awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
inst_1 : HasPullbacks.{v, u} C
f g : E.Hom F
i j : (cylinder f g).I₀
k : (cylinder f g).I₁ i j
⊢ pullback.snd
        (pullback.map (cylinderf f g (i.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (cylinderf f g (j.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (E.f (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (E.f (j.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (pullback.fst
            (pullback.lift (f.h₀ (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
              (g.h₀ (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ⋯)
            (F.toPullback (i.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
          (pullback.fst
            (pullback.lift (f.h₀ (j.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
              (g.h₀ (j.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ⋯)
            (F.toPullback (j.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
          (𝟙 S) ⋯ ⋯)
        (pullback.lift (E.p₁ k.down) (E.p₂ k.down) ⋯) ≫
      E.p₁ k.down =
    (cylinder f g).p₁ k ≫
      pullback.fst
        (pullback.lift (f.h₀ (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (g.h₀ (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ⋯)
        (F.toPullback (i.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
have this : E.p₁ k.down =
  CategoryStruct.comp (Y :=
    pullback (X := E.X (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) (Y :=
      E.X (j.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
      (E.f (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
      (E.f (j.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
    (pullback.lift (E.p₁ k.down) (E.p₂ k.down) sorry)
    (pullback.fst (E.f (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
      (E.f (j.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
Tactic: have : E.p₁ k.down = pullback.lift _ _ (E.w k.down) ≫ pullback.fst _ _ := by simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
inst_1 : HasPullbacks.{v, u} C
f g : E.Hom F
p q : (cylinder f g).I₀
k : (cylinder f g).I₁ p q
⊢ pullback.snd
        (pullback.map (cylinderf f g (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (cylinderf f g (q.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (E.f (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (E.f (q.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (pullback.fst
            (pullback.lift (f.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
              (g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ⋯)
            (F.toPullback (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
          (pullback.fst
            (pullback.lift (f.h₀ (q.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
              (g.h₀ (q.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ⋯)
            (F.toPullback (q.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
          (𝟙 S) ⋯ ⋯)
        (pullback.lift (E.p₁ k.down) (E.p₂ k.down) ⋯) ≫
      E.p₂ k.down =
    (cylinder f g).p₂ k ≫
      pullback.fst
        (pullback.lift (f.h₀ (q.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (g.h₀ (q.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ⋯)
        (F.toPullback (q.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
have this : E.p₂ k.down =
  CategoryStruct.comp (Y :=
    pullback (X := E.X (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) (Y :=
      E.X (q.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
      (E.f (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
      (E.f (q.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
    (pullback.lift (E.p₁ k.down) (E.p₂ k.down) sorry)
    (pullback.snd (E.f (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
      (E.f (q.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
Tactic: have : E.p₂ k.down = pullback.lift _ _ (E.w k.down) ≫ pullback.snd _ _ := by simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
inst_1 : HasPullbacks.{v, u} C
f g : E.Hom F
p : (cylinder f g).I₀
⊢ Eq (α := (cylinder f g).X p ⟶ F.X (PreZeroHypercover.Hom.s₀ (Hom.toHom (Hom.comp (cylinderHom f g) f)) p))
    (pullback.snd
        (pullback.lift (f.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ⋯)
        (F.toPullback (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ≫
