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
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
u₀ : E.I₀ → F.I₀
u₁ : ⦃i j : E.I₀⦄ → E.I₁ i j → F.I₁ (u₀ i) (u₀ j)
k : E.I₁ i j
⊢ Eq.{w' + 1} (α := F.I₁ (u₀ i') (u₀ j')) (u₁ ((congrIndexOneOfEq hii' hjj' : (a : E.I₁ i j) → E.I₁ i' j') k))
    ((congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ (u₀ i) (u₀ j)) → F.I₁ (u₀ i') (u₀ j')) (u₁ k))
have subst : Eq.{w' + 1} (α := F.I₁ (u₀ i) (u₀ j)) (u₁ ((congrIndexOneOfEq sorry sorry : (a : E.I₁ i j) → E.I₁ i j) k))
  ((congrIndexOneOfEq sorry sorry : (a : F.I₁ (u₀ i) (u₀ j)) → F.I₁ (u₀ i) (u₀ j)) (u₁ k))
Tactic: subst hii' hjj'
Cleared hypotheses: i', j', hii', hjj'
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
u₀ v₀ : E.I₀ → F.I₀
u₁ : ⦃i j : E.I₀⦄ → E.I₁ i j → F.I₁ (u₀ i) (u₀ j)
v₁ : ⦃i j : E.I₀⦄ → E.I₁ i j → F.I₁ (v₀ i) (v₀ j)
h₀ : u₀ = v₀
h₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j), u₁ k = (congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ (v₀ i) (v₀ j)) → F.I₁ (u₀ i) (u₀ j)) (v₁ k)
i j : E.I₀
k : E.I₁ i j
⊢ (congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ (v₀ i) (v₀ j)) → F.I₁ (u₀ i) (u₀ j)) (v₁ k) = u₁ k
have subst : ∀ {v₁ : ⦃i j : E.I₀⦄ → E.I₁ i j → F.I₁ (u₀ i) (u₀ j)}
  (h₁ :
    ∀ (i j : E.I₀) (k : E.I₁ i j),
      u₁ k = (congrIndexOneOfEq sorry sorry : (a : F.I₁ (u₀ i) (u₀ j)) → F.I₁ (u₀ i) (u₀ j)) (v₁ k)),
  (congrIndexOneOfEq sorry sorry : (a : F.I₁ (u₀ i) (u₀ j)) → F.I₁ (u₀ i) (u₀ j)) (v₁ k) = u₁ k
Tactic: subst h₀
Cleared hypotheses: v₀, v₁, h₀, h₁
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
a b : E.I₁'
left : a.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).1 = b.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).1
right : a.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).2 = b.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).2
h :
  (congrIndexOneOfEq left right : (a : E.I₁ a.fst.1 a.fst.2) → E.I₁ b.fst.1 b.fst.2)
      (a.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)) =
    b.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)
⊢ a = b
have obtain : ∀ (i j : E.I₀) (k : E.I₁ (i, j).1 (i, j).2)
  (left :
    (Sigma.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)
          (Sigma.mk (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2) (i, j) k)).1 =
      b.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).1)
  (right :
    (Sigma.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)
          (Sigma.mk (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2) (i, j) k)).2 =
      b.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).2)
  (h :
    (congrIndexOneOfEq left right : (a : E.I₁ ⟨(i, j), k⟩.fst.1 ⟨(i, j), k⟩.fst.2) → E.I₁ b.fst.1 b.fst.2)
        (Sigma.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)
          (Sigma.mk (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2) (i, j) k)) =
      b.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)),
  Sigma.mk (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2) (i, j) k = b
Tactic: obtain ⟨⟨i, j⟩, k⟩ := a
Cleared hypotheses: a, left, right, h
