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.{u_2, v, u} S
F : PreOneHypercover.{u_3, v, u} S
f g : E.Hom F
⊢ f = g ↔
    ∃ (hs₀ : f.s₀ = g.s₀) (_ : ∀ (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.h₁ k = g.h₁ k ≫ Iso.inv (congrIndexOneOfEqIso ⋯ ⋯ (g.s₁ k)) ≫ eqToHom ⋯
have refine : ∀ (h : f = g),
  ∃ (hs₀ : f.s₀ = g.s₀) (_ : ∀ (i : E.I₀), f.h₀ i = g.h₀ i ≫ eqToHom sorry) (hs₁ :
    ∀ (i j : E.I₀) (k : E.I₁ i j),
      f.s₁ k = (congrIndexOneOfEq sorry sorry : (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.h₁ k = g.h₁ k ≫ Iso.inv (congrIndexOneOfEqIso sorry sorry (g.s₁ k)) ≫ eqToHom sorry
Tactic: refine ⟨fun h ↦ ?_, fun ⟨hs₀, hh₀, hs₁, hh₁⟩ ↦ Hom.ext' 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
f g : E.Hom F
h : f = g
⊢ ∃ (hs₀ : f.s₀ = g.s₀) (_ : ∀ (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.h₁ k = g.h₁ k ≫ Iso.inv (congrIndexOneOfEqIso ⋯ ⋯ (g.s₁ k)) ≫ eqToHom ⋯
have subst : ∃ (hs₀ : f.s₀ = f.s₀) (_ : ∀ (i : E.I₀), f.h₀ i = f.h₀ i ≫ eqToHom sorry) (hs₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    f.s₁ k = (congrIndexOneOfEq sorry sorry : (a : F.I₁ (f.s₀ i) (f.s₀ j)) → F.I₁ (f.s₀ i) (f.s₀ j)) (f.s₁ k)),
  ∀ (i j : E.I₀) (k : E.I₁ i j), f.h₁ k = f.h₁ k ≫ Iso.inv (congrIndexOneOfEqIso sorry sorry (f.s₁ k)) ≫ eqToHom sorry
Tactic: subst h
Cleared hypotheses: g, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
⊢ e.inv.s₁ (e.hom.s₁ k) =
    (congrIndexOneOfEq ⋯ ⋯ : (a : E.I₁ i j) → E.I₁ (e.inv.s₀ (e.hom.s₀ i)) (e.inv.s₀ (e.hom.s₀ j))) k
have hs₀ : (e.hom ≫ e.inv).s₀ = (Hom.toHom (𝟙 E)).s₀
Tactic: obtain ⟨hs₀, hh₀, hs₁, hh₁⟩ := PreOneHypercover.Hom.ext'_iff.mp e.hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
hs₀ : (e.hom ≫ e.inv).s₀ = (Hom.toHom (𝟙 E)).s₀
⊢ e.inv.s₁ (e.hom.s₁ k) =
    (congrIndexOneOfEq ⋯ ⋯ : (a : E.I₁ i j) → E.I₁ (e.inv.s₀ (e.hom.s₀ i)) (e.inv.s₀ (e.hom.s₀ j))) k
have hh₀ : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (Hom.toHom (𝟙 E)).h₀ i ≫ eqToHom sorry
Tactic: obtain ⟨hs₀, hh₀, hs₁, hh₁⟩ := PreOneHypercover.Hom.ext'_iff.mp e.hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
hs₀ : (e.hom ≫ e.inv).s₀ = (Hom.toHom (𝟙 E)).s₀
hh₀ : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (Hom.toHom (𝟙 E)).h₀ i ≫ eqToHom ⋯
⊢ e.inv.s₁ (e.hom.s₁ k) =
    (congrIndexOneOfEq ⋯ ⋯ : (a : E.I₁ i j) → E.I₁ (e.inv.s₀ (e.hom.s₀ i)) (e.inv.s₀ (e.hom.s₀ j))) k
have hs₁ : ∀ (i j : E.I₀) (k : E.I₁ i j),
  (e.hom ≫ e.inv).s₁ k =
    (congrIndexOneOfEq sorry sorry : (a : E.I₁ ((𝟙 E).s₀ i) ((𝟙 E).s₀ j)) → E.I₁ ((e.hom ≫ e.inv).s₀ i) ((e.hom ≫ e.inv).s₀ j))
      (Hom.s₁ (𝟙 E) k)