      F.p₁ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
    (PreZeroHypercover.Hom.h₀ (Hom.toHom (Hom.comp (cylinderHom f g) f)) p)
have this : F.p₁ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))) =
  CategoryStruct.comp (Y :=
    pullback (X := F.X (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))) (Y :=
      F.X (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
      (F.f (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
      (F.f (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))))
    (pullback.lift (F.p₁ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
      (F.p₂ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) sorry)
    (pullback.fst (F.f (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
      (F.f (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))))
Tactic: have : F.p₁ p.snd = pullback.lift _ _ (F.w p.2) ≫ pullback.fst _ _ := by simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
inst_1 : HasPullbacks.{v, u} C
f g : E.Hom F
p : (cylinder f g).I₀
this :
  F.p₁ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))) =
    CategoryStruct.comp (Y :=
      pullback (X := F.X (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))) (Y :=
        F.X (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
        (F.f (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
        (F.f (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))))
      (pullback.lift (F.p₁ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
        (F.p₂ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ⋯)
      (pullback.fst (F.f (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
        (F.f (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))))
⊢ Eq (α := (cylinder f g).X p ⟶ F.X (PreZeroHypercover.Hom.s₀ (Hom.toHom (Hom.comp (cylinderHom f g) f)) p))
    (pullback.snd
        (pullback.lift (f.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ⋯)
        (F.toPullback (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ≫
      F.p₁ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
    (PreZeroHypercover.Hom.h₀ (Hom.toHom (Hom.comp (cylinderHom f g) f)) p)
have nth_rw : Eq (α := (cylinder f g).X p ⟶ F.X (PreZeroHypercover.Hom.s₀ (Hom.toHom (Hom.comp (cylinderHom f g) f)) p))
  (pullback.snd
      (pullback.lift (f.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
        (g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) sorry)
      (F.toPullback (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ≫
    CategoryStruct.comp (Y :=
      pullback (X := F.X (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))) (Y :=
        F.X (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
        (F.f (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
        (F.f (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))))
      (pullback.lift (F.p₁ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
        (F.p₂ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) sorry)
      (pullback.fst (F.f (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
        (F.f (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))))
  (PreZeroHypercover.Hom.h₀ (Hom.toHom (Hom.comp (cylinderHom f g) f)) p)
Tactic: nth_rw 1 [this]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
inst_1 : HasPullbacks.{v, u} C
f g : E.Hom F
p : (cylinder f g).I₀
⊢ Eq (α := (cylinder f g).X p ⟶ F.X (PreZeroHypercover.Hom.s₀ (Hom.toHom (Hom.comp (cylinderHom f g) g)) p))
    (pullback.snd
        (pullback.lift (f.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ⋯)
        (F.toPullback (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ≫
      F.p₂ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
    (PreZeroHypercover.Hom.h₀ (Hom.toHom (Hom.comp (cylinderHom f g) g)) p)
have this : Eq (α :=
  E.X (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))) ⟶
    F.X (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
  (g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
  (CategoryStruct.comp (Y :=
    pullback (X := F.X (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))) (Y :=
      F.X (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
      (F.f (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
      (F.f (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))))
