awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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 ⋯))
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 ⋯)))).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
⊢ ∃ (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 hj : Over.Hom.left j.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 hj := Over.w j.hom
-- 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 : unop (op Z) ⟶ unop (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π ⋯))
hi :
  Over.Hom.left i.hom ≫ π =
    Z.obj (P := fun (f : Over.{v_1, u_1} B) ↦
        ∃ (Y : C) (h : f.left ⟶ Y) (g : Y ⟶ B),
          ofArrows (fun (x : Unit) ↦ X) (fun (x : Unit) ↦ π) g ∧ h ≫ g = f.hom).hom
hj :
  Over.Hom.left j.hom ≫ π =
    Z.obj (P := fun (f : Over.{v_1, u_1} B) ↦
        ∃ (Y : C) (h : f.left ⟶ Y) (g : Y ⟶ B),
          ofArrows (fun (x : Unit) ↦ X) (fun (x : Unit) ↦ π) g ∧ h ≫ g = f.hom).hom
ij : (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).left ⟶
  c.pt :=
  PullbackCone.IsLimit.lift hc (Over.Hom.left i.hom) (Over.Hom.left j.hom) ⋯
⊢ ∃ (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 h₂ : op i =
  CategoryStruct.comp (Y :=
    op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) (c.fst ≫ π) sorry))
    (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)))
    (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 ij sorry)))
Tactic: refine ⟨Quiver.Hom.op (ObjectProperty.homMk (Over.homMk ij)), ?_, ?_⟩
-- 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 : unop (op Z) ⟶ unop (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π ⋯))
hi :
  Over.Hom.left i.hom ≫ π =
    Z.obj (P := fun (f : Over.{v_1, u_1} B) ↦
        ∃ (Y : C) (h : f.left ⟶ Y) (g : Y ⟶ B),
          ofArrows (fun (x : Unit) ↦ X) (fun (x : Unit) ↦ π) g ∧ h ≫ g = f.hom).hom
hj :
  Over.Hom.left j.hom ≫ π =
    Z.obj (P := fun (f : Over.{v_1, u_1} B) ↦
        ∃ (Y : C) (h : f.left ⟶ Y) (g : Y ⟶ B),
          ofArrows (fun (x : Unit) ↦ X) (fun (x : Unit) ↦ π) g ∧ h ≫ g = f.hom).hom
ij : (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).left ⟶
  c.pt :=
  PullbackCone.IsLimit.lift hc (Over.Hom.left i.hom) (Over.Hom.left j.hom) ⋯
h₂ :
  op i =
    CategoryStruct.comp (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 ij ⋯)))
⊢ ∃ (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 h₃ : op j =
  CategoryStruct.comp (Y :=
    op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) (c.fst ≫ π) sorry))
    (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)))
    (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 ij sorry)))
Tactic: refine ⟨Quiver.Hom.op (ObjectProperty.homMk (Over.homMk ij)), ?_, ?_⟩
-- 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 : unop (op Z) ⟶ unop (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π ⋯))
hi :
  Over.Hom.left i.hom ≫ π =
    Z.obj (P := fun (f : Over.{v_1, u_1} B) ↦
        ∃ (Y : C) (h : f.left ⟶ Y) (g : Y ⟶ B),
          ofArrows (fun (x : Unit) ↦ X) (fun (x : Unit) ↦ π) g ∧ h ≫ g = f.hom).hom
hj :
  Over.Hom.left j.hom ≫ π =
    Z.obj (P := fun (f : Over.{v_1, u_1} B) ↦
        ∃ (Y : C) (h : f.left ⟶ Y) (g : Y ⟶ B),
          ofArrows (fun (x : Unit) ↦ X) (fun (x : Unit) ↦ π) g ∧ h ≫ g = f.hom).hom
ij : (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).left ⟶
  c.pt :=
  PullbackCone.IsLimit.lift hc (Over.Hom.left i.hom) (Over.Hom.left j.hom) ⋯
⊢ op i =
    CategoryStruct.comp (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 ij ⋯)))
have h₂ : i =
  CategoryStruct.comp (Y :=
    unop (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) (c.fst ≫ π) sorry)))
    (Quiver.Hom.unop
      (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 ij sorry))))
    (Quiver.Hom.unop
      (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))))
Tactic: congr
-- 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 : unop (op Z) ⟶ unop (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) π ⋯))
hi :
  Over.Hom.left i.hom ≫ π =
