mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Functor.Flat

Goal state before:
pp: case refine_1
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} E
F : C ⥤ D
G : D ⥤ E
X : E
inst✝¹ : RepresentablyFlat F
inst✝ : IsCofiltered (StructuredArrow X G)
T : StructuredArrow X (F ⋙ G) ⥤ StructuredArrow X G := StructuredArrow.pre X F G
Y : StructuredArrow X G
A✝ : StructuredArrow Y.right F
this : Nonempty (StructuredArrow X (F ⋙ G))
A B : StructuredArrow X (F ⋙ G)
U : StructuredArrow X G := IsCofiltered.min (T.obj A) (T.obj B)
A' : StructuredArrow U.right F :=
  StructuredArrow.mk (StructuredArrow.Hom.right (IsCofiltered.minToLeft (T.obj A) (T.obj B)))
B' : StructuredArrow U.right F :=
  StructuredArrow.mk (StructuredArrow.Hom.right (IsCofiltered.minToRight (T.obj A) (T.obj B)))
⊢ ∃ W x x, True

Tactic:
refine ⟨.mk <| U.hom ≫ G.map (IsCofiltered.min A' B').hom,
      StructuredArrow.homMk (IsCofiltered.minToLeft A' B').right ?_,
      StructuredArrow.homMk (IsCofiltered.minToRight A' B').right ?_, trivial⟩

Goal state after:
case refine_1.refine_1
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} E
F : C ⥤ D
G : D ⥤ E
X : E
inst✝¹ : RepresentablyFlat F
inst✝ : IsCofiltered (StructuredArrow X G)
T : StructuredArrow X (F ⋙ G) ⥤ StructuredArrow X G := StructuredArrow.pre X F G
Y : StructuredArrow X G
A✝ : StructuredArrow Y.right F
this : Nonempty (StructuredArrow X (F ⋙ G))
A B : StructuredArrow X (F ⋙ G)
U : StructuredArrow X G := IsCofiltered.min (T.obj A) (T.obj B)
A' : StructuredArrow U.right F :=
  StructuredArrow.mk (StructuredArrow.Hom.right (IsCofiltered.minToLeft (T.obj A) (T.obj B)))
B' : StructuredArrow U.right F :=
  StructuredArrow.mk (StructuredArrow.Hom.right (IsCofiltered.minToRight (T.obj A) (T.obj B)))
⊢ (StructuredArrow.mk (U.hom ≫ G.map (IsCofiltered.min A' B').hom)).hom ≫
      (F ⋙ G).map (StructuredArrow.Hom.right (IsCofiltered.minToLeft A' B')) =
    A.hom

case refine_1.refine_2
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} E
F : C ⥤ D
G : D ⥤ E
X : E
inst✝¹ : RepresentablyFlat F
inst✝ : IsCofiltered (StructuredArrow X G)
T : StructuredArrow X (F ⋙ G) ⥤ StructuredArrow X G := StructuredArrow.pre X F G
Y : StructuredArrow X G
A✝ : StructuredArrow Y.right F
this : Nonempty (StructuredArrow X (F ⋙ G))
A B : StructuredArrow X (F ⋙ G)
U : StructuredArrow X G := IsCofiltered.min (T.obj A) (T.obj B)
A' : StructuredArrow U.right F :=
  StructuredArrow.mk (StructuredArrow.Hom.right (IsCofiltered.minToLeft (T.obj A) (T.obj B)))
B' : StructuredArrow U.right F :=
  StructuredArrow.mk (StructuredArrow.Hom.right (IsCofiltered.minToRight (T.obj A) (T.obj B)))
⊢ (StructuredArrow.mk (U.hom ≫ G.map (IsCofiltered.min A' B').hom)).hom ≫
      (F ⋙ G).map (StructuredArrow.Hom.right (IsCofiltered.minToRight A' B')) =
    B.hom
Module: Mathlib.CategoryTheory.Functor.Flat