Tactic: obtain ⟨hs₀, hh₀, hs₁, hh₁⟩ := PreOneHypercover.Hom.ext'_iff.mp e.hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
hs₀ : (e.hom ≫ e.inv).s₀ = (Hom.toHom (𝟙 E)).s₀
hh₀ : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (Hom.toHom (𝟙 E)).h₀ i ≫ eqToHom ⋯
hs₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    (e.hom ≫ e.inv).s₁ k =
      (congrIndexOneOfEq ⋯ ⋯ :
          (a : E.I₁ ((𝟙 E).s₀ i) ((𝟙 E).s₀ j)) → E.I₁ ((e.hom ≫ e.inv).s₀ i) ((e.hom ≫ e.inv).s₀ j))
        (Hom.s₁ (𝟙 E) k)
⊢ e.inv.s₁ (e.hom.s₁ k) =
    (congrIndexOneOfEq ⋯ ⋯ : (a : E.I₁ i j) → E.I₁ (e.inv.s₀ (e.hom.s₀ i)) (e.inv.s₀ (e.hom.s₀ j))) k
have hh₁ : ∀ (i j : E.I₀) (k : E.I₁ i j),
  (e.hom ≫ e.inv).h₁ k = Hom.h₁ (𝟙 E) k ≫ Iso.inv (congrIndexOneOfEqIso sorry sorry (Hom.s₁ (𝟙 E) k)) ≫ eqToHom sorry
Tactic: obtain ⟨hs₀, hh₀, hs₁, hh₁⟩ := PreOneHypercover.Hom.ext'_iff.mp e.hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : F.I₀
k : F.I₁ i j
⊢ e.hom.s₁ (e.inv.s₁ k) =
    (congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ i j) → F.I₁ (e.hom.s₀ (e.inv.s₀ i)) (e.hom.s₀ (e.inv.s₀ j))) k
have hs₀ : (e.inv ≫ e.hom).s₀ = (Hom.toHom (𝟙 F)).s₀
Tactic: obtain ⟨hs₀, hh₀, hs₁, hh₁⟩ := PreOneHypercover.Hom.ext'_iff.mp e.inv_hom_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : F.I₀
k : F.I₁ i j
hs₀ : (e.inv ≫ e.hom).s₀ = (Hom.toHom (𝟙 F)).s₀
⊢ e.hom.s₁ (e.inv.s₁ k) =
    (congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ i j) → F.I₁ (e.hom.s₀ (e.inv.s₀ i)) (e.hom.s₀ (e.inv.s₀ j))) k
have hh₀ : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (Hom.toHom (𝟙 F)).h₀ i ≫ eqToHom sorry
Tactic: obtain ⟨hs₀, hh₀, hs₁, hh₁⟩ := PreOneHypercover.Hom.ext'_iff.mp e.inv_hom_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : F.I₀
k : F.I₁ i j
hs₀ : (e.inv ≫ e.hom).s₀ = (Hom.toHom (𝟙 F)).s₀
hh₀ : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (Hom.toHom (𝟙 F)).h₀ i ≫ eqToHom ⋯
⊢ e.hom.s₁ (e.inv.s₁ k) =
    (congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ i j) → F.I₁ (e.hom.s₀ (e.inv.s₀ i)) (e.hom.s₀ (e.inv.s₀ j))) k
have hs₁ : ∀ (i j : F.I₀) (k : F.I₁ i j),
  (e.inv ≫ e.hom).s₁ k =
    (congrIndexOneOfEq sorry sorry : (a : F.I₁ ((𝟙 F).s₀ i) ((𝟙 F).s₀ j)) → F.I₁ ((e.inv ≫ e.hom).s₀ i) ((e.inv ≫ e.hom).s₀ j))
      (Hom.s₁ (𝟙 F) k)
Tactic: obtain ⟨hs₀, hh₀, hs₁, hh₁⟩ := PreOneHypercover.Hom.ext'_iff.mp e.inv_hom_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : F.I₀
k : F.I₁ i j
hs₀ : (e.inv ≫ e.hom).s₀ = (Hom.toHom (𝟙 F)).s₀
hh₀ : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (Hom.toHom (𝟙 F)).h₀ i ≫ eqToHom ⋯
hs₁ :
  ∀ (i j : F.I₀) (k : F.I₁ i j),
