awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (goal)
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
refine_1 : (a : J) → D.obj a ⟶ Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a
⊢ Cocone D
have refine_2 : ∀ {a b : J} (f : a ⟶ b),
  Eq (α := D.obj a ⟶ Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) b)
    (D.map f ≫
      Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
        (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 b)
          (fun (i : (𝒱 b).I₀) ↦
            ((d.cocone (P := P) i).ι.app b).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
              colimit.ι d.relativeGluingData (P := P).functor i)
          sorry sorry)
        trivial trivial)
    (Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
        (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 a)
          (fun (i : (𝒱 a).I₀) ↦
            ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
              colimit.ι d.relativeGluingData (P := P).functor i)
          sorry sorry)
        trivial trivial ≫
      CategoryTheory.Functor.map (Functor.obj (Functor.const J) (d.glued (P := P))) f)
Tactic: refine { pt := d.glued, ι.app a := ?_, ι.naturality {a b} f := ?_ }
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (goal)
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
a : J
⊢ D.obj a ⟶ Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a
have refine_1 : (i : (𝒱 a).I₀) →
  (𝒱 a).X i ⟶
    Over.left
      ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P) (Q := ⊤)
        (W := ⊤))
Tactic: refine ⟨(𝒱 a).glueMorphismsOverOfLocallyDirected ?_ ?_ ?_, trivial, trivial⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (goal)
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
a : J
refine_1 :
  (i : (𝒱 a).I₀) →
    (𝒱 a).X i ⟶
      Over.left
        ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P) (Q :=
          ⊤) (W := ⊤))
⊢ D.obj a ⟶ Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a
have refine_1₁ : ∀ {i j : (𝒱 a).I₀} (hij : i ⟶ j),
  Eq (α :=
    (𝒱 a).X i ⟶
      Over.left
        ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P) (Q :=
          ⊤) (W := ⊤)))
    (Cover.trans (P := IsOpenImmersion) (𝒱 a) hij ≫
      ((d.cocone (P := P) j).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
        colimit.ι d.relativeGluingData (P := P).functor j)
    (((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
      colimit.ι d.relativeGluingData (P := P).functor i)
Tactic: refine ⟨(𝒱 a).glueMorphismsOverOfLocallyDirected ?_ ?_ ?_, trivial, trivial⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (goal)
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
a : J
refine_1 :
  (i : (𝒱 a).I₀) →
    (𝒱 a).X i ⟶
      Over.left
        ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P) (Q :=
          ⊤) (W := ⊤))
refine_1₁ :
  ∀ {i j : (𝒱 a).I₀} (hij : i ⟶ j),
    Eq (α :=
      (𝒱 a).X i ⟶
        Over.left
          ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P)
            (Q := ⊤) (W := ⊤)))
      (Cover.trans (P := IsOpenImmersion) (𝒱 a) hij ≫
        ((d.cocone (P := P) j).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
          colimit.ι d.relativeGluingData (P := P).functor j)
      (((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
        colimit.ι d.relativeGluingData (P := P).functor i)
⊢ D.obj a ⟶ Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a
have refine_1₂ : ∀ (i : (𝒱 a).I₀),
  (((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
        colimit.ι d.relativeGluingData (P := P).functor i) ≫
      Over.hom
        ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P) (Q :=
          ⊤) (W := ⊤)) =
    (𝒱 a).f i ≫ Over.hom ((D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤))
Tactic: refine ⟨(𝒱 a).glueMorphismsOverOfLocallyDirected ?_ ?_ ?_, trivial, trivial⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (goal)
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
a : J
i j : (𝒱 a).I₀
hij : i ⟶ j
| ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
    colimit.ι d.relativeGluingData (P := P).functor i
have rw : ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
  d.relativeGluingData (P := P).functor.map hij ≫ colimit.ι d.relativeGluingData (P := P).functor j
Tactic: rw [← colimit.w d.relativeGluingData.functor hij]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (goal)
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
a : J
i j : (𝒱 a).I₀
hij : i ⟶ j
⊢ Eq (α :=
    (𝒱 a).X i ⟶
      Over.left
        ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P) (Q :=
          ⊤) (W := ⊤)))
    (Cover.trans (P := IsOpenImmersion) (𝒱 a) hij ≫
      ((d.cocone (P := P) j).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
        colimit.ι d.relativeGluingData (P := P).functor j)
    (((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
