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 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
⊢ IsColimit (d.gluedCocone (P := P))
have refine_1 : (s : Cocone D) → d.gluedCocone (P := P).pt ⟶ s.pt
Tactic: refine { desc s := ?_, fac := ?_, uniq := ?_ }
-- 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 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
refine_1 : (s : Cocone D) → d.gluedCocone (P := P).pt ⟶ s.pt
⊢ IsColimit (d.gluedCocone (P := P))
have refine_2 : ∀ (s : Cocone D) (j : J),
  d.gluedCocone (P := P).ι.app j ≫
      Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
        (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
          (fun (i : 𝒱.I₀) ↦
            CategoryStruct.comp (Y :=
              (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                      (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                  (P := P) (Q := ⊤) (W := ⊤).left)
              (CommaMorphism.left
                (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                  ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
              (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
          sorry sorry)
        trivial trivial =
    s.ι.app j
Tactic: refine { desc s := ?_, fac := ?_, uniq := ?_ }
-- 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 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
refine_1 : (s : Cocone D) → d.gluedCocone (P := P).pt ⟶ s.pt
refine_2 :
  ∀ (s : Cocone D) (j : J),
    d.gluedCocone (P := P).ι.app j ≫
        Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
          (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
            (fun (i : 𝒱.I₀) ↦
              CategoryStruct.comp (Y :=
                (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                        (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                    (P := P) (Q := ⊤) (W := ⊤).left)
                (CommaMorphism.left
                  (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                    ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
                (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
            ⋯ ⋯)
          trivial trivial =
      s.ι.app j
⊢ IsColimit (d.gluedCocone (P := P))
have refine_3 : ∀ (s : Cocone D) (m : d.gluedCocone (P := P).pt ⟶ s.pt),
  (∀ (j : J), d.gluedCocone (P := P).ι.app j ≫ m = s.ι.app j) →
    m =
      Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
        (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
          (fun (i : 𝒱.I₀) ↦
            CategoryStruct.comp (Y :=
              (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                      (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                  (P := P) (Q := ⊤) (W := ⊤).left)
              (CommaMorphism.left
                (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                  ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
              (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
          sorry sorry)
        trivial trivial
Tactic: refine { desc s := ?_, fac := ?_, uniq := ?_ }
-- 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 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
⊢ d.gluedCocone (P := P).pt ⟶ s.pt
have refine_1 : (i : 𝒱.I₀) → 𝒱.X i ⟶ Over.left (s.pt.toComma (P := P) (Q := ⊤) (W := ⊤))
Tactic: refine ⟨𝒱.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 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
refine_1 : (i : 𝒱.I₀) → 𝒱.X i ⟶ Over.left (s.pt.toComma (P := P) (Q := ⊤) (W := ⊤))
⊢ d.gluedCocone (P := P).pt ⟶ s.pt
have refine_1₁ : ∀ {i j : 𝒱.I₀} (hij : i ⟶ j),
  Eq (α := 𝒱.X i ⟶ Over.left (s.pt.toComma (P := P) (Q := ⊤) (W := ⊤)))
    (Cover.trans (P := IsOpenImmersion) 𝒱 hij ≫
      CategoryStruct.comp (Y :=
        (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
                (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s)).toComma
            (P := P) (Q := ⊤) (W := ⊤).left)
        (CommaMorphism.left
          (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
            ((d.isColimit (P := P) j).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s))))
        (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f j)))
    (CategoryStruct.comp (Y :=
      (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
              (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
          (P := P) (Q := ⊤) (W := ⊤).left)
      (CommaMorphism.left
        (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
          ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
      (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
Tactic: refine ⟨𝒱.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 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
refine_1 : (i : 𝒱.I₀) → 𝒱.X i ⟶ Over.left (s.pt.toComma (P := P) (Q := ⊤) (W := ⊤))
refine_1₁ :
  ∀ {i j : 𝒱.I₀} (hij : i ⟶ j),
    Eq (α := 𝒱.X i ⟶ Over.left (s.pt.toComma (P := P) (Q := ⊤) (W := ⊤)))
      (Cover.trans (P := IsOpenImmersion) 𝒱 hij ≫
        CategoryStruct.comp (Y :=
          (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
                  (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s)).toComma
              (P := P) (Q := ⊤) (W := ⊤).left)
          (CommaMorphism.left
            (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
              ((d.isColimit (P := P) j).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f j)) s))))
          (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f j)))
      (CategoryStruct.comp (Y :=
        (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
            (P := P) (Q := ⊤) (W := ⊤).left)
        (CommaMorphism.left
          (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
            ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
        (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
⊢ d.gluedCocone (P := P).pt ⟶ s.pt
have refine_1₂ : ∀ (i : 𝒱.I₀),
  CategoryStruct.comp (Y :=
        (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
            (P := P) (Q := ⊤) (W := ⊤).left)
        (CommaMorphism.left
          (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
            ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
        (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)) ≫
      Over.hom (s.pt.toComma (P := P) (Q := ⊤) (W := ⊤)) =
    𝒱.f i ≫ Over.hom (d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤))
Tactic: refine ⟨𝒱.glueMorphismsOverOfLocallyDirected ?_ ?_ ?_, trivial, trivial⟩
-- 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 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
i : 𝒱.I₀
⊢ CategoryStruct.comp (Y :=
        (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