-- 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_1 : C
E_1 E_2 : PreOneHypercover.{w, v, u} S_1
F_1 : PreOneHypercover.{w', v, u} S_1
G : PreOneHypercover.{?u.248471, v, u} S_1
S : C
E_3 : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
i_1 i'_1 j_1 j'_1 : E_3.I₀
hii'_1 : i_1 = i'_1
hjj'_1 : j_1 = j'_1
E : PreOneHypercover.{u_2, v, u} S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ E.Y (i₁ := i') (i₂ := j') ((congrIndexOneOfEq hii' hjj' : (a : E.I₁ i j) → E.I₁ i' j') k) = E.Y k
have subst : E.Y (i₁ := i) (i₂ := j) ((congrIndexOneOfEq sorry sorry : (a : E.I₁ i j) → E.I₁ i j) k) = E.Y k
Tactic: subst hii' hjj'
Cleared hypotheses: i', j', hii', hjj'
-- 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_1 : C
E_1 E_2 : PreOneHypercover.{w, v, u} S_1
F_1 : PreOneHypercover.{w', v, u} S_1
G : PreOneHypercover.{?u.252120, v, u} S_1
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ E.X i' = E.X i
have rw : E.X i' = E.X i'
Tactic: rw [hii']
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ (congrIndexOneOfEqIso hii' hjj' k).hom ≫ E.p₁ k =
    E.p₁ ((congrIndexOneOfEq hii' hjj' : (a : E.I₁ i j) → E.I₁ i' j') k) ≫ eqToHom ⋯
have subst : (congrIndexOneOfEqIso sorry sorry k).hom ≫ E.p₁ k = E.p₁ ((congrIndexOneOfEq sorry sorry : (a : E.I₁ i j) → E.I₁ i j) k) ≫ eqToHom sorry
Tactic: subst hii' hjj'
Cleared hypotheses: i', j', hii', hjj'
-- 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_1 : C
E_1 E_2 : PreOneHypercover.{w, v, u} S_1
F_1 : PreOneHypercover.{w', v, u} S_1
G : PreOneHypercover.{?u.253910, v, u} S_1
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ E.X i = E.X i'
have rw : E.X i' = E.X i'
Tactic: rw [hii']
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ (congrIndexOneOfEqIso hii' hjj' k).inv ≫ E.p₁ ((congrIndexOneOfEq hii' hjj' : (a : E.I₁ i j) → E.I₁ i' j') k) =
    E.p₁ k ≫ eqToHom ⋯
have subst : (congrIndexOneOfEqIso sorry sorry k).inv ≫ E.p₁ ((congrIndexOneOfEq sorry sorry : (a : E.I₁ i j) → E.I₁ i j) k) = E.p₁ k ≫ eqToHom sorry
Tactic: subst hii' hjj'
Cleared hypotheses: i', j', hii', hjj'
-- 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_1 : C
E_1 E_2 : PreOneHypercover.{w, v, u} S_1
F_1 : PreOneHypercover.{w', v, u} S_1
G : PreOneHypercover.{?u.255701, v, u} S_1
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ E.X j = E.X j'
have rw : E.X j' = E.X j'
Tactic: rw [hjj']
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
i i' j j' : E.I₀
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ (congrIndexOneOfEqIso hii' hjj' k).inv ≫ E.p₂ ((congrIndexOneOfEq hii' hjj' : (a : E.I₁ i j) → E.I₁ i' j') k) =
    E.p₂ k ≫ eqToHom ⋯
have subst : (congrIndexOneOfEqIso sorry sorry k).inv ≫ E.p₂ ((congrIndexOneOfEq sorry sorry : (a : E.I₁ i j) → E.I₁ i j) k) = E.p₂ k ≫ eqToHom sorry
Tactic: subst hii' hjj'
Cleared hypotheses: i', j', hii', hjj'
-- 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_1 : C
E_1 E_2 : PreOneHypercover.{w, v, u} S_1
F_1 : PreOneHypercover.{w', v, u} S_1
G : PreOneHypercover.{?u.258677, v, u} S_1
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
i_1 i'_1 j_1 j'_1 : E.I₀
