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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hP :
  ∀ (y : P.obj (op X)),
    (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →
      ∃! x, (ConcreteCategory.hom (P.map π.op)) x = y
hc : IsLimit c
⊢ Function.Bijective ⇑(ConcreteCategory.hom (mapToEqualizer P π c.fst c.snd ⋯))

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hP :
  ∀ (y : P.obj (op X)),
    (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →
      ∃! x, (ConcreteCategory.hom (P.map π.op)) x = y
hc : IsLimit c
⊢ Function.Bijective ⇑(ConcreteCategory.hom (mapToEqualizer P π c.fst c.snd ⋯))
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hP :
  ∀ (y : P.obj (op X)),
    (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →
      ∃! x, (ConcreteCategory.hom (P.map π.op)) x = y
hc : IsLimit c
⊢ ∀ (b : (fun X ↦ X) ↑{x | (ConcreteCategory.hom (P.map c.fst.op)) x = (ConcreteCategory.hom (P.map c.snd.op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π c.fst c.snd ⋯)) a = b

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hP :
  ∀ (y : P.obj (op X)),
    (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →
      ∃! x, (ConcreteCategory.hom (P.map π.op)) x = y
hc : IsLimit c
⊢ ∀ (b : (fun X ↦ X) ↑{x | (ConcreteCategory.hom (P.map c.fst.op)) x = (ConcreteCategory.hom (P.map c.snd.op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π c.fst c.snd ⋯)) a = b
Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits

Goal state before:
pp: A : Type u_1
C : Type u_2
J : Type u_3
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : Category.{v_2, u_2} C
inst✝³ : Category.{v_3, u_3} J
inst✝² : FinitaryExtensive C
inst✝¹ : HasColimitsOfShape J A
inst✝ : PreservesFiniteProducts colim
G : J ⥤ Sheaf (extensiveTopology C) A
⊢ PreservesFiniteProducts (pointwiseCocone (G ⋙ sheafToPresheaf (extensiveTopology C) A)).pt

Tactic:
]

Goal state after:
A : Type u_1
C : Type u_2
J : Type u_3
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : Category.{v_2, u_2} C
inst✝³ : Category.{v_3, u_3} J
inst✝² : FinitaryExtensive C
inst✝¹ : HasColimitsOfShape J A
inst✝ : PreservesFiniteProducts colim
G : J ⥤ Sheaf (extensiveTopology C) A
⊢ PreservesFiniteProducts (pointwiseCocone (G ⋙ sheafToPresheaf (extensiveTopology C) A)).pt
Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits

Goal state before:
pp: A : Type u_1
C : Type u_2
J : Type u_3
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : Category.{v_2, u_2} C
inst✝³ : Category.{v_3, u_3} J
inst✝² : FinitaryExtensive C
inst✝¹ : HasColimitsOfShape J A
inst✝ : PreservesFiniteProducts colim
G : J ⥤ Sheaf (extensiveTopology C) A
i : J
⊢ Presheaf.IsSheaf (extensiveTopology C) ((G ⋙ sheafToPresheaf (extensiveTopology C) A).obj i)

Tactic:
]

Goal state after:
A : Type u_1
C : Type u_2
J : Type u_3
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : Category.{v_2, u_2} C
inst✝³ : Category.{v_3, u_3} J
inst✝² : FinitaryExtensive C
inst✝¹ : HasColimitsOfShape J A
inst✝ : PreservesFiniteProducts colim
G : J ⥤ Sheaf (extensiveTopology C) A
i : J
⊢ Presheaf.IsSheaf (extensiveTopology C) ((G ⋙ sheafToPresheaf (extensiveTopology C) A).obj i)
Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits

Goal state before:
pp: A : Type u_1
C : Type u_2
J : Type u_3
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : Category.{v_2, u_2} C
inst✝³ : Category.{v_3, u_3} J
inst✝² : FinitaryExtensive C
inst✝¹ : HasColimitsOfShape J A
inst✝ : PreservesFiniteProducts colim
G : J ⥤ Sheaf (extensiveTopology C) A
c : Cocone (G ⋙ sheafToPresheaf (extensiveTopology C) A)
hc : IsColimit c
i : c.pt ≅ (G ⋙ sheafToPresheaf (extensiveTopology C) A).flip ⋙ colim :=
  hc.coconePointUniqueUpToIso (pointwiseIsColimit (G ⋙ sheafToPresheaf (extensiveTopology C) A))
⊢ Presheaf.IsSheaf (extensiveTopology C) ((G ⋙ sheafToPresheaf (extensiveTopology C) A).flip ⋙ colim)

