awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits (goal)
Goal:
A : Type u_1
C : Type u_2
J : Type u_3
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} C
inst_2 : Category.{v_3, u_3} J
inst_3 : FinitaryExtensive.{v_2, u_2} C
inst_4 : HasColimitsOfShape.{v_3, u_3, v_1, u_1} J A
inst_5 : PreservesFiniteProducts (C := Functor.{v_3, v_1, u_3, u_1} J A) colim
G : J ⥤ Sheaf.{v_2, v_1, u_2, u_1} (extensiveTopology C) A
⊢ PreservesFiniteProducts ((G ⋙ sheafToPresheaf (extensiveTopology C) A).flip ⋙ colim)
have inst : PreservesFiniteProducts (G ⋙ sheafToPresheaf (extensiveTopology C) A).flip
Tactic: apply +allowSynthFailures comp_preservesFiniteProducts
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits (goal)
Goal:
A : Type u_1
C : Type u_2
J : Type u_3
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} C
inst_2 : Category.{v_3, u_3} J
inst_3 : FinitaryExtensive.{v_2, u_2} C
inst_4 : HasColimitsOfShape.{v_3, u_3, v_1, u_1} J A
inst_5 : PreservesFiniteProducts (C := Functor.{v_3, v_1, u_3, u_1} J A) colim
G : J ⥤ Sheaf.{v_2, v_1, u_2, u_1} (extensiveTopology C) A
i : J
⊢ PreservesFiniteProducts ((G ⋙ sheafToPresheaf (extensiveTopology C) A).obj i)
have rw : Presheaf.IsSheaf (extensiveTopology C) ((G ⋙ sheafToPresheaf (extensiveTopology C) A).obj i)
Tactic: rw [← Presheaf.isSheaf_iff_preservesFiniteProducts]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits (hypothesis)
Goal:
A : Type u_1
C : Type u_2
J : Type u_3
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} C
inst_2 : Category.{v_3, u_3} J
inst_3 : FinitaryExtensive.{v_2, u_2} C
inst_4 : HasColimitsOfShape.{v_3, u_3, v_1, u_1} J A
inst_5 : PreservesFiniteProducts (C := Functor.{v_3, v_1, u_3, u_1} J A) colim
G : J ⥤ Sheaf.{v_2, v_1, u_2, u_1} (extensiveTopology C) A
⊢ PreservesFiniteProducts (G ⋙ sheafToPresheaf (extensiveTopology C) A).flip
have this : ∀ (i : J), PreservesFiniteProducts ((G ⋙ sheafToPresheaf (extensiveTopology C) A).obj i)
Tactic: have : ∀ (i : J), PreservesFiniteProducts ((G ⋙ sheafToPresheaf _ A).obj i) := fun i ↦ by
    rw [← Presheaf.isSheaf_iff_preservesFiniteProducts]
    exact (G.obj i).property
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits (goal)
Goal:
A : Type u_1
C : Type u_2
J : Type u_3
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} C
inst_2 : Category.{v_3, u_3} J
inst_3 : FinitaryExtensive.{v_2, u_2} C
inst_4 : HasColimitsOfShape.{v_3, u_3, v_1, u_1} J A
inst_5 : Preadditive.{v_1, u_1} A
x : ℕ
⊢ PreservesLimitsOfShape (C := Functor.{v_3, v_1, u_3, u_1} J A) (Discrete (Fin x)) colim
have inst : PreservesBiproductsOfShape (C := Functor.{v_3, v_1, u_3, u_1} J A) (Fin x) colim
Tactic: apply +allowSynthFailures preservesProductsOfShape_of_preservesBiproductsOfShape
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits (goal)
Goal:
A : Type u_1
C : Type u_2
J : Type u_3
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} C
inst_2 : Category.{v_3, u_3} J
inst_3 : FinitaryExtensive.{v_2, u_2} C
inst_4 : HasColimitsOfShape.{v_3, u_3, v_1, u_1} J A
inst_5 : PreservesFiniteProducts (C := Functor.{v_3, v_1, u_3, u_1} J A) colim
G : J ⥤ Sheaf.{v_2, v_1, u_2, u_1} (extensiveTopology C) A
⊢ PreservesColimit G (sheafToPresheaf (extensiveTopology C) A)
have suffices : CreatesColimit G (sheafToPresheaf (extensiveTopology C) A)
Tactic: suffices CreatesColimit G (sheafToPresheaf (extensiveTopology C) A) from inferInstance
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits (goal)
Goal:
A : Type u_1
C : Type u_2
J : Type u_3
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} C
inst_2 : Category.{v_3, u_3} J
inst_3 : FinitaryExtensive.{v_2, u_2} C
inst_4 : HasColimitsOfShape.{v_3, u_3, v_1, u_1} J A
inst_5 : PreservesFiniteProducts (C := Functor.{v_3, v_1, u_3, u_1} J A) colim
G : J ⥤ Sheaf.{v_2, v_1, u_2, u_1} (extensiveTopology C) A
⊢ CreatesColimit G (sheafToPresheaf (extensiveTopology C) A)
have refine : ∀ (c : Cocone (G ⋙ sheafToPresheaf (extensiveTopology C) A)) (hc : IsColimit c),
  Presheaf.IsSheaf (extensiveTopology C) c.pt
Tactic: refine createsColimitOfIsSheaf _ (fun c hc ↦ ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits (goal)
Goal:
A : Type u_1
C : Type u_2
J : Type u_3
inst : Category.{v_1, u_1} A
inst_1 : Category.{v_2, u_2} C
inst_2 : Category.{v_3, u_3} J
inst_3 : FinitaryExtensive.{v_2, u_2} C
inst_4 : HasColimitsOfShape.{v_3, u_3, v_1, u_1} J A
inst_5 : PreservesFiniteProducts (C := Functor.{v_3, v_1, u_3, u_1} J A) colim
G : J ⥤ Sheaf.{v_2, v_1, u_2, u_1} (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) c.pt
have rw : Presheaf.IsSheaf (extensiveTopology C) ((G ⋙ sheafToPresheaf (extensiveTopology C) A).flip ⋙ colim)
Tactic: rw [Presheaf.isSheaf_of_iso_iff i]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
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
X : C
S : Presieve.{v_1, u_1} X
inst_3 : S.Extensive
⊢ ∀ {Y Z : C} {f : Y ⟶ X}, S f → ∀ {g : Z ⟶ X}, S g → HasPullback f g
have obtain : ∀ (w : Type),
  Finite w →
    ∀ (w_2 : w → C) (w_3 : (a : w) → w_2 a ⟶ X) [(ofArrows w_2 w_3).Extensive] (hc : IsColimit (Cofan.mk X w_3))
      {Y Z : C} {f : Y ⟶ X}, ofArrows w_2 w_3 f → ∀ {g : Z ⟶ X}, ofArrows w_2 w_3 g → HasPullback f g
Tactic: obtain ⟨_, _, _, _, rfl, ⟨hc⟩⟩ := Presieve.Extensive.arrows_nonempty_isColimit (R := S)
Cleared hypotheses: S, inst_3
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
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
X : C
w : Type
w_1 : Finite w
w_2 : w → C
w_3 : (a : w) → w_2 a ⟶ X
inst_3 : (ofArrows w_2 w_3).Extensive
hc : IsColimit (Cofan.mk X w_3)
Y Z : C
f : Y ⟶ X
h : ofArrows w_2 w_3 f
g : Z ⟶ X
hg : ofArrows w_2 w_3 g
⊢ HasPullback f g
have mk : ∀ (i : w), HasPullback f (w_3 i)
Tactic: cases hg
Cleared hypotheses: Z, g, hg
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X : C
S : Presieve.{v_1, u_1} X
inst_2 : S.Extensive
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
inst_3 : PreservesFiniteProducts F
⊢ IsSheafFor F S
have obtain : ∀ (α : Type),
  Finite α →
    ∀ (Z : α → C) (π : (a : α) → Z a ⟶ X) [(ofArrows Z π).Extensive] (hc : IsColimit (Cofan.mk X π)),
      IsSheafFor F (ofArrows Z π)
Tactic: obtain ⟨α, _, Z, π, rfl, ⟨hc⟩⟩ := Extensive.arrows_nonempty_isColimit (R := S)
Cleared hypotheses: S, inst_2
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X : C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
inst_2 : PreservesFiniteProducts F
α : Type
w : Finite α
Z : α → C
π : (a : α) → Z a ⟶ X
inst_3 : (ofArrows Z π).Extensive
hc : IsColimit (Cofan.mk X π)
⊢ IsSheafFor F (ofArrows Z π)
have this : (ofArrows Z (Cofan.mk X π).inj).HasPairwisePullbacks
Tactic: have : (ofArrows Z (Cofan.mk X π).inj).HasPairwisePullbacks :=
    (inferInstance : (ofArrows Z π).HasPairwisePullbacks)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
X : C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
inst_2 : PreservesFiniteProducts F
α : Type
w : Finite α
Z : α → C
π : (a : α) → Z a ⟶ X
inst_3 : (ofArrows Z π).Extensive
hc : IsColimit (Cofan.mk X π)
this : (ofArrows Z (Cofan.mk X π).inj).HasPairwisePullbacks
⊢ IsSheafFor F (ofArrows Z π)
have intro : ∀ (val : Fintype α), IsSheafFor F (ofArrows Z π)
Tactic: cases nonempty_fintype α
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
W : C
⊢ IsSheaf.{v_1, v_1, u_1} (extensiveTopology C) (Functor.obj yoneda W)
have rw : ∀ {X : C} (R : Presieve.{v_1, u_1} X),
  Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
      (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) R →
    IsSheafFor (Functor.obj yoneda W) R
Tactic: rw [extensiveTopology, isSheaf_coverage]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
W X : C
R : Presieve.{v_1, u_1} X
⊢ Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
      (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) R →
    IsSheafFor (Functor.obj yoneda W) R
have this : ∀ (Y : Type) (α : Finite Y) (Z : Y → C) (π : (a : Y) → Z a ⟶ X) (hR : R = ofArrows Z π)
  (hi : IsIso (X := sigmaObj.{0, v_1, u_1} Z) (Sigma.desc π)), IsSheafFor (Functor.obj yoneda W) R
Tactic: ⟨Y, α, Z, π, hR, hi⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
W X : C
R : Presieve.{v_1, u_1} X
Y : Type
α : Finite Y
Z : Y → C
π : (a : Y) → Z a ⟶ X
hR : R = ofArrows Z π
hi : IsIso (X := sigmaObj.{0, v_1, u_1} Z) (Sigma.desc π)
⊢ IsSheafFor (Functor.obj yoneda W) R
have this : IsIso (X := sigmaObj.{0, v_1, u_1} Z) (Sigma.desc (Cofan.mk X π).inj)
Tactic: have : IsIso (Sigma.desc (Cofan.inj (Cofan.mk X π))) := hi
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
W X : C
R : Presieve.{v_1, u_1} X
Y : Type
α : Finite Y
Z : Y → C
π : (a : Y) → Z a ⟶ X
hR : R = ofArrows Z π
hi : IsIso (X := sigmaObj.{0, v_1, u_1} Z) (Sigma.desc π)
this : IsIso (X := sigmaObj.{0, v_1, u_1} Z) (Sigma.desc (Cofan.mk X π).inj)
⊢ IsSheafFor (Functor.obj yoneda W) R
have this₁ : R.Extensive
Tactic: have : R.Extensive := ⟨Y, α, Z, π, hR, ⟨Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X π)⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
inst_2 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
⊢ IsSheaf (extensiveTopology C) F ↔ PreservesFiniteProducts F
have refine_1 : ∀ (hF : IsSheaf (extensiveTopology C) F) (n : ℕ) {K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ},
  PreservesLimit K F
Tactic: refine ⟨fun hF ↦ ⟨fun n ↦ ⟨fun {K} ↦ ?_⟩⟩, fun hF ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
inst_2 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
refine_1 :
  ∀ (hF : IsSheaf (extensiveTopology C) F) (n : ℕ) {K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ},
    PreservesLimit K F
⊢ IsSheaf (extensiveTopology C) F ↔ PreservesFiniteProducts F
have refine_2 : ∀ (hF : PreservesFiniteProducts F), IsSheaf (extensiveTopology C) F
Tactic: refine ⟨fun hF ↦ ⟨fun n ↦ ⟨fun {K} ↦ ?_⟩⟩, fun hF ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
inst_2 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
hF : IsSheaf (extensiveTopology C) F
n : ℕ
K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ
⊢ PreservesLimit K F
have refine_1 : ∀
  (hF :
    ∀ {X : C} (R : Presieve.{v_1, u_1} X),
      Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
          (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) R →
        IsSheafFor F R)
  (n : ℕ) {K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ}, PreservesLimit K F
Tactic: rw [extensiveTopology, isSheaf_coverage] at hF
Cleared hypotheses: hF, n, K
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
inst_2 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
hF :
  ∀ {X : C} (R : Presieve.{v_1, u_1} X),
    Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
        (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) R →
      IsSheafFor F R
n : ℕ
K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ
Z : Fin n → C := fun (i : Fin n) ↦ unop (K.obj { as := i })
this : (ofArrows Z (Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj).HasPairwisePullbacks
⊢ PreservesLimit K F
have this₁ : ∀ (i : Fin n), Mono ((Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj i)
Tactic: have : ∀ (i : Fin n), Mono (Cofan.inj (Cofan.mk (∐ Z) (Sigma.ι Z)) i) :=
      inferInstanceAs <| ∀ (i : Fin n), Mono (Sigma.ι Z i)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
inst_2 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
hF :
  ∀ {X : C} (R : Presieve.{v_1, u_1} X),
    Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
        (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) R →
      IsSheafFor F R
n : ℕ
K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ
Z : Fin n → C := fun (i : Fin n) ↦ unop (K.obj { as := i })
this_1 : (ofArrows Z (Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj).HasPairwisePullbacks
this : ∀ (i : Fin n), Mono ((Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj i)
i : K ≅ Discrete.functor fun (i : Fin n) ↦ op (Z i) := Discrete.natIsoFunctor
refine_1 : PreservesLimit K F
⊢ PreservesLimit K F
have refine_1₁ : IsSheafFor (X := initial.{v_1, u_1} C) F
  (ofArrows Empty.elim fun (a : Empty) ↦
    Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ initial.{v_1, u_1} C) a)
Tactic: let _ : PreservesLimit (Discrete.functor (fun i ↦ op (Z i))) F :=
        Presieve.preservesProduct_of_isSheafFor F ?_ initialIsInitial _ (coproductIsCoproduct Z)
        (FinitaryExtensive.isPullback_initial_to_sigma_ι Z)
        (hF (Presieve.ofArrows Z (fun i ↦ Sigma.ι Z i)) ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
inst_2 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
hF :
  ∀ {X : C} (R : Presieve.{v_1, u_1} X),
    Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
        (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) R →
      IsSheafFor F R
n : ℕ
K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ
Z : Fin n → C := fun (i : Fin n) ↦ unop (K.obj { as := i })
this_1 : (ofArrows Z (Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj).HasPairwisePullbacks
this : ∀ (i : Fin n), Mono ((Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj i)
i : K ≅ Discrete.functor fun (i : Fin n) ↦ op (Z i) := Discrete.natIsoFunctor
refine_1 : PreservesLimit K F
refine_1₁ :
  IsSheafFor (X := initial.{v_1, u_1} C) F
    (ofArrows Empty.elim fun (a : Empty) ↦
      Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ initial.{v_1, u_1} C) a)
⊢ PreservesLimit K F
have refine_1₂ : Membership.mem (γ := Set (Presieve.{v_1, u_1} (sigmaObj.{0, v_1, u_1} Z)))
  (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) (sigmaObj.{0, v_1, u_1} Z))
  (ofArrows Z fun (i : Fin n) ↦ Sigma.ι Z i)
Tactic: let _ : PreservesLimit (Discrete.functor (fun i ↦ op (Z i))) F :=
        Presieve.preservesProduct_of_isSheafFor F ?_ initialIsInitial _ (coproductIsCoproduct Z)
        (FinitaryExtensive.isPullback_initial_to_sigma_ι Z)
        (hF (Presieve.ofArrows Z (fun i ↦ Sigma.ι Z i)) ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
inst_2 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
hF :
  ∀ {X : C} (R : Presieve.{v_1, u_1} X),
    Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
        (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) R →
      IsSheafFor F R
n : ℕ
K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ
Z : Fin n → C := fun (i : Fin n) ↦ unop (K.obj { as := i })
this_1 : (ofArrows Z (Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj).HasPairwisePullbacks
this : ∀ (i : Fin n), Mono ((Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj i)
i : K ≅ Discrete.functor fun (i : Fin n) ↦ op (Z i) := Discrete.natIsoFunctor
⊢ IsSheafFor (X := initial.{v_1, u_1} C) F
    (ofArrows Empty.elim fun (a : Empty) ↦
      Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ initial.{v_1, u_1} C) a)
have refine_1 : Membership.mem (γ := Set (Presieve.{v_1, u_1} (initial.{v_1, u_1} C)))
  (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) (initial.{v_1, u_1} C))
  (ofArrows Empty.elim fun (a : Empty) ↦
    Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ initial.{v_1, u_1} C) a)
Tactic: apply hF
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
inst_2 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
hF :
  ∀ {X : C} (R : Presieve.{v_1, u_1} X),
    Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
        (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) R →
      IsSheafFor F R
n : ℕ
K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ
Z : Fin n → C := fun (i : Fin n) ↦ unop (K.obj { as := i })
this_1 : (ofArrows Z (Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj).HasPairwisePullbacks
this : ∀ (i : Fin n), Mono ((Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj i)
i : K ≅ Discrete.functor fun (i : Fin n) ↦ op (Z i) := Discrete.natIsoFunctor
⊢ Membership.mem (γ := Set (Presieve.{v_1, u_1} (initial.{v_1, u_1} C)))
    (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) (initial.{v_1, u_1} C))
    (ofArrows Empty.elim fun (a : Empty) ↦
      Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ initial.{v_1, u_1} C) a)
have refine_1 : Eq.{v_1 + 1} (α :=
  Quiver.Hom.{v_1, u_1} (V := C) (sigmaObj.{0, v_1, u_1} Empty.elim) (sigmaObj.{0, v_1, u_1} Empty.elim))
  (CategoryStruct.comp (Y := initial.{v_1, u_1} C)
    (Sigma.desc fun (a : Empty) ↦ IsEmpty.elim sorry (p := fun (x : Empty) ↦ x.elim ⟶ initial.{v_1, u_1} C) a) default)
  (𝟙 (sigmaObj.{0, v_1, u_1} Empty.elim))
Tactic: refine ⟨Empty, inferInstance, Empty.elim, IsEmpty.elim inferInstance, rfl, ⟨default,?_, ?_⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
inst_2 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
hF :
  ∀ {X : C} (R : Presieve.{v_1, u_1} X),
    Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
        (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) R →
      IsSheafFor F R
n : ℕ
K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ
Z : Fin n → C := fun (i : Fin n) ↦ unop (K.obj { as := i })
this_1 : (ofArrows Z (Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj).HasPairwisePullbacks
this : ∀ (i : Fin n), Mono ((Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj i)
i : K ≅ Discrete.functor fun (i : Fin n) ↦ op (Z i) := Discrete.natIsoFunctor
refine_1 :
  Eq.{v_1 + 1} (α :=
    Quiver.Hom.{v_1, u_1} (V := C) (sigmaObj.{0, v_1, u_1} Empty.elim) (sigmaObj.{0, v_1, u_1} Empty.elim))
    (CategoryStruct.comp (Y := initial.{v_1, u_1} C)
      (Sigma.desc fun (a : Empty) ↦ IsEmpty.elim ⋯ (p := fun (x : Empty) ↦ x.elim ⟶ initial.{v_1, u_1} C) a) default)
    (𝟙 (sigmaObj.{0, v_1, u_1} Empty.elim))
⊢ Membership.mem (γ := Set (Presieve.{v_1, u_1} (initial.{v_1, u_1} C)))
    (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) (initial.{v_1, u_1} C))
    (ofArrows Empty.elim fun (a : Empty) ↦
      Empty.instIsEmpty.elim (p := fun (x : Empty) ↦ x.elim ⟶ initial.{v_1, u_1} C) a)
have refine_1₁ : Eq.{v_1 + 1} (α := initial.{v_1, u_1} C ⟶ initial.{v_1, u_1} C)
  (CategoryStruct.comp (Y := sigmaObj.{0, v_1, u_1} Empty.elim) default
    (Sigma.desc fun (a : Empty) ↦ IsEmpty.elim sorry (p := fun (x : Empty) ↦ x.elim ⟶ initial.{v_1, u_1} C) a))
  (𝟙 (initial.{v_1, u_1} C))
Tactic: refine ⟨Empty, inferInstance, Empty.elim, IsEmpty.elim inferInstance, rfl, ⟨default,?_, ?_⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
inst_2 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
hF :
  ∀ {X : C} (R : Presieve.{v_1, u_1} X),
    Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
        (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) R →
      IsSheafFor F R
n : ℕ
K : Functor.{0, v_1, 0, u_1} (Discrete (Fin n)) Cᵒᵖ
Z : Fin n → C := fun (i : Fin n) ↦ unop (K.obj { as := i })
this_1 : (ofArrows Z (Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj).HasPairwisePullbacks
this : ∀ (i : Fin n), Mono ((Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)).inj i)
i : K ≅ Discrete.functor fun (i : Fin n) ↦ op (Z i) := Discrete.natIsoFunctor
⊢ Eq.{v_1 + 1} (α :=
    Quiver.Hom.{v_1, u_1} (V := C) (sigmaObj.{0, v_1, u_1} Empty.elim) (sigmaObj.{0, v_1, u_1} Empty.elim))
    (CategoryStruct.comp (Y := initial.{v_1, u_1} C)
      (Sigma.desc fun (a : Empty) ↦ IsEmpty.elim ⋯ (p := fun (x : Empty) ↦ x.elim ⟶ initial.{v_1, u_1} C) a) default)
    (𝟙 (sigmaObj.{0, v_1, u_1} Empty.elim))
have refine_1 : ∀ (b : Empty),
  Eq.{v_1 + 1} (α := Quiver.Hom.{v_1, u_1} (V := C) b.elim (sigmaObj.{0, v_1, u_1} Empty.elim))
    (CategoryStruct.comp (Y := sigmaObj.{0, v_1, u_1} Empty.elim) (Sigma.ι Empty.elim b)
      (CategoryStruct.comp (Y := initial.{v_1, u_1} C)
        (Sigma.desc fun (a : Empty) ↦ IsEmpty.elim sorry (p := fun (x : Empty) ↦ x.elim ⟶ initial.{v_1, u_1} C) a) default))
    (Sigma.ι Empty.elim b ≫ 𝟙 (sigmaObj.{0, v_1, u_1} Empty.elim))
Tactic: ext b
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
inst_2 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
hF : PreservesFiniteProducts F
⊢ IsSheaf (extensiveTopology C) F
have refine_2 : ∀ {X : C} (R : Presieve.{v_1, u_1} X),
  Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
      (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) R →
    IsSheafFor F R
Tactic: rw [extensiveTopology, Presieve.isSheaf_coverage]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : FinitaryPreExtensive.{v_1, u_1} C
inst_2 : FinitaryExtensive.{v_1, u_1} C
F : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
hF : PreservesFiniteProducts F
X : C
R : Presieve.{v_1, u_1} X
⊢ Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
      (Precoverage.coverings (Coverage.toPrecoverage (extensiveCoverage C)) X) R →
    IsSheafFor F R
have refine_2 : ∀ (Y : Type) (α : Finite Y) (Z : Y → C) (π : (a : Y) → Z a ⟶ X) (hR : R = ofArrows Z π)
  (hi : IsIso (X := sigmaObj.{0, v_1, u_1} Z) (Sigma.desc π)), IsSheafFor F R
Tactic: ⟨Y, α, Z, π, hR, hi⟩