awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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
i : 𝒰.toPreZeroHypercover.1
rw : P (d.relativeGluingData (P := P).natTrans.app i)
⊢ P (X :=
    PreZeroHypercover.X
      (Precoverage.ZeroHypercover.toPreZeroHypercover
        (Precoverage.ZeroHypercover.pullback₁ d.relativeGluingData (P := P).toBase 𝒰))
      i)
    (pullbackHom (P := IsOpenImmersion) 𝒰 d.relativeGluingData (P := P).toBase i :
      PreZeroHypercover.X
          (Precoverage.ZeroHypercover.toPreZeroHypercover
            (Precoverage.ZeroHypercover.pullback₁ d.relativeGluingData (P := P).toBase 𝒰))
          i ⟶
        𝒰.X i)

Tactic:
rw [Scheme.Cover.pullbackHom,
      ← (d.relativeGluingData.isPullback_natTrans_ι_toBase i).flip.isoPullback_inv_snd,
      P.cancel_left_of_respectsIso]

Premises used: AlgebraicGeometry.Scheme.Cover.pullbackHom, CategoryTheory.IsPullback.isoPullback_inv_snd, CategoryTheory.IsPullback.flip, CategoryTheory.MorphismProperty.cancel_left_of_respectsIso, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.relativeGluingData, AlgebraicGeometry.Scheme.Cover.RelativeGluingData.isPullback_natTrans_ι_toBase
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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
i : 𝒰.toPreZeroHypercover.1
⊢ P (d.relativeGluingData (P := P).natTrans.app i)

Tactic:
(d.cocone i).pt.prop

Premises used: AlgebraicGeometry.Scheme.Cover.ColimitGluingData.cocone, CategoryTheory.MorphismProperty.Comma.prop, CategoryTheory.Limits.Cocone.pt
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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
i : 𝒰.toPreZeroHypercover.1
⊢ P (d.relativeGluingData (P := P).natTrans.app i)

Tactic:
exact (d.cocone i).pt.prop

Premises used: AlgebraicGeometry.Scheme.Cover.ColimitGluingData.cocone, CategoryTheory.MorphismProperty.Comma.prop, CategoryTheory.Limits.Cocone.pt
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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
i : 𝒰.I₀
⊢ CommaMorphism.left (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤) (Iso.inv (d.pullbackGluedIso (P := P) i))) ≫
      pullback.fst d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) =
    colimit.ι d.relativeGluingData (P := P).functor i

Tactic:
simp [pullbackGluedIso, glued]

Premises used: AlgebraicGeometry.Scheme.Cover.ColimitGluingData.pullbackGluedIso, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.glued
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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
i : 𝒰.I₀
⊢ CommaMorphism.left (Comma.Hom.toCommaMorphism (P := P) (Q := ⊤) (W := ⊤) (Iso.inv (d.pullbackGluedIso (P := P) i))) ≫
      pullback.snd d.glued (P := P).toComma (P := P) (Q := ⊤) (W := ⊤).hom (𝒰.f i) =
    (d.cocone (P := P) i).pt.toComma (P := P) (Q := ⊤) (W := ⊤).hom

Tactic:
simp [pullbackGluedIso, glued]

Premises used: AlgebraicGeometry.Scheme.Cover.ColimitGluingData.pullbackGluedIso, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.glued
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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
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)
            ⋯ ⋯)
          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)
⊢ Cocone D

Tactic:
refine { pt := d.glued, ι.app a := ?_, ι.naturality {a b} f := ?_ }
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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)
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 := ⊤))
⊢ D.obj a ⟶ Functor.obj.{v_1, u, u_1, u + 1} (Functor.obj (Functor.const J) (d.glued (P := P))) a

Tactic:
refine ⟨(𝒱 a).glueMorphismsOverOfLocallyDirected ?_ ?_ ?_, trivial, trivial⟩
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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 : (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:
exact ((d.cocone i).ι.app a).left ≫ colimit.ι d.relativeGluingData.functor i

Premises used: AlgebraicGeometry.Scheme.Cover.ColimitGluingData.cocone, CategoryTheory.Limits.Cocone.ι, CategoryTheory.CommaMorphism.left, CategoryTheory.MorphismProperty.Comma.Hom.toCommaMorphism, AlgebraicGeometry.Scheme.Cover.RelativeGluingData.functor, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.relativeGluingData, CategoryTheory.NatTrans.app, CategoryTheory.Limits.colimit.ι
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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
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
| ((d.cocone (P := P) i).ι.app a).toCommaMorphism (P := P) (Q := ⊤) (W := ⊤).left ≫
    colimit.ι d.relativeGluingData (P := P).functor i

Tactic:
rw [← colimit.w d.relativeGluingData.functor hij]

Premises used: CategoryTheory.Limits.colimit.w, AlgebraicGeometry.Scheme.Cover.RelativeGluingData.functor, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.relativeGluingData
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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
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)
⊢ 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:
conv_rhs => rw [← colimit.w d.relativeGluingData.functor hij]

Premises used: CategoryTheory.Limits.colimit.w, AlgebraicGeometry.Scheme.Cover.RelativeGluingData.functor, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.relativeGluingData
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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
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)
⊢ 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]

Premises used: CategoryTheory.Category.assoc
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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
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
⊢ 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:
congr 1
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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
⊢ 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($(d.cocone_ι_transitionMap hij a).left).symm

Premises used: congrArg, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.cocone_ι_transitionMap, id, CategoryTheory.Limits.limit, CategoryTheory.CommaMorphism.left, CategoryTheory.MorphismProperty.Comma.Hom.toCommaMorphism, Eq.symm
-- Module: Mathlib.AlgebraicGeometry.ColimitsOver
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
⊢ 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:
exact congr($(d.cocone_ι_transitionMap hij a).left).symm

Premises used: congrArg, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.cocone_ι_transitionMap, id, CategoryTheory.CommaMorphism.left, CategoryTheory.MorphismProperty.Comma.Hom.toCommaMorphism, Eq.symm