awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
P : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
X B : C
π : X ⟶ B
inst_1 : HasPullback π π
⊢ Eq (α :=
    P.obj (op B) ⟶
      equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
        (P.map (Quiver.Hom.op (pullback.snd π π))))
    (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯ ≫
      Iso.inv
        (Types.equalizerIso (P.map (Quiver.Hom.op (pullback.fst π π))) (P.map (Quiver.Hom.op (pullback.snd π π)))))
    (equalizer.lift (P.map π.op) ⋯)
have h : CategoryStruct.comp (Y :=
    equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
      (P.map (Quiver.Hom.op (pullback.snd π π))))
    (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) sorry ≫
      Iso.inv
        (Types.equalizerIso (P.map (Quiver.Hom.op (pullback.fst π π))) (P.map (Quiver.Hom.op (pullback.snd π π)))))
    (equalizer.ι (P.map (Quiver.Hom.op (pullback.fst π π))) (P.map (Quiver.Hom.op (pullback.snd π π)))) =
  CategoryStruct.comp (Y :=
    equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
      (P.map (Quiver.Hom.op (pullback.snd π π))))
    (equalizer.lift (P.map π.op) sorry)
    (equalizer.ι (P.map (Quiver.Hom.op (pullback.fst π π))) (P.map (Quiver.Hom.op (pullback.snd π π))))
Tactic: apply equalizer.hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
P : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
⊢ EqualizerCondition P ↔
    ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst_2 : HasPullback π π],
      IsIso (Y :=
        equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
          (P.map (Quiver.Hom.op (pullback.snd π π))))
        (equalizer.lift (P.map π.op) ⋯)
have mp : EqualizerCondition P →
  ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst_2 : HasPullback π π],
    IsIso (Y :=
      equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
        (P.map (Quiver.Hom.op (pullback.snd π π))))
      (equalizer.lift (P.map π.op) sorry)
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
P : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
mp :
  EqualizerCondition P →
    ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst_2 : HasPullback π π],
      IsIso (Y :=
        equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
          (P.map (Quiver.Hom.op (pullback.snd π π))))
        (equalizer.lift (P.map π.op) ⋯)
⊢ EqualizerCondition P ↔
    ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst_2 : HasPullback π π],
      IsIso (Y :=
        equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
          (P.map (Quiver.Hom.op (pullback.snd π π))))
        (equalizer.lift (P.map π.op) ⋯)
have mpr : (∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst_2 : HasPullback π π],
    IsIso (Y :=
      equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
        (P.map (Quiver.Hom.op (pullback.snd π π))))
      (equalizer.lift (P.map π.op) sorry)) →
  EqualizerCondition P
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
P : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
hP : EqualizerCondition P
X B : C
π : X ⟶ B
inst_1 : EffectiveEpi π
inst_2 : HasPullback π π
⊢ IsIso (Y :=
    equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
      (P.map (Quiver.Hom.op (pullback.snd π π))))
    (equalizer.lift (P.map π.op) ⋯)
have h : Function.Bijective (α := (fun (X : Type u_4) ↦ X) (P.obj (op B))) (β :=
  (fun (X : Type u_4) ↦ X)
    (↑{x : P.obj (op X) |
          Eq (α := (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.fst π π))) :
                  (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) → (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                x :
              (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.snd π π))) :
                  (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) → (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                x :
              (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))} :
      Type u_4))
  (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) sorry)) :
    (a : (fun (X : Type u_4) ↦ X) (P.obj (op B))) →
      (fun (X : Type u_4) ↦ X)
        (↑{x : P.obj (op X) |
              (ConcreteCategory.hom (P.map (pullback.fst π π).op) :
                    (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                      (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                  x =
                (ConcreteCategory.hom (P.map (pullback.snd π π).op) :
                    (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                      (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                  x} :
          Type u_4))
Tactic: have h := hP.bijective_mapToEqualizer_pullback π
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
P : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
hP : EqualizerCondition P
X B : C
π : X ⟶ B
inst_1 : EffectiveEpi π
inst_2 : HasPullback π π
h :
  Function.Bijective (α := (fun (X : Type u_4) ↦ X) (P.obj (op B))) (β :=
    (fun (X : Type u_4) ↦ X)
      (↑{x : P.obj (op X) |
            Eq (α := (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
              ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.fst π π))) :
                    (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                      (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                  x :
                (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
              ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.snd π π))) :
                    (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                      (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                  x :
                (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))} :
        Type u_4))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) :
      (a : (fun (X : Type u_4) ↦ X) (P.obj (op B))) →
        (fun (X : Type u_4) ↦ X)
          (↑{x : P.obj (op X) |
                (ConcreteCategory.hom (P.map (pullback.fst π π).op) :
                      (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                        (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                    x =
                  (ConcreteCategory.hom (P.map (pullback.snd π π).op) :
                      (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                        (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                    x} :
            Type u_4))
⊢ IsIso (Y :=
    equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