      colimit.ι d.relativeGluingData (P := P).functor i)
have refine_1 : Eq (α :=
  (𝒱 a).X i ⟶
    Over.left
      ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P) (Q := ⊤)
        (W := ⊤)))
  (Cover.trans (P := IsOpenImmersion) (𝒱 a) hij ≫
    ((d.cocone (P := P) j).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
      colimit.ι d.relativeGluingData (P := P).functor j)
  (((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
    d.relativeGluingData (P := P).functor.map hij ≫ colimit.ι d.relativeGluingData (P := P).functor j)
Tactic: conv_rhs => rw [← colimit.w d.relativeGluingData.functor hij]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (goal)
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
a : J
i j : (𝒱 a).I₀
hij : i ⟶ j
⊢ Eq (α :=
    (𝒱 a).X i ⟶
      Over.left
        ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P) (Q :=
          ⊤) (W := ⊤)))
    (Cover.trans (P := IsOpenImmersion) (𝒱 a) hij ≫
      ((d.cocone (P := P) j).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
        colimit.ι d.relativeGluingData (P := P).functor j)
    (((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
      d.relativeGluingData (P := P).functor.map hij ≫ colimit.ι d.relativeGluingData (P := P).functor j)
have refine_1 : Eq (α :=
  (𝒱 a).X i ⟶
    Over.left
      ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P) (Q := ⊤)
        (W := ⊤)))
  ((Cover.trans (P := IsOpenImmersion) (𝒱 a) hij ≫
      ((d.cocone (P := P) j).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left) ≫
    colimit.ι d.relativeGluingData (P := P).functor j)
  ((((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
      d.relativeGluingData (P := P).functor.map hij) ≫
    colimit.ι d.relativeGluingData (P := P).functor j)
Tactic: simp only [← Category.assoc]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (goal)
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
a : J
i j : (𝒱 a).I₀
hij : i ⟶ j
⊢ Eq (α :=
    (𝒱 a).X i ⟶
      Over.left
        ((Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a).toComma (P := P) (Q :=
          ⊤) (W := ⊤)))
    ((Cover.trans (P := IsOpenImmersion) (𝒱 a) hij ≫
        ((d.cocone (P := P) j).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left) ≫
      colimit.ι d.relativeGluingData (P := P).functor j)
    ((((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
        d.relativeGluingData (P := P).functor.map hij) ≫
      colimit.ι d.relativeGluingData (P := P).functor j)
have refine_1 : Cover.trans (P := IsOpenImmersion) (𝒱 a) hij ≫
    ((d.cocone (P := P) j).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left =
  ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
    d.relativeGluingData (P := P).functor.map hij
Tactic: congr 1
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (goal)
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
a b : J
f : a ⟶ b
⊢ Eq (α := D.obj a ⟶ Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) b)
    (D.map f ≫
      Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
        (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 b)
          (fun (i : (𝒱 b).I₀) ↦
            ((d.cocone (P := P) i).ι.app b).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
              colimit.ι d.relativeGluingData (P := P).functor i)
          ⋯ ⋯)
        trivial trivial)
    (Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
        (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 a)
          (fun (i : (𝒱 a).I₀) ↦
            ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
              colimit.ι d.relativeGluingData (P := P).functor i)
          ⋯ ⋯)
        trivial trivial ≫
      CategoryTheory.Functor.map (Functor.obj (Functor.const J) (d.glued (P := P))) f)
have refine_2 : Eq (α :=
  (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left ⟶
    (Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) b).toComma (P := P) (Q := ⊤)
        (W := ⊤).left)
  (CommaMorphism.left
    (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
      (D.map f ≫
        Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
          (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 b)
            (fun (i : (𝒱 b).I₀) ↦
              ((d.cocone (P := P) i).ι.app b).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
                colimit.ι d.relativeGluingData (P := P).functor i)
            sorry sorry)
          trivial trivial)))
  (CommaMorphism.left
    (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
      (Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
          (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 a)
            (fun (i : (𝒱 a).I₀) ↦
              ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
                colimit.ι d.relativeGluingData (P := P).functor i)
            sorry sorry)
          trivial trivial ≫