hii'_1 : i_1 = i'_1
hjj'_1 : j_1 = j'_1
i i' j j' : E.I₀
u₀ : E.I₀ → F.I₀
u₁ : (i j : E.I₀) → E.I₁ i j → F.I₁ (u₀ i) (u₀ j)
z : (i j : E.I₀) → (k : E.I₁ i j) → E.Y k ⟶ F.Y (u₁ i j k)
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ F.Y (u₁ i' j' ((congrIndexOneOfEq hii' hjj' : (a : E.I₁ i j) → E.I₁ i' j') k)) =
    F.Y (i₁ := u₀ i') (i₂ := u₀ j')
      ((congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ (u₀ i) (u₀ j)) → F.I₁ (u₀ i') (u₀ j')) (u₁ i j k))
have subst : F.Y (u₁ i j ((congrIndexOneOfEq sorry sorry : (a : E.I₁ i j) → E.I₁ i j) k)) =
  F.Y (i₁ := u₀ i) (i₂ := u₀ j) ((congrIndexOneOfEq sorry sorry : (a : F.I₁ (u₀ i) (u₀ j)) → F.I₁ (u₀ i) (u₀ j)) (u₁ i j k))
Tactic: subst hii' hjj'
Cleared hypotheses: i', j', hii', hjj'
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
i i' j j' : E.I₀
u₀ : E.I₀ → F.I₀
u₁ : (i j : E.I₀) → E.I₁ i j → F.I₁ (u₀ i) (u₀ j)
z : (i j : E.I₀) → (k : E.I₁ i j) → E.Y k ⟶ F.Y (u₁ i j k)
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ (congrIndexOneOfEqIso hii' hjj' k).hom ≫ z i j k =
    z i' j' ((congrIndexOneOfEq hii' hjj' : (a : E.I₁ i j) → E.I₁ i' j') k) ≫
      eqToHom ⋯ ≫ Iso.hom (congrIndexOneOfEqIso ⋯ ⋯ (u₁ i j k))
have subst : (congrIndexOneOfEqIso sorry sorry k).hom ≫ z i j k =
  z i j ((congrIndexOneOfEq sorry sorry : (a : E.I₁ i j) → E.I₁ i j) k) ≫
    eqToHom sorry ≫ Iso.hom (congrIndexOneOfEqIso sorry sorry (u₁ i j k))
Tactic: subst hii' hjj'
Cleared hypotheses: i', j', hii', hjj'
-- 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_1 : C
E_1 E_2 : PreOneHypercover.{w, v, u} S_1
F_1 : PreOneHypercover.{w', v, u} S_1
G : PreOneHypercover.{?u.261170, v, u} S_1
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
i_1 i'_1 j_1 j'_1 : E.I₀
hii'_1 : i_1 = i'_1
hjj'_1 : j_1 = j'_1
i i' j j' : E.I₀
u₀ : E.I₀ → F.I₀
u₁ : (i j : E.I₀) → E.I₁ i j → F.I₁ (u₀ i) (u₀ j)
z : (i j : E.I₀) → (k : E.I₁ i j) → E.Y k ⟶ F.Y (u₁ i j k)
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ F.Y (u₁ i' j' ((congrIndexOneOfEq hii' hjj' : (a : E.I₁ i j) → E.I₁ i' j') k)) =
    F.Y (i₁ := u₀ i') (i₂ := u₀ j')
      ((congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ (u₀ i) (u₀ j)) → F.I₁ (u₀ i') (u₀ j')) (u₁ i j k))
have subst : F.Y (u₁ i j ((congrIndexOneOfEq sorry sorry : (a : E.I₁ i j) → E.I₁ i j) k)) =
  F.Y (i₁ := u₀ i) (i₂ := u₀ j) ((congrIndexOneOfEq sorry sorry : (a : F.I₁ (u₀ i) (u₀ j)) → F.I₁ (u₀ i) (u₀ j)) (u₁ i j k))
Tactic: subst hii' hjj'
Cleared hypotheses: i', j', hii', hjj'
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
i i' j j' : E.I₀
u₀ : E.I₀ → F.I₀
u₁ : (i j : E.I₀) → E.I₁ i j → F.I₁ (u₀ i) (u₀ j)
z : (i j : E.I₀) → (k : E.I₁ i j) → E.Y k ⟶ F.Y (u₁ i j k)
hii' : i = i'
hjj' : j = j'
k : E.I₁ i j
⊢ (congrIndexOneOfEqIso hii' hjj' k).inv ≫
      z i' j' ((congrIndexOneOfEq hii' hjj' : (a : E.I₁ i j) → E.I₁ i' j') k) ≫ eqToHom ⋯ =
    z i j k ≫ Iso.inv (congrIndexOneOfEqIso ⋯ ⋯ (u₁ i j k))
have subst : (congrIndexOneOfEqIso sorry sorry k).inv ≫ z i j ((congrIndexOneOfEq sorry sorry : (a : E.I₁ i j) → E.I₁ i j) k) ≫ eqToHom sorry =
  z i j k ≫ Iso.inv (congrIndexOneOfEqIso sorry sorry (u₁ i j k))