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Category.{v_3, u_3} E
F : C ⥤ D
G : D ⥤ E
X : E
inst✝ : RepresentablyCoflat F
h : IsCofiltered (CostructuredArrow G X)ᵒᵖ
⊢ IsCofiltered (CostructuredArrow (F ⋙ G) X)ᵒᵖ

Tactic:
IsCofiltered.iff_of_equivalence
    (costructuredArrowOpEquivalence _ _)

Goal state after:
C : Type u_1
D : Type u_2
E : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Category.{v_3, u_3} E
F : C ⥤ D
G : D ⥤ E
X : E
inst✝ : RepresentablyCoflat F
h : IsCofiltered (StructuredArrow (op X) G.op)
⊢ IsCofiltered (StructuredArrow (op X) (F ⋙ G).op)
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: case mpr.transitive.refine_2
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X Y : C} {f : Y ⟶ X},
    ∀ R ∈ J.coverings X,
      Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
        ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
          x.Compatible → ∃ t, x.IsAmalgamation t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
    x.Compatible → ∃ t, x.IsAmalgamation t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
H2' :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y) (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
        x.Compatible → ∃ t, x.IsAmalgamation t
x : Presieve.FamilyOfElements P (Sieve.pullback f S).arrows
hx : x.Compatible
y : ⦃Z : C⦄ → (g : Z ⟶ Y) → Presieve.FamilyOfElements P (Sieve.pullback (𝟙 Z) (Sieve.pullback (g ≫ f) S)).arrows :=
  fun Z g ZZ gg hgg ↦ x (gg ≫ g) ⋯
⊢ ∃ t, x.IsAmalgamation t

Tactic:
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]

Goal state after:
case mpr.transitive.refine_2
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
hS : J.Saturate X R
h : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → J.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
  ∀ {X Y : C} {f : Y ⟶ X},
    ∀ R ∈ J.coverings X,
      Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.generate R)).arrows ∧
        ∀ (x : Presieve.FamilyOfElements P (Sieve.pullback f (Sieve.generate R)).arrows),
          x.Compatible → ∃ t, x.IsAmalgamation t
H1 : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor P (Sieve.pullback f R).arrows
H1' :
  ∀ ⦃Y : C⦄ (f : Y ⟶ X) (x : Presieve.FamilyOfElements P (Sieve.pullback f R).arrows),
    x.Compatible → ∃ t, x.IsAmalgamation t
H2 :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f → ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y), Presieve.IsSeparatedFor P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows
H2' :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y) (x : Presieve.FamilyOfElements P (Sieve.pullback f_1 (Sieve.pullback f S)).arrows),
        x.Compatible → ∃ t, x.IsAmalgamation t
x : Presieve.FamilyOfElements P (Sieve.pullback f S).arrows
hx : x.Compatible
y : ⦃Z : C⦄ → (g : Z ⟶ Y) → Presieve.FamilyOfElements P (Sieve.pullback (𝟙 Z) (Sieve.pullback (g ≫ f) S)).arrows :=
  fun Z g ZZ gg hgg ↦ x (gg ≫ g) ⋯
hy : ∀ ⦃Z : C⦄ (g : Z ⟶ Y), (y g).Compatible
⊢ ∃ t, x.IsAmalgamation t
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
J : Precoverage C
inst✝ : J.RespectsIso
S : C
E F : PreZeroHypercover S
e : E ≅ F
hE : E.presieve₀ ∈ J.coverings S
Z : C
⊢ ∀ (g : Z ⟶ S), E.presieve₀ g → ∃ Y e, F.presieve₀ (e.hom ≫ g)

Tactic:
_

Goal state after:
case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
J : Precoverage C
inst✝ : J.RespectsIso
S : C
E F : PreZeroHypercover S
e : E ≅ F
hE : E.presieve₀ ∈ J.coverings S
Z : C
g✝ : Z ⟶ S
⊢ E.presieve₀ g✝ → ∃ Y e, F.presieve₀ (e.hom ≫ g✝)
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
J : Precoverage C
inst✝ : J.RespectsIso
S : C
E F : PreZeroHypercover S
e : E ≅ F
hE : E.presieve₀ ∈ J.coverings S
Z : C
⊢ ∀ (g : Z ⟶ S), F.presieve₀ g → ∃ Y e, E.presieve₀ (e.hom ≫ g)

Tactic:
_

Goal state after:
case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
J : Precoverage C
inst✝ : J.RespectsIso
S : C
E F : PreZeroHypercover S
e : E ≅ F
hE : E.presieve₀ ∈ J.coverings S
Z : C
g✝ : Z ⟶ S
⊢ F.presieve₀ g✝ → ∃ Y e, E.presieve₀ (e.hom ≫ g✝)
Module: Mathlib.CategoryTheory.Sites.Coverage

Goal state before:
pp: case mp
C : Type ?u.2
D : Type ?u.4
inst✝¹ : Category.{v_1, ?u.2} C
inst✝ : Category.{v_2, ?u.4} D
K : Coverage C
J : GrothendieckTopology C
H : K.toGrothendieck ≤ J
⊢ K ≤ J.toCoverage

Tactic:
X

Goal state after:
case mp
C : Type ?u.2
D : Type ?u.4
inst✝¹ : Category.{v_1, ?u.2} C
inst✝ : Category.{v_2, ?u.4} D
K : Coverage C
J : GrothendieckTopology C
H : K.toGrothendieck ≤ J
X : C
⊢ K.coverings X ⊆ J.toCoverage.coverings X
Module: Mathlib.CategoryTheory.Sites.Coverage

Goal state before:
pp: case mpr
C : Type ?u.2
D : Type ?u.4
inst✝¹ : Category.{v_1, ?u.2} C
inst✝ : Category.{v_2, ?u.4} D
K : Coverage C
J : GrothendieckTopology C
H : K ≤ J.toCoverage
⊢ K.toGrothendieck ≤ J

Tactic:
X

Goal state after:
case mpr
C : Type ?u.2
D : Type ?u.4
inst✝¹ : Category.{v_1, ?u.2} C
inst✝ : Category.{v_2, ?u.4} D
K : Coverage C
J : GrothendieckTopology C
H : K ≤ J.toCoverage
X : C
⊢ K.toGrothendieck X ⊆ J X
Module: Mathlib.CategoryTheory.Sites.Coverage

Goal state before:
pp: case a
C : Type u_1
inst✝ : Category.{v_1, u_1} C
K : Coverage C
J : GrothendieckTopology C
hJ : J ∈ {J | K ≤ J.toCoverage}
⊢ K.toGrothendieck ≤ J

Tactic:
X

Goal state after:
case a
C : Type u_1
inst✝ : Category.{v_1, u_1} C
K : Coverage C
J : GrothendieckTopology C
hJ : J ∈ {J | K ≤ J.toCoverage}
X : C
⊢ K.toGrothendieck X ⊆ J X
Module: Mathlib.CategoryTheory.Sites.Coverage

Goal state before:
pp: case mp
C : Type u_2
inst✝ : Category.{v_1, u_2} C
K : Coverage C
P : Cᵒᵖ ⥤ Type u_1
H : ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ K.coverings X, IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
⊢ ∀ {X : C}, ∀ R ∈ K.coverings X, IsSheafFor P R

Tactic:
X

Goal state after:
case mp
C : Type u_2
inst✝ : Category.{v_1, u_2} C
K : Coverage C
P : Cᵒᵖ ⥤ Type u_1
H : ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ K.coverings X, IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
X : C
⊢ ∀ R ∈ K.coverings X, IsSheafFor P R
Module: Mathlib.CategoryTheory.Sites.Coverage

Goal state before:
pp: case mpr
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}, ∀ R ∈ K.coverings X, IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows

Tactic:
X

Goal state after:
case mpr
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 : C
⊢ ∀ {Y : C} {f : Y ⟶ X}, ∀ R ∈ K.coverings X, IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
Module: Mathlib.CategoryTheory.Sites.Coverage

Goal state before:
pp: case mpr.h
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
⊢ (Sieve.pullback f (Sieve.generate S)).arrows g → ∃ R, IsSeparatedFor P R ∧ R.FactorsThruAlong T g

Tactic:
_

Goal state after:
case mpr.h
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
a✝ : (Sieve.pullback f (Sieve.generate S)).arrows g
⊢ ∃ R, IsSeparatedFor P R ∧ R.FactorsThruAlong T g
Module: Mathlib.CategoryTheory.Sites.Hypercover.IsSheaf

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
H : J.OneHypercoverFamily
P : Cᵒᵖ ⥤ A
hP : ∀ ⦃X : C⦄ (E : J.OneHypercover X), H E → Nonempty (IsLimit (E.multifork P))
X : C
S : Sieve X
E : J.OneHypercover X
hE : H E
le : E.sieve₀ ≤ S
F : Multifork (Cover.index ⟨S, ⋯⟩ P)
inst✝ : H.IsGenerating
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ Y
⊢ (Sieve.pullback f E.sieve₀).arrows g →
    (lift hP hE le F ≫ P.map f.op) ≫ P.map g.op = F.ι { Y := Y, f := f, hf := hf } ≫ P.map g.op

Tactic:
rintro ⟨T, a, b, ⟨i⟩, fac⟩

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
H : J.OneHypercoverFamily
P : Cᵒᵖ ⥤ A
hP : ∀ ⦃X : C⦄ (E : J.OneHypercover X), H E → Nonempty (IsLimit (E.multifork P))
X : C
S : Sieve X
E : J.OneHypercover X
hE : H E
le : E.sieve₀ ≤ S
F : Multifork (Cover.index ⟨S, ⋯⟩ P)
inst✝ : H.IsGenerating
Y✝ : C
f : Y✝ ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ Y✝
Y : C
i : E.I₀
a : Z ⟶ E.X i
fac : a ≫ E.f i = g ≫ f
⊢ (lift hP hE le F ≫ P.map f.op) ≫ P.map g.op = F.ι { Y := Y✝, f := f, hf := hf } ≫ P.map g.op
Module: Mathlib.CategoryTheory.Sites.Hypercover.IsSheaf

Goal state before:
pp: case mp
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
H : J.OneHypercoverFamily
P : Cᵒᵖ ⥤ A
inst✝ : H.IsGenerating
hP : Presheaf.IsSheaf J P
⊢ ∀ ⦃X : C⦄ (E : J.OneHypercover X), H E → Nonempty (IsLimit (E.multifork P))

Tactic:
X

Goal state after:
case mp
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
H : J.OneHypercoverFamily
P : Cᵒᵖ ⥤ A
inst✝ : H.IsGenerating
hP : Presheaf.IsSheaf J P
X : C
⊢ ∀ (E : J.OneHypercover X), H E → Nonempty (IsLimit (E.multifork P))
Module: Mathlib.CategoryTheory.Sites.Hypercover.IsSheaf

Goal state before:
pp: case mp
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
H : J.OneHypercoverFamily
P : Cᵒᵖ ⥤ A
inst✝ : H.IsGenerating
hP : Presheaf.IsSheaf J P
X : C
E : J.OneHypercover X
⊢ H E → Nonempty (IsLimit (E.multifork P))

Tactic:
_

Goal state after:
case mp
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
H : J.OneHypercoverFamily
P : Cᵒᵖ ⥤ A
inst✝ : H.IsGenerating
hP : Presheaf.IsSheaf J P
X : C
E : J.OneHypercover X
x✝ : H E
⊢ Nonempty (IsLimit (E.multifork P))
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),
    f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂ →
      (ConcreteCategory.hom (ℱ.obj.map f₁.op)) (x g₁ hg₁) = (ConcreteCategory.hom (ℱ.obj.map f₂.op)) (x g₂ hg₂)

Tactic:
X

Goal state after:
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),
    f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂ →
      (ConcreteCategory.hom (ℱ.obj.map f₁.op)) (x g₁ hg₁) = (ConcreteCategory.hom (ℱ.obj.map f₂.op)) (x g₂ hg₂)
Module: Mathlib.CategoryTheory.Sites.CoverLifting

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
F : C ⥤ D
G : D ⥤ E
h : CoverPreserving J L (F ⋙ G)
inst✝² : G.IsCocontinuous K L
inst✝¹ : G.Full
inst✝ : G.Faithful
U : C
S : Sieve U
hS : S ∈ J U
⊢ Sieve.functorPushforward F S ∈ K (F.obj U)

Tactic:
refine K.superset_covering ?_ (G.cover_lift K _ (h.cover_preserve hS))
    rw [Sieve.functorPushforward_comp, Sieve.functorPullback_functorPushforward_eq G]

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

Goal state before:
pp: C : Type uC
inst✝² : Category.{vC, uC} C
D : Type uD
inst✝¹ : Category.{vD, uD} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝ : G.IsLocallyFull K
ℱ : Sheaf K (Type u_2)
X Y : C
i : G.obj X ⟶ G.obj Y
s t : ℱ.obj.obj (op (G.obj X))
h :
  ∀ ⦃Z : C⦄ (j : Z ⟶ X) (f : Z ⟶ Y),
    G.map f = G.map j ≫ i →
      (ConcreteCategory.hom (ℱ.obj.map (G.map j).op)) s = (ConcreteCategory.hom (ℱ.obj.map (G.map j).op)) t
⊢ s = t

Tactic:
apply (((isSheaf_iff_isSheaf_of_type _ _).1 ℱ.property) _
    (G.functorPushforward_imageSieve_mem K i)).isSeparatedFor.ext
  rintro Z _ ⟨W, iWX, iZW, ⟨iWY, e⟩, rfl⟩
  simp [h iWX iWY e]

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
⊢ IsIso ((yoneda.map ((G.op.ranCounit.app Y.obj).app (op U))).app (op X))

Tactic:
rw [isIso_iff_bijective]

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 : A
⊢ Function.Bijective ⇑(ConcreteCategory.hom ((yoneda.map ((G.op.ranCounit.app Y.obj).app (op U))).app (op 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
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),
    f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂ →
      (ConcreteCategory.hom (ℱ.obj.map f₁.op)) (x g₁ hg₁) = (ConcreteCategory.hom (ℱ.obj.map f₂.op)) (x g₂ hg₂)

Tactic:
X

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
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),
    f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂ →
      (ConcreteCategory.hom (ℱ.obj.map f₁.op)) (x g₁ hg₁) = (ConcreteCategory.hom (ℱ.obj.map f₂.op)) (x g₂ hg₂)
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:
rintro ⟨T₀, a, ⟨b, fac₁⟩⟩
      apply Presheaf.IsSheaf.hom_ext F.property
        ⟨_, J.pullback_stable b (imageSieve_mem J K G g)⟩
      rintro ⟨U₀, c, ⟨d, fac₂⟩⟩
      dsimp
      simp only [Category.assoc, ← Functor.map_comp, ← op_comp]
      rw [mapPreimage_map_of_fac K G F (f ≫ g) (c ≫ a) d,
        mapPreimage_map_of_fac K G F f (c ≫ a) (c ≫ b),
        mapPreimage_map_of_fac K G F g (c ≫ b) d]
      all_goals
        simp only [Functor.map_comp, Category.assoc, fac₁, fac₂]

