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 : 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 ⟶ ⊤ := homOfLE ⋯
⊢ Eq (α := Functor.obj (J.diagram P (unop X)) S ⟶ Functor.obj (J.diagram (J.plusObj P) (unop X)) S)
    (NatTrans.app
      (J.diagramNatTrans
        {
          app := fun (X : Cᵒᵖ) ↦
            CategoryStruct.comp (Y :=
              multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
              (Cover.toMultiequalizer ⊤ P) (colimit.ι (J.diagram P (unop X)) (op ⊤)),
          naturality := ⋯ }
        (unop X))
      S)
    (CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ (J.plusObj P)))
      (colimit.ι (J.diagram P (unop X)) S ≫ Cover.toMultiequalizer ⊤ (J.plusObj P))
      (Functor.map (J.diagram (J.plusObj P) (unop X)) e.op))
have w : ∀ (I : (unop S).shape.L),
  NatTrans.app
        (J.diagramNatTrans
          {
            app := fun (X : Cᵒᵖ) ↦
              CategoryStruct.comp (Y :=
                multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
                (Cover.toMultiequalizer ⊤ P) (colimit.ι (J.diagram P (unop X)) (op ⊤)),
            naturality := sorry }
          (unop X))
        S ≫
      Multiequalizer.ι ((unop S).index (J.plusObj P)) I =
    CategoryStruct.comp (Y := multiequalizer ((unop S).index (J.plusObj P)))
      (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ (J.plusObj P)))
        (colimit.ι (J.diagram P (unop X)) S ≫ Cover.toMultiequalizer ⊤ (J.plusObj P))
        (Functor.map (J.diagram (J.plusObj P) (unop X)) e.op))
      (Multiequalizer.ι ((unop S).index (J.plusObj 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 ⟶ ⊤ := homOfLE ⋯
I : (unop S).shape.L
⊢ NatTrans.app
        (J.diagramNatTrans
          {
            app := fun (X : Cᵒᵖ) ↦
              CategoryStruct.comp (Y :=
                multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
                (Cover.toMultiequalizer ⊤ P) (colimit.ι (J.diagram P (unop X)) (op ⊤)),
            naturality := ⋯ }
          (unop X))
        S ≫
      Multiequalizer.ι ((unop S).index (J.plusObj P)) I =
    CategoryStruct.comp (Y := multiequalizer ((unop S).index (J.plusObj P)))
      (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ (J.plusObj P)))
        (colimit.ι (J.diagram P (unop X)) S ≫ Cover.toMultiequalizer ⊤ (J.plusObj P))
        (Functor.map (J.diagram (J.plusObj P) (unop X)) e.op))
      (Multiequalizer.ι ((unop S).index (J.plusObj P)) I)
have w : Multiequalizer.ι ((unop S).index P) I ≫
    NatTrans.app
      {
        app := fun (X : Cᵒᵖ) ↦
          CategoryStruct.comp (Y :=
            multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
            (Cover.toMultiequalizer ⊤ P) (colimit.ι (J.diagram P (unop X)) (op ⊤)),
        naturality := sorry }
      (op I.Y) =
  CategoryStruct.comp (Y := multiequalizer ((unop S).index (J.plusObj P)))
    (CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ (J.plusObj P)))
      (colimit.ι (J.diagram P (unop X)) S ≫ Cover.toMultiequalizer ⊤ (J.plusObj P))
      (Functor.map (J.diagram (J.plusObj P) (unop X)) e.op))
    (Multiequalizer.ι ((unop S).index (J.plusObj P)) I)
Tactic: erw [Multiequalizer.lift_ι]
-- 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 ⟶ ⊤ := homOfLE ⋯
I : (unop S).shape.L
⊢ Multiequalizer.ι ((unop S).index P) I ≫
      NatTrans.app
        {
          app := fun (X : Cᵒᵖ) ↦
            CategoryStruct.comp (Y :=
              multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
              (Cover.toMultiequalizer ⊤ P) (colimit.ι (J.diagram P (unop X)) (op ⊤)),
          naturality := ⋯ }