Tactic: subst hii' hjj'
Cleared hypotheses: i', j', hii', hjj'
-- 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.263665, v, u} S
E : PreOneHypercover.{u_2, v, u} S
F : PreOneHypercover.{u_3, v, u} S
f g : E.Hom F
hs₀ : f.s₀ = g.s₀
hh₀ : ∀ (i : E.I₀), f.h₀ i = g.h₀ i ≫ eqToHom ⋯
hs₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    f.s₁ k = (congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ (g.s₀ i) (g.s₀ j)) → F.I₁ (f.s₀ i) (f.s₀ j)) (g.s₁ k)
i j : E.I₀
k : E.I₁ i j
⊢ F.Y (i₁ := f.s₀ i) (i₂ := f.s₀ j)
      ((congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ (g.s₀ i) (g.s₀ j)) → F.I₁ (f.s₀ i) (f.s₀ j)) (g.s₁ k)) =
    F.Y (f.s₁ k)
have rw : F.Y (f.s₁ k) = F.Y (f.s₁ k)
Tactic: rw [PreOneHypercover.congrIndexOneOfEq_congrFun hs₀ hs₁]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{u_2, v, u} S
F : PreOneHypercover.{u_3, v, u} S
f g : E.Hom F
hs₀ : f.s₀ = g.s₀
hh₀ : ∀ (i : E.I₀), f.h₀ i = g.h₀ i ≫ eqToHom ⋯
hs₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    f.s₁ k = (congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ (g.s₀ i) (g.s₀ j)) → F.I₁ (f.s₀ i) (f.s₀ j)) (g.s₁ k)
hh₁ : ∀ (i j : E.I₀) (k : E.I₁ i j), f.h₁ k = g.h₁ k ≫ Iso.inv (congrIndexOneOfEqIso ⋯ ⋯ (g.s₁ k)) ≫ eqToHom ⋯
⊢ f = g
have obtain : ∀ (toHomf : PreZeroHypercover.Hom.{u_3, u_2, v, u} E.toPreZeroHypercover F.toPreZeroHypercover)
  (fs₁ : {i j : E.I₀} → (k : E.I₁ i j) → F.I₁ (toHomf.s₀ i) (toHomf.s₀ j))
  (fh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (fs₁ k))
  (w₁₁ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₁ (fs₁ k) = E.p₁ k ≫ toHomf.h₀ i)
  (w₁₂ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₂ (fs₁ k) = E.p₂ k ≫ toHomf.h₀ j)
  (hs₀ : { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.s₀ = g.s₀)
  (hh₀ :
    ∀ (i : E.I₀),
      { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.h₀ i = g.h₀ i ≫ eqToHom sorry)
  (hs₁ :
    ∀ (i j : E.I₀) (k : E.I₁ i j),
      { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.s₁ k =
        (congrIndexOneOfEq sorry sorry :
            (a : F.I₁ (g.s₀ i) (g.s₀ j)) →
              F.I₁ ({ toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁, w₁₂ := w₁₂ }.s₀ i)
                ({ toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁, w₁₂ := w₁₂ }.s₀ j))
          (g.s₁ k))
  (hh₁ :
    ∀ (i j : E.I₀) (k : E.I₁ i j),
      { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.h₁ k =
        g.h₁ k ≫ Iso.inv (congrIndexOneOfEqIso sorry sorry (g.s₁ k)) ≫ eqToHom sorry),
  { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F } = g
Tactic: obtain ⟨toHomf, fs₁, fh₁⟩ := f
Cleared hypotheses: f, hs₀, hh₀, hs₁, hh₁
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{u_2, v, u} S
F : PreOneHypercover.{u_3, v, u} S
toHomf : PreZeroHypercover.Hom.{u_3, u_2, v, u} E.toPreZeroHypercover F.toPreZeroHypercover
fs₁ : {i j : E.I₀} → (k : E.I₁ i j) → F.I₁ (toHomf.s₀ i) (toHomf.s₀ j)
fh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (fs₁ k)
w₁₁ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₁ (fs₁ k) = E.p₁ k ≫ toHomf.h₀ i
