awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P Q R : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
γ : Q ⟶ R
X : C
x : (J.Cover X)ᵒᵖ
⊢ Eq (α := (J.diagram P X).obj x ⟶ (J.diagram R X).obj x) (NatTrans.app (J.diagramNatTrans (η ≫ γ) X) x)
    (NatTrans.app (J.diagramNatTrans η X ≫ J.diagramNatTrans γ X) x)
have w : ∀ (i : (unop x).shape.L),
  Eq (α := (J.diagram P X).obj x ⟶ ((unop x).index R).left i)
    (NatTrans.app (J.diagramNatTrans (η ≫ γ) X) x ≫ Multiequalizer.ι ((unop x).index R) i)
    (NatTrans.app (J.diagramNatTrans η X ≫ J.diagramNatTrans γ X) x ≫ Multiequalizer.ι ((unop x).index R) i)
Tactic: refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
⊢ CategoryStruct.comp (Y :=
      colimit (J := (J.Cover (unop X))ᵒᵖ)
        (Functor.comp (D := (J.Cover (unop X))ᵒᵖ) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X))))
          (J.diagram P (unop X))))
      (colimMap (J.diagramPullback P (Quiver.Hom.unop (𝟙 X))))
      (colimit.pre (J.diagram P (unop X)) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X))))) =
    𝟙 (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
have refine : ∀ (S : (J.Cover (unop X))ᵒᵖ),
  CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
      (colimit.ι (J.diagram P (unop X)) S)
      (CategoryStruct.comp (Y :=
        colimit (J := (J.Cover (unop X))ᵒᵖ)
          (Functor.comp (D := (J.Cover (unop X))ᵒᵖ) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X))))
            (J.diagram P (unop X))))
        (colimMap (J.diagramPullback P (Quiver.Hom.unop (𝟙 X))))
        (colimit.pre (J.diagram P (unop X)) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X)))))) =
    colimit.ι (J.diagram P (unop X)) S ≫ 𝟙 (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
Tactic: refine colimit.hom_ext (fun S => ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
⊢ CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
      (colimit.ι (J.diagram P (unop X)) S)
      (CategoryStruct.comp (Y :=
        colimit (J := (J.Cover (unop X))ᵒᵖ)
          (Functor.comp (D := (J.Cover (unop X))ᵒᵖ) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X))))
            (J.diagram P (unop X))))
        (colimMap (J.diagramPullback P (Quiver.Hom.unop (𝟙 X))))
        (colimit.pre (J.diagram P (unop X)) (Functor.op (J.pullback (Quiver.Hom.unop (𝟙 X)))))) =
    colimit.ι (J.diagram P (unop X)) S ≫ 𝟙 (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
have dsimp : CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (colimit.ι (J.diagram P (unop X)) S)
    (CategoryStruct.comp (Y := colimit ((J.pullback (𝟙 (unop X))).op ⋙ J.diagram P (unop X)))
      (colimMap (J.diagramPullback P (𝟙 (unop X)))) (colimit.pre (J.diagram P (unop X)) (J.pullback (𝟙 (unop X))).op)) =
  colimit.ι (J.diagram P (unop X)) S ≫ 𝟙 (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
⊢ CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
      (colimit.ι (J.diagram P (unop X)) S)
      (CategoryStruct.comp (Y := colimit ((J.pullback (𝟙 (unop X))).op ⋙ J.diagram P (unop X)))
        (colimMap (J.diagramPullback P (𝟙 (unop X))))
        (colimit.pre (J.diagram P (unop X)) (J.pullback (𝟙 (unop X))).op)) =
    colimit.ι (J.diagram P (unop X)) S ≫ 𝟙 (colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
have simp : Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
  (CategoryStruct.comp (Y := ((J.pullback (𝟙 (unop X))).op ⋙ J.diagram P (unop X)).obj S)
      (Multiequalizer.lift ((unop ((J.pullback (𝟙 (unop X))).op.obj S)).index P) (Functor.obj (J.diagram P (unop X)) S)
        (fun (I : (unop ((J.pullback (𝟙 (unop X))).op.obj S)).shape.L) ↦
          Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
        sorry)
      (colimit.ι (J.diagram P (unop X)) ((J.pullback (𝟙 (unop X))).op.obj S)) :
    Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
  (colimit.ι (J.diagram P (unop X)) S)
Tactic: simp only [diagramPullback_app, colimit.ι_pre, ι_colimMap_assoc, Category.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (CategoryStruct.comp (Y := ((J.pullback (𝟙 (unop X))).op ⋙ J.diagram P (unop X)).obj S)
        (Multiequalizer.lift ((unop ((J.pullback (𝟙 (unop X))).op.obj S)).index P)
          (Functor.obj (J.diagram P (unop X)) S)
          (fun (I : (unop ((J.pullback (𝟙 (unop X))).op.obj S)).shape.L) ↦
            Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
          ⋯)
        (colimit.ι (J.diagram P (unop X)) ((J.pullback (𝟙 (unop X))).op.obj S)) :
      Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (colimit.ι (J.diagram P (unop X)) S)
have dsimp : Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
  (CategoryStruct.comp (Y :=
      Functor.obj
        (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
          { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
            map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
            map_id := sorry, map_comp := sorry }
          (J.diagram P (unop X)))
        S)
      (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