    (pullback.lift (f.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
      (g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) sorry)
    (pullback.snd (F.f (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
      (F.f (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))))
Tactic: have : g.h₀ p.fst = pullback.lift (f.h₀ p.fst) (g.h₀ p.fst) (by simp) ≫
        pullback.snd (F.f _) (F.f _) := by simp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (goal)
Goal:
C : Type u
inst : Category.{v, u} C
S : C
E : PreOneHypercover.{w, v, u} S
F : PreOneHypercover.{w', v, u} S
inst_1 : HasPullbacks.{v, u} C
f g : E.Hom F
p : (cylinder f g).I₀
this :
  Eq (α :=
    E.X (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))) ⟶
      F.X (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
    (g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
    (CategoryStruct.comp (Y :=
      pullback (X := F.X (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))) (Y :=
        F.X (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
        (F.f (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
        (F.f (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))))
      (pullback.lift (f.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
        (g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ⋯)
      (pullback.snd (F.f (f.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
        (F.f (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))))
⊢ Eq (α := (cylinder f g).X p ⟶ F.X (PreZeroHypercover.Hom.s₀ (Hom.toHom (Hom.comp (cylinderHom f g) g)) p))
    (pullback.snd
        (pullback.lift (f.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ⋯)
        (F.toPullback (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ≫
      F.p₂ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
    (PreZeroHypercover.Hom.h₀ (Hom.toHom (Hom.comp (cylinderHom f g) g)) p)
have dsimp : Eq (α :=
  cylinderX f g (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))) ⟶
    F.X (g.s₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
  (pullback.snd
      (pullback.lift (f.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
        (g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) sorry)
      (F.toPullback (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ≫
    F.p₂ (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
  (pullback.fst
      (pullback.lift (f.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
        (g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) sorry)
      (F.toPullback (p.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) ≫
    g.h₀ (p.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
Tactic: dsimp only [cylinder_X, Hom.comp_s₀, cylinder_I₀, Function.comp_apply, cylinderHom_s₀,
      Hom.comp_h₀, cylinderHom_h₀]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
S : C
E : OneHypercover.{w, v, u} J S
F : OneHypercover.{w', v, u} J S
inst_1 : HasPullbacks.{v, u} C
f g : E.Hom F
⊢ (PreOneHypercover.cylinder f g).sieve₀ ∈ (J : (a : C) → Set (Sieve a)) S
have rw : Sieve.generate
    (Presieve.bindOfArrows E.X E.f fun (i : E.I₀) ↦
      (Sieve.pullback (X := Limits.pullback (X := F.X (f.s₀ i)) (Y := F.X (g.s₀ i)) (F.f (f.s₀ i)) (F.f (g.s₀ i)))
          (pullback.lift (f.h₀ i) (g.h₀ i) sorry) (F.sieve₁' (f.s₀ i) (g.s₀ i))).arrows) ∈
  (J : (a : C) → Set (Sieve a)) S
Tactic: rw [PreOneHypercover.sieve₀_cylinder]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
S : C
E : OneHypercover.{w, v, u} J S
F : OneHypercover.{w', v, u} J S
inst_1 : HasPullbacks.{v, u} C
f g : E.Hom F
⊢ Sieve.generate
      (Presieve.bindOfArrows E.X E.f fun (i : E.I₀) ↦
        (Sieve.pullback (X := Limits.pullback (X := F.X (f.s₀ i)) (Y := F.X (g.s₀ i)) (F.f (f.s₀ i)) (F.f (g.s₀ i)))
            (pullback.lift (f.h₀ i) (g.h₀ i) ⋯) (F.sieve₁' (f.s₀ i) (g.s₀ i))).arrows) ∈
    (J : (a : C) → Set (Sieve a)) S
have refine : ∀ (i : E.I₀),
  Membership.mem (γ := Set (Sieve.{v, u} (E.X i))) ((J : (a : C) → Set (Sieve a)) (E.X i))
    (Sieve.generate
      (Sieve.pullback (X := Limits.pullback (X := F.X (f.s₀ i)) (Y := F.X (g.s₀ i)) (F.f (f.s₀ i)) (F.f (g.s₀ i)))
          (pullback.lift (f.h₀ i) (g.h₀ i) sorry) (F.sieve₁' (f.s₀ i) (g.s₀ i))).arrows)
Tactic: refine J.bindOfArrows E.mem₀ fun i ↦ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