w₁₂ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₂ (fs₁ k) = E.p₂ k ≫ toHomf.h₀ j
gs₁ : {i j : E.I₀} → (k : E.I₁ i j) → F.I₁ (toHomf.s₀ i) (toHomf.s₀ j)
gh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (gs₁ k)
w₁₁_1 : ∀ {i j : E.I₀} (k : E.I₁ i j), gh₁ k ≫ F.p₁ (gs₁ k) = E.p₁ k ≫ toHomf.h₀ i
w₁₂_1 : ∀ {i j : E.I₀} (k : E.I₁ i j), gh₁ k ≫ F.p₂ (gs₁ k) = E.p₂ k ≫ toHomf.h₀ j
hs₀ :
  { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.s₀ =
    { toHom := toHomf, s₁ := @gs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 }.s₀
hh₀ :
  ∀ (i : E.I₀),
    { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.h₀ i =
      { toHom := toHomf, s₁ := @gs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 : E.Hom F }.h₀ i ≫ eqToHom ⋯
hs₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.s₁ k =
      (congrIndexOneOfEq ⋯ ⋯ :
          (a :
              F.I₁ ({ toHom := toHomf, s₁ := gs₁, h₁ := gh₁, w₁₁ := w₁₁_1, w₁₂ := w₁₂_1 }.s₀ i)
                ({ toHom := toHomf, s₁ := gs₁, h₁ := gh₁, w₁₁ := w₁₁_1, w₁₂ := w₁₂_1 }.s₀ j)) →
            F.I₁ ({ toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁, w₁₂ := w₁₂ }.s₀ i)
              ({ toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁, w₁₂ := w₁₂ }.s₀ j))
        ({ toHom := toHomf, s₁ := @gs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 }.s₁ k)
hh₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.h₁ k =
      { toHom := toHomf, s₁ := @gs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 : E.Hom F }.h₁ k ≫
        Iso.inv
            (congrIndexOneOfEqIso ⋯ ⋯
              ({ toHom := toHomf, s₁ := @gs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 : E.Hom F }.s₁ k)) ≫
          eqToHom ⋯
⊢ @fs₁ = @gs₁
have h : ∀ (i j : E.I₀) (k : E.I₁ i j), fs₁ k = gs₁ k
Tactic: ext i j k
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{u_2, v, u} S
F : PreOneHypercover.{u_3, v, u} S
toHomf : PreZeroHypercover.Hom.{u_3, u_2, v, u} E.toPreZeroHypercover F.toPreZeroHypercover
fs₁ : {i j : E.I₀} → (k : E.I₁ i j) → F.I₁ (toHomf.s₀ i) (toHomf.s₀ j)
fh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (fs₁ k)
w₁₁ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₁ (fs₁ k) = E.p₁ k ≫ toHomf.h₀ i
w₁₂ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₂ (fs₁ k) = E.p₂ k ≫ toHomf.h₀ j
gh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (fs₁ k)
w₁₁_1 : ∀ {i j : E.I₀} (k : E.I₁ i j), gh₁ k ≫ F.p₁ (fs₁ k) = E.p₁ k ≫ toHomf.h₀ i
w₁₂_1 : ∀ {i j : E.I₀} (k : E.I₁ i j), gh₁ k ≫ F.p₂ (fs₁ k) = E.p₂ k ≫ toHomf.h₀ j
hs₀ :
  { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.s₀ =
    { toHom := toHomf, s₁ := @fs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 }.s₀
hh₀ :
  ∀ (i : E.I₀),
    { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.h₀ i =
      { toHom := toHomf, s₁ := @fs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 : E.Hom F }.h₀ i ≫ eqToHom ⋯
hs₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.s₁ k =
      (congrIndexOneOfEq ⋯ ⋯ :
          (a :
              F.I₁ ({ toHom := toHomf, s₁ := fs₁, h₁ := gh₁, w₁₁ := w₁₁_1, w₁₂ := w₁₂_1 }.s₀ i)