    (e.inv ≫ e.hom).s₁ k =
      (congrIndexOneOfEq ⋯ ⋯ :
          (a : F.I₁ ((𝟙 F).s₀ i) ((𝟙 F).s₀ j)) → F.I₁ ((e.inv ≫ e.hom).s₀ i) ((e.inv ≫ e.hom).s₀ j))
        (Hom.s₁ (𝟙 F) k)
⊢ e.hom.s₁ (e.inv.s₁ k) =
    (congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ i j) → F.I₁ (e.hom.s₀ (e.inv.s₀ i)) (e.hom.s₀ (e.inv.s₀ j))) k
have hh₁ : ∀ (i j : F.I₀) (k : F.I₁ i j),
  (e.inv ≫ e.hom).h₁ k = Hom.h₁ (𝟙 F) k ≫ Iso.inv (congrIndexOneOfEqIso sorry sorry (Hom.s₁ (𝟙 F) k)) ≫ eqToHom sorry
Tactic: obtain ⟨hs₀, hh₀, hs₁, hh₁⟩ := PreOneHypercover.Hom.ext'_iff.mp e.inv_hom_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i : E.I₀
⊢ e.hom.h₀ i ≫ e.inv.h₀ (e.hom.s₀ i) = eqToHom ⋯
have hs : (e.hom ≫ e.inv).s₀ = (Hom.toHom (𝟙 E)).s₀
Tactic: obtain ⟨hs, hh, _⟩ := Hom.ext'_iff.mp e.hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i : E.I₀
hs : (e.hom ≫ e.inv).s₀ = (Hom.toHom (𝟙 E)).s₀
⊢ e.hom.h₀ i ≫ e.inv.h₀ (e.hom.s₀ i) = eqToHom ⋯
have hh : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (Hom.toHom (𝟙 E)).h₀ i ≫ eqToHom sorry
Tactic: obtain ⟨hs, hh, _⟩ := Hom.ext'_iff.mp e.hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i : E.I₀
hs : (e.hom ≫ e.inv).s₀ = (Hom.toHom (𝟙 E)).s₀
hh : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (Hom.toHom (𝟙 E)).h₀ i ≫ eqToHom ⋯
⊢ e.hom.h₀ i ≫ e.inv.h₀ (e.hom.s₀ i) = eqToHom ⋯
have h : ∃ (hs₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    (e.hom ≫ e.inv).s₁ k =
      (congrIndexOneOfEq sorry sorry :
          (a : E.I₁ ((𝟙 E).s₀ i) ((𝟙 E).s₀ j)) → E.I₁ ((e.hom ≫ e.inv).s₀ i) ((e.hom ≫ e.inv).s₀ j))
        (Hom.s₁ (𝟙 E) k)),
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    (e.hom ≫ e.inv).h₁ k = Hom.h₁ (𝟙 E) k ≫ Iso.inv (congrIndexOneOfEqIso sorry sorry (Hom.s₁ (𝟙 E) k)) ≫ eqToHom sorry
Tactic: obtain ⟨hs, hh, _⟩ := Hom.ext'_iff.mp e.hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i : F.I₀
⊢ e.inv.h₀ i ≫ e.hom.h₀ (e.inv.s₀ i) = eqToHom ⋯
have hs : (e.inv ≫ e.hom).s₀ = (Hom.toHom (𝟙 F)).s₀
Tactic: obtain ⟨hs, hh, _⟩ := Hom.ext'_iff.mp e.inv_hom_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i : F.I₀
hs : (e.inv ≫ e.hom).s₀ = (Hom.toHom (𝟙 F)).s₀
⊢ e.inv.h₀ i ≫ e.hom.h₀ (e.inv.s₀ i) = eqToHom ⋯
have hh : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (Hom.toHom (𝟙 F)).h₀ i ≫ eqToHom sorry
Tactic: obtain ⟨hs, hh, _⟩ := Hom.ext'_iff.mp e.inv_hom_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i : F.I₀
hs : (e.inv ≫ e.hom).s₀ = (Hom.toHom (𝟙 F)).s₀
hh : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (Hom.toHom (𝟙 F)).h₀ i ≫ eqToHom ⋯