        (op I.Y) =
    CategoryStruct.comp (Y := multiequalizer ((unop S).index (J.plusObj P)))
      (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ (J.plusObj P)))
        (colimit.ι (J.diagram P (unop X)) S ≫ Cover.toMultiequalizer ⊤ (J.plusObj P))
        (Functor.map (J.diagram (J.plusObj P) (unop X)) e.op))
      (Multiequalizer.ι ((unop S).index (J.plusObj P)) I)
have w : Multiequalizer.ι ((unop S).index P) I ≫
    CategoryStruct.comp (Y := multiequalizer (J := Cover.shape.{v, u} (C := C) (X := I.Y) (J := J) ⊤) (Cover.index ⊤ P))
      (Cover.toMultiequalizer ⊤ P) (colimit.ι (J.diagram P I.Y) (op ⊤)) =
  colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op
Tactic: simp only [unop_op, op_unop, diagram_map, Category.assoc, limit.lift_π,
    Multifork.ofι_π_app]
-- 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 ⟶ ⊤ := homOfLE ⋯
I : (unop S).shape.L
ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯
⊢ Multiequalizer.ι ((unop S).index P) I ≫
      CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := I.Y) (J := J) ⊤) (Cover.index ⊤ P))
        (Cover.toMultiequalizer ⊤ P) (colimit.ι (J.diagram P I.Y) (op ⊤)) =
    colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op
have w : Eq (α := Functor.obj (J.diagram P (unop X)) S ⟶ ((unop S).index (J.plusObj P)).left I)
  (CategoryStruct.comp (Y := multiequalizer (J := Cover.shape.{v, u} (C := C) (X := I.Y) (J := J) ⊤) (Cover.index ⊤ P))
      (Multiequalizer.ι ((unop S).index P) I ≫ Cover.toMultiequalizer ⊤ P) (Functor.map (J.diagram P I.Y) ee.op) ≫
    colimit.ι (J.diagram P I.Y) (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S))))
  (CategoryStruct.comp (Y :=
    ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op ⋙
          J.diagram P (unop (op (Cover.Arrow.map I e.op.unop).Y))).obj
      S)
    (Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)
      (Functor.obj (J.diagram P (unop X)) S)
      (fun (I_1 : (unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).shape.L) ↦
        Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1))
      sorry)
    (colimit.ι (J.diagram P (unop (op (Cover.Arrow.map I e.op.unop).Y)))
      ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)))
Tactic: erw [← colimit.w _ ee.op, ι_colimMap_assoc, colimit.ι_pre, diagramPullback_app,
    ← Category.assoc, ← 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 ⟶ ⊤ := homOfLE ⋯
I : (unop S).shape.L
ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯
⊢ Eq (α := Functor.obj (J.diagram P (unop X)) S ⟶ ((unop S).index (J.plusObj P)).left I)
    (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := I.Y) (J := J) ⊤) (Cover.index ⊤ P))
        (Multiequalizer.ι ((unop S).index P) I ≫ Cover.toMultiequalizer ⊤ P) (Functor.map (J.diagram P I.Y) ee.op) ≫
      colimit.ι (J.diagram P I.Y) (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S))))
    (CategoryStruct.comp (Y :=
      ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op ⋙
            J.diagram P (unop (op (Cover.Arrow.map I e.op.unop).Y))).obj
        S)
      (Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)
        (Functor.obj (J.diagram P (unop X)) S)
        (fun (I_1 : (unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).shape.L) ↦
          Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1))
        ⋯)
      (colimit.ι (J.diagram P (unop (op (Cover.Arrow.map I e.op.unop).Y)))
        ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)))
have w : Eq (α :=
  Functor.obj (J.diagram P (unop X)) S ⟶
    Functor.obj (J.diagram P I.Y) (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S))))