Tactic:
]

Goal state after:
A : Type u_1
C : Type u_2
J : Type u_3
inst✝⁵ : Category.{v_1, u_1} A
inst✝⁴ : Category.{v_2, u_2} C
inst✝³ : Category.{v_3, u_3} J
inst✝² : FinitaryExtensive C
inst✝¹ : HasColimitsOfShape J A
inst✝ : PreservesFiniteProducts colim
G : J ⥤ Sheaf (extensiveTopology C) A
c : Cocone (G ⋙ sheafToPresheaf (extensiveTopology C) A)
hc : IsColimit c
i : c.pt ≅ (G ⋙ sheafToPresheaf (extensiveTopology C) A).flip ⋙ colim :=
  hc.coconePointUniqueUpToIso (pointwiseIsColimit (G ⋙ sheafToPresheaf (extensiveTopology C) A))
⊢ Presheaf.IsSheaf (extensiveTopology C) ((G ⋙ sheafToPresheaf (extensiveTopology C) A).flip ⋙ colim)
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hP : Function.Bijective ⇑(ConcreteCategory.hom (mapToEqualizer P π c.fst c.snd ⋯))
hc : IsLimit c
⊢ ∀ (y : P.obj (op X)),
    (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →
      ∃! x, (ConcreteCategory.hom (P.map π.op)) x = y

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hP : Function.Bijective ⇑(ConcreteCategory.hom (mapToEqualizer P π c.fst c.snd ⋯))
hc : IsLimit c
⊢ ∀ (y : P.obj (op X)),
    (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →
      ∃! x, (ConcreteCategory.hom (P.map π.op)) x = y
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hP :
  ∀ (b : (fun X ↦ X) ↑{x | (ConcreteCategory.hom (P.map c.fst.op)) x = (ConcreteCategory.hom (P.map c.snd.op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π c.fst c.snd ⋯)) a = b
hc : IsLimit c
⊢ ∀ (y : P.obj (op X)),
    (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →
      ∃! x, (ConcreteCategory.hom (P.map π.op)) x = y

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hP :
  ∀ (b : (fun X ↦ X) ↑{x | (ConcreteCategory.hom (P.map c.fst.op)) x = (ConcreteCategory.hom (P.map c.snd.op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π c.fst c.snd ⋯)) a = b
hc : IsLimit c
⊢ ∀ (y : P.obj (op X)),
    (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →
      ∃! x, (ConcreteCategory.hom (P.map π.op)) x = y
Module: Mathlib.CategoryTheory.Sites.Coherent.Equivalence

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
inst✝¹ : Precoherent C
A : Type u_3
inst✝ : Category.{v_3, u_3} A
e : C ≌ D
F : Cᵒᵖ ⥤ A
hF : IsSheaf (coherentTopology D) (e.inverse.op ⋙ F)
⊢ IsSheaf (coherentTopology C) (e.functor.op ⋙ e.inverse.op ⋙ F)

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
inst✝¹ : Precoherent C
A : Type u_3
inst✝ : Category.{v_3, u_3} A
e : C ≌ D
F : Cᵒᵖ ⥤ A
hF : IsSheaf (coherentTopology D) (e.inverse.op ⋙ F)
⊢ F ≅ e.functor.op ⋙ e.inverse.op ⋙ F

Tactic:
]

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Precoherent C
A : Type u_3
inst✝ : Category.{v_3, u_3} A
e : C ≌ D
F : Cᵒᵖ ⥤ A
hF : IsSheaf (coherentTopology D) (e.inverse.op ⋙ F)
⊢ IsSheaf (coherentTopology C) (e.functor.op ⋙ e.inverse.op ⋙ F)