        (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦ Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
        sorry)
      (colimit.ι (J.diagram P (unop X)) (op ((unop S).pullback (𝟙 (unop X))))) :
    Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
  (colimit.ι (J.diagram P (unop X)) S)
Tactic: dsimp only [Functor.op, pullback_obj]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (CategoryStruct.comp (Y :=
        Functor.obj
          (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
            { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
              map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
              map_id := ⋯, map_comp := ⋯ }
            (J.diagram P (unop X)))
          S)
        (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
          (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦
            Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
          ⋯)
        (colimit.ι (J.diagram P (unop X)) (op ((unop S).pullback (𝟙 (unop X))))) :
      Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (colimit.ι (J.diagram P (unop X)) S)
have rw : Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
  (CategoryStruct.comp (Y :=
        Functor.obj
          (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
            { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
              map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
              map_id := sorry, map_comp := sorry }
            (J.diagram P (unop X)))
          S)
        (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
          (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦
            Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
          sorry)
        (Functor.map (J.diagram P (unop X)) e.inv.op) ≫
      colimit.ι (J.diagram P (unop X)) (op (unop S)) :
    Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
  (colimit.ι (J.diagram P (unop X)) S)
Tactic: rw [← colimit.w _ e.inv.op, ← Category.assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (CategoryStruct.comp (Y :=
          Functor.obj
            (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
              { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
                map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
                map_id := ⋯, map_comp := ⋯ }
              (J.diagram P (unop X)))
            S)
          (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
            (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦
              Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
            ⋯)
          (Functor.map (J.diagram P (unop X)) e.inv.op) ≫
        colimit.ι (J.diagram P (unop X)) (op (unop S)) :
      Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
    (colimit.ι (J.diagram P (unop X)) S)
have h : (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) →
  CategoryStruct.comp (Y :=
      Functor.obj
        (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
          { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
            map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
            map_id := sorry, map_comp := sorry }
          (J.diagram P (unop X)))
        S)
      (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
        (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦ Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
        sorry)
      (Functor.map (J.diagram P (unop X)) e.inv.op) =
    𝟙 (Functor.obj (J.diagram P (unop X)) S)
Tactic: convert! Category.id_comp (colimit.ι (diagram J P (unop X)) S)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
⊢ CategoryStruct.comp (Y :=
      Functor.obj
        (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
          { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
            map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
            map_id := ⋯, map_comp := ⋯ }
          (J.diagram P (unop X)))
        S)
      (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
        (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦ Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
        ⋯)
      (Functor.map (J.diagram P (unop X)) e.inv.op) =
    𝟙 (Functor.obj (J.diagram P (unop X)) S)
have h : ∀ (I : (unop (op (unop S))).shape.L),
  CategoryStruct.comp (Y := multiequalizer ((unop (op (unop S))).index P))
      (CategoryStruct.comp (Y :=
        Functor.obj
          (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
            { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
              map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
              map_id := sorry, map_comp := sorry }
            (J.diagram P (unop X)))
          S)
        (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
          (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦
            Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
          sorry)
        (Functor.map (J.diagram P (unop X)) e.inv.op))
      (Multiequalizer.ι ((unop (op (unop S))).index P) I) =