  (CategoryStruct.comp (Y := multiequalizer (J := Cover.shape.{v, u} (C := C) (X := I.Y) (J := J) ⊤) (Cover.index ⊤ P))
    (Multiequalizer.ι ((unop S).index P) I ≫ Cover.toMultiequalizer ⊤ P) (Functor.map (J.diagram P I.Y) ee.op))
  (Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)
    (Functor.obj (J.diagram P (unop X)) S)
    (fun (I_1 : (unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).shape.L) ↦
      Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1))
    sorry)
Tactic: congr 1
-- 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 ⟶ ⊤ := homOfLE ⋯
I : (unop S).shape.L
ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯
⊢ Eq (α :=
    Functor.obj (J.diagram P (unop X)) S ⟶
      Functor.obj (J.diagram P I.Y) (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S))))
    (CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := I.Y) (J := J) ⊤) (Cover.index ⊤ P))
      (Multiequalizer.ι ((unop S).index P) I ≫ Cover.toMultiequalizer ⊤ P) (Functor.map (J.diagram P I.Y) ee.op))
    (Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)
      (Functor.obj (J.diagram P (unop X)) S)
      (fun (I_1 : (unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).shape.L) ↦
        Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1))
      ⋯)
have w : ∀ (II : (unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).shape.L),
  Eq (α :=
    Functor.obj (J.diagram P (unop X)) S ⟶
      ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P).left II)
    (CategoryStruct.comp (Y := multiequalizer ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P))
      (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := I.Y) (J := J) ⊤) (Cover.index ⊤ P))
        (Multiequalizer.ι ((unop S).index P) I ≫ Cover.toMultiequalizer ⊤ P) (Functor.map (J.diagram P I.Y) ee.op))
      (Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II))
    (CategoryStruct.comp (Y := multiequalizer ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P))
      (Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)
        (Functor.obj (J.diagram P (unop X)) S)
        (fun (I_1 : (unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).shape.L) ↦
          Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1))
        sorry)
      (Multiequalizer.ι ((unop (op ((J.pullback (Cover.Arrow.map I e).f).obj (unop S)))).index P) II))
Tactic: refine Multiequalizer.hom_ext _ _ _ (fun II => ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (hypothesis)
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
hP : Presheaf.IsSheaf J P
⊢ IsIso (Y := J.plusObj P) (J.toPlus P)
have hP₁ : ∀ (X : C) (S : J.Cover X), IsIso (Y := multiequalizer (S.index P)) (S.toMultiequalizer P)
Tactic: rw [Presheaf.isSheaf_iff_multiequalizer] at hP
Cleared hypotheses: hP
-- 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
hP : ∀ (X : C) (S : J.Cover X), IsIso (Y := multiequalizer (S.index P)) (S.toMultiequalizer P)
⊢ IsIso (Y := J.plusObj P) (J.toPlus P)
have suffices : ∀ (X : Cᵒᵖ), IsIso (Y := (J.plusObj P).obj X) (NatTrans.app (J.toPlus P) X)
Tactic: suffices ∀ X, IsIso ((J.toPlus P).app X) from NatIso.isIso_of_isIso_app _
-- 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
hP : ∀ (X : C) (S : J.Cover X), IsIso (Y := multiequalizer (S.index P)) (S.toMultiequalizer P)
X : Cᵒᵖ
⊢ IsIso (Y := (J.plusObj P).obj X) (NatTrans.app (J.toPlus P) X)
have refine : IsIso (X := multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P)) (Y :=
  (J.plusObj P).obj X)
  (colimit.ι (J.diagram P (unop X)) (op ⊤) :