        CategoryTheory.Functor.map (Functor.obj (Functor.const J) (d.glued (P := P))) f)))
Tactic: ext
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (hypothesis)
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
a b : J
f : a ⟶ b
i : (𝒱 a).I₀
⊢ Eq (α :=
    (𝒱 a).X i ⟶
      (Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) b).toComma (P := P) (Q := ⊤)
          (W := ⊤).left)
    ((𝒱 a).f i ≫
      CommaMorphism.left
        (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
          (D.map f ≫
            Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
              (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 b)
                (fun (i : (𝒱 b).I₀) ↦
                  ((d.cocone (P := P) i).ι.app b).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
                    colimit.ι d.relativeGluingData (P := P).functor i)
                ⋯ ⋯)
              trivial trivial)))
    ((𝒱 a).f i ≫
      CommaMorphism.left
        (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
          (Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
              (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 a)
                (fun (i : (𝒱 a).I₀) ↦
                  ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
                    colimit.ι d.relativeGluingData (P := P).functor i)
                ⋯ ⋯)
              trivial trivial ≫
            CategoryTheory.Functor.map (Functor.obj (Functor.const J) (d.glued (P := P))) f)))
have this : (𝒱 a).f i ≫ (D.map f).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left =
  ((D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).map f).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫ (𝒱 b).f i
Tactic: have : (𝒱 a).f i ≫ (D.map f).left =
        ((D ⋙ Over.pullback _ _ (𝒰.f i)).map f).left ≫ (𝒱 b).f i := by
      simp [𝒱]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ColimitsOver (goal)
Goal:
P : MorphismProperty Scheme
inst : P.IsStableUnderBaseChange
inst_1 : P.IsMultiplicative
S : Scheme
J : Type u_1
inst_2 : Category.{v_1, u_1} J
D : CategoryTheory.Functor.{v_1, u, u_1, u + 1} J (P.Over ⊤ S)
𝒰 : OpenCover.{u_2, u} S
inst_3 : Category.{v_2, u_2} 𝒰.I₀
inst_4 : LocallyDirected.{v_2, u, u_2} (P := IsOpenImmersion) 𝒰
d : ColimitGluingData.{v_2, u, v_1, u_1, u_2} (P := P) D 𝒰
inst_5 :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j),
    PreservesColimitsOfShape.{v_1, u_1, u, u, u + 1, u + 1} (C := P.Over ⊤ (𝒰.X j)) J
      (MorphismProperty.Over.pullback P ⊤ (Cover.trans (P := IsOpenImmersion) 𝒰 hij))
inst_6 : Quiver.IsThin.{u_2, v_2} 𝒰.I₀
inst_7 : Small.{u, u_2} 𝒰.I₀
inst_8 : IsZariskiLocalAtTarget P
𝒱 : (a : J) → OpenCover.{u_2, u} (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).left :=
  fun (a : J) ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).toComma (P := P) (Q := ⊤) (W := ⊤).hom 𝒰
a b : J
f : a ⟶ b
i : (𝒱 a).I₀
this :
  (𝒱 a).f i ≫ (D.map f).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left =
    ((D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).map f).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫ (𝒱 b).f i
⊢ (𝒱 a).f i ≫
      (Comma.Hom.hom (P := P) (Q := ⊤) (W := ⊤) (D.map f)).left ≫
        CommaMorphism.left
          (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 b)
            (fun (i : (𝒱 b).I₀) ↦
              ((d.cocone (P := P) i).ι.app b).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
                colimit.ι d.relativeGluingData (P := P).functor i)
            ⋯ ⋯) =
    (𝒱 a).f i ≫
      CommaMorphism.left
          (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 a)
            (fun (i : (𝒱 a).I₀) ↦
              ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
                colimit.ι d.relativeGluingData (P := P).functor i)
            ⋯ ⋯) ≫
        𝟙 d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left
have refine_2 : Eq (α := (𝒱 a).X i ⟶ d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left)
  ((𝒱 a).f i ≫
    (Comma.Hom.hom (P := P) (Q := ⊤) (W := ⊤) (D.map f)).left ≫
      CommaMorphism.left
        (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 b)
          (fun (i : (𝒱 b).I₀) ↦
            ((d.cocone (P := P) i).ι.app b).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
              colimit.ι d.relativeGluingData (P := P).functor i)
          sorry sorry))
  ((𝒱 a).f i ≫
    CommaMorphism.left
      (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} (𝒱 a)
        (fun (i : (𝒱 a).I₀) ↦
          ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
            colimit.ι d.relativeGluingData (P := P).functor i)
        sorry sorry))
Tactic: rw [Category.comp_id, Scheme.OpenCover.map_glueMorphismsOverOfLocallyDirected_left,
      reassoc_of% this, Scheme.OpenCover.map_glueMorphismsOverOfLocallyDirected_left,
      ← Over.comp_left_assoc, ← Comma.comp_hom, ← Functor.comp_map, Cocone.w]