awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Gluing (goal)
Goal:
J : Type w
inst : Category.{v, w} J
F : CategoryTheory.Functor.{v, u, w, u + 1} J Scheme
inst_1 : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst_2 : (F ⋙ forget).IsLocallyDirected
inst_3 : Quiver.IsThin.{w, v} J
i j : J
k₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)
U : (F.obj i).Opens
h₁ : Hom.opensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1) ≤ U
h₂ : Hom.opensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1) ≤ U
x : ↥(pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂))
⊢ Eq (α := ↥(F.obj i))
    (((pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
            Iso.inv (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1))) ≫
          F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)
        x :
      ↥(F.obj i))
    (((pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
            Iso.inv (Hom.isoOpensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1))) ≫
          F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)
        x :
      ↥(F.obj i))
have e_a : Eq (α := (pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂)).Hom (F.obj i))
  ((pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
      Iso.inv (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1))) ≫
    F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)
  ((pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
      Iso.inv (Hom.isoOpensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1))) ≫
    F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)
Tactic: congr 5
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Gluing (goal)
Goal:
J : Type w
inst : Category.{v, w} J
F : CategoryTheory.Functor.{v, u, w, u + 1} J Scheme
inst_1 : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst_2 : (F ⋙ forget).IsLocallyDirected
inst_3 : Quiver.IsThin.{w, v} J
i j : J
k₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)
U : (F.obj i).Opens
h₁ : Hom.opensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1) ≤ U
h₂ : Hom.opensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1) ≤ U
x : ↥(pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂))
l : J
hli : l ⟶ k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j))
hlj : l ⟶ k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j))
y : (F ⋙ forget).obj l
hy₁ :
  Eq (α := (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F ⋙ forget).map hli) :
          (a : (fun (X : Type u) ↦ X) ((F ⋙ forget).obj l)) → (fun (X : Type u) ↦ X) ((F ⋙ forget).obj k₁.fst))
        y :
      (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
    ((pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
          Iso.inv (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
        x :
      ↥(F.obj (k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
hy₂ :
  Eq (α := (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F ⋙ forget).map hlj) :
          (a : (fun (X : Type u) ↦ X) ((F ⋙ forget).obj l)) → (fun (X : Type u) ↦ X) ((F ⋙ forget).obj k₂.fst))
        y :
      (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
    ((pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
          Iso.inv (Hom.isoOpensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
        x :
      ↥(F.obj (k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
α : F.obj l ⟶ pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) :=
  pullback.lift (F.map hli ≫ Iso.hom (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
    (F.map hlj ≫ Iso.hom (Hom.isoOpensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1))) ⋯
⊢ IsOpenImmersion (α ≫ pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂))
have simp : IsOpenImmersion (Y := (↑(Hom.opensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) : Scheme))
