awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreZeroHypercover.{w, v, u} S
F : PreZeroHypercover.{w', v, u} S
g : E.Hom F
s₀ : (i : E.I₀) → F.I₀
h₀ : (i : E.I₀) → E.X i ⟶ F.X (s₀ i)
w₀ : ∀ (i : E.I₀), h₀ i ≫ F.f (s₀ i) = E.f i
hs : { s₀ := s₀, h₀ := h₀, w₀ := w₀ : E.Hom F }.s₀ = g.s₀
hh : ∀ (i : E.I₀), { s₀ := s₀, h₀ := h₀, w₀ := w₀ : E.Hom F }.h₀ i = g.h₀ i ≫ eqToHom ⋯
⊢ { s₀ := s₀, h₀ := h₀, w₀ := w₀ : E.Hom F } = g
have mk : ∀ (s₀_1 : (i : E.I₀) → F.I₀) (h₀_1 : (i : E.I₀) → E.X i ⟶ F.X (s₀_1 i))
  (w₀_1 : ∀ (i : E.I₀), h₀_1 i ≫ F.f (s₀_1 i) = E.f i)
  (hs : { s₀ := s₀, h₀ := h₀, w₀ := w₀ : E.Hom F }.s₀ = { s₀ := s₀_1, h₀ := h₀_1, w₀ := w₀_1 }.s₀)
  (hh :
    ∀ (i : E.I₀),
      { s₀ := s₀, h₀ := h₀, w₀ := w₀ : E.Hom F }.h₀ i =
        { s₀ := s₀_1, h₀ := h₀_1, w₀ := w₀_1 : E.Hom F }.h₀ i ≫ eqToHom sorry),
  { s₀ := s₀, h₀ := h₀, w₀ := w₀ : E.Hom F } = { s₀ := s₀_1, h₀ := h₀_1, w₀ := w₀_1 }
Tactic: cases g
Cleared hypotheses: g, hs, hh
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S T : C
E_1 : PreZeroHypercover.{w, v, u} S
F_1 : PreZeroHypercover.{w', v, u} S
E_2 : PreZeroHypercover.{w, v, u} S
F_2 : PreZeroHypercover.{w', v, u} S
G : PreZeroHypercover.{?u.68465, v, u} S
E : PreZeroHypercover.{w, v, u} S
F : PreZeroHypercover.{w', v, u} S
f g : E.Hom F
hs : f.s₀ = g.s₀
i : E.I₀
⊢ F.X (g.s₀ i) = F.X (f.s₀ i)
have rw : F.X (g.s₀ i) = F.X (g.s₀ i)
Tactic: rw [hs]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S_1 T : C
E_1 : PreZeroHypercover.{w, v, u} S_1
F_1 : PreZeroHypercover.{w', v, u} S_1
E_2 : PreZeroHypercover.{w, v, u} S_1
F_2 : PreZeroHypercover.{w', v, u} S_1
G : PreZeroHypercover.{?u.70083, v, u} S_1
S : C
E F : PreZeroHypercover.{w, v, u} S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X ((s₀ : (a : E.I₀) → F.I₀) i)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f ((s₀ : (a : E.I₀) → F.I₀) i) = E.f i
i : F.I₀
⊢ (eqToHom ⋯ ≫ Iso.inv (h₀ ((s₀.symm : (a : F.I₀) → E.I₀) i))) ≫ E.f ((s₀.symm : (a : F.I₀) → E.I₀) i) = F.f i
have obtain : ∀ (i : E.I₀),
  Eq (α := F.X ((s₀ : (a : E.I₀) → F.I₀) i) ⟶ S)
    ((eqToHom sorry ≫ Iso.inv (h₀ ((s₀.symm : (a : F.I₀) → E.I₀) ((s₀ : (a : E.I₀) → F.I₀) i : F.I₀)))) ≫
      E.f ((s₀.symm : (a : F.I₀) → E.I₀) ((s₀ : (a : E.I₀) → F.I₀) i : F.I₀)))
    (F.f ((s₀ : (a : E.I₀) → F.I₀) i))
Tactic: obtain ⟨i, rfl⟩ := s₀.surjective i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S_1 T : C
E_1 : PreZeroHypercover.{w, v, u} S_1
F_1 : PreZeroHypercover.{w', v, u} S_1
E_2 : PreZeroHypercover.{w, v, u} S_1
F_2 : PreZeroHypercover.{w', v, u} S_1
G : PreZeroHypercover.{?u.70083, v, u} S_1
S : C
E F : PreZeroHypercover.{w, v, u} S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X ((s₀ : (a : E.I₀) → F.I₀) i)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f ((s₀ : (a : E.I₀) → F.I₀) i) = E.f i
i : E.I₀
⊢ Eq (α := F.X ((s₀ : (a : E.I₀) → F.I₀) i) ⟶ S)
    ((eqToHom ⋯ ≫ Iso.inv (h₀ ((s₀.symm : (a : F.I₀) → E.I₀) ((s₀ : (a : E.I₀) → F.I₀) i : F.I₀)))) ≫
      E.f ((s₀.symm : (a : F.I₀) → E.I₀) ((s₀ : (a : E.I₀) → F.I₀) i : F.I₀)))
    (F.f ((s₀ : (a : E.I₀) → F.I₀) i))