                ({ toHom := toHomf, s₁ := fs₁, h₁ := gh₁, w₁₁ := w₁₁_1, w₁₂ := w₁₂_1 }.s₀ j)) →
            F.I₁ ({ toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁, w₁₂ := w₁₂ }.s₀ i)
              ({ toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁, w₁₂ := w₁₂ }.s₀ j))
        ({ toHom := toHomf, s₁ := @fs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 }.s₁ k)
hh₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.h₁ k =
      { toHom := toHomf, s₁ := @fs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 : E.Hom F }.h₁ k ≫
        Iso.inv
            (congrIndexOneOfEqIso ⋯ ⋯
              ({ toHom := toHomf, s₁ := @fs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 : E.Hom F }.s₁ k)) ≫
          eqToHom ⋯
⊢ { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F } =
    { toHom := toHomf, s₁ := @fs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 }
have simp_all : ∀
  (hh₁ :
    ∀ (i j : E.I₀) (k : E.I₁ i j),
      fh₁ k =
        gh₁ k ≫
          𝟙
            (F.Y (i₁ := toHomf.s₀ i) (i₂ := toHomf.s₀ j)
              ((congrIndexOneOfEq sorry sorry : (a : F.I₁ (toHomf.s₀ i) (toHomf.s₀ j)) → F.I₁ (toHomf.s₀ i) (toHomf.s₀ j))
                (fs₁ k)))),
  @fh₁ = @gh₁
Tactic: simp_all only [eqToHom_refl, Category.comp_id, implies_true, congrIndexOneOfEqIso_refl,
    Iso.refl_inv, mk.injEq, heq_eq_eq, true_and]
Cleared hypotheses: hh₀, hh₁
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{u_2, v, u} S
F : PreOneHypercover.{u_3, v, u} S
toHomf : PreZeroHypercover.Hom.{u_3, u_2, v, u} E.toPreZeroHypercover F.toPreZeroHypercover
fs₁ : {i j : E.I₀} → (k : E.I₁ i j) → F.I₁ (toHomf.s₀ i) (toHomf.s₀ j)
fh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (fs₁ k)
w₁₁ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₁ (fs₁ k) = E.p₁ k ≫ toHomf.h₀ i
w₁₂ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₂ (fs₁ k) = E.p₂ k ≫ toHomf.h₀ j
gh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (fs₁ k)
w₁₁_1 : ∀ {i j : E.I₀} (k : E.I₁ i j), gh₁ k ≫ F.p₁ (fs₁ k) = E.p₁ k ≫ toHomf.h₀ i
w₁₂_1 : ∀ {i j : E.I₀} (k : E.I₁ i j), gh₁ k ≫ F.p₂ (fs₁ k) = E.p₂ k ≫ toHomf.h₀ j
hs₀ :
  { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.s₀ =
    { toHom := toHomf, s₁ := @fs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 }.s₀
hs₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.s₁ k =
      (congrIndexOneOfEq ⋯ ⋯ :
          (a :
              F.I₁ ({ toHom := toHomf, s₁ := fs₁, h₁ := gh₁, w₁₁ := w₁₁_1, w₁₂ := w₁₂_1 }.s₀ i)
                ({ toHom := toHomf, s₁ := fs₁, h₁ := gh₁, w₁₁ := w₁₁_1, w₁₂ := w₁₂_1 }.s₀ j)) →
            F.I₁ ({ toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁, w₁₂ := w₁₂ }.s₀ i)
              ({ toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁, w₁₂ := w₁₂ }.s₀ j))
        ({ toHom := toHomf, s₁ := @fs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 }.s₁ k)
hh₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    fh₁ k =
      gh₁ k ≫
        𝟙
          (F.Y (i₁ := toHomf.s₀ i) (i₂ := toHomf.s₀ j)
            ((congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ (toHomf.s₀ i) (toHomf.s₀ j)) → F.I₁ (toHomf.s₀ i) (toHomf.s₀ j))