  (F.map hli ≫ Iso.hom (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
Tactic: simp only [pullback.lift_fst, α]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Gluing (hypothesis)
Goal:
J : Type w
inst : Category.{v, w} J
F : CategoryTheory.Functor.{v, u, w, u + 1} J Scheme
inst_1 : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst_2 : (F ⋙ forget).IsLocallyDirected
inst_3 : Quiver.IsThin.{w, v} J
i j : J
k₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)
U : (F.obj i).Opens
h₁ : Hom.opensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1) ≤ U
h₂ : Hom.opensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1) ≤ U
x : ↥(pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂))
l : J
hli : l ⟶ k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j))
hlj : l ⟶ k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j))
y : (F ⋙ forget).obj l
hy₁ :
  Eq (α := (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F ⋙ forget).map hli) :
          (a : (fun (X : Type u) ↦ X) ((F ⋙ forget).obj l)) → (fun (X : Type u) ↦ X) ((F ⋙ forget).obj k₁.fst))
        y :
      (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
    ((pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
          Iso.inv (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
        x :
      ↥(F.obj (k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
hy₂ :
  Eq (α := (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F ⋙ forget).map hlj) :
          (a : (fun (X : Type u) ↦ X) ((F ⋙ forget).obj l)) → (fun (X : Type u) ↦ X) ((F ⋙ forget).obj k₂.fst))
        y :
      (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
    ((pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
          Iso.inv (Hom.isoOpensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
        x :
      ↥(F.obj (k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
α : F.obj l ⟶ pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) :=
  pullback.lift (F.map hli ≫ Iso.hom (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
    (F.map hlj ≫ Iso.hom (Hom.isoOpensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1))) ⋯
⊢ IsOpenImmersion α
have this : IsOpenImmersion (α ≫ pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂))
Tactic: have : IsOpenImmersion (α ≫ pullback.fst _ _) := by
      simp only [pullback.lift_fst, α]; infer_instance
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Gluing (hypothesis)
Goal:
J : Type w
inst : Category.{v, w} J
F : CategoryTheory.Functor.{v, u, w, u + 1} J Scheme
inst_1 : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst_2 : (F ⋙ forget).IsLocallyDirected
inst_3 : Quiver.IsThin.{w, v} J
i j : J
k₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)
U : (F.obj i).Opens
h₁ : Hom.opensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1) ≤ U
h₂ : Hom.opensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1) ≤ U
x : ↥(pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂))
l : J
hli : l ⟶ k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j))
hlj : l ⟶ k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j))
y : (F ⋙ forget).obj l
hy₁ :
  Eq (α := (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F ⋙ forget).map hli) :
          (a : (fun (X : Type u) ↦ X) ((F ⋙ forget).obj l)) → (fun (X : Type u) ↦ X) ((F ⋙ forget).obj k₁.fst))
        y :
      (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
    ((pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
          Iso.inv (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
        x :
      ↥(F.obj (k₁.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
hy₂ :
  Eq (α := (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((F ⋙ forget).map hlj) :
          (a : (fun (X : Type u) ↦ X) ((F ⋙ forget).obj l)) → (fun (X : Type u) ↦ X) ((F ⋙ forget).obj k₂.fst))