⊢ e.inv.h₀ i ≫ e.hom.h₀ (e.inv.s₀ i) = eqToHom ⋯
have h : ∃ (hs₁ :
  ∀ (i j : F.I₀) (k : F.I₁ i j),
    (e.inv ≫ e.hom).s₁ k =
      (congrIndexOneOfEq sorry sorry :
          (a : F.I₁ ((𝟙 F).s₀ i) ((𝟙 F).s₀ j)) → F.I₁ ((e.inv ≫ e.hom).s₀ i) ((e.inv ≫ e.hom).s₀ j))
        (Hom.s₁ (𝟙 F) k)),
  ∀ (i j : F.I₀) (k : F.I₁ i j),
    (e.inv ≫ e.hom).h₁ k = Hom.h₁ (𝟙 F) k ≫ Iso.inv (congrIndexOneOfEqIso sorry sorry (Hom.s₁ (𝟙 F) k)) ≫ eqToHom sorry
Tactic: obtain ⟨hs, hh, _⟩ := Hom.ext'_iff.mp e.inv_hom_id
-- 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.339607, v, u} S_1
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
⊢ E.Y (i₁ := e.inv.s₀ (e.hom.s₀ i)) (i₂ := e.inv.s₀ (e.hom.s₀ j))
      ((congrIndexOneOfEq ⋯ ⋯ : (a : E.I₁ i j) → E.I₁ (e.inv.s₀ (e.hom.s₀ i)) (e.inv.s₀ (e.hom.s₀ j))) k) =
    E.Y (e.inv.s₁ (e.hom.s₁ k))
have e_j : (congrIndexOneOfEq sorry sorry : (a : E.I₁ i j) → E.I₁ (e.inv.s₀ (e.hom.s₀ i)) (e.inv.s₀ (e.hom.s₀ j))) k =
  e.inv.s₁ (e.hom.s₁ k)
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
⊢ e.hom.h₁ k ≫ e.inv.h₁ (e.hom.s₁ k) = (congrIndexOneOfEqIso ⋯ ⋯ k).inv ≫ eqToHom ⋯
have hs : (e.hom ≫ e.inv).s₀ = (Hom.toHom (𝟙 E)).s₀
Tactic: obtain ⟨hs, _, _, hh⟩ := Hom.ext'_iff.mp e.hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
hs : (e.hom ≫ e.inv).s₀ = (Hom.toHom (𝟙 E)).s₀
⊢ e.hom.h₁ k ≫ e.inv.h₁ (e.hom.s₁ k) = (congrIndexOneOfEqIso ⋯ ⋯ k).inv ≫ eqToHom ⋯
have w : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (Hom.toHom (𝟙 E)).h₀ i ≫ eqToHom sorry
Tactic: obtain ⟨hs, _, _, hh⟩ := Hom.ext'_iff.mp e.hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
hs : (e.hom ≫ e.inv).s₀ = (Hom.toHom (𝟙 E)).s₀
w : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (Hom.toHom (𝟙 E)).h₀ i ≫ eqToHom ⋯
⊢ e.hom.h₁ k ≫ e.inv.h₁ (e.hom.s₁ k) = (congrIndexOneOfEqIso ⋯ ⋯ k).inv ≫ eqToHom ⋯
have w₁ : ∀ (i j : E.I₀) (k : E.I₁ i j),
  (e.hom ≫ e.inv).s₁ k =
    (congrIndexOneOfEq sorry sorry : (a : E.I₁ ((𝟙 E).s₀ i) ((𝟙 E).s₀ j)) → E.I₁ ((e.hom ≫ e.inv).s₀ i) ((e.hom ≫ e.inv).s₀ j))
      (Hom.s₁ (𝟙 E) k)
Tactic: obtain ⟨hs, _, _, hh⟩ := Hom.ext'_iff.mp e.hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
hs : (e.hom ≫ e.inv).s₀ = (Hom.toHom (𝟙 E)).s₀
w : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (Hom.toHom (𝟙 E)).h₀ i ≫ eqToHom ⋯
w₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    (e.hom ≫ e.inv).s₁ k =
      (congrIndexOneOfEq ⋯ ⋯ :
          (a : E.I₁ ((𝟙 E).s₀ i) ((𝟙 E).s₀ j)) → E.I₁ ((e.hom ≫ e.inv).s₀ i) ((e.hom ≫ e.inv).s₀ j))
        (Hom.s₁ (𝟙 E) k)
⊢ e.hom.h₁ k ≫ e.inv.h₁ (e.hom.s₁ k) = (congrIndexOneOfEqIso ⋯ ⋯ k).inv ≫ eqToHom ⋯
