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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝ : ConcreteCategory A FA
F : Cᵒᵖ ⥤ Type (max u v)
⊢ (Subfunctor.sheafify J (Subfunctor.range (J.toSheafify F))).ι ≫
      J.sheafifyLift (Subfunctor.toRangeSheafify J (J.toSheafify F)) ⋯ ≫
        (Subfunctor.sheafify J (Subfunctor.range (J.toSheafify F))).ι =
    (Subfunctor.sheafify J (Subfunctor.range (J.toSheafify F))).ι

Tactic:
refine Eq.trans ?_ (Category.comp_id _)

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝ : ConcreteCategory A FA
F : Cᵒᵖ ⥤ Type (max u v)
⊢ (Subfunctor.sheafify J (Subfunctor.range (J.toSheafify F))).ι ≫
      J.sheafifyLift (Subfunctor.toRangeSheafify J (J.toSheafify F)) ⋯ ≫
        (Subfunctor.sheafify J (Subfunctor.range (J.toSheafify F))).ι =
    (Subfunctor.sheafify J (Subfunctor.range (J.toSheafify F))).ι ≫ 𝟙 (J.sheafify F)
Module: Mathlib.CategoryTheory.Abelian.Projective.Dimension

Goal state before:
pp: case succ
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
inst✝ : EnoughInjectives C
X : C
n : ℕ
hX : ∀ (Y : C), Subsingleton (Ext X Y n)
k : ℕ
hk : ∀ ⦃d : ℕ⦄ ⦃Y : C⦄ (e : Ext X Y d), d = n + k → e = 0
d : ℕ
Y : C
e : Ext X Y d
hd✝ : d = n + (k + 1)
p : InjectivePresentation Y
h : { X₁ := Y, X₂ := p.J, X₃ := cokernel p.f, f := p.f, g := cokernel.π p.f, zero := ⋯ }.ShortExact
hd : n + k + 1 = d
⊢ e = 0

Tactic:
obtain ⟨x, rfl⟩ := Ext.covariant_sequence_exact₁ X h e
      (by subst hd; apply Ext.eq_zero_of_injective) hd

Goal state after:
case succ
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
inst✝ : EnoughInjectives C
X : C
n : ℕ
hX : ∀ (Y : C), Subsingleton (Ext X Y n)
k : ℕ
hk : ∀ ⦃d : ℕ⦄ ⦃Y : C⦄ (e : Ext X Y d), d = n + k → e = 0
d : ℕ
Y : C
hd✝ : d = n + (k + 1)
p : InjectivePresentation Y
h : { X₁ := Y, X₂ := p.J, X₃ := cokernel p.f, f := p.f, g := cokernel.π p.f, zero := ⋯ }.ShortExact
hd : n + k + 1 = d
x : Ext X { X₁ := Y, X₂ := p.J, X₃ := cokernel p.f, f := p.f, g := cokernel.π p.f, zero := ⋯ }.X₃ (n + k)
⊢ x.comp h.extClass hd = 0
Module: Mathlib.CategoryTheory.Bicategory.Functor.LocallyDiscrete

Goal state before:
pp: B : Type u_1
C : Type u_2
inst✝² : Bicategory B
inst✝¹ : IsLocallyDiscrete B
inst✝ : Bicategory C
obj : B → C
map : {b b' : B} → (b ⟶ b') → (obj b ⟶ obj b')
mapId : (b : B) → map (𝟙 b) ≅ 𝟙 (obj b)
mapComp : {b₀ b₁ b₂ : B} → (f : b₀ ⟶ b₁) → (g : b₁ ⟶ b₂) → map (f ≫ g) ≅ map f ≫ map g
map₂_associator :
  ∀ {b₀ b₁ b₂ b₃ : B} (f : b₀ ⟶ b₁) (g : b₁ ⟶ b₂) (h : b₂ ⟶ b₃),
    (mapComp (f ≫ g) h).hom ≫
        (mapComp f g).hom ▷ map h ≫
          (α_ (map f) (map g) (map h)).hom ≫ map f ◁ (mapComp g h).inv ≫ (mapComp f (g ≫ h)).inv =
      eqToHom ⋯
map₂_left_unitor :
  ∀ {b₀ b₁ : B} (f : b₀ ⟶ b₁), (mapComp (𝟙 b₀) f).hom ≫ (mapId b₀).hom ▷ map f ≫ (λ_ (map f)).hom = eqToHom ⋯
map₂_right_unitor :
  ∀ {b₀ b₁ : B} (f : b₀ ⟶ b₁), (mapComp f (𝟙 b₁)).hom ≫ map f ◁ (mapId b₁).hom ≫ (ρ_ (map f)).hom = eqToHom ⋯
a✝ b✝ : B
f✝ g✝ : a✝ ⟶ b✝
φ : f✝ ⟶ g✝
⊢ map f✝ = map g✝