        y :
      (fun (X : Type u) ↦ X) ((F ⋙ forget).obj (k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
    ((pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
          Iso.inv (Hom.isoOpensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
        x :
      ↥(F.obj (k₂.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)))))
α : F.obj l ⟶ pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) :=
  pullback.lift (F.map hli ≫ Iso.hom (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
    (F.map hlj ≫ Iso.hom (Hom.isoOpensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1))) ⋯
⊢ ∃ (U_1 : (pullback ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂)).Opens),
    x ∈ U_1 ∧
      U_1.ι ≫
          pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
            Iso.inv (Hom.isoOpensRange (F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) ≫
              F.map k₁.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).2 =
        U_1.ι ≫
          pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂) ≫
            Iso.inv (Hom.isoOpensRange (F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) ≫
              F.map k₂.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).2
have this : IsOpenImmersion α
Tactic: have : IsOpenImmersion α := by
    have : IsOpenImmersion (α ≫ pullback.fst _ _) := by
      simp only [pullback.lift_fst, α]; infer_instance
    exact .of_comp _ (pullback.fst _ _)
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Gluing (goal)
Goal:
J : Type w
inst : Category.{v, w} J
F : CategoryTheory.Functor.{v, u, w, u + 1} J Scheme
inst_1 : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst_2 : (F ⋙ forget).IsLocallyDirected
inst_3 : Quiver.IsThin.{w, v} J
i j : J
x : ↥(↑(V F i j) : Scheme)
⊢ Membership.mem (α := ↥(F.obj j)) (γ := Set ↥(F.obj j)) (Set.range (ι := ↥(↑(V F j i) : Scheme)) ⇑(V F j i).ι)
    ((tAux F i j) x : ↥(F.obj j))
have obtain : ∀
  (l :
    PreZeroHypercover.I₀ (S :=
      (↑(⨆ (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)),
            Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) :
        Scheme))
      (Precoverage.ZeroHypercover.toPreZeroHypercover
        (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)) ↦
          Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1))))
  (x :
    ↥(PreZeroHypercover.X
        (Precoverage.ZeroHypercover.toPreZeroHypercover
          (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)) ↦
            Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
        l)),
  Membership.mem (α := ↥(F.obj j)) (γ := Set ↥(F.obj j)) (Set.range (ι := ↥(↑(V F j i) : Scheme)) ⇑(V F j i).ι)
    ((tAux F i j)
        ((PreZeroHypercover.f
              (Precoverage.ZeroHypercover.toPreZeroHypercover
                (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)) ↦
                  Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
              l)
            x :
          ↥(↑(⨆ (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)),
                  Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) :
              Scheme)) :
      ↥(F.obj j))
Tactic: obtain ⟨l, x, rfl⟩ := (Scheme.Opens.iSupOpenCover _).exists_eq x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Gluing (goal)
Goal:
J : Type w
inst : Category.{v, w} J
F : CategoryTheory.Functor.{v, u, w, u + 1} J Scheme
inst_1 : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst_2 : (F ⋙ forget).IsLocallyDirected
inst_3 : Quiver.IsThin.{w, v} J
i j : J
l :
  PreZeroHypercover.I₀ (S :=
    (↑(⨆ (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)), Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) :
      Scheme))
    (Precoverage.ZeroHypercover.toPreZeroHypercover
      (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)) ↦
        Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
x :
  ↥(PreZeroHypercover.X
      (Precoverage.ZeroHypercover.toPreZeroHypercover
        (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)) ↦
          Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
      l)
⊢ Membership.mem (α := ↥(F.obj j)) (γ := Set ↥(F.obj j)) (Set.range (ι := ↥(↑(V F j i) : Scheme)) ⇑(V F j i).ι)
    ((tAux F i j)
        ((PreZeroHypercover.f
              (Precoverage.ZeroHypercover.toPreZeroHypercover
                (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)) ↦
                  Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
              l)
            x :
          ↥(↑(⨆ (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)),
                  Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) :
              Scheme)) :
      ↥(F.obj j))
have simp : Membership.mem (α := ↥(F.obj j)) (γ := Set ↥(F.obj j))
  (Set.range (ι :=
    ↥(↑(⨆ (k : (k : J) × (k ⟶ j) × (k ⟶ i)), Hom.opensRange (F.map k.snd (β := fun (k : J) ↦ (k ⟶ j) × (k ⟶ i)).1)) :
        Scheme))
    ⇑(Opens.ι
        (⨆ (k : (k : J) × (k ⟶ j) × (k ⟶ i)), Hom.opensRange (F.map k.snd (β := fun (k : J) ↦ (k ⟶ j) × (k ⟶ i)).1))))
  ((Iso.inv (Hom.isoOpensRange (F.map l.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) ≫
        F.map l.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).2)
      x :
    ↥(F.obj j))
Tactic: simp only [V, tAux, ← Scheme.Hom.comp_apply, Cover.ι_glueMorphisms]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Gluing (goal)
Goal:
J : Type w
inst : Category.{v, w} J
F : CategoryTheory.Functor.{v, u, w, u + 1} J Scheme
inst_1 : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst_2 : (F ⋙ forget).IsLocallyDirected
inst_3 : Quiver.IsThin.{w, v} J
i j : J
l :
  PreZeroHypercover.I₀ (S :=
    (↑(⨆ (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)), Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) :
      Scheme))
    (Precoverage.ZeroHypercover.toPreZeroHypercover
      (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)) ↦
        Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
x :
  ↥(PreZeroHypercover.X
      (Precoverage.ZeroHypercover.toPreZeroHypercover
        (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)) ↦
          Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))
      l)
⊢ Membership.mem (α := ↥(F.obj j)) (γ := Set ↥(F.obj j))
    (Set.range (ι :=
      ↥(↑(⨆ (k : (k : J) × (k ⟶ j) × (k ⟶ i)), Hom.opensRange (F.map k.snd (β := fun (k : J) ↦ (k ⟶ j) × (k ⟶ i)).1)) :
          Scheme))
      ⇑(Opens.ι
          (⨆ (k : (k : J) × (k ⟶ j) × (k ⟶ i)), Hom.opensRange (F.map k.snd (β := fun (k : J) ↦ (k ⟶ j) × (k ⟶ i)).1))))
    ((Iso.inv (Hom.isoOpensRange (F.map l.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)) ≫
          F.map l.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).2)
        x :
      ↥(F.obj j))
have simp : Membership.mem (α := ↥(F.obj j)) (γ := Set ↥(F.obj j))
  (⋃ (i_1 : (k : J) × (k ⟶ j) × (k ⟶ i)),
    Set.range (ι := ↥(F.obj (i_1.fst (β := fun (k : J) ↦ (k ⟶ j) × (k ⟶ i)))))
      ⇑(F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ j) × (k ⟶ i)).1))
  ((TopCat.Hom.hom (F.map l.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).2).base : (a : ↥(F.obj l.fst)) → ↥(F.obj j))
      ((TopCat.Hom.hom
              (Hom.base
                (LocallyRingedSpace.Hom.toHom
                  (Hom.toLRSHom'
                    (Iso.inv (Hom.isoOpensRange (F.map l.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j)).1)))))) :
            (a :
                ↥((Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ j)) ↦ Hom.opensRange (F.map i_1.snd.1)).X
                    l)) →
              ↥(F.obj l.fst))
          x :