have hh : ∀ (i j : E.I₀) (k : E.I₁ i j),
  (e.hom ≫ e.inv).h₁ k = Hom.h₁ (𝟙 E) k ≫ Iso.inv (congrIndexOneOfEqIso sorry sorry (Hom.s₁ (𝟙 E) k)) ≫ eqToHom sorry
Tactic: obtain ⟨hs, _, _, hh⟩ := Hom.ext'_iff.mp e.hom_inv_id
-- 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.344064, v, u} S_1
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : F.I₀
k : F.I₁ i j
⊢ F.Y (i₁ := e.hom.s₀ (e.inv.s₀ i)) (i₂ := e.hom.s₀ (e.inv.s₀ j))
      ((congrIndexOneOfEq ⋯ ⋯ : (a : F.I₁ i j) → F.I₁ (e.hom.s₀ (e.inv.s₀ i)) (e.hom.s₀ (e.inv.s₀ j))) k) =
    F.Y (e.hom.s₁ (e.inv.s₁ k))
have e_j : (congrIndexOneOfEq sorry sorry : (a : F.I₁ i j) → F.I₁ (e.hom.s₀ (e.inv.s₀ i)) (e.hom.s₀ (e.inv.s₀ j))) k =
  e.hom.s₁ (e.inv.s₁ k)
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : F.I₀
k : F.I₁ i j
⊢ e.inv.h₁ k ≫ e.hom.h₁ (e.inv.s₁ k) = (congrIndexOneOfEqIso ⋯ ⋯ k).inv ≫ eqToHom ⋯
have hs : (e.inv ≫ e.hom).s₀ = (Hom.toHom (𝟙 F)).s₀
Tactic: obtain ⟨hs, _, _, hh⟩ := Hom.ext'_iff.mp e.inv_hom_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : F.I₀
k : F.I₁ i j
hs : (e.inv ≫ e.hom).s₀ = (Hom.toHom (𝟙 F)).s₀
⊢ e.inv.h₁ k ≫ e.hom.h₁ (e.inv.s₁ k) = (congrIndexOneOfEqIso ⋯ ⋯ k).inv ≫ eqToHom ⋯
have w : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (Hom.toHom (𝟙 F)).h₀ i ≫ eqToHom sorry
Tactic: obtain ⟨hs, _, _, hh⟩ := Hom.ext'_iff.mp e.inv_hom_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : F.I₀
k : F.I₁ i j
hs : (e.inv ≫ e.hom).s₀ = (Hom.toHom (𝟙 F)).s₀
w : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (Hom.toHom (𝟙 F)).h₀ i ≫ eqToHom ⋯
⊢ e.inv.h₁ k ≫ e.hom.h₁ (e.inv.s₁ k) = (congrIndexOneOfEqIso ⋯ ⋯ k).inv ≫ eqToHom ⋯
have w₁ : ∀ (i j : F.I₀) (k : F.I₁ i j),
  (e.inv ≫ e.hom).s₁ k =
    (congrIndexOneOfEq sorry sorry : (a : F.I₁ ((𝟙 F).s₀ i) ((𝟙 F).s₀ j)) → F.I₁ ((e.inv ≫ e.hom).s₀ i) ((e.inv ≫ e.hom).s₀ j))
      (Hom.s₁ (𝟙 F) k)
Tactic: obtain ⟨hs, _, _, hh⟩ := Hom.ext'_iff.mp e.inv_hom_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.One (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : F.I₀
k : F.I₁ i j
hs : (e.inv ≫ e.hom).s₀ = (Hom.toHom (𝟙 F)).s₀
w : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (Hom.toHom (𝟙 F)).h₀ i ≫ eqToHom ⋯
w₁ :
  ∀ (i j : F.I₀) (k : F.I₁ i j),
    (e.inv ≫ e.hom).s₁ k =
      (congrIndexOneOfEq ⋯ ⋯ :
          (a : F.I₁ ((𝟙 F).s₀ i) ((𝟙 F).s₀ j)) → F.I₁ ((e.inv ≫ e.hom).s₀ i) ((e.inv ≫ e.hom).s₀ j))
        (Hom.s₁ (𝟙 F) k)
⊢ e.inv.h₁ k ≫ e.hom.h₁ (e.inv.s₁ k) = (congrIndexOneOfEqIso ⋯ ⋯ k).inv ≫ eqToHom ⋯
have hh : ∀ (i j : F.I₀) (k : F.I₁ i j),
  (e.inv ≫ e.hom).h₁ k = Hom.h₁ (𝟙 F) k ≫ Iso.inv (congrIndexOneOfEqIso sorry sorry (Hom.s₁ (𝟙 F) k)) ≫ eqToHom sorry
Tactic: obtain ⟨hs, _, _, hh⟩ := Hom.ext'_iff.mp e.inv_hom_id
-- 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.348521, v, u} S_1
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i : E.I₀
⊢ IsIso (e.hom.h₀ i)
have h : e.hom.h₀ i ≫ e.inv.h₀ (e.hom.s₀ i) ≫ eqToHom sorry = 𝟙 (E.X i) ∧
  (e.inv.h₀ (e.hom.s₀ i) ≫ eqToHom sorry) ≫ e.hom.h₀ i = 𝟙 (F.X (e.hom.s₀ i))
Tactic: use e.inv.h₀ (e.hom.s₀ i) ≫ eqToHom (by simp)
-- 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.348521, v, u} S_1
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i : E.I₀
⊢ e.hom.h₀ i ≫ e.inv.h₀ (e.hom.s₀ i) ≫ eqToHom ⋯ = 𝟙 (E.X i) ∧
    (e.inv.h₀ (e.hom.s₀ i) ≫ eqToHom ⋯) ≫ e.hom.h₀ i = 𝟙 (F.X (e.hom.s₀ i))
have h : Eq.{v + 1} (α := E.X i ⟶ E.X i) (𝟙 (E.X i)) (𝟙 (E.X i)) ∧
  Eq.{v + 1} (α := F.X (e.hom.s₀ i) ⟶ F.X (e.hom.s₀ i)) (eqToHom sorry ≫ eqToHom sorry) (𝟙 (F.X (e.hom.s₀ i)))
Tactic: rw [PreOneHypercover.hom_inv_h₀_assoc, eqToHom_trans, eqToHom_refl, Category.assoc,
    ← eqToHom_naturality _ (by simp), PreOneHypercover.inv_hom_h₀_assoc]
-- 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.354132, v, u} S_1
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
⊢ E.Y (e.inv.s₁ (e.hom.s₁ k)) =
    E.Y (i₁ := e.inv.s₀ (e.hom.s₀ i)) (i₂ := e.inv.s₀ (e.hom.s₀ j))
      ((congrIndexOneOfEq ⋯ ⋯ : (a : E.I₁ i j) → E.I₁ (e.inv.s₀ (e.hom.s₀ i)) (e.inv.s₀ (e.hom.s₀ j))) k)
have e_j : e.inv.s₁ (e.hom.s₁ k) =
  (congrIndexOneOfEq sorry sorry : (a : E.I₁ i j) → E.I₁ (e.inv.s₀ (e.hom.s₀ i)) (e.inv.s₀ (e.hom.s₀ j))) k
Tactic: congr 1
-- 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.354132, v, u} S_1
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
⊢ IsIso (e.hom.h₁ k)
have h : e.hom.h₁ k ≫ e.inv.h₁ (e.hom.s₁ k) ≫ eqToHom sorry ≫ (congrIndexOneOfEqIso sorry sorry k).hom = 𝟙 (E.Y k) ∧
  (e.inv.h₁ (e.hom.s₁ k) ≫ eqToHom sorry ≫ (congrIndexOneOfEqIso sorry sorry k).hom) ≫ e.hom.h₁ k = 𝟙 (F.Y (e.hom.s₁ k))
Tactic: use e.inv.h₁ _ ≫ eqToHom (by congr 1; simp) ≫ (E.congrIndexOneOfEqIso (by simp) (by simp) k).hom
-- 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.354132, v, u} S_1
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
⊢ e.hom.h₁ k ≫ e.inv.h₁ (e.hom.s₁ k) ≫ eqToHom ⋯ ≫ (congrIndexOneOfEqIso ⋯ ⋯ k).hom = 𝟙 (E.Y k) ∧
    (e.inv.h₁ (e.hom.s₁ k) ≫ eqToHom ⋯ ≫ (congrIndexOneOfEqIso ⋯ ⋯ k).hom) ≫ e.hom.h₁ k = 𝟙 (F.Y (e.hom.s₁ k))
have h : e.inv.h₁ (e.hom.s₁ k) ≫
    eqToHom sorry ≫
      e.hom.h₁ ((congrIndexOneOfEq sorry sorry : (a : E.I₁ i j) → E.I₁ (e.inv.s₀ (e.hom.s₀ i)) (e.inv.s₀ (e.hom.s₀ j))) k) ≫