have simp : eqToHom sorry ≫ E.f ((s₀.symm : (a : F.I₀) → E.I₀) ((s₀ : (a : E.I₀) → F.I₀) i : F.I₀)) = E.f i
Tactic: simp only [← cancel_epi (h₀ i).hom, w₀, Category.assoc, Equiv.symm_apply_apply,
      eqToHom_iso_hom_naturality_assoc, Iso.hom_inv_id_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S_1 T : C
E_1 : PreZeroHypercover.{w, v, u} S_1
F_1 : PreZeroHypercover.{w', v, u} S_1
E_2 : PreZeroHypercover.{w, v, u} S_1
F_2 : PreZeroHypercover.{w', v, u} S_1
G : PreZeroHypercover.{?u.70083, v, u} S_1
S : C
E F : PreZeroHypercover.{w, v, u} S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X ((s₀ : (a : E.I₀) → F.I₀) i)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f ((s₀ : (a : E.I₀) → F.I₀) i) = E.f i
i : E.I₀
⊢ eqToHom ⋯ ≫ E.f ((s₀.symm : (a : F.I₀) → E.I₀) ((s₀ : (a : E.I₀) → F.I₀) i : F.I₀)) = E.f i
have rw : E.f i ≫ eqToHom sorry = E.f i
Tactic: rw [← CategoryTheory.eqToHom_naturality _ (by simp)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S_1 T : C
E_1 : PreZeroHypercover.{w, v, u} S_1
F_1 : PreZeroHypercover.{w', v, u} S_1
E_2 : PreZeroHypercover.{w, v, u} S_1
F_2 : PreZeroHypercover.{w', v, u} S_1
G : PreZeroHypercover.{?u.70083, v, u} S_1
S : C
E F : PreZeroHypercover.{w, v, u} S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X ((s₀ : (a : E.I₀) → F.I₀) i)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f ((s₀ : (a : E.I₀) → F.I₀) i) = E.f i
⊢ Hom.s₀
      ({ s₀ := (⇑s₀ : (a : E.I₀) → F.I₀), h₀ := fun (i : E.I₀) ↦ (h₀ i).hom, w₀ := ⋯ : E.Hom F } ≫
        { s₀ := (⇑s₀.symm : (a : F.I₀) → E.I₀),
          h₀ := fun (i : F.I₀) ↦ eqToHom ⋯ ≫ Iso.inv (h₀ ((s₀.symm : (a : F.I₀) → E.I₀) i)), w₀ := ⋯ }) =
    Hom.s₀ (𝟙 E)
have h : ∀ (x : E.I₀),
  Hom.s₀
      ({ s₀ := (⇑s₀ : (a : E.I₀) → F.I₀), h₀ := fun (i : E.I₀) ↦ (h₀ i).hom, w₀ := sorry : E.Hom F } ≫
        { s₀ := (⇑s₀.symm : (a : F.I₀) → E.I₀),
          h₀ := fun (i : F.I₀) ↦ eqToHom sorry ≫ Iso.inv (h₀ ((s₀.symm : (a : F.I₀) → E.I₀) i)), w₀ := sorry })
      x =
    Hom.s₀ (𝟙 E) x
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S_1 T : C
E_1 : PreZeroHypercover.{w, v, u} S_1
F_1 : PreZeroHypercover.{w', v, u} S_1
E_2 : PreZeroHypercover.{w, v, u} S_1
F_2 : PreZeroHypercover.{w', v, u} S_1
G : PreZeroHypercover.{?u.70083, v, u} S_1
S : C
E F : PreZeroHypercover.{w, v, u} S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X ((s₀ : (a : E.I₀) → F.I₀) i)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f ((s₀ : (a : E.I₀) → F.I₀) i) = E.f i
⊢ Hom.s₀
      ({ s₀ := (⇑s₀.symm : (a : F.I₀) → E.I₀),
          h₀ := fun (i : F.I₀) ↦ eqToHom ⋯ ≫ Iso.inv (h₀ ((s₀.symm : (a : F.I₀) → E.I₀) i)), w₀ := ⋯ } ≫
        { s₀ := (⇑s₀ : (a : E.I₀) → F.I₀), h₀ := fun (i : E.I₀) ↦ (h₀ i).hom, w₀ := ⋯ : E.Hom F }) =
    Hom.s₀ (𝟙 F)
have h : ∀ (x : F.I₀),
  Hom.s₀
      ({ s₀ := (⇑s₀.symm : (a : F.I₀) → E.I₀),
          h₀ := fun (i : F.I₀) ↦ eqToHom sorry ≫ Iso.inv (h₀ ((s₀.symm : (a : F.I₀) → E.I₀) i)), w₀ := sorry } ≫
        { s₀ := (⇑s₀ : (a : E.I₀) → F.I₀), h₀ := fun (i : E.I₀) ↦ (h₀ i).hom, w₀ := sorry : E.Hom F })
      x =
    Hom.s₀ (𝟙 F) x
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreZeroHypercover.{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.s₀ (𝟙 E)
Tactic: obtain ⟨hs, hh⟩ := Hom.ext'_iff.mp e.hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreZeroHypercover.{w, v, u} S
e : E ≅ F
i : E.I₀
hs : (e.hom ≫ e.inv).s₀ = Hom.s₀ (𝟙 E)
⊢ e.hom.h₀ i ≫ e.inv.h₀ (e.hom.s₀ i) = eqToHom ⋯
have hh : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = Hom.h₀ (𝟙 E) i ≫ eqToHom sorry
Tactic: obtain ⟨hs, hh⟩ := Hom.ext'_iff.mp e.hom_inv_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreZeroHypercover.{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.s₀ (𝟙 F)
Tactic: obtain ⟨hs, hh⟩ := Hom.ext'_iff.mp e.inv_hom_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreZeroHypercover.{w, v, u} S
e : E ≅ F
i : F.I₀
hs : (e.inv ≫ e.hom).s₀ = Hom.s₀ (𝟙 F)
⊢ e.inv.h₀ i ≫ e.hom.h₀ (e.inv.s₀ i) = eqToHom ⋯
have hh : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = Hom.h₀ (𝟙 F) i ≫ eqToHom sorry
Tactic: obtain ⟨hs, hh⟩ := Hom.ext'_iff.mp e.inv_hom_id
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S T : C
E_1 : PreZeroHypercover.{w, v, u} S
F_1 : PreZeroHypercover.{w', v, u} S
E_2 : PreZeroHypercover.{w, v, u} S
F_2 : PreZeroHypercover.{w', v, u} S
G : PreZeroHypercover.{?u.84161, v, u} S
E F : PreZeroHypercover.{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.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S T : C
E_1 : PreZeroHypercover.{w, v, u} S
F_1 : PreZeroHypercover.{w', v, u} S
E_2 : PreZeroHypercover.{w, v, u} S
F_2 : PreZeroHypercover.{w', v, u} S
G : PreZeroHypercover.{?u.84161, v, u} S
E F : PreZeroHypercover.{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 [hom_inv_h₀_assoc, eqToHom_trans, eqToHom_refl, Category.assoc,
    ← eqToHom_naturality _ (by simp), inv_hom_h₀_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreZeroHypercover.{u_1, v, u} S
F : PreZeroHypercover.{u_2, v, u} S
f : E.Hom F
⊢ E.sieve₀ ≤ F.sieve₀
have rw : ∀ (i : E.I₀), F.sieve₀.arrows (E.f i)
Tactic: rw [Sieve.generate_le_iff, Presieve.ofArrows_le_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreZeroHypercover.{u_1, v, u} S
F : PreZeroHypercover.{u_2, v, u} S
f : E.Hom F
i : E.I₀
⊢ F.sieve₀.arrows (E.f i)
have rw : F.sieve₀.arrows (f.h₀ i ≫ F.f (f.s₀ i))
Tactic: rw [← f.w₀ i]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreZeroHypercover.{u_1, v, u} S
F : PreZeroHypercover.{u_2, v, u} S
f : E.Hom F
i : E.I₀
⊢ F.sieve₀.arrows (f.h₀ i ≫ F.f (f.s₀ i))
have x : F.sieve₀.arrows (Y := F.X (f.s₀ i)) (F.f (f.s₀ i))
Tactic: apply Sieve.downward_closed
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S_1 T_1 : C
E_1 : PreZeroHypercover.{w, v, u} S_1
F : PreZeroHypercover.{w', v, u} S_1
S T : C
f : S ⟶ T
E : PreZeroHypercover.{w, v, u} T
inst_1 : ∀ (i : E.I₀), HasPullback f (E.f i)
inst_2 : ∀ (i : E.I₀), HasPullback (E.f i) f
⊢ { s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := ⋯ :
        (pullback₁ f E).Hom (pullback₂ f E) } ≫
      { s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := ⋯ } =
    𝟙 (pullback₁ f E)
have apply : HEq (α :=
  (i : (pullback₁ f E).I₀) →
    (pullback₁ f E).X i ⟶
      (pullback₁ f E).X
        (Hom.s₀
          ({ s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := sorry :
              (pullback₁ f E).Hom (pullback₂ f E) } ≫
            { s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := sorry })
          i))
  (Hom.h₀
    ({ s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := sorry :
        (pullback₁ f E).Hom (pullback₂ f E) } ≫
      { s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := sorry }))
  (Hom.h₀ (𝟙 (pullback₁ f E)))
Tactic: apply Hom.ext (by rfl)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S_1 T_1 : C
E_1 : PreZeroHypercover.{w, v, u} S_1
F : PreZeroHypercover.{w', v, u} S_1
S T : C
f : S ⟶ T
E : PreZeroHypercover.{w, v, u} T
inst_1 : ∀ (i : E.I₀), HasPullback f (E.f i)
inst_2 : ∀ (i : E.I₀), HasPullback (E.f i) f
⊢ HEq (α :=
    (i : (pullback₁ f E).I₀) →
      (pullback₁ f E).X i ⟶
        (pullback₁ f E).X
          (Hom.s₀
            ({ s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := ⋯ :
                (pullback₁ f E).Hom (pullback₂ f E) } ≫
              { s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := ⋯ })
            i))
    (Hom.h₀
      ({ s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := ⋯ :
          (pullback₁ f E).Hom (pullback₂ f E) } ≫
        { s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := ⋯ }))
    (Hom.h₀ (𝟙 (pullback₁ f E)))
have simp : Hom.h₀
    ({ s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := sorry :
        (pullback₁ f E).Hom (pullback₂ f E) } ≫
      { s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := sorry }) =
  Hom.h₀ (𝟙 (pullback₁ f E))
Tactic: simp only [heq_eq_eq]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S_1 T_1 : C
E_1 : PreZeroHypercover.{w, v, u} S_1
F : PreZeroHypercover.{w', v, u} S_1
S T : C
f : S ⟶ T
E : PreZeroHypercover.{w, v, u} T
inst_1 : ∀ (i : E.I₀), HasPullback f (E.f i)
inst_2 : ∀ (i : E.I₀), HasPullback (E.f i) f
⊢ Hom.h₀
      ({ s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := ⋯ :
          (pullback₁ f E).Hom (pullback₂ f E) } ≫
        { s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := ⋯ }) =
    Hom.h₀ (𝟙 (pullback₁ f E))
have h : ∀ (i : (pullback₁ f E).I₀),
  Hom.h₀
      ({ s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := sorry :
          (pullback₁ f E).Hom (pullback₂ f E) } ≫
        { s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := sorry })
      i =
    Hom.h₀ (𝟙 (pullback₁ f E)) i
Tactic: ext i
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S_1 T_1 : C
E_1 : PreZeroHypercover.{w, v, u} S_1
F : PreZeroHypercover.{w', v, u} S_1
S T : C
f : S ⟶ T
E : PreZeroHypercover.{w, v, u} T
inst_1 : ∀ (i : E.I₀), HasPullback f (E.f i)
inst_2 : ∀ (i : E.I₀), HasPullback (E.f i) f
⊢ { s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := ⋯ :
        (pullback₂ f E).Hom (pullback₁ f E) } ≫
      { s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := ⋯ } =
    𝟙 (pullback₂ f E)
have apply : HEq (α :=
  (i : (pullback₂ f E).I₀) →
    (pullback₂ f E).X i ⟶
      (pullback₂ f E).X
        (Hom.s₀
          ({ s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := sorry :
              (pullback₂ f E).Hom (pullback₁ f E) } ≫
            { s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := sorry })
          i))
  (Hom.h₀
    ({ s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := sorry :
        (pullback₂ f E).Hom (pullback₁ f E) } ≫
      { s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := sorry }))
  (Hom.h₀ (𝟙 (pullback₂ f E)))
Tactic: apply Hom.ext (by rfl)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S_1 T_1 : C
E_1 : PreZeroHypercover.{w, v, u} S_1
F : PreZeroHypercover.{w', v, u} S_1
S T : C
f : S ⟶ T
E : PreZeroHypercover.{w, v, u} T
inst_1 : ∀ (i : E.I₀), HasPullback f (E.f i)
inst_2 : ∀ (i : E.I₀), HasPullback (E.f i) f
⊢ HEq (α :=
    (i : (pullback₂ f E).I₀) →
      (pullback₂ f E).X i ⟶
        (pullback₂ f E).X
          (Hom.s₀
            ({ s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := ⋯ :
                (pullback₂ f E).Hom (pullback₁ f E) } ≫
              { s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := ⋯ })
            i))
    (Hom.h₀
      ({ s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := ⋯ :
          (pullback₂ f E).Hom (pullback₁ f E) } ≫
        { s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := ⋯ }))
    (Hom.h₀ (𝟙 (pullback₂ f E)))
have simp : Hom.h₀
    ({ s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := sorry :
        (pullback₂ f E).Hom (pullback₁ f E) } ≫
      { s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := sorry }) =
  Hom.h₀ (𝟙 (pullback₂ f E))
Tactic: simp only [heq_eq_eq]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S_1 T_1 : C
E_1 : PreZeroHypercover.{w, v, u} S_1
F : PreZeroHypercover.{w', v, u} S_1
S T : C
f : S ⟶ T
E : PreZeroHypercover.{w, v, u} T
inst_1 : ∀ (i : E.I₀), HasPullback f (E.f i)
inst_2 : ∀ (i : E.I₀), HasPullback (E.f i) f
⊢ Hom.h₀
      ({ s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := ⋯ :
          (pullback₂ f E).Hom (pullback₁ f E) } ≫
        { s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := ⋯ }) =
    Hom.h₀ (𝟙 (pullback₂ f E))
have h : ∀ (i : (pullback₂ f E).I₀),
  Hom.h₀
      ({ s₀ := id, h₀ := fun (i : (pullback₂ f E).I₀) ↦ Iso.inv (pullbackSymmetry f (E.f i)), w₀ := sorry :
          (pullback₂ f E).Hom (pullback₁ f E) } ≫
        { s₀ := id, h₀ := fun (i : (pullback₁ f E).I₀) ↦ Iso.hom (pullbackSymmetry f (E.f i)), w₀ := sorry })
      i =
    Hom.h₀ (𝟙 (pullback₂ f E)) i
Tactic: ext i
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
R : Presieve.{v, u} S
⊢ R.preZeroHypercover.presieve₀ = R
have refine_1 : R.preZeroHypercover.presieve₀ ≤ R
Tactic: refine le_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
R : Presieve.{v, u} S
refine_1 : R.preZeroHypercover.presieve₀ ≤ R
⊢ R.preZeroHypercover.presieve₀ = R
have refine_2 : R ≤ R.preZeroHypercover.presieve₀
Tactic: refine le_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreZeroHypercover.{w, v, u} S
h : E.presieve₀ = F.presieve₀
⊢ E.shrink = F.shrink
have rw : E.presieve₀.preZeroHypercover = F.shrink
Tactic: rw [shrink, shrink, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreZeroHypercover.{w, v, u} S
h : E.presieve₀ = F.presieve₀
rw : E.presieve₀.preZeroHypercover = F.shrink
⊢ E.shrink = F.shrink
have rw₁ : E.presieve₀.preZeroHypercover = F.presieve₀.preZeroHypercover
Tactic: rw [shrink, shrink, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreZeroHypercover.{w, v, u} S
h : E.presieve₀ = F.presieve₀
rw : E.presieve₀.preZeroHypercover = F.shrink
rw₁ : E.presieve₀.preZeroHypercover = F.presieve₀.preZeroHypercover
⊢ E.shrink = F.shrink
have rw₂ : F.presieve₀.preZeroHypercover = F.presieve₀.preZeroHypercover
Tactic: rw [shrink, shrink, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreZeroHypercover.{w, v, u} S
⊢ E.presieve₀ = F.presieve₀ ↔ E.shrink = F.shrink
have refine : ∀ (h : E.shrink = F.shrink), E.presieve₀ = F.presieve₀
Tactic: refine ⟨fun h ↦ shrink_eq_shrink_of_presieve₀_eq_presieve₀ h, fun h ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreZeroHypercover.{w, v, u} S
h : E.shrink = F.shrink
⊢ E.presieve₀ = F.presieve₀
have rw : E.shrink.presieve₀ = F.presieve₀
Tactic: rw [← E.presieve₀_shrink, ← F.presieve₀_shrink, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreZeroHypercover.{w, v, u} S
h : E.shrink = F.shrink
rw : E.shrink.presieve₀ = F.presieve₀
⊢ E.presieve₀ = F.presieve₀
have rw₁ : E.shrink.presieve₀ = F.shrink.presieve₀
Tactic: rw [← E.presieve₀_shrink, ← F.presieve₀_shrink, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E F : PreZeroHypercover.{w, v, u} S
h : E.shrink = F.shrink
rw : E.shrink.presieve₀ = F.presieve₀
rw₁ : E.shrink.presieve₀ = F.shrink.presieve₀
⊢ E.presieve₀ = F.presieve₀
have rw₂ : F.shrink.presieve₀ = F.shrink.presieve₀
Tactic: rw [← E.presieve₀_shrink, ← F.presieve₀_shrink, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Precoverage.{v, u} C
inst_1 : J.RespectsIso
S : C
R T : Presieve.{v, u} S
hRT : ∀ ⦃Z : C⦄ (g : Z ⟶ S), R g → ∃ (Y : C) (e : Y ≅ Z), T (e.hom ≫ g)
hTR : ∀ ⦃Z : C⦄ (g : Z ⟶ S), T g → ∃ (Y : C) (e : Y ≅ Z), R (e.hom ≫ g)
hR : R ∈ J.coverings S
⊢ T ∈ J.coverings S
have choose : ∀ (YR : ⦃Z : C⦄ → (g : Z ⟶ S) → R g → C) (eR : ⦃Z : C⦄ → (g : Z ⟶ S) → (a : R g) → YR g a ≅ Z)
  (hTeg : ∀ ⦃Z : C⦄ (g : Z ⟶ S) (a : R g), T ((eR g a).hom ≫ g)), T ∈ J.coverings S
Tactic: choose YR eR hTeg using hRT
Cleared hypotheses: hRT
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Zero (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : Precoverage.{v, u} C
inst_1 : J.RespectsIso
S : C
R T : Presieve.{v, u} S
hTR : ∀ ⦃Z : C⦄ (g : Z ⟶ S), T g → ∃ (Y : C) (e : Y ≅ Z), R (e.hom ≫ g)
hR : R ∈ J.coverings S
YR : ⦃Z : C⦄ → (g : Z ⟶ S) → R g → C
eR : ⦃Z : C⦄ → (g : Z ⟶ S) → (a : R g) → YR g a ≅ Z
hTeg : ∀ ⦃Z : C⦄ (g : Z ⟶ S) (a : R g), T ((eR g a).hom ≫ g)
⊢ T ∈ J.coverings S
have choose : ∀ (YT : ⦃Z : C⦄ → (g : Z ⟶ S) → T g → C) (eT : ⦃Z : C⦄ → (g : Z ⟶ S) → (a : T g) → YT g a ≅ Z)
  (hReg : ∀ ⦃Z : C⦄ (g : Z ⟶ S) (a : T g), R ((eT g a).hom ≫ g)), T ∈ J.coverings S
Tactic: choose YT eT hReg using hTR
Cleared hypotheses: hTR