        ↥(F.obj (l.fst (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ j))))) :
    ↥(F.obj j))
Tactic: simp only [Opens.range_ι, iSup_mk, carrier_eq_coe, Hom.coe_opensRange, coe_mk, Hom.comp_base,
      TopCat.hom_comp, ContinuousMap.comp_apply]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Gluing (goal)
Goal:
J : Type w
inst : Category.{v, w} J
F : CategoryTheory.Functor.{v, u, w, u + 1} J Scheme
inst_1 : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst_2 : (F ⋙ forget).IsLocallyDirected
inst_3 : Quiver.IsThin.{w, v} J
i : J
⊢ t F i i = 𝟙 (↑(V F i i) : Scheme)
have refine : ∀
  (k :
    PreZeroHypercover.I₀ (S :=
      (↑(⨆ (i_1 : (k : J) × (k ⟶ i) × (k ⟶ i)),
            Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ i)).1)) :
        Scheme))
      (Precoverage.ZeroHypercover.toPreZeroHypercover
        (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ i)) ↦
          Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ i)).1)))),
  PreZeroHypercover.f
        (Precoverage.ZeroHypercover.toPreZeroHypercover
          (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ i)) ↦
            Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ i)).1)))
        k ≫
      t F i i =
    PreZeroHypercover.f
        (Precoverage.ZeroHypercover.toPreZeroHypercover
          (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ i)) ↦
            Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ i)).1)))
        k ≫
      𝟙 (↑(V F i i) : Scheme)
Tactic: refine (Scheme.Opens.iSupOpenCover _).hom_ext _ _ fun k ↦ ?_
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Gluing (goal)
Goal:
J : Type w
inst : Category.{v, w} J
F : CategoryTheory.Functor.{v, u, w, u + 1} J Scheme
inst_1 : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst_2 : (F ⋙ forget).IsLocallyDirected
inst_3 : Quiver.IsThin.{w, v} J
i : J
k :
  PreZeroHypercover.I₀ (S :=
    (↑(⨆ (i_1 : (k : J) × (k ⟶ i) × (k ⟶ i)), Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ i)).1)) :
      Scheme))
    (Precoverage.ZeroHypercover.toPreZeroHypercover
      (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ i)) ↦
        Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ i)).1)))
⊢ PreZeroHypercover.f
        (Precoverage.ZeroHypercover.toPreZeroHypercover
          (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ i)) ↦
            Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ i)).1)))
        k ≫
      t F i i =
    PreZeroHypercover.f
        (Precoverage.ZeroHypercover.toPreZeroHypercover
          (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ i)) ↦
            Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ i)).1)))
        k ≫
      𝟙 (↑(V F i i) : Scheme)
have simp : Iso.inv (Hom.isoOpensRange (F.map k.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ i)).1)) ≫
    F.map k.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ i)).2 =
  PreZeroHypercover.f
      (Precoverage.ZeroHypercover.toPreZeroHypercover
        (Opens.iSupOpenCover fun (i_1 : (k : J) × (k ⟶ i) × (k ⟶ i)) ↦
          Hom.opensRange (F.map i_1.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ i)).1)))
      k ≫
    Opens.ι (⨆ (k : (k : J) × (k ⟶ i) × (k ⟶ i)), Hom.opensRange (F.map k.snd (β := fun (k : J) ↦ (k ⟶ i) × (k ⟶ i)).1))
Tactic: simp only [Category.comp_id, ← cancel_mono (Scheme.Opens.ι _), Category.assoc,
    IsOpenImmersion.lift_fac, Scheme.Cover.ι_glueMorphisms, t, tAux, V]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Gluing (goal)
Goal:
J : Type w
inst : Category.{v, w} J
F : CategoryTheory.Functor.{v, u, w, u + 1} J Scheme
inst_1 : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst_2 : (F ⋙ forget).IsLocallyDirected
inst_3 : Quiver.IsThin.{w, v} J
inst_4 : Small.{u, w} J
i j k : Shrink.{u, w} J
x :