        eqToHom sorry ≫ Iso.hom (congrIndexOneOfEqIso sorry sorry (e.hom.s₁ k)) =
  𝟙 (F.Y (e.hom.s₁ k))
Tactic: simp only [PreOneHypercover.hom_inv_h₁_assoc, eqToHom_trans_assoc, eqToHom_refl, Category.id_comp,
    Iso.inv_hom_id, Category.assoc, true_and, PreOneHypercover.congrIndexOneOfEqIso_hom_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_1 : C
E_1 E_2 : PreOneHypercover.{w, v, u} S_1
F_1 : PreOneHypercover.{w', v, u} S_1
G : PreOneHypercover.{?u.354132, v, u} S_1
S : C
E F : PreOneHypercover.{w, v, u} S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
⊢ e.inv.h₁ (e.hom.s₁ k) ≫
      eqToHom ⋯ ≫
        e.hom.h₁ ((congrIndexOneOfEq ⋯ ⋯ : (a : E.I₁ i j) → E.I₁ (e.inv.s₀ (e.hom.s₀ i)) (e.inv.s₀ (e.hom.s₀ j))) k) ≫
          eqToHom ⋯ ≫ Iso.hom (congrIndexOneOfEqIso ⋯ ⋯ (e.hom.s₁ k)) =
    𝟙 (F.Y (e.hom.s₁ k))
have h : e.inv.h₁ (e.hom.s₁ k) ≫
    e.hom.h₁ (e.inv.s₁ (e.hom.s₁ k)) ≫ eqToHom sorry ≫ eqToHom sorry ≫ Iso.hom (congrIndexOneOfEqIso sorry sorry (e.hom.s₁ k)) =
  𝟙 (F.Y (e.hom.s₁ k))
Tactic: rw [← eqToHom_naturality_assoc _ (by simp)]
-- 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 : E.multicospanShape.R
⊢ Functor.obj (𝟭 (WalkingMulticospan E.multicospanShape)) (WalkingMulticospan.right x) =
    (Hom.mapMulticospan f.hom ⋙ Hom.mapMulticospan f.inv).obj (WalkingMulticospan.right x)
have dsimp : WalkingMulticospan.right x =
  WalkingMulticospan.right
    (Sigma.mk (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)
      (f.inv.s₀ (f.hom.s₀ x.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).1),
        f.inv.s₀ (f.hom.s₀ x.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).2))
      (f.inv.s₁ (f.hom.s₁ (x.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)))))
Tactic: dsimp
-- 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 : E.multicospanShape.R
⊢ WalkingMulticospan.right x =
    WalkingMulticospan.right
      (Sigma.mk (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)
        (f.inv.s₀ (f.hom.s₀ x.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).1),
          f.inv.s₀ (f.hom.s₀ x.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).2))
        (f.inv.s₁ (f.hom.s₁ (x.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)))))
have e_a : x =
  Sigma.mk (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2)
    (f.inv.s₀ (f.hom.s₀ x.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).1),
      f.inv.s₀ (f.hom.s₀ x.fst (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2).2))
    (f.inv.s₁ (f.hom.s₁ (x.snd (β := fun (i : E.I₀ × E.I₀) ↦ E.I₁ i.1 i.2))))
Tactic: congr