              (fs₁ k)))
⊢ @fh₁ = @gh₁
have h : ∀ (i j : E.I₀) (k : E.I₁ i j), fh₁ k = gh₁ k
Tactic: ext i j k
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{u_2, v, u} S
F : PreOneHypercover.{u_3, v, u} S
toHomf : PreZeroHypercover.Hom.{u_3, u_2, v, u} E.toPreZeroHypercover F.toPreZeroHypercover
fs₁ : {i j : E.I₀} → (k : E.I₁ i j) → F.I₁ (toHomf.s₀ i) (toHomf.s₀ j)
fh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (fs₁ k)
w₁₁ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₁ (fs₁ k) = E.p₁ k ≫ toHomf.h₀ i
w₁₂ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₂ (fs₁ k) = E.p₂ k ≫ toHomf.h₀ j
gh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (fs₁ k)
w₁₁_1 : ∀ {i j : E.I₀} (k : E.I₁ i j), gh₁ k ≫ F.p₁ (fs₁ k) = E.p₁ k ≫ toHomf.h₀ i
w₁₂_1 : ∀ {i j : E.I₀} (k : E.I₁ i j), gh₁ k ≫ F.p₂ (fs₁ k) = E.p₂ k ≫ toHomf.h₀ j
hs₀ :
  { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.s₀ =
    { toHom := toHomf, s₁ := @fs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 }.s₀
hs₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    { toHom := toHomf, s₁ := @fs₁, h₁ := @fh₁, w₁₁ := @w₁₁, w₁₂ := @w₁₂ : E.Hom F }.s₁ k =
      (congrIndexOneOfEq ⋯ ⋯ :
          (a :
              F.I₁ ({ toHom := toHomf, s₁ := fs₁, h₁ := gh₁, w₁₁ := w₁₁_1, w₁₂ := w₁₂_1 }.s₀ i)
                ({ toHom := toHomf, s₁ := fs₁, h₁ := gh₁, w₁₁ := w₁₁_1, w₁₂ := w₁₂_1 }.s₀ j)) →
            F.I₁ ({ toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁, w₁₂ := w₁₂ }.s₀ i)
              ({ toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁, w₁₂ := w₁₂ }.s₀ j))
        ({ toHom := toHomf, s₁ := @fs₁, h₁ := @gh₁, w₁₁ := @w₁₁_1, w₁₂ := @w₁₂_1 }.s₁ k)
hh₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    fh₁ k =
      gh₁ k ≫
        𝟙
          (F.Y (i₁ := toHomf.s₀ i) (i₂ := toHomf.s₀ j)
            ((congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ (toHomf.s₀ i) (toHomf.s₀ j)) → F.I₁ (toHomf.s₀ i) (toHomf.s₀ j))
              (fs₁ k)))
i j : E.I₀
k : E.I₁ i j
⊢ fh₁ k = gh₁ k
have h : gh₁ k ≫
    𝟙
      (F.Y (i₁ := toHomf.s₀ i) (i₂ := toHomf.s₀ j)
        ((congrIndexOneOfEq sorry sorry : (a : F.I₁ (toHomf.s₀ i) (toHomf.s₀ j)) → F.I₁ (toHomf.s₀ i) (toHomf.s₀ j)) (fs₁ k))) =
  gh₁ k
Tactic: rw [hh₁ i j 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_1 E_2 : PreOneHypercover.{w, v, u} S
F_1 : PreOneHypercover.{w', v, u} S
G : PreOneHypercover.{?u.269827, v, u} S
E : PreOneHypercover.{u_2, v, u} S
F : PreOneHypercover.{u_3, v, u} S
f g : E.Hom F
hs₀ : f.s₀ = g.s₀
h : ∀ (i : E.I₀), f.h₀ i = g.h₀ i ≫ eqToHom ⋯
hs₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    f.s₁ k = (congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ (g.s₀ i) (g.s₀ j)) → F.I₁ (f.s₀ i) (f.s₀ j)) (g.s₁ k)
i j : E.I₀
k : E.I₁ i j
⊢ F.Y (i₁ := f.s₀ i) (i₂ := f.s₀ j)
      ((congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ (g.s₀ i) (g.s₀ j)) → F.I₁ (f.s₀ i) (f.s₀ j)) (g.s₁ k)) =
    F.Y (f.s₁ k)
have rw : F.Y (f.s₁ k) = F.Y (f.s₁ k)
Tactic: rw [PreOneHypercover.congrIndexOneOfEq_congrFun hs₀ hs₁]