    Z.obj (P := fun (f : Over.{v_1, u_1} B) ↦
        ∃ (Y : C) (h : f.left ⟶ Y) (g : Y ⟶ B),
          ofArrows (fun (x : Unit) ↦ X) (fun (x : Unit) ↦ π) g ∧ h ≫ g = f.hom).hom
hj :
  Over.Hom.left j.hom ≫ π =
    Z.obj (P := fun (f : Over.{v_1, u_1} B) ↦
        ∃ (Y : C) (h : f.left ⟶ Y) (g : Y ⟶ B),
          ofArrows (fun (x : Unit) ↦ X) (fun (x : Unit) ↦ π) g ∧ h ≫ g = f.hom).hom
ij : (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).left ⟶
  c.pt :=
  PullbackCone.IsLimit.lift hc (Over.Hom.left i.hom) (Over.Hom.left j.hom) ⋯
⊢ op j =
    CategoryStruct.comp (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.snd ⋯)))
      (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 ij ⋯)))
have h₂ : j =
  CategoryStruct.comp (Y :=
    unop (op (categoryMk (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)) (c.fst ≫ π) sorry)))
    (Quiver.Hom.unop
      (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 ij sorry))))
    (Quiver.Hom.unop
      (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))))
Tactic: congr
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (hypothesis)
Goal:
C : Type u_1
D : Type u_2
E : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Category.{v_3, u_3} E
P : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
S : Presieve.{v_1, u_1} B := Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)
X' : S.category := S.categoryMk π ⋯
P' : S.category := S.categoryMk (c.fst ≫ π) ⋯
fst : P' ⟶ X' := ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦ S f.hom) (Over.homMk c.fst ⋯)
snd : P' ⟶ X' := ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦ S f.hom) (Over.homMk c.snd ⋯)
F : S.categoryᵒᵖ ⥤ D := S.diagram.op ⋙ P
G : Functor.{0, v_2, 0, u_2} WalkingParallelPair D := parallelPair (P.map c.fst.op) (P.map c.snd.op)
H : WalkingParallelPair ⥤ S.categoryᵒᵖ := parallelPair fst.op snd.op
⊢ IsLimit (Fork.ofι (P.map π.op) ⋯) ≃
    IsLimit (J :=
      (Presieve.category.{v_1, u_1} (C := C) (X := B)
          (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))ᵒᵖ)
      (P.mapCone (Cocone.op (cocone (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))))
have this : H.Initial
Tactic: have : H.Initial := parallelPair_pullback_initial π c hc
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
D : Type u_2
E : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Category.{v_3, u_3} E
P : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
S : Presieve.{v_1, u_1} B := Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)
X' : S.category := S.categoryMk π ⋯
P' : S.category := S.categoryMk (c.fst ≫ π) ⋯
fst : P' ⟶ X' := ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦ S f.hom) (Over.homMk c.fst ⋯)
snd : P' ⟶ X' := ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦ S f.hom) (Over.homMk c.snd ⋯)
F : S.categoryᵒᵖ ⥤ D := S.diagram.op ⋙ P
G : Functor.{0, v_2, 0, u_2} WalkingParallelPair D := parallelPair (P.map c.fst.op) (P.map c.snd.op)
H : WalkingParallelPair ⥤ S.categoryᵒᵖ := parallelPair fst.op snd.op
this : H.Initial
i : H ⋙ F ≅ G :=
  parallelPair.ext (Iso.refl ((H ⋙ F).obj WalkingParallelPair.zero)) (Iso.refl ((H ⋙ F).obj WalkingParallelPair.one)) ⋯
    ⋯
⊢ IsLimit (Fork.ofι (P.map π.op) ⋯) ≃
    IsLimit (J :=
      (Presieve.category.{v_1, u_1} (C := C) (X := B)
          (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))ᵒᵖ)
      (P.mapCone (Cocone.op (cocone (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))))
have refine : (Cone.postcompose i.symm.hom).obj (Fork.ofι (P.map π.op) sorry) ≅
  Cone.whisker H
    (P.mapCone (Cocone.op (cocone (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))))
Tactic: refine (IsLimit.equivOfNatIsoOfIso i.symm _ _ ?_).trans (Functor.Initial.isLimitWhiskerEquiv H _)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
D : Type u_2
E : Type u_3
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Category.{v_3, u_3} E
P : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
S : Presieve.{v_1, u_1} B := Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)
X' : S.category := S.categoryMk π ⋯
P' : S.category := S.categoryMk (c.fst ≫ π) ⋯
fst : P' ⟶ X' := ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦ S f.hom) (Over.homMk c.fst ⋯)
snd : P' ⟶ X' := ObjectProperty.homMk (P := fun (f : Over.{v_1, u_1} B) ↦ S f.hom) (Over.homMk c.snd ⋯)
F : S.categoryᵒᵖ ⥤ D := S.diagram.op ⋙ P
G : Functor.{0, v_2, 0, u_2} WalkingParallelPair D := parallelPair (P.map c.fst.op) (P.map c.snd.op)
H : WalkingParallelPair ⥤ S.categoryᵒᵖ := parallelPair fst.op snd.op
this : H.Initial
i : H ⋙ F ≅ G :=
  parallelPair.ext (Iso.refl ((H ⋙ F).obj WalkingParallelPair.zero)) (Iso.refl ((H ⋙ F).obj WalkingParallelPair.one)) ⋯
    ⋯
⊢ (Cone.postcompose i.symm.hom).obj (Fork.ofι (P.map π.op) ⋯) ≅
    Cone.whisker H
      (P.mapCone (Cocone.op (cocone (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))))
have refine : ∀ (j : WalkingParallelPair),
  ((Cone.postcompose i.symm.hom).obj (Fork.ofι (P.map π.op) sorry)).π.app j =
    Iso.hom (Iso.refl ((Cone.postcompose i.symm.hom).obj (Fork.ofι (P.map π.op) sorry)).pt) ≫
      (Cone.whisker H
              (P.mapCone
                (Cocone.op (cocone (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))))).π.app
        j
Tactic: refine Cone.ext (Iso.refl _) ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
P : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
X B : C
π : X ⟶ B
inst_2 : HasPullback π π
⊢ SingleEqualizerCondition P π ↔
    Nonempty
      (IsLimit (J :=
        (Presieve.category.{v_1, u_1} (C := C) (X := B)
            (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))ᵒᵖ)
        (P.mapCone (Cocone.op (cocone (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π))))))
have mp : SingleEqualizerCondition P π →
  Nonempty
    (IsLimit (J :=
      (Presieve.category.{v_1, u_1} (C := C) (X := B)
          (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))ᵒᵖ)
      (P.mapCone (Cocone.op (cocone (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π))))))
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
P : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
X B : C
π : X ⟶ B
inst_2 : HasPullback π π
mp :
  SingleEqualizerCondition P π →
    Nonempty
      (IsLimit (J :=
        (Presieve.category.{v_1, u_1} (C := C) (X := B)
            (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))ᵒᵖ)
        (P.mapCone (Cocone.op (cocone (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π))))))
⊢ SingleEqualizerCondition P π ↔
    Nonempty
      (IsLimit (J :=
        (Presieve.category.{v_1, u_1} (C := C) (X := B)
            (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))ᵒᵖ)
        (P.mapCone (Cocone.op (cocone (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π))))))
have mpr : Nonempty
    (IsLimit (J :=
      (Presieve.category.{v_1, u_1} (C := C) (X := B)
          (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))ᵒᵖ)
      (P.mapCone (Cocone.op (cocone (Sieve.arrows (Sieve.ofArrows (fun (x : Unit) ↦ X) fun (x : Unit) ↦ π)))))) →
  SingleEqualizerCondition P π
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
inst_2 : Preregular.{v_1, u_1} C
inst_3 : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f
⊢ EqualizerCondition F ↔ Presheaf.IsSheaf (Coverage.toGrothendieck (regularCoverage C)) F
have rw : EqualizerCondition F ↔
  ∀ ⦃X : C⦄ (R : Presieve.{v_1, u_1} X),
    Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
        (Precoverage.coverings (Coverage.toPrecoverage (regularCoverage C)) X) R →
      Nonempty (IsLimit (F.mapCone (Sieve.generate R).arrows.cocone.op))
Tactic: rw [Presheaf.isSheaf_iff_isLimit_coverage]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
inst_2 : Preregular.{v_1, u_1} C
inst_3 : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f
⊢ EqualizerCondition F ↔
    ∀ ⦃X : C⦄ (R : Presieve.{v_1, u_1} X),
      Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
          (Precoverage.coverings (Coverage.toPrecoverage (regularCoverage C)) X) R →
        Nonempty (IsLimit (F.mapCone (Sieve.generate R).arrows.cocone.op))
have mp : EqualizerCondition F →
  ∀ ⦃X : C⦄ (R : Presieve.{v_1, u_1} X),
    Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
        (Precoverage.coverings (Coverage.toPrecoverage (regularCoverage C)) X) R →
      Nonempty (IsLimit (F.mapCone (Sieve.generate R).arrows.cocone.op))
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
F : Functor.{v_1, v_2, u_1, u_2} Cᵒᵖ D
inst_2 : Preregular.{v_1, u_1} C
inst_3 : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f
mp :
  EqualizerCondition F →
    ∀ ⦃X : C⦄ (R : Presieve.{v_1, u_1} X),
      Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
          (Precoverage.coverings (Coverage.toPrecoverage (regularCoverage C)) X) R →
        Nonempty (IsLimit (F.mapCone (Sieve.generate R).arrows.cocone.op))
⊢ EqualizerCondition F ↔
    ∀ ⦃X : C⦄ (R : Presieve.{v_1, u_1} X),
      Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
          (Precoverage.coverings (Coverage.toPrecoverage (regularCoverage C)) X) R →
        Nonempty (IsLimit (F.mapCone (Sieve.generate R).arrows.cocone.op))
have mpr : (∀ ⦃X : C⦄ (R : Presieve.{v_1, u_1} X),
    Membership.mem (γ := Set (Presieve.{v_1, u_1} X))
        (Precoverage.coverings (Coverage.toPrecoverage (regularCoverage C)) X) R →
      Nonempty (IsLimit (F.mapCone (Sieve.generate R).arrows.cocone.op))) →
  EqualizerCondition F
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
X : C
S : Presieve.{v_1, u_1} X
inst_1 : S.regular
inst_2 : Projective.{v_1, u_1} X
F : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
⊢ IsSheafFor F S
have obtain : ∀ (Y : C) (f : Y ⟶ X) (hf : EffectiveEpi f)
  [regular.{v_1, u_1} (C := C) (X := X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)],
  IsSheafFor (X := X) F (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
Tactic: obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.single_epi (R := S)
Cleared hypotheses: S, inst_1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : C
inst_1 : Projective.{v_1, u_1} X
F : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
Y : C
f : Y ⟶ X
hf : EffectiveEpi f
inst_2 : regular.{v_1, u_1} (C := C) (X := X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
⊢ IsSheafFor (X := X) F (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
have rw : ∀ (x : (i : Unit) → F.obj (op Y)),
  Arrows.Compatible F (B := X) (fun (x : Unit) ↦ f) x →
    ∃! t : (fun (X : Type u_4) ↦ X) (F.obj (op X)),
      ∀ (i : Unit),
        (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
              (a : (fun (X : Type u_4) ↦ X) (F.obj (op X))) → (fun (X : Type u_4) ↦ X) (F.obj (op Y)))
            t =
          x i
Tactic: rw [isSheafFor_arrows_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : C
inst_1 : Projective.{v_1, u_1} X
F : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
Y : C
f : Y ⟶ X
hf : EffectiveEpi f
inst_2 : regular.{v_1, u_1} (C := C) (X := X) (ofArrows (fun (x : Unit) ↦ Y) fun (x : Unit) ↦ f)
⊢ ∀ (x : (i : Unit) → F.obj (op Y)),
    Arrows.Compatible F (B := X) (fun (x : Unit) ↦ f) x →
      ∃! t : (fun (X : Type u_4) ↦ X) (F.obj (op X)),
        ∀ (i : Unit),
          (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
                (a : (fun (X : Type u_4) ↦ X) (F.obj (op X))) → (fun (X : Type u_4) ↦ X) (F.obj (op Y)))
              t =
            x i
have refine_1 : ∀ (x : (i : Unit) → F.obj (op Y)) (hx : Arrows.Compatible F (B := X) (fun (x : Unit) ↦ f) x) (x_1 : Unit),
  (ConcreteCategory.hom (FC := TypeCat.Fun) (F.map f.op) :
        (a : (fun (X : Type u_4) ↦ X) (F.obj (op X))) → (fun (X : Type u_4) ↦ X) (F.obj (op Y)))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map (Projective.factorThru (𝟙 X) f).op) :
            (a : (fun (X : Type u_4) ↦ X) (F.obj (op Y))) → (fun (X : Type u_4) ↦ X) (F.obj (op X)))
          (x ()) :
        (fun (X : Type u_4) ↦ X) (F.obj (op X))) =
    x x_1
Tactic: refine fun x hx ↦ ⟨F.map (Projective.factorThru (𝟙 _) f).op <| x (), fun _ ↦ ?_, fun y h ↦ ?_⟩