C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Precoherent C
A : Type u_3
inst✝ : Category.{v_3, u_3} A
e : C ≌ D
F : Cᵒᵖ ⥤ A
hF : IsSheaf (coherentTopology D) (e.inverse.op ⋙ F)
⊢ F ≅ e.functor.op ⋙ e.inverse.op ⋙ F
Module: Mathlib.CategoryTheory.Sites.Coherent.Equivalence

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
inst✝¹ : Precoherent C
A : Type u_3
inst✝ : Category.{v_3, u_3} A
e : C ≌ D
F : Cᵒᵖ ⥤ A
⊢ IsSheaf (coherentTopology C) F ↔ IsSheaf (coherentTopology D) (e.inverse.op ⋙ F)

Tactic:
by
  refine ⟨fun hF ↦ ((e.sheafCongrPrecoherent A).functor.obj ⟨F, hF⟩).property, fun hF ↦ ?_⟩
  rw [isSheaf_of_iso_iff (P' := e.functor.op ⋙ e.inverse.op ⋙ F)]
  · exact (e.sheafCongrPrecoherent A).inverse.obj ⟨e.inverse.op ⋙ F, hF⟩ |>.property
  · exact Functor.isoWhiskerRight e.op.unitIso F

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Coherent.Equivalence

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
inst✝¹ : Preregular C
A : Type u_3
inst✝ : Category.{v_3, u_3} A
e : C ≌ D
F : Cᵒᵖ ⥤ A
hF : IsSheaf (regularTopology D) (e.inverse.op ⋙ F)
⊢ IsSheaf (regularTopology C) (e.functor.op ⋙ e.inverse.op ⋙ F)

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
inst✝¹ : Preregular C
A : Type u_3
inst✝ : Category.{v_3, u_3} A
e : C ≌ D
F : Cᵒᵖ ⥤ A
hF : IsSheaf (regularTopology D) (e.inverse.op ⋙ F)
⊢ F ≅ e.functor.op ⋙ e.inverse.op ⋙ F

Tactic:
]

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Preregular C
A : Type u_3
inst✝ : Category.{v_3, u_3} A
e : C ≌ D
F : Cᵒᵖ ⥤ A
hF : IsSheaf (regularTopology D) (e.inverse.op ⋙ F)
⊢ IsSheaf (regularTopology C) (e.functor.op ⋙ e.inverse.op ⋙ F)

C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
inst✝¹ : Preregular C
A : Type u_3
inst✝ : Category.{v_3, u_3} A
e : C ≌ D
F : Cᵒᵖ ⥤ A
hF : IsSheaf (regularTopology D) (e.inverse.op ⋙ F)
⊢ F ≅ e.functor.op ⋙ e.inverse.op ⋙ F
Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology

Goal state before:
pp: case mpr.hR
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : FinitaryPreExtensive C
X : C
S : Sieve X
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), S.arrows (π a)
⊢ Nonempty (IsColimit (Cofan.mk X π))

Tactic:
]

Goal state after:
case mpr.hR
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : FinitaryPreExtensive C
X : C
S : Sieve X
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), S.arrows (π a)
⊢ Nonempty (IsColimit (Cofan.mk X π))
Module: Mathlib.CategoryTheory.Sites.Coherent.SheafComparison

Goal state before:
pp: case refine_2
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies
inst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Precoherent D
B : D
⊢ EffectiveEpi (F.effectiveEpiOver B)

Tactic:
]

Goal state after:
case refine_2
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies
inst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Precoherent D
B : D
⊢ EffectiveEpi (F.effectiveEpiOver B)
Module: Mathlib.CategoryTheory.Sites.Coherent.SheafComparison

Goal state before:
pp: case refine_1
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies
inst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Precoherent D
X : C
S : Sieve X
x✝ : ∃ α, ∃ (_ : Finite α), ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ Sieve.functorPushforward F S ∈ (coherentTopology D) (F.obj X)

Tactic:
]

Goal state after:
case refine_1
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies
inst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Precoherent D
X : C
S : Sieve X
x✝ : ∃ α, ∃ (_ : Finite α), ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), S.arrows (π a)
⊢ Sieve.functorPushforward F S ∈ (coherentTopology D) (F.obj X)
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP : Function.Bijective ⇑(ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯))
⊢ ∀ (y : P.obj (op X)),
    (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →
      ∃! x, (ConcreteCategory.hom (P.map π.op)) x = y