            (P := P) (Q := ⊤) (W := ⊤).left)
        (CommaMorphism.left
          (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
            ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
        (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)) ≫
      Over.hom (s.pt.toComma (P := P) (Q := ⊤) (W := ⊤)) =
    𝒱.f i ≫ Over.hom (d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤))
have this : CategoryStruct.comp (Y := pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i))
    (CommaMorphism.left
        (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
          ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))) :
      (d.cocone (P := P) i).pt.toComma (P := P) (Q := ⊤) (W := ⊤).left ⟶
        (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
            (P := P) (Q := ⊤) (W := ⊤).left)
    (pullback.snd s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)) =
  (d.cocone (P := P) i).pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom
Tactic: have : _ ≫ pullback.snd _ _ = _ := Over.w ((d.isColimit i).desc (Functor.mapCocone _ s))
-- 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 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
i : 𝒱.I₀
this :
  CategoryStruct.comp (Y := pullback s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i))
      (CommaMorphism.left
          (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
            ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))) :
        (d.cocone (P := P) i).pt.toComma (P := P) (Q := ⊤) (W := ⊤).left ⟶
          (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                  (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
              (P := P) (Q := ⊤) (W := ⊤).left)
      (pullback.snd s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)) =
    (d.cocone (P := P) i).pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom
⊢ CategoryStruct.comp (Y :=
      (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
              (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
          (P := P) (Q := ⊤) (W := ⊤).left)
      (CommaMorphism.left
        (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
          ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
      (pullback.snd s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) ≫ 𝒰.f i) =
    PreZeroHypercover.f (Precoverage.ZeroHypercover.toPreZeroHypercover d.relativeGluingData (P := P).cover) i ≫
      Over.hom
        (Comma.toComma (P := P) (Q := ⊤) (W := ⊤)
          (MorphismProperty.Over.mk (P := P) ⊤ (A := d.relativeGluingData (P := P).glued)
            d.relativeGluingData (P := P).toBase ⋯))
have refine_1 : (d.cocone (P := P) i).pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom ≫ 𝒰.f i =
  PreZeroHypercover.f (Precoverage.ZeroHypercover.toPreZeroHypercover d.relativeGluingData (P := P).cover) i ≫
    Over.hom
      (Comma.toComma (P := P) (Q := ⊤) (W := ⊤)
        (MorphismProperty.Over.mk (P := P) ⊤ (A := d.relativeGluingData (P := P).glued)
          d.relativeGluingData (P := P).toBase sorry))
Tactic: rw [reassoc_of% this]
-- 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 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
a : J
𝒲 : (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
⊢ d.gluedCocone (P := P).ι.app a ≫
      Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
        (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
          (fun (i : 𝒱.I₀) ↦
            CategoryStruct.comp (Y :=
              (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                      (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                  (P := P) (Q := ⊤) (W := ⊤).left)
              (CommaMorphism.left
                (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                  ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
              (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
          ⋯ ⋯)
        trivial trivial =
    s.ι.app a
have refine_2 : CommaMorphism.left
    (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
      (d.gluedCocone (P := P).ι.app a ≫
        Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
          (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
            (fun (i : 𝒱.I₀) ↦
              CategoryStruct.comp (Y :=
                (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                        (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                    (P := P) (Q := ⊤) (W := ⊤).left)
                (CommaMorphism.left
                  (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                    ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
                (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
            sorry sorry)
          trivial trivial)) =
  (s.ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left
Tactic: ext
-- 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 : OpenCover.{u_2, u} d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).left := d.relativeGluingData (P := P).cover
s : Cocone D
m : d.gluedCocone (P := P).pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone (P := P).ι.app j ≫ m = s.ι.app j
⊢ m =
    Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
      (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
        (fun (i : 𝒱.I₀) ↦
          CategoryStruct.comp (Y :=
            (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                    (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                (P := P) (Q := ⊤) (W := ⊤).left)
            (CommaMorphism.left
              (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
            (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
        ⋯ ⋯)
      trivial trivial
have refine_3 : m.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left =
  CommaMorphism.left
    (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
      (Comma.Hom.mk (P := P) (Q := ⊤) (W := ⊤)
        (OpenCover.glueMorphismsOverOfLocallyDirected.{v_2, u, u_2} 𝒱
          (fun (i : 𝒱.I₀) ↦
            CategoryStruct.comp (Y :=
              (Cocone.pt (F := D ⋙ MorphismProperty.Over.pullback P ⊤ (𝒰.f i))
                      (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s)).toComma
                  (P := P) (Q := ⊤) (W := ⊤).left)
              (CommaMorphism.left
                (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤)
                  ((d.isColimit (P := P) i).desc (Functor.mapCocone (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)) s))))
              (pullback.fst s.pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i)))
          sorry sorry)
        trivial trivial))
Tactic: ext