
inst✝ : J.HasIsos
X✝ : C
S : Sieve X✝
X : C
⊢ Presieve.singleton (𝟙 X) ≤ ⊤.arrows

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
h : F.IsCocontinuous J K
U : D
S : Sieve U
hS : S ∈ K U
X : C
f : X ⟶ G.obj U
hf : (Sieve.functorPullback F (Sieve.pullback (adj.counit.app U) S)).arrows f
⊢ f = adj.unit.app X ≫ G.map (F.map f ≫ adj.counit.app U)

Tactic:
by simp

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

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝ : RepresentablyFlat G
ℱ : Sheaf K (Type u_1)
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
e : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂
c : Cone (cospan g₁ g₂ ⋙ G) :=
  (Cone.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
c' : Cone (c.toStructuredArrow ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=
  IsCofiltered.cone (c.toStructuredArrow ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)
⊢ G.obj ((c.toStructuredArrow ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).obj left).right = G.obj Y₁

Tactic:
by simp

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

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝ : RepresentablyFlat G
ℱ : Sheaf K (Type u_1)
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
e : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂
c : Cone (cospan g₁ g₂ ⋙ G) :=
  (Cone.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
c' : Cone (c.toStructuredArrow ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G) :=
  IsCofiltered.cone (c.toStructuredArrow ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G)
eq₁ : f₁ = (c'.pt.hom ≫ G.map (StructuredArrow.Hom.right (c'.π.app left))) ≫ eqToHom ⋯
⊢ G.obj ((c.toStructuredArrow ⋙ StructuredArrow.pre c.pt (cospan g₁ g₂) G).obj right).right = G.obj Y₂

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
G : C ⥤ D
A : Type w
inst✝¹ : Category.{w', w} A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : G.IsCocontinuous J K
F : Cᵒᵖ ⥤ A
hF : Presheaf.IsSheaf J F
R : Dᵒᵖ ⥤ A
α : G.op ⋙ R ⟶ F
X : D
S : K.Cover X
s : Multifork (S.index R)
Y Y' : C
f : G.obj Y ⟶ X
f' : G.obj Y' ⟶ X
W : C
a : W ⟶ Y
b : W ⟶ Y'
w : G.map a ≫ f = G.map b ≫ f'
T : C
g : T ⟶ W
hg : (↑⟨Sieve.functorPullback G (Sieve.pullback (G.map a ≫ f) ↑S), ⋯⟩).arrows g
⊢ 𝟙 (G.obj T) ≫ { Y := G.obj T, f := G.map g ≫ G.map a ≫ f, hf := hg }.f = G.map (g ≫ a) ≫ f

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
G : C ⥤ D
A : Type w
inst✝¹ : Category.{w', w} A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : G.IsCocontinuous J K
F : Cᵒᵖ ⥤ A
hF : Presheaf.IsSheaf J F
R : Dᵒᵖ ⥤ A
α : G.op ⋙ R ⟶ F
hR : (Functor.RightExtension.mk R α).IsPointwiseRightKanExtension
X : D
S : K.Cover X
s : Multifork (S.index R)
i : S.Arrow
j : StructuredArrow (op i.Y) G.op
eq : lift hF hR s ≫ R.map i.f.op ≫ R.map j.hom ≫ α.app j.right = liftAux hF α s (j.hom.unop ≫ i.f)
⊢ j.hom.unop ≫ i.f = G.map (𝟙 (unop j.right)) ≫ j.hom.unop ≫ i.f

Tactic:
by simp

Goal state after:
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
⊢ (G.op.ranAdjunction A).unit.app F.obj ≫ 𝟙 (G.op.ran.obj (G.op ⋙ F.obj)) ≫ 𝟙 (G.op.ran.obj (G.op ⋙ F.obj)) =
    (G.op.ranAdjunction A).unit.app F.obj

Tactic:
simp only [Category.comp_id]

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

Goal state before:
pp: C : Type uC
inst✝¹ : Category.{vC, uC} C
D : Type uD
inst✝ : Category.{vD, uD} D
G : C ⥤ D
U V : C
f : U ⟶ V
W : C
g : W ⟶ U
⊢ G.map (g ≫ f) = G.map g ≫ G.map f