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP : Function.Bijective ⇑(ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯))
⊢ ∀ (y : P.obj (op X)),
    (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →
      ∃! x, (ConcreteCategory.hom (P.map π.op)) x = y
Module: Mathlib.CategoryTheory.Sites.Coherent.SheafComparison

Goal state before:
pp: case refine_2
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies
inst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Precoherent D
X : C
S : Sieve X
hS : Sieve.functorPushforward F S ∈ (coherentTopology D) (F.obj X)
⊢ ∃ α, ∃ (_ : Finite α), ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)

Tactic:
]

Goal state after:
case refine_2
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies
inst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Precoherent D
X : C
S : Sieve X
hS : Sieve.functorPushforward F S ∈ (coherentTopology D) (F.obj X)
⊢ ∃ α, ∃ (_ : Finite α), ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP :
  ∀
    (b :
      (fun X ↦ X)
        ↑{x |
            (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =
              (ConcreteCategory.hom (P.map (pullback.snd π π).op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = b
⊢ ∀ (y : P.obj (op X)),
    (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →
      ∃! x, (ConcreteCategory.hom (P.map π.op)) x = y

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP :
  ∀
    (b :
      (fun X ↦ X)
        ↑{x |
            (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =
              (ConcreteCategory.hom (P.map (pullback.snd π π).op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = b
⊢ ∀ (y : P.obj (op X)),
    (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →
      ∃! x, (ConcreteCategory.hom (P.map π.op)) x = y
Module: Mathlib.CategoryTheory.Sites.Coherent.SheafComparison

Goal state before:
pp: case refine_2.refine_2
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies
inst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Precoherent D
X : C
S : Sieve X
hS : Sieve.functorPushforward F S ∈ (coherentTopology D) (F.obj X)
α : Type
w✝ : Finite α
Y : α → D
π : (a : α) → Y a ⟶ F.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), (Sieve.functorPushforward F S).arrows (π a)
Z : α → C := fun a ↦ ⋯.some.p
g₀ : (a : α) → F.obj (Z a) ⟶ Y a := fun a ↦ F.effectiveEpiOver (Y a)
this : EffectiveEpiFamily (fun a ↦ F.obj (Z a)) fun a ↦ g₀ a ≫ π a
a : α
W : C
g₁ : W ⟶ X
g₂ : Y a ⟶ F.obj W
h₁ : S.arrows g₁
h₂ : π a = g₂ ≫ F.map g₁
⊢ S.arrows (F.preimage (g₀ a ≫ g₂ ≫ F.map g₁))

Tactic:
]

Goal state after:
case refine_2.refine_2
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies
inst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Precoherent D
X : C
S : Sieve X
hS : Sieve.functorPushforward F S ∈ (coherentTopology D) (F.obj X)
α : Type
w✝ : Finite α
Y : α → D
π : (a : α) → Y a ⟶ F.obj X
H₁ : EffectiveEpiFamily Y π
H₂ : ∀ (a : α), (Sieve.functorPushforward F S).arrows (π a)
Z : α → C := fun a ↦ ⋯.some.p
g₀ : (a : α) → F.obj (Z a) ⟶ Y a := fun a ↦ F.effectiveEpiOver (Y a)
this : EffectiveEpiFamily (fun a ↦ F.obj (Z a)) fun a ↦ g₀ a ≫ π a
a : α
W : C
g₁ : W ⟶ X
g₂ : Y a ⟶ F.obj W
h₁ : S.arrows g₁
h₂ : π a = g₂ ≫ F.map g₁
⊢ S.arrows (F.preimage (g₀ a ≫ g₂ ≫ F.map g₁))
Module: Mathlib.CategoryTheory.Sites.EpiMono

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁵ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w
inst✝⁴ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝³ : ConcreteCategory A FA
inst✝² : HasFunctorialSurjectiveInjectiveFactorization A
inst✝¹ : J.WEqualsLocallyBijective A
data : FunctorialSurjectiveInjectiveFactorizationData A
inst✝ : HasWeakSheafify J A
x✝ : Arrow (Sheaf J A)
⊢ Presheaf.IsLocallySurjective J
    ((MorphismProperty.FunctorialFactorizationData.functorCategory data Cᵒᵖ).i.app (Arrow.mk x✝.hom.hom))