Tactic:
by
    obtain rfl := obj_ext_of_isDiscrete φ
    dsimp

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.Functor.LocallyDiscrete

Goal state before:
pp: B : Type u_1
C : Type u_2
inst✝² : Bicategory B
inst✝¹ : IsLocallyDiscrete B
inst✝ : Bicategory C
obj : B → C
map : {b b' : B} → (b ⟶ b') → (obj b ⟶ obj b')
mapId : (b : B) → map (𝟙 b) ⟶ 𝟙 (obj b)
mapComp : {b₀ b₁ b₂ : B} → (f : b₀ ⟶ b₁) → (g : b₁ ⟶ b₂) → map (f ≫ g) ⟶ map f ≫ map g
map₂_associator :
  ∀ {b₀ b₁ b₂ b₃ : B} (f : b₀ ⟶ b₁) (g : b₁ ⟶ b₂) (h : b₂ ⟶ b₃),
    eqToHom ⋯ ≫ mapComp f (g ≫ h) ≫ map f ◁ mapComp g h =
      mapComp (f ≫ g) h ≫ mapComp f g ▷ map h ≫ (α_ (map f) (map g) (map h)).hom
map₂_left_unitor : ∀ {b₀ b₁ : B} (f : b₀ ⟶ b₁), mapComp (𝟙 b₀) f ≫ mapId b₀ ▷ map f ≫ (λ_ (map f)).hom = eqToHom ⋯
map₂_right_unitor : ∀ {b₀ b₁ : B} (f : b₀ ⟶ b₁), mapComp f (𝟙 b₁) ≫ map f ◁ mapId b₁ ≫ (ρ_ (map f)).hom = eqToHom ⋯
a✝ b✝ : B
f✝ g✝ : a✝ ⟶ b✝
φ : f✝ ⟶ g✝
⊢ map f✝ = map g✝

Tactic:
by
    obtain rfl := obj_ext_of_isDiscrete φ
    dsimp

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrpCat
φ : M₀.presheaf ⟶ A.obj
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X : Cᵒᵖ
r : ↑(R.obj.obj X)
Y : C
f : Y ⟶ Opposite.unop X
r₀ : ToType (R₀.obj (Opposite.op Y))
hr₀ : (ConcreteCategory.hom (α.app (Opposite.op Y))) r₀ = (ConcreteCategory.hom (R.obj.map f.op)) r
⊢ (ConcreteCategory.hom (A.obj.map f.op)) (smul α φ r 0) = 0

Tactic:
map_smul_eq α φ r 0 f.op r₀ hr₀ 0 (by simp),

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrpCat
φ : M₀.presheaf ⟶ A.obj
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X : Cᵒᵖ
r : ↑(R.obj.obj X)
Y : C
f : Y ⟶ Opposite.unop X
r₀ : ToType (R₀.obj (Opposite.op Y))
hr₀ : (ConcreteCategory.hom (α.app (Opposite.op Y))) r₀ = (ConcreteCategory.hom (R.obj.map f.op)) r
⊢ (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀ • 0) = 0
Module: Mathlib.CategoryTheory.Limits.Preserves.Opposites

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : Cᵒᵖ ⥤ D
inst✝ : ReflectsFiniteCoproducts F
x✝ : ℕ
⊢ ReflectsLimitsOfShape (Discrete (Fin x✝)) F.rightOp

Tactic:
by
    apply +allowSynthFailures reflectsLimitsOfShape_rightOp
    exact reflectsColimitsOfShape_of_equiv (Discrete.opposite _).symm _

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj
inst✝⁶ : Presheaf.IsLocallyInjective J α
inst✝⁵ : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrpCat
φ : M₀.presheaf ⟶ A.obj
inst✝⁴ : Presheaf.IsLocallyInjective J φ
inst✝³ : Presheaf.IsLocallySurjective J φ
inst✝² : J.WEqualsLocallyBijective AddCommGrpCat
M₀' : PresheafOfModules R₀
A' : Sheaf J AddCommGrpCat
φ' : M₀'.presheaf ⟶ A'.obj
inst✝¹ : Presheaf.IsLocallyInjective J φ'
inst✝ : Presheaf.IsLocallySurjective J φ'
τ₀ : M₀ ⟶ M₀'
τ : A ⟶ A'
fac : (toPresheaf R₀).map τ₀ ≫ φ' = φ ≫ τ.hom
X : Cᵒᵖ
r : ↑(R.obj.obj X)
m : ↑((sheafify α φ).val.obj X)
f : (sheafify α φ).val ⟶ (sheafify α φ').val := (sheafifyHomEquiv' α φ ⋯).symm (τ₀ ≫ toSheafify α φ')
⊢ φ ≫ τ.hom = (toPresheaf R₀).map (τ₀ ≫ toSheafify α φ')

Tactic:
rw [← fac, Functor.map_comp, toPresheaf_map_toSheafify]

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

Goal state before:
pp: case a
C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
F : PreZeroHypercover S
Z : C
g : Z ⟶ S
i : (E.sum F).I₀
⊢ (Presieve.ofArrows E.X E.f ⊔ Presieve.ofArrows F.X F.f) ((E.sum F).f i)

Tactic:
cases i

Goal state after:
case a.inl
C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
F : PreZeroHypercover S
Z : C
g : Z ⟶ S
val✝ : E.I₀
⊢ (Presieve.ofArrows E.X E.f ⊔ Presieve.ofArrows F.X F.f) ((E.sum F).f (Sum.inl val✝))

case a.inr
C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
F : PreZeroHypercover S
Z : C
g : Z ⟶ S
val✝ : F.I₀
⊢ (Presieve.ofArrows E.X E.f ⊔ Presieve.ofArrows F.X F.f) ((E.sum F).f (Sum.inr val✝))
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: case refine_1
C : Type u_2
inst✝ : Category.{u_1, u_2} C
J : Precoverage C
K : GrothendieckTopology C
hK : K ∈ {K | ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ K X}
i✝ : C
a✝ : Sieve i✝
hS : a✝ ∈ J.toGrothendieck i✝
⊢ a✝ ∈ K i✝

Tactic:
induction hS with
    | of _ _ hS => exact hK _ hS
    | top => exact K.top_mem _
    | pullback _ _ _ _ _ ih => exact K.pullback_stable _ ih
    | transitive _ _ _ _ _ ih1 ih2 => exact K.transitive ih1 _ ih2

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

Goal state before:
pp: C : Type u_2
inst✝³ : Category.{v_1, u_2} C
J : Precoverage C
inst✝² : J.IsStableUnderBaseChange
inst✝¹ : J.HasPullbacks
inst✝ : J.Small
P : Cᵒᵖ ⥤ Type u_1
h : ∀ ⦃X : C⦄ (E : J.ZeroHypercover X), Presieve.IsSheafFor P E.presieve₀
X : C
E₀ : PreZeroHypercover X
hR : E₀.presieve₀ ∈ J.coverings X
E : J.ZeroHypercover X := { toPreZeroHypercover := E₀, mem₀ := hR }
⊢ Presieve.IsSheafFor P (Sieve.generate E₀.presieve₀).arrows

Tactic:
apply Presieve.isSheafFor_subsieve
      (S := .generate <| (ZeroHypercover.restrictIndexOfSmall.{w} E).presieve₀)

Goal state after:
case h
C : Type u_2
inst✝³ : Category.{v_1, u_2} C
J : Precoverage C
inst✝² : J.IsStableUnderBaseChange
inst✝¹ : J.HasPullbacks
inst✝ : J.Small
P : Cᵒᵖ ⥤ Type u_1
h : ∀ ⦃X : C⦄ (E : J.ZeroHypercover X), Presieve.IsSheafFor P E.presieve₀
X : C
E₀ : PreZeroHypercover X
hR : E₀.presieve₀ ∈ J.coverings X
E : J.ZeroHypercover X := ⋯
⊢ (Sieve.generate E.restrictIndexOfSmall.presieve₀).arrows ≤ (Sieve.generate E₀.presieve₀).arrows

case trans
C : Type u_2
inst✝³ : Category.{v_1, u_2} C
J : Precoverage C
inst✝² : J.IsStableUnderBaseChange
inst✝¹ : J.HasPullbacks
inst✝ : J.Small
P : Cᵒᵖ ⥤ Type u_1
h : ∀ ⦃X : C⦄ (E : J.ZeroHypercover X), Presieve.IsSheafFor P E.presieve₀
X : C
E₀ : PreZeroHypercover X
hR : E₀.presieve₀ ∈ J.coverings X
E : J.ZeroHypercover X := ⋯
⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X),
    Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate E.restrictIndexOfSmall.presieve₀)).arrows
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: C : Type u_3
inst✝¹ : Category.{u_2, u_3} C
D : Type u_1
inst✝ : Category.{v_1, u_1} D
J : GrothendieckTopology C
S✝ T✝ : C
f : S✝ ⟶ T✝
hf : IsIso f
⊢ Presieve.singleton f ∈ J.toPrecoverage.coverings T✝

Tactic:
by simp [mem_toPrecoverage_iff]

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

Goal state before:
pp: C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
F : C ⥤ D
J : Precoverage C
K : Precoverage D
inst✝⁴ : J.IsStableUnderBaseChange
inst✝³ : J.HasPullbacks
inst✝² : K.IsStableUnderBaseChange
inst✝¹ : K.HasPullbacks
inst✝ : J.PullbacksPreservedBy F
h : J ≤ Precoverage.comap F K
x✝ : Sheaf K.toGrothendieck (Type (max u₁ v₁ u₂ v₂))
G : Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
H✝ : Presheaf.IsSheaf K.toGrothendieck G
H : ∀ {X : D}, ∀ R ∈ K.toCoverage.coverings X, Presieve.IsSheafFor G R
X : C
R : Presieve X
hR : R ∈ J.toCoverage.coverings X
this✝ : F.PreservesPairwisePullbacks R
this : R.HasPairwisePullbacks
⊢ Presieve.IsSheafFor (F.op ⋙ { obj := G, property := H✝ }.obj) R

Tactic:
Presieve.IsSheafFor.comp_iff_of_preservesPairwisePullbacks

Goal state after:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
F : C ⥤ D
J : Precoverage C
K : Precoverage D
inst✝⁴ : J.IsStableUnderBaseChange
inst✝³ : J.HasPullbacks
inst✝² : K.IsStableUnderBaseChange
inst✝¹ : K.HasPullbacks
inst✝ : J.PullbacksPreservedBy F
h : J ≤ Precoverage.comap F K
x✝ : Sheaf K.toGrothendieck (Type (max u₁ v₁ u₂ v₂))
G : Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
H✝ : Presheaf.IsSheaf K.toGrothendieck G
H : ∀ {X : D}, ∀ R ∈ K.toCoverage.coverings X, Presieve.IsSheafFor G R
X : C
R : Presieve X
hR : R ∈ J.toCoverage.coverings X
this✝ : F.PreservesPairwisePullbacks R
this : R.HasPairwisePullbacks
⊢ Presieve.IsSheafFor { obj := G, property := H✝ }.obj (Presieve.map F R)
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
i₁ i₂ : E.I₀
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
⊢ ∀ {Y Z : C} {f : Y ⟶ W},
    (∃ j h, f ≫ p₁ = h ≫ E.p₁ j ∧ f ≫ p₂ = h ≫ E.p₂ j) →
      ∀ (g : Z ⟶ Y), ∃ j h, (g ≫ f) ≫ p₁ = h ≫ E.p₁ j ∧ (g ≫ f) ≫ p₂ = h ≫ E.p₂ j

Tactic:
rintro Z Z' g ⟨j, h, fac₁, fac₂⟩ φ

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 : PreOneHypercover S
i₁ i₂ : E.I₀
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
Z Z' : C
g : Z ⟶ W
j : E.I₁ i₁ i₂
h : Z ⟶ E.Y j
fac₁ : g ≫ p₁ = h ≫ E.p₁ j
fac₂ : g ≫ p₂ = h ≫ E.p₂ j
φ : Z' ⟶ Z
⊢ ∃ j h, (φ ≫ g) ≫ p₁ = h ≫ E.p₁ j ∧ (φ ≫ g) ≫ p₂ = h ≫ E.p₂ j
Module: Mathlib.CategoryTheory.Sites.CoverPreserving

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
K : GrothendieckTopology D
F : C ⥤ D
inst✝¹ : F.Full
inst✝ : F.Faithful
hF : {c : C} → {d : D} → (d ⟶ F.obj c) → (c' : C) × (F.obj c' ≅ d)
⊢ CompatiblePreserving K F

Tactic:
constructor
  introv hx he
  obtain ⟨X', e⟩ := hF f₁
  apply (ℱ.1.mapIso e.op).toEquiv.injective
  simp only [Iso.op_hom, Iso.toEquiv_fun, ℱ.1.mapIso_hom, ← Functor.map_comp_apply]
  simpa using!
    hx (F.preimage <| e.hom ≫ f₁) (F.preimage <| e.hom ≫ f₂) hg₁ hg₂
      (F.map_injective <| by simpa using! he)

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
hF₁ : CompatiblePreserving K F
hF₂ : CoverPreserving J K F
G : Sheaf K (Type (max u₁ v₁ u₂ v₂))
X : C
S : Sieve X
hS : S ∈ J X
x : FamilyOfElements (F.op ⋙ G.obj) S.arrows
hx : x.Compatible
⊢ ∃! t, x.IsAmalgamation t

Tactic:
by
    apply existsUnique_of_exists_of_unique
    · have H := (isSheaf_iff_isSheaf_of_type _ _).1 G.2 _ (hF₂.cover_preserve hS)
      exact ⟨H.amalgamate (x.functorPushforward F) (hx.functorPushforward hF₁),
        fun V f hf => (H.isAmalgamation (hx.functorPushforward hF₁) (F.map f) _).trans
          (hF₁.apply_map _ hx hf)⟩
    · intro y₁ y₂ hy₁ hy₂
      apply (((isSheaf_iff_isSheaf_of_type _ _).1 G.2).isSeparated _ (hF₂.cover_preserve hS)).ext
      rintro Y _ ⟨Z, g, h, hg, rfl⟩
      simpa using! congrArg _ ((hy₁ g hg).trans (hy₂ g hg).symm)

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

Goal state before:
pp: case refine_3
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
c : Cofan E.X
hc : IsColimit c
d : Cofan E.Y'
hd : IsColimit d
F : Cᵒᵖ ⥤ A
inst✝¹ : PreservesLimit (Discrete.functor fun i ↦ op (E.X i)) F
inst✝ : PreservesLimit (Discrete.functor fun i ↦ op (E.Y' i)) F
c' : Fan (E.multicospanIndex F).left := Fan.mk (F.obj (op c.pt)) fun i ↦ F.map (c.inj i).op
hc' : IsLimit c' := isLimitFanMkObjOfIsLimit F (fun i ↦ op (E.X i)) (fun i ↦ (c.inj i).op) (Cofan.IsColimit.op hc)
d' : Fan (E.multicospanIndex F).right := Fan.mk (F.obj (op d.pt)) fun i ↦ F.map (d.inj i).op
hd' : IsLimit d' := isLimitFanMkObjOfIsLimit F (fun i ↦ op (E.Y' i)) (fun i ↦ (d.inj i).op) (Cofan.IsColimit.op hd)
⊢ (Iso.refl (((E.multicospanIndex F).multiforkEquivPiForkOfIsLimit hc' hd').functor.obj (E.multifork F)).pt).hom ≫
      (E.forkOfIsColimit hc hd F).ι =
    Fork.ι (((E.multicospanIndex F).multiforkEquivPiForkOfIsLimit hc' hd').functor.obj (E.multifork F)) ≫
      (Iso.refl c'.pt).hom

