awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : FinitaryPreExtensive.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
h : ∀ (E : D), Presieve.IsSheaf (extensiveTopology C) (F ⋙ Functor.obj coyoneda (op E))
n : ℕ
K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ
c : Cone K
hc : IsLimit c
E : D
⊢ Presieve.IsSheaf (extensiveTopology C) (F ⋙ Functor.obj coyoneda (op E))

Tactic:
specialize h E
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : FinitaryPreExtensive.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
n : ℕ
K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ
c : Cone K
hc : IsLimit c
E : D
h : Presieve.IsSheaf (extensiveTopology C) (F ⋙ Functor.obj coyoneda (op E))
⊢ PreservesFiniteProducts (F ⋙ Functor.obj coyoneda (op E))

Tactic:
rw [Presieve.isSheaf_iff_preservesFiniteProducts] at h

Premises used: CategoryTheory.Presieve.isSheaf_iff_preservesFiniteProducts
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : FinitaryPreExtensive.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
n : ℕ
K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ
c : Cone K
hc : IsLimit c
E : D
h : PreservesFiniteProducts (F ⋙ Functor.obj coyoneda (op E))
⊢ IsLimit (F := (K ⋙ F) ⋙ Functor.obj coyoneda (op E)) (Functor.mapCone (Functor.obj coyoneda (op E)) (F.mapCone c))

Tactic:
exact isLimitOfPreserves (F.comp (coyoneda.obj ⟨E⟩)) hc

Premises used: CategoryTheory.coyoneda, CategoryTheory.Functor.comp, Opposite.op, CategoryTheory.Limits.isLimitOfPreserves, CategoryTheory.Functor.obj
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : FinitaryPreExtensive.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
h : PreservesFiniteProducts F
E : D
mpr : PreservesFiniteProducts (F ⋙ Functor.obj coyoneda (op E))
⊢ Presieve.IsSheaf (extensiveTopology C) (F ⋙ Functor.obj coyoneda (op E))

Tactic:
rw [Presieve.isSheaf_iff_preservesFiniteProducts]

Premises used: CategoryTheory.Presieve.isSheaf_iff_preservesFiniteProducts
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : FinitaryPreExtensive.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
h : PreservesFiniteProducts F
E : D
⊢ PreservesFiniteProducts (F ⋙ Functor.obj coyoneda (op E))

Tactic:
⟨inferInstance⟩

Premises used: inferInstance, CategoryTheory.Limits.PreservesFiniteProducts.mk
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : FinitaryPreExtensive.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
h : PreservesFiniteProducts F
E : D
⊢ PreservesFiniteProducts (F ⋙ Functor.obj coyoneda (op E))

Tactic:
exact ⟨inferInstance⟩
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
inst_2 : FinitaryPreExtensive.{v_1, u_1} C
inst_3 : FinitaryExtensive.{v_1, u_1} C
F : Sheaf.{v_1, v_2, u_1, u_2} (extensiveTopology C) D
⊢ PreservesFiniteProducts (F.obj (P := Presheaf.IsSheaf (extensiveTopology C)))

Tactic:
(Presheaf.isSheaf_iff_preservesFiniteProducts F.obj).mp F.property

Premises used: CategoryTheory.Presheaf.isSheaf_iff_preservesFiniteProducts, CategoryTheory.ObjectProperty.FullSubcategory.property, CategoryTheory.ObjectProperty.FullSubcategory.obj, Iff.mp
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X : C
S : Sieve.{v_1, u_1} X
mp :
  S ∈ (extensiveTopology C : (a : C) → Set (Sieve a)) X →
    ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
      Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)
mpr :
  (∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
      Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)) →
    S ∈ (extensiveTopology C : (a : C) → Set (Sieve a)) X
⊢ S ∈ (extensiveTopology C : (a : C) → Set (Sieve a)) X ↔
    ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
      Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)

Tactic:
constructor
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S_1 : Sieve.{v_1, u_1} X_1
X : C
S : Presieve.{v_1, u_1} X
hS :
  Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
    (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) S
mp :
  ∀ (α : Type) (w : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X) (h : S = Presieve.ofArrows Y π)
    (h' : IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc π)),
    ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
      Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), (Sieve.generate S).arrows (π a)
⊢ ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
    Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), (Sieve.generate S).arrows (π a)

Tactic:
obtain ⟨α, _, Y, π, h, h'⟩ := hS
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S_1 : Sieve.{v_1, u_1} X_1
X : C
S : Presieve.{v_1, u_1} X
α : Type
w : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : S = Presieve.ofArrows Y π
h' : IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc π)
⊢ ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
    Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), (Sieve.generate S).arrows (π a)

Tactic:
⟨α, inferInstance, Y, π, ?_, fun a ↦ ?_⟩

Premises used: inferInstance, And.intro, Exists.intro
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S_1 : Sieve.{v_1, u_1} X_1
X : C
S : Presieve.{v_1, u_1} X
α : Type
w : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : S = Presieve.ofArrows Y π
h' : IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc π)
mp : Nonempty (IsColimit (Cofan.mk X π))
mp₁ : ∀ (a : α), (Sieve.generate S).arrows (π a)
⊢ ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
    Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), (Sieve.generate S).arrows (π a)

Tactic:
refine ⟨α, inferInstance, Y, π, ?_, fun a ↦ ?_⟩
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S_1 : Sieve.{v_1, u_1} X_1
X : C
S : Presieve.{v_1, u_1} X
α : Type
w : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : S = Presieve.ofArrows Y π
h' : IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc π)
⊢ IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc (Cofan.mk X π).inj)

Tactic:
simpa using h'
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S_1 : Sieve.{v_1, u_1} X_1
X : C
S : Presieve.{v_1, u_1} X
α : Type
w : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : S = Presieve.ofArrows Y π
h' : IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc π)
⊢ IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc (Cofan.mk X π).inj)

Tactic:
have : IsIso (Sigma.desc (Cofan.mk X π).inj) := by simpa using h'
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S_1 : Sieve.{v_1, u_1} X_1
X : C
S : Presieve.{v_1, u_1} X
α : Type
w : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : S = Presieve.ofArrows Y π
h' : IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc π)
this : IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc (Cofan.mk X π).inj)
⊢ Nonempty (IsColimit (Cofan.mk X π))

Tactic:
⟨Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X π)⟩

Premises used: Nonempty.intro, CategoryTheory.Limits.Cofan.mk, CategoryTheory.Limits.Cofan.isColimitOfIsIsoSigmaDesc
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S_1 : Sieve.{v_1, u_1} X_1
X : C
S : Presieve.{v_1, u_1} X
α : Type
w : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : S = Presieve.ofArrows Y π
h' : IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc π)
this : IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc (Cofan.mk X π).inj)
⊢ Nonempty (IsColimit (Cofan.mk X π))

Tactic:
exact ⟨Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X π)⟩
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S_1 : Sieve.{v_1, u_1} X_1
X : C
S : Presieve.{v_1, u_1} X
α : Type
w : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : S = Presieve.ofArrows Y π
h' : IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc π)
a : α
mp : (Sieve.generate (Presieve.ofArrows Y π)).arrows (π a)
⊢ (Sieve.generate S).arrows (π a)

Tactic:
obtain ⟨rfl, _⟩ := h

Premises used: rfl
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S : Sieve.{v_1, u_1} X_1
X : C
α : Type
w : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h' : IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc π)
a : α
⊢ 𝟙 Y a ≫ π a = π a

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S : Sieve.{v_1, u_1} X_1
X : C
α : Type
w : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h' : IsIso (X := sigmaObj.{0, v_1, u_1} Y) (Sigma.desc π)
a : α
⊢ (Sieve.generate (Presieve.ofArrows Y π)).arrows (π a)

Tactic:
exact ⟨Y a, 𝟙 Y a, π a, Presieve.ofArrows.mk a, by simp⟩
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S : Sieve.{v_1, u_1} X_1
X : C
⊢ ∀ (a : Unit),
    Sieve.arrows.{v_1, u_1} (C := C) (X := X) ⊤ (Y := (fun (x : Unit) ↦ X) a) ((fun (x : Unit) ↦ (𝟙 X : X ⟶ X)) a)

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S : Sieve.{v_1, u_1} X_1
X : C
mp :
  IsColimit.{0, v_1, 0, u_1} (J := Discrete Unit) (F := Discrete.functor fun (x : Unit) ↦ X)
    (Cofan.mk X fun (x : Unit) ↦ 𝟙 X)
⊢ ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
    Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), Sieve.arrows ⊤ (π a)

Tactic:
refine ⟨Unit, inferInstance, fun _ => X, fun _ => (𝟙 X), ⟨?_⟩, by simp⟩
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S : Sieve.{v_1, u_1} X_1
X : C
⊢ IsIso.{v_1, u_1} (C := C) (X := sigmaObj.{0, v_1, u_1} fun (x : Unit) ↦ X)
    (Iso.hom (coproductUniqueIso fun (x : Unit) ↦ X) : (sigmaObj.{0, v_1, u_1} fun (x : Unit) ↦ X) ⟶ X)

Tactic:
have : IsIso (coproductUniqueIso (fun () => X)).hom := inferInstance
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S : Sieve.{v_1, u_1} X_1
X : C
this :
  IsIso.{v_1, u_1} (C := C) (X := sigmaObj.{0, v_1, u_1} fun (x : Unit) ↦ X)
    (Iso.hom (coproductUniqueIso fun (x : Unit) ↦ X) : (sigmaObj.{0, v_1, u_1} fun (x : Unit) ↦ X) ⟶ X)
⊢ IsIso.{v_1, u_1} (X := sigmaObj.{0, v_1, u_1} fun (b : Unit) ↦ X) (Y :=
    Cocone.pt.{0, v_1, 0, u_1} (J := Discrete Unit) (F := Discrete.functor fun (x : Unit) ↦ X)
      (Cofan.mk X fun (x : Unit) ↦ 𝟙 X))
    (Sigma.desc (Cofan.inj (Cofan.mk X fun (x : Unit) ↦ 𝟙 X)))

Tactic:
this

Premises used: Eq.mpr, CategoryTheory.eqToHom, CategoryTheory.Limits.Cofan.inj, CategoryTheory.Limits.colimit.desc, CategoryTheory.Limits.Cofan.mk
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S : Sieve.{v_1, u_1} X_1
X : C
this :
  IsIso.{v_1, u_1} (C := C) (X := sigmaObj.{0, v_1, u_1} fun (x : Unit) ↦ X)
    (Iso.hom (coproductUniqueIso fun (x : Unit) ↦ X) : (sigmaObj.{0, v_1, u_1} fun (x : Unit) ↦ X) ⟶ X)
⊢ IsIso.{v_1, u_1} (X := sigmaObj.{0, v_1, u_1} fun (b : Unit) ↦ X) (Y :=
    Cocone.pt.{0, v_1, 0, u_1} (J := Discrete Unit) (F := Discrete.functor fun (x : Unit) ↦ X)
      (Cofan.mk X fun (x : Unit) ↦ 𝟙 X))
    (Sigma.desc (Cofan.inj (Cofan.mk X fun (x : Unit) ↦ 𝟙 X)))

Tactic:
exact this
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S : Sieve.{v_1, u_1} X_1
X : C
⊢ IsIso.{v_1, u_1} (X := sigmaObj.{0, v_1, u_1} fun (b : Unit) ↦ X) (Y :=
    Cocone.pt.{0, v_1, 0, u_1} (J := Discrete Unit) (F := Discrete.functor fun (x : Unit) ↦ X)
      (Cofan.mk X fun (x : Unit) ↦ 𝟙 X))
    (Sigma.desc (Cofan.inj (Cofan.mk X fun (x : Unit) ↦ 𝟙 X)))

Tactic:
have : IsIso (Sigma.desc (Cofan.mk X fun (_ : Unit) ↦ 𝟙 X).inj) := by
        have : IsIso (coproductUniqueIso (fun () => X)).hom := inferInstance
        exact this
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S : Sieve.{v_1, u_1} X_1
X : C
this :
  IsIso.{v_1, u_1} (X := sigmaObj.{0, v_1, u_1} fun (b : Unit) ↦ X) (Y :=
    Cocone.pt.{0, v_1, 0, u_1} (J := Discrete Unit) (F := Discrete.functor fun (x : Unit) ↦ X)
      (Cofan.mk X fun (x : Unit) ↦ 𝟙 X))
    (Sigma.desc (Cofan.inj (Cofan.mk X fun (x : Unit) ↦ 𝟙 X)))
⊢ IsColimit.{0, v_1, 0, u_1} (J := Discrete Unit) (F := Discrete.functor fun (x : Unit) ↦ X)
    (Cofan.mk X fun (x : Unit) ↦ 𝟙 X)

Tactic:
exact Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X _)

Premises used: CategoryTheory.Limits.Cofan.mk, CategoryTheory.Limits.Cofan.isColimitOfIsIsoSigmaDesc
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S_1 : Sieve.{v_1, u_1} X_1
X : C
R S : Sieve.{v_1, u_1} X
h : Coverage.Saturate (extensiveCoverage C) X R
h_1 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Coverage.Saturate (extensiveCoverage C) Y (Sieve.pullback f S)
a :
  ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
    Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), R.arrows (π a)
b :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∃ (α : Type) (_ : Finite α) (Y_1 : α → C) (π : (a : α) → Y_1 a ⟶ Y),
        Nonempty (IsColimit (Cofan.mk Y π)) ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
mp :
  ∀ (α : Type) (w : Finite α) (Y₁ : α → C) (π : (a : α) → Y₁ a ⟶ X) (h : Nonempty (IsColimit (Cofan.mk X π)))
    (h' : ∀ (a : α), R.arrows (π a)),
    ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
      Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)
⊢ ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
    Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)

Tactic:
obtain ⟨α, w, Y₁, π, h, h'⟩ := a
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S_1 : Sieve.{v_1, u_1} X_1
X : C
R S : Sieve.{v_1, u_1} X
h_1 : Coverage.Saturate (extensiveCoverage C) X R
h_2 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Coverage.Saturate (extensiveCoverage C) Y (Sieve.pullback f S)
b :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∃ (α : Type) (_ : Finite α) (Y_1 : α → C) (π : (a : α) → Y_1 a ⟶ Y),
        Nonempty (IsColimit (Cofan.mk Y π)) ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Finite α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), R.arrows (π a)
mp :
  ∀ (β : (a : α) → Type),
    (∀ (a : α), Finite (β a)) →
      ∀ (Y_n : (a : α) → β a → C) (π_n : (a : α) → (a_1 : β a) → Y_n a a_1 ⟶ Y₁ a)
        (H :
          ∀ (a : α),
            Nonempty (IsColimit (Cofan.mk (Y₁ a) (π_n a))) ∧
              ∀ (a_1 : β a), (Sieve.pullback (π a) S).arrows (π_n a a_1)),
        ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
          Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)
⊢ ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
    Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)

Tactic:
choose β _ Y_n π_n H using fun a => b (h' a)
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S_1 : Sieve.{v_1, u_1} X_1
X : C
R S : Sieve.{v_1, u_1} X
h_1 : Coverage.Saturate (extensiveCoverage C) X R
h_2 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Coverage.Saturate (extensiveCoverage C) Y (Sieve.pullback f S)
b :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∃ (α : Type) (_ : Finite α) (Y_1 : α → C) (π : (a : α) → Y_1 a ⟶ Y),
        Nonempty (IsColimit (Cofan.mk Y π)) ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Finite α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), R.arrows (π a)
β : (a : α) → Type
h_3 : ∀ (a : α), Finite (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (a_1 : β a) → Y_n a a_1 ⟶ Y₁ a
H :
  ∀ (a : α), Nonempty (IsColimit (Cofan.mk (Y₁ a) (π_n a))) ∧ ∀ (a_1 : β a), (Sieve.pullback (π a) S).arrows (π_n a a_1)
⊢ ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
    Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)

Tactic:
⟨(Σ a, β a), inferInstance, fun ⟨a,b⟩ => Y_n a b, fun ⟨a, b⟩ => (π_n a b) ≫ (π a),
        ⟨Limits.Cofan.isColimitTrans _ h.some _ (fun a ↦ (H a).1.some)⟩,
        fun c => (H c.fst).2 c.snd⟩

Premises used: Quiver.Hom, CategoryTheory.Limits.Cofan.isColimitTrans, _private.0.CategoryTheory.extensiveTopology.mem_sieves_iff_contains_colimit_cofan.match_1, inferInstance, Nonempty.some, Sigma.fst, Nonempty.intro, And.right, And.left, And.intro, Exists.intro, CategoryTheory.CategoryStruct.comp, Sigma.snd, Sigma
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X_1 : C
S_1 : Sieve.{v_1, u_1} X_1
X : C
R S : Sieve.{v_1, u_1} X
h_1 : Coverage.Saturate (extensiveCoverage C) X R
h_2 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Coverage.Saturate (extensiveCoverage C) Y (Sieve.pullback f S)
b :
  ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
    R.arrows f →
      ∃ (α : Type) (_ : Finite α) (Y_1 : α → C) (π : (a : α) → Y_1 a ⟶ Y),
        Nonempty (IsColimit (Cofan.mk Y π)) ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)
α : Type
w : Finite α
Y₁ : α → C
π : (a : α) → Y₁ a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), R.arrows (π a)
β : (a : α) → Type
h_3 : ∀ (a : α), Finite (β a)
Y_n : (a : α) → β a → C
π_n : (a : α) → (a_1 : β a) → Y_n a a_1 ⟶ Y₁ a
H :
  ∀ (a : α), Nonempty (IsColimit (Cofan.mk (Y₁ a) (π_n a))) ∧ ∀ (a_1 : β a), (Sieve.pullback (π a) S).arrows (π_n a a_1)
⊢ ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
    Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)

Tactic:
exact ⟨(Σ a, β a), inferInstance, fun ⟨a,b⟩ => Y_n a b, fun ⟨a, b⟩ => (π_n a b) ≫ (π a),
        ⟨Limits.Cofan.isColimitTrans _ h.some _ (fun a ↦ (H a).1.some)⟩,
        fun c => (H c.fst).2 c.snd⟩
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X : C
S : Sieve.{v_1, u_1} X
h : S ∈ (extensiveTopology C : (a : C) → Set (Sieve a)) X
mp :
  ∀ (X : C) (S : Presieve.{v_1, u_1} X)
    (hS :
      Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
        (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) S),
    ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
      Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), (Sieve.generate S).arrows (π a)
mp₁ :
  ∀ (X : C),
    ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
      Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), Sieve.arrows ⊤ (π a)
mp₂ :
  ∀ (X : C) (R S : Sieve.{v_1, u_1} X),
    Coverage.Saturate (extensiveCoverage C) X R →
      (∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → Coverage.Saturate (extensiveCoverage C) Y (Sieve.pullback f S)) →
        ∀
          (a :
            ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
              Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), R.arrows (π a))
          (b :
            ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,
              R.arrows f →
                ∃ (α : Type) (_ : Finite α) (Y_1 : α → C) (π : (a : α) → Y_1 a ⟶ Y),
                  Nonempty (IsColimit (Cofan.mk Y π)) ∧ ∀ (a : α), (Sieve.pullback f S).arrows (π a)),
          ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
            Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)
⊢ ∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → Y a ⟶ X),
    Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)

Tactic:
induction h with
    | of X S hS =>
      obtain ⟨α, _, Y, π, h, h'⟩ := hS
      refine ⟨α, inferInstance, Y, π, ?_, fun a ↦ ?_⟩
      · have : IsIso (Sigma.desc (Cofan.mk X π).inj) := by simpa using h'
        exact ⟨Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X π)⟩
      · obtain ⟨rfl, _⟩ := h
        exact ⟨Y a, 𝟙 Y a, π a, Presieve.ofArrows.mk a, by simp⟩
    | top X =>
      refine ⟨Unit, inferInstance, fun _ => X, fun _ => (𝟙 X), ⟨?_⟩, by simp⟩
      have : IsIso (Sigma.desc (Cofan.mk X fun (_ : Unit) ↦ 𝟙 X).inj) := by
        have : IsIso (coproductUniqueIso (fun () => X)).hom := inferInstance
        exact this
      exact Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X _)
    | transitive X R S _ _ a b =>
      obtain ⟨α, w, Y₁, π, h, h'⟩ := a
      choose β _ Y_n π_n H using fun a => b (h' a)
      exact ⟨(Σ a, β a), inferInstance, fun ⟨a,b⟩ => Y_n a b, fun ⟨a, b⟩ => (π_n a b) ≫ (π a),
        ⟨Limits.Cofan.isColimitTrans _ h.some _ (fun a ↦ (H a).1.some)⟩,
        fun c => (H c.fst).2 c.snd⟩

Premises used: CategoryTheory.Limits.Cofan.mk, rfl, CategoryTheory.Limits.Cofan.isColimitOfIsIsoSigmaDesc
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X : C
S : Sieve.{v_1, u_1} 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)
mpr : Presieve.ofArrows Y π ≤ S.arrows
mpr₁ :
  Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
    (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) (Presieve.ofArrows Y π)
⊢ S ∈ (extensiveTopology C : (a : C) → Set (Sieve a)) X

Tactic:
apply (extensiveCoverage C).mem_toGrothendieck_sieves_of_superset (R := Presieve.ofArrows Y π)

Premises used: CategoryTheory.Coverage.mem_toGrothendieck_sieves_of_superset, CategoryTheory.extensiveCoverage
-- Module: Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X : C
S : Sieve.{v_1, u_1} 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)
x : C
x_1 : x ⟶ X
hh : Presieve.ofArrows Y π x_1
mk : ∀ (i : α), S.arrows (π i)
⊢ S.arrows x_1

Tactic:
cases hh