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

Goal state before:
pp: case refine_1
C : Type u
inst✝ : Category.{v, u} C
P✝ : Cᵒᵖ ⥤ Type w
X✝ : C
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ Type w
X : C
S : Sieve X
hS : S ∈ {S | ∀ (Y : C) (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows}
R : Sieve X
hR :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    S.arrows f → Sieve.pullback f R ∈ {S | ∀ (Y_1 : C) (f : Y_1 ⟶ Y), Presieve.IsSheafFor P (Sieve.pullback f S).arrows}
Z : C
g : Z ⟶ X
Y : C
f : Y ⟶ Z
⊢ (Sieve.pullback g R).arrows f → Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.pullback g S)).arrows

Tactic:
_

Goal state after:
case refine_1
C : Type u
inst✝ : Category.{v, u} C
P✝ : Cᵒᵖ ⥤ Type w
X✝ : C
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ Type w
X : C
S : Sieve X
hS : S ∈ {S | ∀ (Y : C) (f : Y ⟶ X), Presieve.IsSheafFor P (Sieve.pullback f S).arrows}
R : Sieve X
hR :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    S.arrows f → Sieve.pullback f R ∈ {S | ∀ (Y_1 : C) (f : Y_1 ⟶ Y), Presieve.IsSheafFor P (Sieve.pullback f S).arrows}
Z : C
g : Z ⟶ X
Y : C
f : Y ⟶ Z
x✝ : (Sieve.pullback g R).arrows f
⊢ Presieve.IsSeparatedFor P (Sieve.pullback f (Sieve.pullback g S)).arrows
Module: Mathlib.CategoryTheory.Sites.Equivalence

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
K : GrothendieckTopology D
e✝ : C ≌ D
G : D ⥤ C
A : Type u₃
inst✝¹ : Category.{v₃, u₃} A
inst✝ : G.IsEquivalence
U : C
e : C ≌ D := G.asEquivalence.symm
Y : C
f : Y ⟶ U
g : e.inverse.obj (e.functor.obj Y) ⟶ U := e.unitInv.app Y ≫ f
this : (Sieve.coverByImage e.inverse U).arrows g
⊢ (Sieve.coverByImage G U).arrows f

Tactic:
replace := Sieve.downward_closed _ this (e.unit.app Y)

Goal state after:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
K : GrothendieckTopology D
e✝ : C ≌ D
G : D ⥤ C
A : Type u₃
inst✝¹ : Category.{v₃, u₃} A
inst✝ : G.IsEquivalence
U : C
e : C ≌ D := G.asEquivalence.symm
Y : C
f : Y ⟶ U
g : e.inverse.obj (e.functor.obj Y) ⟶ U := e.unitInv.app Y ≫ f
this : (Sieve.coverByImage e.inverse U).arrows (e.unit.app Y ≫ g)
⊢ (Sieve.coverByImage G U).arrows f
Module: Mathlib.CategoryTheory.Sites.Equivalence

Goal state before:
pp: case mp
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
h₁ : K.W = ObjectProperty.isLocal fun x ↦ x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj
h₂ : ∀ (R : Sheaf J A), (Function.Bijective fun g ↦ G.op.whiskerLeft f ≫ g) ↔ Function.Bijective fun g ↦ f ≫ g
⊢ ObjectProperty.isLocal (fun x ↦ x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj)
      (((whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).map f) →
    ObjectProperty.isLocal (fun x ↦ x ∈ Set.range (sheafToPresheaf J A).obj) f

Tactic:
rintro h _ ⟨R, rfl⟩

Goal state after:
case mp
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
h₁ : K.W = ObjectProperty.isLocal fun x ↦ x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj
h₂ : ∀ (R : Sheaf J A), (Function.Bijective fun g ↦ G.op.whiskerLeft f ≫ g) ↔ Function.Bijective fun g ↦ f ≫ g
h :
  ObjectProperty.isLocal (fun x ↦ x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj)
    (((whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).map f)
R : Sheaf J A
⊢ Function.Bijective fun g ↦ f ≫ g