Tactic:
by simp

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
G : C ⥤ D
X : D
Y : C
f : G.obj Y ⟶ X
⊢ 𝟙 (G.obj Y) ≫ f = f

Tactic:
by simp

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✝
S : C
x✝² x✝¹ : (PreZeroHypercover.singleton (𝟙 S)).I₀
x✝ : PUnit.{w + 1}
⊢ 𝟙 S ≫ (PreZeroHypercover.singleton (𝟙 S)).f x✝² = 𝟙 S ≫ (PreZeroHypercover.singleton (𝟙 S)).f x✝¹

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
A : Type u_1
inst✝² : Category.{v_1, u_1} A
S : C
E✝ : PreOneHypercover S
E : PreOneHypercover S
F : PreOneHypercover S
inst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)
inst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullback (E.p₁ k ≫ E.f i) (F.p₁ l ≫ F.f a)
i j : __spread✝⁻⁰.I₀
k : E.I₁ i.1 j.1 × F.I₁ i.2 j.2
⊢ (E.p₁ k.1 ≫ E.f i.1) ≫ 𝟙 S = E.p₁ k.1 ≫ E.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).fst

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
A : Type u_1
inst✝² : Category.{v_1, u_1} A
S : C
E✝ : PreOneHypercover S
E : PreOneHypercover S
F : PreOneHypercover S
inst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)
inst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullback (E.p₁ k ≫ E.f i) (F.p₁ l ≫ F.f a)
i j : __spread✝⁻⁰.I₀
k : E.I₁ i.1 j.1 × F.I₁ i.2 j.2
⊢ (F.p₁ k.2 ≫ F.f i.2) ≫ 𝟙 S = F.p₁ k.2 ≫ F.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i).snd

Tactic:
by simp

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
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
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 Y : Aᵒᵖ
f : X ⟶ Y
U : Dᵒᵖ
x : ((coyoneda ⋙ (whiskeringLeft Dᵒᵖ A (Type v_4)).obj ℱ).obj X).obj U
Y' : D
f' : Y' ⟶ unop U
hf' : (Sieve.coverByImage G (unop U)).arrows f'
⊢ (ConcreteCategory.hom ((homOver α (unop X)).app (op (Nonempty.some hf').obj)))
      ((ConcreteCategory.hom ((ℱ ⋙ coyoneda.obj (op (unop X))).map (Nonempty.some hf').map.op)) x) =
    (x ≫ ℱ.map (Nonempty.some hf').map.op) ≫ α.app (op (Nonempty.some hf').1)

Tactic:
by simp

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
X : Dᵒᵖ
U : Aᵒᵖ
x✝ : (yoneda.obj (ℱ.obj X)).obj U
⊢ (ConcreteCategory.hom ((yoneda.map (α.app X)).app U)) x✝ =
    (ConcreteCategory.hom (((sheafYonedaHom (G.op.whiskerLeft α)).app X).app U)) x✝

Tactic:
change (show (ℱ'.obj ⋙ coyoneda.obj (op (unop U))).obj (op (unop X)) from _) = _

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
⊢ (have this := (ConcreteCategory.hom ((yoneda.map (α.app X)).app U)) x✝;
    this) =
    (ConcreteCategory.hom (((sheafYonedaHom (G.op.whiskerLeft α)).app X).app U)) x✝
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
  simp [congrIndexOneOfEq]

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).1 (i, j).2
i' j' : E.I₀
k' : E.I₁ (i', j').1 (i', j').2
left : ⟨(i, j), k⟩.fst.1 = ⟨(i', j'), k'⟩.fst.1
right : ⟨(i, j), k⟩.fst.2 = ⟨(i', j'), k'⟩.fst.2
h : (congrIndexOneOfEq left right) ⟨(i, j), k⟩.snd = ⟨(i', j'), k'⟩.snd
⊢ ⟨(i, j), k⟩ = ⟨(i', j'), k'⟩