S : C
E : OneHypercover.{w, v, u} J S
F : OneHypercover.{w', v, u} J S
inst_1 : HasPullbacks.{v, u} C
f g : E.Hom F
i : E.I₀
⊢ Membership.mem (γ := Set (Sieve.{v, u} (E.X i))) ((J : (a : C) → Set (Sieve a)) (E.X i))
    (Sieve.generate
      (Sieve.pullback (X := Limits.pullback (X := F.X (f.s₀ i)) (Y := F.X (g.s₀ i)) (F.f (f.s₀ i)) (F.f (g.s₀ i)))
          (pullback.lift (f.h₀ i) (g.h₀ i) ⋯) (F.sieve₁' (f.s₀ i) (g.s₀ i))).arrows)
have rw : Membership.mem (γ := Set (Sieve.{v, u} (E.X i))) ((J : (a : C) → Set (Sieve a)) (E.X i))
  (Sieve.pullback (X := Limits.pullback (X := F.X (f.s₀ i)) (Y := F.X (g.s₀ i)) (F.f (f.s₀ i)) (F.f (g.s₀ i)))
    (pullback.lift (f.h₀ i) (g.h₀ i) sorry) (F.sieve₁' (f.s₀ i) (g.s₀ i)))
Tactic: rw [Sieve.generate_sieve]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
S : C
E : OneHypercover.{w, v, u} J S
F : OneHypercover.{w', v, u} J S
inst_1 : HasPullbacks.{v, u} C
f g : E.Hom F
i j : (PreOneHypercover.cylinder f g).I₀
⊢ Membership.mem (γ :=
    Set (Sieve.{v, u} (Limits.pullback ((PreOneHypercover.cylinder f g).f i) ((PreOneHypercover.cylinder f g).f j))))
    ((J : (a : C) → Set (Sieve a))
      (Limits.pullback ((PreOneHypercover.cylinder f g).f i) ((PreOneHypercover.cylinder f g).f j)))
    ((PreOneHypercover.cylinder f g).sieve₁' i j)
have rw : Membership.mem (γ :=
  Set (Sieve.{v, u} (Limits.pullback ((PreOneHypercover.cylinder f g).f i) ((PreOneHypercover.cylinder f g).f j))))
  ((J : (a : C) → Set (Sieve a))
    (Limits.pullback ((PreOneHypercover.cylinder f g).f i) ((PreOneHypercover.cylinder f g).f j)))
  (Sieve.pullback (X :=
    Limits.pullback (X := E.X (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) (Y :=
      E.X (j.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
      (E.f (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
      (E.f (j.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
    (pullback.map ((PreOneHypercover.cylinder f g).f i) ((PreOneHypercover.cylinder f g).f j)
      (E.f (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
      (E.f (j.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
      (pullback.fst
        (pullback.lift (f.h₀ (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (g.h₀ (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) sorry)
        (F.toPullback (i.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
      (pullback.fst
        (pullback.lift (f.h₀ (j.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i))))
          (g.h₀ (j.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))) sorry)
        (F.toPullback (j.snd (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
      (𝟙 S) sorry sorry)
    (E.sieve₁' (i.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))
      (j.fst (β := fun (i : E.I₀) ↦ F.I₁ (f.s₀ i) (g.s₀ i)))))
Tactic: rw [PreOneHypercover.sieve₁'_cylinder]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
S_1 : ?m.2
S : C
inst_1 : HasPullbacks.{v, u} C
X Y : HOneHypercover.{v, u, u_1} J S
f g : X ⟶ Y
⊢ ∃ (W : HOneHypercover.{v, u, u_1} J S) (h : W ⟶ X), h ≫ f = h ≫ g
have obtain : ∀ (f : X.as (r := homotopicRel J S) ⟶ Y.as (r := homotopicRel J S)),
  ∃ (W : HOneHypercover.{v, u, u_1} J S) (h : W ⟶ X), h ≫ (toHOneHypercover J S).map f = h ≫ g
Tactic: obtain ⟨(f : X.1 ⟶ Y.1), rfl⟩ := (toHOneHypercover J S).map_surjective f
Cleared hypotheses: f
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
S_1 : ?m.2
S : C
inst_1 : HasPullbacks.{v, u} C
X Y : HOneHypercover.{v, u, u_1} J S
g : X ⟶ Y
f : X.as (r := homotopicRel J S) ⟶ Y.as (r := homotopicRel J S)
⊢ ∃ (W : HOneHypercover.{v, u, u_1} J S) (h : W ⟶ X), h ≫ (toHOneHypercover J S).map f = h ≫ g
have obtain : ∀ (g : X.as (r := homotopicRel J S) ⟶ Y.as (r := homotopicRel J S)),
  ∃ (W : HOneHypercover.{v, u, u_1} J S) (h : W ⟶ X),
    h ≫ (toHOneHypercover J S).map f = h ≫ (toHOneHypercover J S).map g
Tactic: obtain ⟨(g : X.1 ⟶ Y.1), rfl⟩ := (toHOneHypercover J S).map_surjective g
Cleared hypotheses: g
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
S_1 : ?m.2
S : C
inst_1 : HasPullbacks.{v, u} C
X Y : HOneHypercover.{v, u, u_1} J S
f g : X.as (r := homotopicRel J S) ⟶ Y.as (r := homotopicRel J S)
⊢ ∃ (W : HOneHypercover.{v, u, u_1} J S) (h : W ⟶ X),
    h ≫ (toHOneHypercover J S).map f = h ≫ (toHOneHypercover J S).map g
have obtain : ∀ (W : OneHypercover.{u_1, v, u} J S) (h : W.Hom (X.as (r := homotopicRel J S)))
  (H : Homotopy (Hom.comp h f) (Hom.comp h g)),
  ∃ (W : HOneHypercover.{v, u, u_1} J S) (h : W ⟶ X),
    h ≫ (toHOneHypercover J S).map f = h ≫ (toHOneHypercover J S).map g
Tactic: obtain ⟨W, h, ⟨H⟩⟩ := OneHypercover.exists_nonempty_homotopy f g
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
S_1 : ?m.2
S : C
inst_1 : HasPullbacks.{v, u} C
X Y : HOneHypercover.{v, u, u_1} J S
f g : X.as (r := homotopicRel J S) ⟶ Y.as (r := homotopicRel J S)
W : OneHypercover.{u_1, v, u} J S
h : W.Hom (X.as (r := homotopicRel J S))
H : Homotopy (Hom.comp h f) (Hom.comp h g)
⊢ ∃ (W : HOneHypercover.{v, u, u_1} J S) (h : W ⟶ X),
    h ≫ (toHOneHypercover J S).map f = h ≫ (toHOneHypercover J S).map g
have h₁ : (toHOneHypercover J S).map h ≫ (toHOneHypercover J S).map f =
  (toHOneHypercover J S).map h ≫ (toHOneHypercover J S).map g
Tactic: use (toHOneHypercover J S).obj W, (toHOneHypercover J S).map h
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.Homotopy (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
S_1 : ?m.2
S : C
inst_1 : HasPullbacks.{v, u} C
X Y : HOneHypercover.{v, u, u_1} J S
f g : X.as (r := homotopicRel J S) ⟶ Y.as (r := homotopicRel J S)
W : OneHypercover.{u_1, v, u} J S
h : W.Hom (X.as (r := homotopicRel J S))
H : Homotopy (Hom.comp h f) (Hom.comp h g)
⊢ (toHOneHypercover J S).map h ≫ (toHOneHypercover J S).map f =
    (toHOneHypercover J S).map h ≫ (toHOneHypercover J S).map g
have h₁ : Eq (α := (toHOneHypercover J S).obj W ⟶ Y) ((toHOneHypercover J S).map (h ≫ f)) ((toHOneHypercover J S).map (h ≫ g))
Tactic: rw [← Functor.map_comp, ← Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.IsSheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
H : OneHypercoverFamily.{w, v, u} J
P : Functor.{v, v', u, u'} Cᵒᵖ A
hP : ∀ ⦃X : C⦄ (E : OneHypercover.{w, v, u} J X), H E → Nonempty (IsLimit (E.multifork P))
inst_2 : H.IsGenerating
X : C
S : Sieve.{v, u} X
hS : S ∈ (J : (a : C) → Set (Sieve a)) X
T : A
x y : T ⟶ P.obj (Opposite.op X)
h : ∀ ⦃Y : C⦄ (f : Y ⟶ X), S.arrows f → x ≫ P.map f.op = y ≫ P.map f.op
⊢ x = y
have obtain : ∀ (E : OneHypercover.{w, v, u} J X) (hE : H E) (le : PreZeroHypercover.sieve₀.{w, v, u} E.toPreZeroHypercover ≤ S),
  x = y
Tactic: obtain ⟨E, hE, le⟩ := H.exists_oneHypercover S hS
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.IsSheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
H : OneHypercoverFamily.{w, v, u} J
P : Functor.{v, v', u, u'} Cᵒᵖ A
hP : ∀ ⦃X : C⦄ (E : OneHypercover.{w, v, u} J X), H E → Nonempty (IsLimit (E.multifork P))
X : C
S : Sieve.{v, u} X
E : OneHypercover.{w, v, u} J X
hE : H E
le : PreZeroHypercover.sieve₀.{w, v, u} E.toPreZeroHypercover ≤ S
F : Multifork (Cover.index (Subtype.mk (p := fun (S : Sieve.{v, u} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) S ⋯) P)
inst_2 : H.IsGenerating
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ lift (H := H) hP hE le F ≫ P.map f.op = F.ι { Y := Y, f := f, hf := hf }
have apply : ∀ ⦃Y_1 : C⦄ (f_1 : Y_1 ⟶ Y),
  (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} E.toPreZeroHypercover)).arrows f_1 →
    (lift (H := H) hP hE le F ≫ P.map f.op) ≫ P.map f_1.op = F.ι { Y := Y, f := f, hf := hf } ≫ P.map f_1.op
Tactic: apply hom_ext H P hP _ (J.pullback_stable f E.mem₀)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Hypercover.IsSheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
H : OneHypercoverFamily.{w, v, u} J
P : Functor.{v, v', u, u'} Cᵒᵖ A
hP : ∀ ⦃X : C⦄ (E : OneHypercover.{w, v, u} J X), H E → Nonempty (IsLimit (E.multifork P))
X : C
S : Sieve.{v, u} X
E : OneHypercover.{w, v, u} J X
hE : H E
le : PreZeroHypercover.sieve₀.{w, v, u} E.toPreZeroHypercover ≤ S
F : Multifork (Cover.index (Subtype.mk (p := fun (S : Sieve.{v, u} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) S ⋯) P)
inst_2 : H.IsGenerating
Y_1 : C
f : Y_1 ⟶ X
hf : S.arrows f
Z : C
g : Z ⟶ Y_1
Y : C
i : E.I₀
a : Z ⟶ E.X i
fac : a ≫ E.f i = g ≫ f
⊢ (lift (H := H) hP hE le F ≫ P.map f.op) ≫ P.map g.op = F.ι { Y := Y_1, f := f, hf := hf } ≫ P.map g.op
have rw : Eq (α := F.pt ⟶ P.obj (Opposite.op Z)) (F.ι { Y := E.X i, f := E.f i, hf := sorry } ≫ P.map a.op)
  (F.ι { Y := Y_1, f := f, hf := hf } ≫ P.map g.op)
Tactic: rw [assoc, ← P.map_comp, ← op_comp, ← fac,
    op_comp, P.map_comp, fac'_assoc]