Tactic:
refine Fan.IsLimit.hom_ext hc' _ _ fun i ↦ ?_

Goal state after:
case refine_3
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
c : Cofan E.X
hc : IsColimit c
d : Cofan E.Y'
hd : IsColimit d
F : Cᵒᵖ ⥤ A
inst✝¹ : PreservesLimit (Discrete.functor fun i ↦ op (E.X i)) F
inst✝ : PreservesLimit (Discrete.functor fun i ↦ op (E.Y' i)) F
c' : Fan (E.multicospanIndex F).left := Fan.mk (F.obj (op c.pt)) fun i ↦ F.map (c.inj i).op
hc' : IsLimit c' := isLimitFanMkObjOfIsLimit F (fun i ↦ op (E.X i)) (fun i ↦ (c.inj i).op) (Cofan.IsColimit.op hc)
d' : Fan (E.multicospanIndex F).right := Fan.mk (F.obj (op d.pt)) fun i ↦ F.map (d.inj i).op
hd' : IsLimit d' := isLimitFanMkObjOfIsLimit F (fun i ↦ op (E.Y' i)) (fun i ↦ (d.inj i).op) (Cofan.IsColimit.op hd)
i : E.multicospanShape.L
⊢ ((Iso.refl (((E.multicospanIndex F).multiforkEquivPiForkOfIsLimit hc' hd').functor.obj (E.multifork F)).pt).hom ≫
        (E.forkOfIsColimit hc hd F).ι) ≫
      c'.proj i =
    (Fork.ι (((E.multicospanIndex F).multiforkEquivPiForkOfIsLimit hc' hd').functor.obj (E.multifork F)) ≫
        (Iso.refl c'.pt).hom) ≫
      c'.proj i
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
  convert! (sheafIso (asIso (whiskerLeft G.op α.hom))).isIso_hom using 1
  ext1
  apply (sheafHom_eq _ _).symm

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:
rintro ⟨W₀, a, ha⟩ ⟨W₀', a', ha'⟩ ⟨T₀, p₁, p₂, fac⟩

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
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
W₀ : C
a : W₀ ⟶ X
ha : (↑⟨G.imageSieve f, ⋯⟩).arrows a
W₀' : C
a' : W₀' ⟶ X
ha' : (↑⟨G.imageSieve f, ⋯⟩).arrows a'
T₀ : C
p₁ : T₀ ⟶ { Y := W₀, f := a, hf := ha }.Y
p₂ : T₀ ⟶ { Y := W₀', f := a', hf := ha' }.Y
fac : p₁ ≫ { Y := W₀, f := a, hf := ha }.f = p₂ ≫ { Y := W₀', f := a', hf := ha' }.f
⊢ (match { Y := W₀, f := a, hf := ha } with
      | { Y := W₀, f := a, hf := ha } => F.obj.map (Exists.choose ha).op) ≫
      F.obj.map { Z := T₀, g₁ := p₁, g₂ := p₂, w := fac }.g₁.op =
    (match { Y := W₀', f := a', hf := ha' } with
      | { Y := W₀, f := a, hf := ha } => F.obj.map (Exists.choose ha).op) ≫
      F.obj.map { Z := T₀, g₁ := p₁, g₂ := p₂, w := fac }.g₂.op
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology

Goal state before:
pp: case transitive
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 R : Sieve X
a✝¹ : (Precoverage.comap G K.toPrecoverage).Saturate X S
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → (Precoverage.comap G K.toPrecoverage).Saturate Y (Sieve.pullback f R)
hS : Sieve.functorPushforward G S ∈ K (G.obj X)
H' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.functorPushforward G (Sieve.pullback f R) ∈ K (G.obj Y)
Y : D
Z : C
g : Z ⟶ X
i : Y ⟶ G.obj Z
hg : S.arrows g
⊢ Sieve.functorPushforward G (Sieve.pullback g R) ≤ Sieve.pullback (G.map g) (Sieve.functorPushforward G R)

Tactic:
rintro W _ ⟨Z', g', i', hg, rfl⟩

Goal state after:
case transitive
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 R : Sieve X
a✝¹ : (Precoverage.comap G K.toPrecoverage).Saturate X S
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → (Precoverage.comap G K.toPrecoverage).Saturate Y (Sieve.pullback f R)
hS : Sieve.functorPushforward G S ∈ K (G.obj X)
H' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.functorPushforward G (Sieve.pullback f R) ∈ K (G.obj Y)
Y : D
Z : C
g : Z ⟶ X
i : Y ⟶ G.obj Z
hg✝ : S.arrows g
W : D
Z' : C
g' : Z' ⟶ Z
i' : W ⟶ G.obj Z'
hg : (Sieve.pullback g R).arrows g'
⊢ (Sieve.pullback (G.map g) (Sieve.functorPushforward G R)).arrows (i' ≫ G.map g')
Module: Mathlib.CategoryTheory.Sites.PreservesLocallyBijective