Tactic:
]

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁵ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w
inst✝⁴ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝³ : ConcreteCategory A FA
inst✝² : HasFunctorialSurjectiveInjectiveFactorization A
inst✝¹ : J.WEqualsLocallyBijective A
data : FunctorialSurjectiveInjectiveFactorizationData A
inst✝ : HasWeakSheafify J A
x✝ : Arrow (Sheaf J A)
⊢ Presheaf.IsLocallySurjective J
    ((MorphismProperty.FunctorialFactorizationData.functorCategory data Cᵒᵖ).i.app (Arrow.mk x✝.hom.hom))
Module: Mathlib.CategoryTheory.Sites.EpiMono

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁵ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w
inst✝⁴ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝³ : ConcreteCategory A FA
inst✝² : HasFunctorialSurjectiveInjectiveFactorization A
inst✝¹ : J.WEqualsLocallyBijective A
data : FunctorialSurjectiveInjectiveFactorizationData A
inst✝ : HasWeakSheafify J A
x✝ : Arrow (Sheaf J A)
⊢ Presheaf.IsLocallyInjective J
    ((MorphismProperty.FunctorialFactorizationData.functorCategory data Cᵒᵖ).p.app (Arrow.mk x✝.hom.hom))

Tactic:
]

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁵ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w
inst✝⁴ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝³ : ConcreteCategory A FA
inst✝² : HasFunctorialSurjectiveInjectiveFactorization A
inst✝¹ : J.WEqualsLocallyBijective A
data : FunctorialSurjectiveInjectiveFactorizationData A
inst✝ : HasWeakSheafify J A
x✝ : Arrow (Sheaf J A)
⊢ Presheaf.IsLocallyInjective J
    ((MorphismProperty.FunctorialFactorizationData.functorCategory data Cᵒᵖ).p.app (Arrow.mk x✝.hom.hom))
Module: Mathlib.CategoryTheory.Sites.Coherent.SheafComparison

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies
inst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Precoherent D
X : C
S : Sieve X
this : Precoherent C
⊢ S ∈ (coherentTopology C) X ↔ ∃ α, ∃ (_ : Finite α), ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)

Tactic:
]

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies
inst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Precoherent D
X : C
S : Sieve X
this : Precoherent C
⊢ S ∈ (coherentTopology C) X ↔ ∃ α, ∃ (_ : Finite α), ∃ Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
Module: Mathlib.CategoryTheory.Sites.Coherent.SheafComparison

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies
inst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Precoherent D
X : C
S : Sieve X
this : Precoherent C
⊢ S ∈ (coherentTopology C) X ↔ S ∈ (coherentTopology C) X

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP :
  ∀
    (b :
      (fun X ↦ X)
        ↑{x |
            (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =
              (ConcreteCategory.hom (P.map (pullback.snd π π).op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = b
b : P.obj (op X)
hb : (ConcreteCategory.hom (P.map c.fst.op)) b = (ConcreteCategory.hom (P.map c.snd.op)) b
h₁ : ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom ≫ c.fst = pullback.fst π π
⊢ (ConcreteCategory.hom (P.map ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom.op))
      ((ConcreteCategory.hom (P.map c.snd.op)) b) =
    (ConcreteCategory.hom (P.map (pullback.snd π π).op)) b

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP :
  ∀
    (b :
      (fun X ↦ X)
        ↑{x |
            (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =
              (ConcreteCategory.hom (P.map (pullback.snd π π).op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = b
b : P.obj (op X)
hb : (ConcreteCategory.hom (P.map c.fst.op)) b = (ConcreteCategory.hom (P.map c.snd.op)) b
h₁ : ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom ≫ c.fst = pullback.fst π π
⊢ (ConcreteCategory.hom (P.map ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom.op))
      ((ConcreteCategory.hom (P.map c.snd.op)) b) =
    (ConcreteCategory.hom (P.map (pullback.snd π π).op)) b