    𝟙 (Functor.obj (J.diagram P (unop X)) S) ≫ Multiequalizer.ι ((unop (op (unop S))).index P) I
Tactic: refine Multiequalizer.hom_ext _ _ _ (fun I => ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
I : (unop (op (unop S))).shape.L
⊢ CategoryStruct.comp (Y := multiequalizer ((unop (op (unop S))).index P))
      (CategoryStruct.comp (Y :=
        Functor.obj
          (Functor.comp (D := (J.Cover (unop X))ᵒᵖ)
            { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (𝟙 (unop X))),
              map := fun {X_1 Y : (J.Cover (unop X))ᵒᵖ} (f : X_1 ⟶ Y) ↦ ((J.pullback (𝟙 (unop X))).map f.unop).op,
              map_id := ⋯, map_comp := ⋯ }
            (J.diagram P (unop X)))
          S)
        (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (Functor.obj (J.diagram P (unop X)) S)
          (fun (I : ((unop S).pullback (𝟙 (unop X))).shape.L) ↦
            Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
          ⋯)
        (Functor.map (J.diagram P (unop X)) e.inv.op))
      (Multiequalizer.ι ((unop (op (unop S))).index P) I) =
    𝟙 (Functor.obj (J.diagram P (unop X)) S) ≫ Multiequalizer.ι ((unop (op (unop S))).index P) I
have h : CategoryStruct.comp (Y := multiequalizer ((unop S).index P))
    (CategoryStruct.comp (Y := multiequalizer (((unop S).pullback (𝟙 (unop X))).index P))
      (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (multiequalizer ((unop S).index P))
        (fun (I : ((unop S).pullback (𝟙 (unop X))).Arrow) ↦ Multiequalizer.ι ((unop S).index P) I.base) sorry)
      (Multiequalizer.lift ((unop S).index P) (multiequalizer (((unop S).pullback (𝟙 (unop X))).index P))
        (fun (I : (unop S).Arrow) ↦ Multiequalizer.ι (((unop S).pullback (𝟙 (unop X))).index P) (I.map e.inv)) sorry))
    (Multiequalizer.ι ((unop S).index P) I) =
  𝟙 (multiequalizer ((unop S).index P)) ≫ Multiequalizer.ι ((unop S).index P) I
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
I : (unop (op (unop S))).shape.L
⊢ CategoryStruct.comp (Y := multiequalizer ((unop S).index P))
      (CategoryStruct.comp (Y := multiequalizer (((unop S).pullback (𝟙 (unop X))).index P))
        (Multiequalizer.lift (((unop S).pullback (𝟙 (unop X))).index P) (multiequalizer ((unop S).index P))
          (fun (I : ((unop S).pullback (𝟙 (unop X))).Arrow) ↦ Multiequalizer.ι ((unop S).index P) I.base) ⋯)
        (Multiequalizer.lift ((unop S).index P) (multiequalizer (((unop S).pullback (𝟙 (unop X))).index P))
          (fun (I : (unop S).Arrow) ↦ Multiequalizer.ι (((unop S).pullback (𝟙 (unop X))).index P) (I.map e.inv)) ⋯))
      (Multiequalizer.ι ((unop S).index P) I) =
    𝟙 (multiequalizer ((unop S).index P)) ≫ Multiequalizer.ι ((unop S).index P) I
have h : Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op I.Y))
  (Multiequalizer.ι ((unop S).index P) (Cover.Arrow.map I e.inv).base) (Multiequalizer.ι ((unop S).index P) I)
Tactic: simp only [Multiequalizer.lift_ι, Category.id_comp, Category.assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
I : (unop (op (unop S))).shape.L
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op I.Y))
    (Multiequalizer.ι ((unop S).index P) (Cover.Arrow.map I e.inv).base) (Multiequalizer.ι ((unop S).index P) I)
have h : Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op I.Y))
  (Multiequalizer.ι ((unop S).index P) { Y := I.Y, f := I.f ≫ 𝟙 (unop X), hf := sorry })
  (Multiequalizer.ι ((unop S).index P) I)
Tactic: dsimp [Cover.Arrow.map, Cover.Arrow.base]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
I : (unop (op (unop S))).shape.L
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op I.Y))
    (Multiequalizer.ι ((unop S).index P) { Y := I.Y, f := I.f ≫ 𝟙 (unop X), hf := ⋯ })
    (Multiequalizer.ι ((unop S).index P) I)
have h : ∀ (Y : C) (f : Y ⟶ unop X) (hf : (↑(unop (op (unop S))) : Sieve (unop X)).arrows f),
  Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.Y))
    (Multiequalizer.ι ((unop S).index P)
      { Y := { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.Y,
        f := { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.f ≫ 𝟙 (unop X), hf := sorry })
    (Multiequalizer.ι ((unop S).index P) { Y := Y, f := f, hf := hf })
Tactic: cases I
Cleared hypotheses: I
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (𝟙 (unop X)) ≅ unop S := (unop S).pullbackId
e_1 :
  (multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X))) =
    (Functor.obj (J.diagram P (unop X)) S ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
Y : C
f : Y ⟶ unop X
hf : (↑(unop (op (unop S))) : Sieve (unop X)).arrows f
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ P.obj (op { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.Y))
    (Multiequalizer.ι ((unop S).index P)
      { Y := { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.Y,
        f := { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.f ≫ 𝟙 (unop X), hf := ⋯ })
    (Multiequalizer.ι ((unop S).index P) { Y := Y, f := f, hf := hf })
have h : { Y := Y, f := f, hf := hf : (unop (op (unop S))).Arrow }.f ≫ 𝟙 (unop X) = f
Tactic: congr