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : Category.{v_3, u_3} A
J : GrothendieckTopology C
K : GrothendieckTopology D
H : C ⥤ D
F G : Dᵒᵖ ⥤ A
f : F ⟶ G
FA : A → A → Type u_4
CA : A → Type u_5
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
hH : CoverPreserving J K H
inst✝¹ : H.IsCoverDense K
inst✝ : IsLocallyInjective J (H.op.whiskerLeft f)
X : Dᵒᵖ
a b : ToType (F.obj X)
h : (ConcreteCategory.hom (f.app X)) a = (ConcreteCategory.hom (f.app X)) b
Y : D
g : Y ⟶ Opposite.unop X
Z : C
lift : Y ⟶ H.obj Z
m : H.obj Z ⟶ Opposite.unop X
fac : lift ≫ m = g
⊢ Sieve.pullback m (equalizerSieve a b) ∈ K (H.obj Z)

Tactic:
refine K.superset_covering (Sieve.functorPullback_pushforward_le H _) ?_

Goal state after:
C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : Category.{v_3, u_3} A
J : GrothendieckTopology C
K : GrothendieckTopology D
H : C ⥤ D
F G : Dᵒᵖ ⥤ A
f : F ⟶ G
FA : A → A → Type u_4
CA : A → Type u_5
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
hH : CoverPreserving J K H
inst✝¹ : H.IsCoverDense K
inst✝ : IsLocallyInjective J (H.op.whiskerLeft f)
X : Dᵒᵖ
a b : ToType (F.obj X)
h : (ConcreteCategory.hom (f.app X)) a = (ConcreteCategory.hom (f.app X)) b
Y : D
g : Y ⟶ Opposite.unop X
Z : C
lift : Y ⟶ H.obj Z
m : H.obj Z ⟶ Opposite.unop X
fac : lift ≫ m = g
⊢ Sieve.functorPushforward H (Sieve.functorPullback H (Sieve.pullback m (equalizerSieve a b))) ∈ K (H.obj Z)
Module: Mathlib.CategoryTheory.Sites.PreservesLocallyBijective

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : Category.{v_3, u_3} A
J : GrothendieckTopology C
K : GrothendieckTopology D
H : C ⥤ D
F G : Dᵒᵖ ⥤ A
f : F ⟶ G
FA : A → A → Type u_4
CA : A → Type u_5
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
hH : CoverPreserving J K H
inst✝¹ : H.IsCoverDense K
inst✝ : IsLocallySurjective J (H.op.whiskerLeft f)
X : D
a : ToType (G.obj (Opposite.op X))
Y : D
g : Y ⟶ X
Z : C
lift : Y ⟶ H.obj Z
m : H.obj Z ⟶ X
fac : lift ≫ m = g
hh :
  Sieve.functorPushforward H (imageSieve (H.op.whiskerLeft f) ((ConcreteCategory.hom (G.map m.op)) a)) ∈
    K (H.obj (Opposite.unop (Opposite.op Z)))
⊢ Sieve.pullback m (imageSieve f a) ∈ K (H.obj Z)

Tactic:
refine K.superset_covering (Sieve.functorPullback_pushforward_le H _) ?_

Goal state after:
C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : Category.{v_3, u_3} A
J : GrothendieckTopology C
K : GrothendieckTopology D
H : C ⥤ D
F G : Dᵒᵖ ⥤ A
f : F ⟶ G
FA : A → A → Type u_4
CA : A → Type u_5
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
hH : CoverPreserving J K H
inst✝¹ : H.IsCoverDense K
inst✝ : IsLocallySurjective J (H.op.whiskerLeft f)
X : D
a : ToType (G.obj (Opposite.op X))
Y : D
g : Y ⟶ X
Z : C
lift : Y ⟶ H.obj Z
m : H.obj Z ⟶ X
fac : lift ≫ m = g
hh :
  Sieve.functorPushforward H (imageSieve (H.op.whiskerLeft f) ((ConcreteCategory.hom (G.map m.op)) a)) ∈
    K (H.obj (Opposite.unop (Opposite.op Z)))
⊢ Sieve.functorPushforward H (Sieve.functorPullback H (Sieve.pullback m (imageSieve f a))) ∈ K (H.obj Z)