      (P.map (Quiver.Hom.op (pullback.snd π π))))
    (equalizer.lift (P.map π.op) ⋯)
have h₁ : IsIso (Y :=
  (↑{x : P.obj (op X) |
        Eq (α := (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.fst π π))) :
                (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) → (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
              x :
            (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.snd π π))) :
                (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) → (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
              x :
            (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))} :
    Type u_4))
  (equalizer.lift (P.map π.op) sorry ≫
    Iso.hom (Types.equalizerIso (P.map (Quiver.Hom.op (pullback.fst π π))) (P.map (Quiver.Hom.op (pullback.snd π π)))))
Tactic: rw [← isIso_iff_bijective, mapToEqualizer_eq_comp] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
P : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
hP :
  ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst_2 : HasPullback π π],
    IsIso (Y :=
      equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
        (P.map (Quiver.Hom.op (pullback.snd π π))))
      (equalizer.lift (P.map π.op) ⋯)
⊢ EqualizerCondition P
have mpr : ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst_2 : HasPullback π π],
  Function.Bijective (α := (fun (X : Type u_4) ↦ X) (P.obj (op B))) (β :=
    (fun (X : Type u_4) ↦ X)
      (↑{x : P.obj (op X) |
            Eq (α := (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
              ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.fst π π))) :
                    (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                      (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                  x :
                (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
              ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.snd π π))) :
                    (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                      (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                  x :
                (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))} :
        Type u_4))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) sorry)) :
      (a : (fun (X : Type u_4) ↦ X) (P.obj (op B))) →
        (fun (X : Type u_4) ↦ X)
          (↑{x : P.obj (op X) |
                (ConcreteCategory.hom (P.map (pullback.fst π π).op) :
                      (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                        (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                    x =
                  (ConcreteCategory.hom (P.map (pullback.snd π π).op) :
                      (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                        (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                    x} :
            Type u_4))
Tactic: apply EqualizerCondition.mk
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
P : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
hP :
  ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst_4 : HasPullback π π],
    IsIso (Y :=
      equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
        (P.map (Quiver.Hom.op (pullback.snd π π))))
      (equalizer.lift (P.map π.op) ⋯)
X B : C
π : X ⟶ B
inst_1 : EffectiveEpi π
inst_2 : HasPullback π π
⊢ Function.Bijective (α := (fun (X : Type u_4) ↦ X) (P.obj (op B))) (β :=
    (fun (X : Type u_4) ↦ X)
      (↑{x : P.obj (op X) |
            Eq (α := (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
              ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.fst π π))) :
                    (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                      (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                  x :
                (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
              ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.snd π π))) :
                    (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                      (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                  x :
                (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))} :
        Type u_4))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) :
      (a : (fun (X : Type u_4) ↦ X) (P.obj (op B))) →
        (fun (X : Type u_4) ↦ X)
          (↑{x : P.obj (op X) |
                (ConcreteCategory.hom (P.map (pullback.fst π π).op) :
                      (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                        (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                    x =
                  (ConcreteCategory.hom (P.map (pullback.snd π π).op) :
                      (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                        (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                    x} :
            Type u_4))
have mpr : IsIso (Y :=
  (↑{x : P.obj (op X) |
        Eq (α := (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.fst π π))) :
                (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) → (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
              x :
            (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.snd π π))) :
                (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) → (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
              x :
            (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))} :
    Type u_4))
  (equalizer.lift (P.map π.op) sorry ≫
    Iso.hom (Types.equalizerIso (P.map (Quiver.Hom.op (pullback.fst π π))) (P.map (Quiver.Hom.op (pullback.snd π π)))))
Tactic: rw [mapToEqualizer_eq_comp, ← isIso_iff_bijective]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
⊢ Initial (D :=
    (Presieve.category.{v_1, u_1} (C := C) (X := B)
        (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))ᵒᵖ)
    (parallelPair (X := op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π ⋯))
      (Y := op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) (c.fst ≫ π) ⋯))
      (Quiver.Hom.op
        (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
          Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.fst ⋯)))
      (Quiver.Hom.op
        (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
          Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.snd ⋯))))
have h₁ : ∀
  (Z :
    (Presieve.category.{v_1, u_1} (C := C) (X := B)
        (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))ᵒᵖ),
  Nonempty.{v_1 + 1} (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π sorry) ⟶ Z)
Tactic: apply Limits.parallelPair_initial_mk
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
h₁ :
  ∀
    (Z :
      (Presieve.category.{v_1, u_1} (C := C) (X := B)
          (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))ᵒᵖ),
    Nonempty.{v_1 + 1} (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π ⋯) ⟶ Z)
⊢ Initial (D :=
    (Presieve.category.{v_1, u_1} (C := C) (X := B)
        (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))ᵒᵖ)
    (parallelPair (X := op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π ⋯))
      (Y := op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) (c.fst ≫ π) ⋯))
      (Quiver.Hom.op
        (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
          Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.fst ⋯)))
      (Quiver.Hom.op
        (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
          Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.snd ⋯))))
have h₂ : ∀
  ⦃Z :
    (Presieve.category.{v_1, u_1} (C := C) (X := B)
        (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))ᵒᵖ⦄
  (i j : op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π sorry) ⟶ Z),
  ∃ (a : op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) (c.fst ≫ π) sorry) ⟶ Z),
    i =
        Quiver.Hom.op
            (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
              Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.fst sorry)) ≫
          a ∧
      j =
        Quiver.Hom.op
            (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
              Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.snd sorry)) ≫
          a
Tactic: apply Limits.parallelPair_initial_mk
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
Z :
  Presieve.category.{v_1, u_1} (C := C) (X := B) (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π))
⊢ Nonempty.{v_1 + 1}
    (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π ⋯) ⟶ op Z)
have h₁ : ∀ ⦃Y : C⦄ (i : Unit)
  (f :
    Z.obj (P := fun (f : Over.{v_1, u_1} B) ↦
          Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom).left ⟶
      X)
  (hh :
    f ≫ π =
      Z.obj (P := fun (f : Over.{v_1, u_1} B) ↦
          Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom).hom),
  Nonempty.{v_1 + 1}
    (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π sorry) ⟶ op Z)
Tactic: obtain ⟨_, f, g, ⟨⟩, hh⟩ := Z.property
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
Z :
  Presieve.category.{v_1, u_1} (C := C) (X := B) (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π))
⊢ ∀ (i j : op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π ⋯) ⟶ op Z),
    ∃ (a :
      op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) (c.fst ≫ π) ⋯) ⟶ op Z),
      i =
          Quiver.Hom.op
              (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
                Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.fst ⋯)) ≫
            a ∧
        j =
          Quiver.Hom.op
              (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
                Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.snd ⋯)) ≫
            a
have h₂ : ∀ (i : unop (op Z) ⟶ unop (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π sorry)))
  (j : op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π sorry) ⟶ op Z),
  ∃ (a : op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) (c.fst ≫ π) sorry) ⟶ op Z),
    op i =
        Quiver.Hom.op
            (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
              Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.fst sorry)) ≫
          a ∧
      j =
        Quiver.Hom.op
            (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
              Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.snd sorry)) ≫
          a
Tactic: ⟨i⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
Z :
  Presieve.category.{v_1, u_1} (C := C) (X := B) (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π))
i : unop (op Z) ⟶ unop (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π ⋯))
⊢ ∀ (j : op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π ⋯) ⟶ op Z),
    ∃ (a :
      op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) (c.fst ≫ π) ⋯) ⟶ op Z),
      op i =
          Quiver.Hom.op
              (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
                Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.fst ⋯)) ≫
            a ∧
        j =
          Quiver.Hom.op
              (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
                Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.snd ⋯)) ≫
            a
have h₂ : ∀
  (j :
    unop (op Z) ⟶ unop (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π sorry))),
  ∃ (a : op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) (c.fst ≫ π) sorry) ⟶ op Z),
    op i =
        Quiver.Hom.op
            (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
              Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.fst sorry)) ≫
          a ∧
      op j =
        Quiver.Hom.op
            (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
              Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.snd sorry)) ≫
          a
Tactic: ⟨j⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
Z :
  Presieve.category.{v_1, u_1} (C := C) (X := B) (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π))
i j : unop (op Z) ⟶ unop (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π ⋯))
⊢ ∃ (a : op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) (c.fst ≫ π) ⋯) ⟶ op Z),
    op i =
        Quiver.Hom.op
            (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
              Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.fst ⋯)) ≫
          a ∧
      op j =
        Quiver.Hom.op
            (ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦
              Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom) (Over.homMk c.snd ⋯)) ≫
          a
have hi : Over.Hom.left i.hom ≫
    (ObjectProperty.FullSubcategory.obj (P := fun (f : Over.{v_1, u_1} B) ↦
        Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom)
        (unop (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π sorry)))).hom =
  (unop (op Z)).obj (P := fun (f : Over.{v_1, u_1} B) ↦
      Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π) f.hom).hom
Tactic: have hi := Over.w i.hom