Tactic:
dsimp at left right

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreOneHypercover S
i j : E.I₀
k : E.I₁ (i, j).1 (i, j).2
i' j' : E.I₀
k' : E.I₁ (i', j').1 (i', j').2
left : i = i'
right : j = j'
h : (congrIndexOneOfEq left right) ⟨(i, j), k⟩.snd = ⟨(i', j'), k'⟩.snd
⊢ ⟨(i, j), k⟩ = ⟨(i', j'), 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✝ E : PreOneHypercover S
F : PreOneHypercover S
G : PreOneHypercover S
s₀ : E.I₀ ≃ F.I₀
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
i j : E.I₀
k : E.I₁ i j
⊢ s₀ i = s₀ (s₀.symm (s₀ i))

Tactic:
by simp

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
s₀ : E.I₀ ≃ F.I₀
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
i j : E.I₀
k : E.I₁ i j
⊢ s₀ j = s₀ (s₀.symm (s₀ j))

Tactic:
by simp

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
s₀ : E.I₀ ≃ F.I₀
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
i j : E.I₀
k✝ : E.I₁ i j
h₁ : ⦃i j : E.I₀⦄ → (k : E.I₁ i j) → E.Y k ≅ F.Y (s₁ k)
k : E.I₁ i j
⊢ F.Y ((congrIndexOneOfEq ⋯ ⋯) (s₁ k)) = F.Y (s₁ (s₁.symm ((congrIndexOneOfEq ⋯ ⋯) (s₁ k))))

Tactic:
by simp

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
s₀ : E.I₀ ≃ F.I₀
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
i j : E.I₀
h₁ : ⦃i j : E.I₀⦄ → (k : E.I₁ i j) → E.Y k ≅ F.Y (s₁ k)
k : E.I₁ i j
⊢ i = s₀.symm (s₀ i)

Tactic:
by simp

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
s₀ : E.I₀ ≃ F.I₀
s₁ : ⦃i j : E.I₀⦄ → E.I₁ i j ≃ F.I₁ (s₀ i) (s₀ j)
i j : E.I₀
h₁ : ⦃i j : E.I₀⦄ → (k : E.I₁ i j) → E.Y k ≅ F.Y (s₁ k)
k : E.I₁ i j
⊢ j = s₀.symm (s₀ j)

Tactic:
by simp

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

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
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
Z : D
U₂ : C
iU₂U₁ : U₂ ⟶ U₁
iZU₂ : Z ⟶ G.obj U₂
e₃ : iU₂U₁ ≫ iU₁U ≫ iUY ≫ f = iU₂U₁ ≫ iU₁W ≫ iWX
⊢ ((iZU₂ ≫ G.map iU₂U₁) ≫ G.map iU₁U) ≫ G.map iUY = iZU₂ ≫ G.map (iU₂U₁ ≫ iU₁U ≫ iUY)

Tactic:
by simp

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 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 : F.I₀
k : F.I₁ i j
⊢ i = s₀ (__Hom✝.s₀ i)

Tactic:
by simp

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
⊢ IsIso ((sheafToPresheaf J A).map ((G.sheafAdjunctionCocontinuous A J K).counit.app Y))

Tactic:
rw [NatTrans.isIso_iff_isIso_app]

Goal state after:
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
⊢ ∀ (X : Cᵒᵖ), IsIso (((sheafToPresheaf J A).map ((G.sheafAdjunctionCocontinuous A J K).counit.app Y)).app X)
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 : F.I₀
k : F.I₁ i j
⊢ j = s₀ (__Hom✝.s₀ j)

Tactic:
by simp

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
⊢ IsIso (yoneda.map (((sheafToPresheaf J A).map ((G.sheafAdjunctionCocontinuous A J K).counit.app Y)).app (op U)))

Tactic:
rw [NatTrans.isIso_iff_isIso_app]

Goal state after:
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 : Aᵒᵖ),
    IsIso
      ((yoneda.map (((sheafToPresheaf J A).map ((G.sheafAdjunctionCocontinuous A J K).counit.app Y)).app (op U))).app X)
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 : F.I₀
k : F.I₁ i j
⊢ F.Y ((congrIndexOneOfEq ⋯ ⋯) k) = F.Y (s₁ (s₁.symm ((congrIndexOneOfEq ⋯ ⋯) k)))

Tactic:
by simp

Goal state after: