awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G : PreOneHypercover.{?u.367710, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
x : F.multicospanShape.R
⊢ CategoryStruct.comp (Y :=
      WalkingMulticospan.right
        (Sigma.mk (β := fun (i : F.I₀ × F.I₀) ↦ F.I₁ i.1 i.2)
          (f.hom.s₀ (f.inv.s₀ x.fst (β := fun (i : F.I₀ × F.I₀) ↦ F.I₁ i.1 i.2).1),
            f.hom.s₀ (f.inv.s₀ x.fst (β := fun (i : F.I₀ × F.I₀) ↦ F.I₁ i.1 i.2).2))
          (f.hom.s₁ (f.inv.s₁ (x.snd (β := fun (i : F.I₀ × F.I₀) ↦ F.I₁ i.1 i.2))))))
      (WalkingMulticospan.Hom.snd
        (Sigma.mk (β := fun (i : F.I₀ × F.I₀) ↦ F.I₁ i.1 i.2)
          (f.hom.s₀ (f.inv.s₀ x.fst (β := fun (i : F.I₀ × F.I₀) ↦ F.I₁ i.1 i.2).1),
            f.hom.s₀ (f.inv.s₀ x.fst (β := fun (i : F.I₀ × F.I₀) ↦ F.I₁ i.1 i.2).2))
          (f.hom.s₁ (f.inv.s₁ (x.snd (β := fun (i : F.I₀ × F.I₀) ↦ F.I₁ i.1 i.2))))))
      (eqToHom ⋯) =
    eqToHom ⋯ ≫ WalkingMulticospan.Hom.snd x
have w : Sigma.mk (β := fun (i : F.I₀ × F.I₀) ↦ F.I₁ i.1 i.2)
    (f.hom.s₀ (f.inv.s₀ x.fst (β := fun (i : F.I₀ × F.I₀) ↦ F.I₁ i.1 i.2).1),
      f.hom.s₀ (f.inv.s₀ x.fst (β := fun (i : F.I₀ × F.I₀) ↦ F.I₁ i.1 i.2).2))
    (f.hom.s₁ (f.inv.s₁ (x.snd (β := fun (i : F.I₀ × F.I₀) ↦ F.I₁ i.1 i.2)))) =
  x
Tactic: rw [eqToHom_naturality]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G : PreOneHypercover.{?u.367710, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
c : WalkingMulticospan E.multicospanShape
⊢ (Hom.mapMulticospan f.hom).map (NatTrans.app (Iso.hom (eqToIso ⋯)) c) ≫
      NatTrans.app (Iso.hom (eqToIso ⋯)) ((Hom.mapMulticospan f.hom).obj c) =
    𝟙 ((Hom.mapMulticospan f.hom).obj c)
have left : ∀ (a : E.multicospanShape.L),
  (Hom.mapMulticospan f.hom).map (NatTrans.app (Iso.hom (eqToIso sorry)) (WalkingMulticospan.left a)) ≫
      NatTrans.app (Iso.hom (eqToIso sorry)) ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.left a)) =
    𝟙 ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.left a))
Tactic: cases c
Cleared hypotheses: c
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G : PreOneHypercover.{?u.367710, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
c : WalkingMulticospan E.multicospanShape
left :
  ∀ (a : E.multicospanShape.L),
    (Hom.mapMulticospan f.hom).map (NatTrans.app (Iso.hom (eqToIso ⋯)) (WalkingMulticospan.left a)) ≫
        NatTrans.app (Iso.hom (eqToIso ⋯)) ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.left a)) =
      𝟙 ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.left a))
⊢ (Hom.mapMulticospan f.hom).map (NatTrans.app (Iso.hom (eqToIso ⋯)) c) ≫
      NatTrans.app (Iso.hom (eqToIso ⋯)) ((Hom.mapMulticospan f.hom).obj c) =
    𝟙 ((Hom.mapMulticospan f.hom).obj c)
have right : ∀ (a : E.multicospanShape.R),
  (Hom.mapMulticospan f.hom).map (NatTrans.app (Iso.hom (eqToIso sorry)) (WalkingMulticospan.right a)) ≫
      NatTrans.app (Iso.hom (eqToIso sorry)) ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.right a)) =
    𝟙 ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.right a))
Tactic: cases c
Cleared hypotheses: c
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G : PreOneHypercover.{?u.367710, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
a : E.multicospanShape.L
⊢ (Hom.mapMulticospan f.hom).map (NatTrans.app (Iso.hom (eqToIso ⋯)) (WalkingMulticospan.left a)) ≫
      NatTrans.app (Iso.hom (eqToIso ⋯)) ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.left a)) =
    𝟙 ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.left a))
have left : eqToHom sorry ≫ NatTrans.app (Iso.hom (eqToIso sorry)) ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.left a)) =
  𝟙 ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.left a))
Tactic: rw [eqToIso.hom, eqToHom_app, eqToHom_map]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G : PreOneHypercover.{?u.367710, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
a : E.multicospanShape.R
⊢ (Hom.mapMulticospan f.hom).map (NatTrans.app (Iso.hom (eqToIso ⋯)) (WalkingMulticospan.right a)) ≫
      NatTrans.app (Iso.hom (eqToIso ⋯)) ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.right a)) =
    𝟙 ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.right a))
have right : eqToHom sorry ≫ NatTrans.app (Iso.hom (eqToIso sorry)) ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.right a)) =
  𝟙 ((Hom.mapMulticospan f.hom).obj (WalkingMulticospan.right a))
Tactic: rw [eqToIso.hom, eqToHom_app, eqToHom_map]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G_1 : PreOneHypercover.{?u.383445, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
G : Functor.{v, v_1, u, u_1} Cᵒᵖ A
⊢ IsLimit (E.multifork G) ≃ IsLimit (F.multifork G)
have refine : IsLimit (E.multifork G) ≃ IsLimit (Cone.whisker (equivalenceMulticospanOfIso f).functor (F.multifork G))
Tactic: refine Equiv.trans ?_
    (IsLimit.whiskerEquivalenceEquiv <| PreOneHypercover.equivalenceMulticospanOfIso f).symm
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G_1 : PreOneHypercover.{?u.383445, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
G : Functor.{v, v_1, u, u_1} Cᵒᵖ A
⊢ IsLimit (E.multifork G) ≃ IsLimit (Cone.whisker (equivalenceMulticospanOfIso f).functor (F.multifork G))
have refine_1 : (E.multicospanIndex G).multicospan ≅ (equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan
Tactic: refine IsLimit.equivOfNatIsoOfIso ?_ _ _ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G_1 : PreOneHypercover.{?u.383445, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
G : Functor.{v, v_1, u, u_1} Cᵒᵖ A
refine_1 :
  (E.multicospanIndex G).multicospan ≅ (equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan
⊢ IsLimit (E.multifork G) ≃ IsLimit (Cone.whisker (equivalenceMulticospanOfIso f).functor (F.multifork G))
have refine_2 : Functor.obj
    (Cone.postcompose
      (Iso.hom
        (WalkingMulticospan.functorExt (fun (i : E.multicospanShape.L) ↦ G.mapIso (asIso (f.hom.h₀ i)).symm.op)
          (fun (i : E.multicospanShape.R) ↦
            G.mapIso (asIso (f.hom.h₁ (i.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)))).symm.op)
          sorry sorry)))
    (E.multifork G) ≅
  Cone.whisker (equivalenceMulticospanOfIso f).functor (F.multifork G)
Tactic: refine IsLimit.equivOfNatIsoOfIso ?_ _ _ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G_1 : PreOneHypercover.{?u.383445, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
G : Functor.{v, v_1, u, u_1} Cᵒᵖ A
⊢ (E.multicospanIndex G).multicospan ≅ (equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan
have refine_1 : (i : E.multicospanShape.L) →
  (E.multicospanIndex G).multicospan.obj (WalkingMulticospan.left i) ≅
    ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).obj (WalkingMulticospan.left i)
Tactic: refine WalkingMulticospan.functorExt ?_ ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G_1 : PreOneHypercover.{?u.383445, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
G : Functor.{v, v_1, u, u_1} Cᵒᵖ A
refine_1 :
  (i : E.multicospanShape.L) →
    (E.multicospanIndex G).multicospan.obj (WalkingMulticospan.left i) ≅
      ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).obj (WalkingMulticospan.left i)
⊢ (E.multicospanIndex G).multicospan ≅ (equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan
have refine_1₁ : (i : E.multicospanShape.R) →
  (E.multicospanIndex G).multicospan.obj (WalkingMulticospan.right i) ≅
    ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).obj (WalkingMulticospan.right i)
Tactic: refine WalkingMulticospan.functorExt ?_ ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G_1 : PreOneHypercover.{?u.383445, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
G : Functor.{v, v_1, u, u_1} Cᵒᵖ A
refine_1 :
  (i : E.multicospanShape.L) →
    (E.multicospanIndex G).multicospan.obj (WalkingMulticospan.left i) ≅
      ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).obj (WalkingMulticospan.left i)
refine_1₁ :
  (i : E.multicospanShape.R) →
    (E.multicospanIndex G).multicospan.obj (WalkingMulticospan.right i) ≅
      ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).obj (WalkingMulticospan.right i)
⊢ (E.multicospanIndex G).multicospan ≅ (equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan
have refine_1₂ : ∀ (i : E.multicospanShape.R),
  (E.multicospanIndex G).multicospan.map (WalkingMulticospan.Hom.fst i) ≫
      (G.mapIso (asIso (f.hom.h₁ (i.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)))).symm.op).hom =
    Iso.hom (G.mapIso (Iso.op (Iso.symm (asIso (f.hom.h₀ (E.multicospanShape.fst i)))))) ≫
      ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).map (WalkingMulticospan.Hom.fst i)
Tactic: refine WalkingMulticospan.functorExt ?_ ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G_1 : PreOneHypercover.{?u.383445, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
G : Functor.{v, v_1, u, u_1} Cᵒᵖ A
refine_1 :
  (i : E.multicospanShape.L) →
    (E.multicospanIndex G).multicospan.obj (WalkingMulticospan.left i) ≅
      ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).obj (WalkingMulticospan.left i)
refine_1₁ :
  (i : E.multicospanShape.R) →
    (E.multicospanIndex G).multicospan.obj (WalkingMulticospan.right i) ≅
      ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).obj (WalkingMulticospan.right i)
refine_1₂ :
  ∀ (i : E.multicospanShape.R),
    (E.multicospanIndex G).multicospan.map (WalkingMulticospan.Hom.fst i) ≫
        (G.mapIso (asIso (f.hom.h₁ (i.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)))).symm.op).hom =
      Iso.hom (G.mapIso (Iso.op (Iso.symm (asIso (f.hom.h₀ (E.multicospanShape.fst i)))))) ≫
        ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).map
          (WalkingMulticospan.Hom.fst i)
⊢ (E.multicospanIndex G).multicospan ≅ (equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan
have refine_1₃ : ∀ (i : E.multicospanShape.R),
  (E.multicospanIndex G).multicospan.map (WalkingMulticospan.Hom.snd i) ≫
      (G.mapIso (asIso (f.hom.h₁ (i.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)))).symm.op).hom =
    Iso.hom (G.mapIso (Iso.op (Iso.symm (asIso (f.hom.h₀ (E.multicospanShape.snd i)))))) ≫
      ((equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan).map (WalkingMulticospan.Hom.snd i)
Tactic: refine WalkingMulticospan.functorExt ?_ ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G_1 : PreOneHypercover.{?u.383445, v, u} S
E F : PreOneHypercover.{w, v, u} S
f : E ≅ F
G : Functor.{v, v_1, u, u_1} Cᵒᵖ A
⊢ Functor.obj
      (Cone.postcompose
        (Iso.hom
          (WalkingMulticospan.functorExt (fun (i : E.multicospanShape.L) ↦ G.mapIso (asIso (f.hom.h₀ i)).symm.op)
            (fun (i : E.multicospanShape.R) ↦
              G.mapIso (asIso (f.hom.h₁ (i.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)))).symm.op)
            ⋯ ⋯)))
      (E.multifork G) ≅
    Cone.whisker (equivalenceMulticospanOfIso f).functor (F.multifork G)
have refine_2 : ∀ (i : WalkingMulticospan E.multicospanShape),
  NatTrans.app
      (Cone.π
        (Functor.obj
          (Cone.postcompose
            (Iso.hom
              (WalkingMulticospan.functorExt (fun (i : E.multicospanShape.L) ↦ G.mapIso (asIso (f.hom.h₀ i)).symm.op)
                (fun (i : E.multicospanShape.R) ↦
                  G.mapIso (asIso (f.hom.h₁ (i.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)))).symm.op)
                sorry sorry)))
          (E.multifork G)))
      i =
    Iso.hom
        (Iso.refl
          (Cone.pt (F := (equivalenceMulticospanOfIso f).functor ⋙ (F.multicospanIndex G).multicospan)
            (Functor.obj
              (Cone.postcompose
                (Iso.hom
                  (WalkingMulticospan.functorExt
                    (fun (i : E.multicospanShape.L) ↦ G.mapIso (asIso (f.hom.h₀ i)).symm.op)
                    (fun (i : E.multicospanShape.R) ↦
                      G.mapIso (asIso (f.hom.h₁ (i.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)))).symm.op)
                    sorry sorry)))
              (E.multifork G)))) ≫
      (Cone.whisker (equivalenceMulticospanOfIso f).functor (F.multifork G)).π.app i
Tactic: refine Cone.ext (Iso.refl _) fun i ↦ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
G_1 : PreOneHypercover.{w'', v, u} S
inst_2 : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)
inst_3 : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullback (E.p₁ k ≫ E.f i) (F.p₁ l ≫ F.f a)
G : PreOneHypercover.{w'', v, u} S
f : G.Hom E
g : G.Hom F
i j : G.I₀
k : G.I₁ i j
⊢ f.h₁ k ≫ E.p₁ (f.s₁ k, g.s₁ k).1 ≫ E.f (__spread.0.s₀ i).1 =
    g.h₁ k ≫ F.p₁ (f.s₁ k, g.s₁ k).2 ≫ F.f (__spread.0.s₀ i).2
have rw : G.p₁ k ≫ f.h₀ i ≫ E.f (__spread.0.s₀ i).1 = G.p₁ k ≫ g.h₀ i ≫ F.f (__spread.0.s₀ i).2
Tactic: rw [f.w₁₁_assoc k, g.w₁₁_assoc k]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
G_1 : PreOneHypercover.{w'', v, u} S
inst_2 : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)
inst_3 : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullback (E.p₁ k ≫ E.f i) (F.p₁ l ≫ F.f a)
G : PreOneHypercover.{w'', v, u} S
f : G.Hom E
g : G.Hom F
i j : G.I₀
k : G.I₁ i j
⊢ pullback.lift (f.h₁ k) (g.h₁ k) ⋯ ≫ (E.inter F).p₁ (f.s₁ k, g.s₁ k) =
    G.p₁ k ≫ PreZeroHypercover.Hom.h₀ (PreZeroHypercover.interLift f.toHom g.toHom) i
have h₀ : (pullback.lift (f.h₁ k) (g.h₁ k) sorry ≫ (E.inter F).p₁ (f.s₁ k, g.s₁ k)) ≫
    pullback.fst
      (E.f
        (Sigma.fst (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{w', v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{w', v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            (PreZeroHypercover.Hom.s₀ (PreZeroHypercover.interLift f.toHom g.toHom) i))))
      (F.f
        (Sigma.snd (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{w', v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{w', v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            (PreZeroHypercover.Hom.s₀ (PreZeroHypercover.interLift f.toHom g.toHom) i)))) =
  (G.p₁ k ≫ PreZeroHypercover.Hom.h₀ (PreZeroHypercover.interLift f.toHom g.toHom) i) ≫
    pullback.fst
      (E.f
        (Sigma.fst (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{w', v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{w', v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            (PreZeroHypercover.Hom.s₀ (PreZeroHypercover.interLift f.toHom g.toHom) i))))
      (F.f
        (Sigma.snd (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{w', v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{w', v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            (PreZeroHypercover.Hom.s₀ (PreZeroHypercover.interLift f.toHom g.toHom) i))))
Tactic: apply pullback.hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
G_1 : PreOneHypercover.{w'', v, u} S
inst_2 : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)
inst_3 : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullback (E.p₁ k ≫ E.f i) (F.p₁ l ≫ F.f a)
G : PreOneHypercover.{w'', v, u} S
f : G.Hom E
g : G.Hom F
i j : G.I₀
k : G.I₁ i j
⊢ pullback.lift (f.h₁ k) (g.h₁ k) ⋯ ≫ (E.inter F).p₂ (f.s₁ k, g.s₁ k) =
    G.p₂ k ≫ PreZeroHypercover.Hom.h₀ (PreZeroHypercover.interLift f.toHom g.toHom) j
have h₀ : (pullback.lift (f.h₁ k) (g.h₁ k) sorry ≫ (E.inter F).p₂ (f.s₁ k, g.s₁ k)) ≫
    pullback.fst
      (E.f
        (Sigma.fst (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{w', v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{w', v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            (PreZeroHypercover.Hom.s₀ (PreZeroHypercover.interLift f.toHom g.toHom) j))))
      (F.f
        (Sigma.snd (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{w', v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{w', v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            (PreZeroHypercover.Hom.s₀ (PreZeroHypercover.interLift f.toHom g.toHom) j)))) =
  (G.p₂ k ≫ PreZeroHypercover.Hom.h₀ (PreZeroHypercover.interLift f.toHom g.toHom) j) ≫
    pullback.fst
      (E.f
        (Sigma.fst (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{w', v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{w', v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            (PreZeroHypercover.Hom.s₀ (PreZeroHypercover.interLift f.toHom g.toHom) j))))
      (F.f
        (Sigma.snd (β := fun (i : E.I₀) ↦
          PreZeroHypercover.I₀.{w', v, u} (C := C) (S := E.X i)
            ((fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁.{w', v, u} (E.f i) F.toPreZeroHypercover) i))
          ((Equiv.symm (Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1) :
              (a : E.I₀ × F.toPreZeroHypercover.1) →
                (E.bind fun (i : E.I₀) ↦ PreZeroHypercover.pullback₁ (E.f i) F.toPreZeroHypercover).I₀)
            (PreZeroHypercover.Hom.s₀ (PreZeroHypercover.interLift f.toHom g.toHom) j))))
Tactic: apply pullback.hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
S : C
E : OneHypercover.{u_2, v, u} J S
i₁ i₂ : E.I₀
inst_1 : HasPullback (E.f i₁) (E.f i₂)
⊢ Membership.mem (γ := Set (Sieve.{v, u} (Limits.pullback (E.f i₁) (E.f i₂))))
    ((J : (a : C) → Set (Sieve a)) (Limits.pullback (E.f i₁) (E.f i₂))) (E.sieve₁' i₁ i₂)
have rw : Membership.mem (γ := Set (Sieve.{v, u} (Limits.pullback (E.f i₁) (E.f i₂))))
  ((J : (a : C) → Set (Sieve a)) (Limits.pullback (E.f i₁) (E.f i₂)))
  (E.sieve₁ (pullback.fst (E.f i₁) (E.f i₂)) (pullback.snd (E.f i₁) (E.f i₂)))
Tactic: rw [E.sieve₁'_eq_sieve₁]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
A : Type u_1
inst_1 : Category.{v_1, u_1} A
J : GrothendieckTopology.{v, u} C
S : C
E : PreOneHypercover.{u_2, v, u} S
inst_2 : E.HasPullbacks
mem₀ : PreZeroHypercover.sieve₀.{u_2, v, u} E.toPreZeroHypercover ∈ (J : (a : C) → Set (Sieve a)) S
mem₁' :
  ∀ (i₁ i₂ : E.I₀),
    Membership.mem (γ := Set (Sieve.{v, u} (Limits.pullback (E.f i₁) (E.f i₂))))
      ((J : (a : C) → Set (Sieve a)) (Limits.pullback (E.f i₁) (E.f i₂))) (E.sieve₁' i₁ i₂)
i₁ i₂ : E.I₀
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
⊢ E.sieve₁ p₁ p₂ ∈ (J : (a : C) → Set (Sieve a)) W
have rw : Sieve.pullback (X := Limits.pullback (E.f i₁) (E.f i₂)) (pullback.lift p₁ p₂ w) (E.sieve₁' i₁ i₂) ∈
  (J : (a : C) → Set (Sieve a)) W
Tactic: rw [E.sieve₁_eq_pullback_sieve₁' _ _ w]