  ↥(pullback (X := (↑(V F (↓(i, j).1) ↓(i, j).2) : Scheme)) (Y := (↑(V F (↓(i, k).1) ↓(i, k).2) : Scheme))
      (Opens.ι (V F (↓(i, j).1) ↓(i, j).2)) (Opens.ι (V F (↓(i, k).1) ↓(i, k).2)))
⊢ Membership.mem (α := ↥(F.obj ↓j)) (γ := Set ↥(F.obj ↓j))
    (Set.range (ι := ↥(↑(V F (↓(j, k).1) ↓(j, k).2) : Scheme)) ⇑(Opens.ι (V F (↓j) ↓k)))
    ((pullback.fst (Opens.ι (V F (↓(i, j).1) ↓(i, j).2)) (Opens.ι (V F (↓(i, k).1) ↓(i, k).2)) ≫ tAux F (↓i) ↓j) x :
      ↥(F.obj ↓j))
have obtain : ∀ (l : J) (fi : l ⟶ ↓(i, j).1) (fj : l ⟶ ↓(i, j).2) (fk : l ⟶ ↓(i, k).2)
  (α :
    F.obj l ⟶
      pullback (X := (↑(V F (↓(i, j).1) ↓(i, j).2) : Scheme)) (Y := (↑(V F (↓(i, j).1) ↓(i, k).2) : Scheme))
        (Opens.ι (V F (↓(i, j).1) ↓(i, j).2)) (Opens.ι (V F (↓(i, j).1) ↓(i, k).2)))
  (z : ↥(F.obj l)) (hα : IsOpenImmersion α)
  (hα₁ :
    α ≫ pullback.fst (Opens.ι (V F (↓(i, j).1) ↓(i, j).2)) (Opens.ι (V F (↓(i, j).1) ↓(i, k).2)) =
      CategoryStruct.comp (Y := (↑(Hom.opensRange (F.map fi)) : Scheme)) (Iso.hom (Hom.isoOpensRange (F.map fi)))
        ((F.obj ↓(i, j).1).homOfLE sorry))
  (hα₂ :
    α ≫ pullback.snd (Opens.ι (V F (↓(i, j).1) ↓(i, j).2)) (Opens.ι (V F (↓(i, j).1) ↓(i, k).2)) =
      CategoryStruct.comp (Y := (↑(Hom.opensRange (F.map fi)) : Scheme)) (Iso.hom (Hom.isoOpensRange (F.map fi)))
        ((F.obj ↓(i, j).1).homOfLE sorry)),
  Membership.mem (α := ↥(F.obj ↓j)) (γ := Set ↥(F.obj ↓j))
    (Set.range (ι := ↥(↑(V F (↓(j, k).1) ↓(j, k).2) : Scheme)) ⇑(Opens.ι (V F (↓j) ↓k)))
    ((pullback.fst (Opens.ι (V F (↓(i, j).1) ↓(i, j).2)) (Opens.ι (V F (↓(i, k).1) ↓(i, k).2)) ≫ tAux F (↓i) ↓j)
        (α z :
          ↥(pullback (X := (↑(V F (↓(i, j).1) ↓(i, j).2) : Scheme)) (Y := (↑(V F (↓(i, j).1) ↓(i, k).2) : Scheme))
              (Opens.ι (V F (↓(i, j).1) ↓(i, j).2)) (Opens.ι (V F (↓(i, j).1) ↓(i, k).2)))) :
      ↥(F.obj ↓j))
Tactic: obtain ⟨l, fi, fj, fk, α, z, hα, hα₁, hα₂, rfl⟩ := exists_of_pullback_V_V F x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Gluing (goal)
Goal:
J : Type w
inst : Category.{v, w} J
F : CategoryTheory.Functor.{v, u, w, u + 1} J Scheme
inst_1 : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)
inst_2 : (F ⋙ forget).IsLocallyDirected
inst_3 : Quiver.IsThin.{w, v} J
inst_4 : Small.{u, w} J
i j k : Shrink.{u, w} J
l : J
fi : l ⟶ ↓(i, j).1
fj : l ⟶ ↓(i, j).2
fk : l ⟶ ↓(i, k).2
α :
  F.obj l ⟶
    pullback (X := (↑(V F (↓(i, j).1) ↓(i, j).2) : Scheme)) (Y := (↑(V F (↓(i, j).1) ↓(i, k).2) : Scheme))
      (Opens.ι (V F (↓(i, j).1) ↓(i, j).2)) (Opens.ι (V F (↓(i, j).1) ↓(i, k).2))
z : ↥(F.obj l)
hα : IsOpenImmersion α
hα₁ :
  α ≫ pullback.fst (Opens.ι (V F (↓(i, j).1) ↓(i, j).2)) (Opens.ι (V F (↓(i, j).1) ↓(i, k).2)) =
    CategoryStruct.comp (Y := (↑(Hom.opensRange (F.map fi)) : Scheme)) (Iso.hom (Hom.isoOpensRange (F.map fi)))
      ((F.obj ↓(i, j).1).homOfLE ⋯)
hα₂ :
  α ≫ pullback.snd (Opens.ι (V F (↓(i, j).1) ↓(i, j).2)) (Opens.ι (V F (↓(i, j).1) ↓(i, k).2)) =
    CategoryStruct.comp (Y := (↑(Hom.opensRange (F.map fi)) : Scheme)) (Iso.hom (Hom.isoOpensRange (F.map fi)))
      ((F.obj ↓(i, j).1).homOfLE ⋯)
⊢ Membership.mem (α := ↥(F.obj ↓j)) (γ := Set ↥(F.obj ↓j))
    (Set.range (ι := ↥(↑(V F (↓(j, k).1) ↓(j, k).2) : Scheme)) ⇑(Opens.ι (V F (↓j) ↓k)))
    ((pullback.fst (Opens.ι (V F (↓(i, j).1) ↓(i, j).2)) (Opens.ι (V F (↓(i, k).1) ↓(i, k).2)) ≫ tAux F (↓i) ↓j)
        (α z :
          ↥(pullback (X := (↑(V F (↓(i, j).1) ↓(i, j).2) : Scheme)) (Y := (↑(V F (↓(i, j).1) ↓(i, k).2) : Scheme))
              (Opens.ι (V F (↓(i, j).1) ↓(i, j).2)) (Opens.ι (V F (↓(i, j).1) ↓(i, k).2)))) :
      ↥(F.obj ↓j))
have rw : Membership.mem (α := ↥(F.obj ↓j)) (γ := (F.obj ↓j).Opens) (V F (↓j) ↓k) ((F.map fj) z : ↥(F.obj ↓j))
Tactic: rw [← Scheme.Hom.comp_apply, reassoc_of% hα₁, homOfLE_tAux F ↓i ↓j fi fj,
        Iso.hom_inv_id_assoc, Scheme.Opens.range_ι, SetLike.mem_coe]