    Functor.obj (C := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)) (op ⊤) ⟶
      colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
Tactic: refine IsIso.comp_isIso' inferInstance ?_
-- 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
hP : ∀ (X : C) (S : J.Cover X), IsIso (Y := multiequalizer (S.index P)) (S.toMultiequalizer P)
X : Cᵒᵖ
⊢ IsIso (X := multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P)) (Y :=
    (J.plusObj P).obj X)
    (colimit.ι (J.diagram P (unop X)) (op ⊤) :
      Functor.obj (C := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)) (op ⊤) ⟶
        colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)))
have suffices : ∀ (S T : (J.Cover (unop X))ᵒᵖ) (f : S ⟶ T),
  IsIso (X := Functor.obj (J.diagram P (unop X)) S) (Functor.map (J.diagram P (unop X)) f)
Tactic: suffices ∀ (S T : (J.Cover X.unop)ᵒᵖ) (f : S ⟶ T), IsIso ((J.diagram P X.unop).map f) from
    isIso_ι_of_isInitial (initialOpOfTerminal isTerminalTop) _
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (hypothesis)
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
hP : ∀ (X : C) (S : J.Cover X), IsIso (Y := multiequalizer (S.index P)) (S.toMultiequalizer P)
X : Cᵒᵖ
S T : (J.Cover (unop X))ᵒᵖ
e : S ⟶ T
⊢ IsIso (X := Functor.obj (J.diagram P (unop X)) S) (Functor.map (J.diagram P (unop X)) e)
have this : Eq (α := P.obj (op (unop X)) ⟶ Functor.obj (J.diagram P (unop X)) T)
  (CategoryStruct.comp (Y := multiequalizer ((unop S).index P)) ((unop S).toMultiequalizer P)
    (Functor.map (J.diagram P (unop X)) e))
  ((unop T).toMultiequalizer P)
Tactic: have : S.unop.toMultiequalizer P ≫ (J.diagram P X.unop).map e = T.unop.toMultiequalizer P :=
    Multiequalizer.hom_ext _ _ _ (fun II => by simp)
-- 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)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
hQ : Presheaf.IsSheaf J Q
⊢ CategoryStruct.comp (Y := J.plusObj P) (J.toPlus P) (J.plusMap η ≫ Iso.inv (J.isoToPlus Q hQ)) = η
have rw : CategoryStruct.comp (Y := J.plusObj Q) (J.toPlus P ≫ J.plusMap η) (Iso.inv (J.isoToPlus Q hQ)) = η
Tactic: rw [← 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)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
hQ : Presheaf.IsSheaf J Q
⊢ CategoryStruct.comp (Y := J.plusObj Q) (J.toPlus P ≫ J.plusMap η) (Iso.inv (J.isoToPlus Q hQ)) = η
have rw : Eq (α := P ⟶ J.plusObj Q) (J.toPlus P ≫ J.plusMap η) (η ≫ Iso.hom (J.isoToPlus Q hQ))
Tactic: rw [Iso.comp_inv_eq]
-- 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)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
hQ : Presheaf.IsSheaf J Q
⊢ Eq (α := P ⟶ J.plusObj Q) (J.toPlus P ≫ J.plusMap η) (η ≫ J.toPlus Q)
have rw : Eq (α := P ⟶ J.plusObj Q) (J.toPlus P ≫ J.plusMap η) (J.toPlus P ≫ J.plusMap η)
Tactic: rw [toPlus_naturality]
-- 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)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
hQ : Presheaf.IsSheaf J Q
γ : J.plusObj P ⟶ Q
hγ : J.toPlus P ≫ γ = η
⊢ γ = J.plusMap η ≫ Iso.inv (J.isoToPlus Q hQ)
have rw : γ ≫ Iso.hom (J.isoToPlus Q hQ) = J.plusMap (J.toPlus P) ≫ J.plusMap γ
Tactic: rw [Iso.eq_comp_inv, ← hγ, plusMap_comp]
-- 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)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η γ : J.plusObj P ⟶ Q
hQ : Presheaf.IsSheaf J Q
h : J.toPlus P ≫ η = J.toPlus P ≫ γ
⊢ γ = J.plusLift (J.toPlus P ≫ γ) hQ
have hγ : J.toPlus P ≫ γ = J.toPlus P ≫ γ
Tactic: apply plusLift_unique
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (hypothesis)
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)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η γ : J.plusObj P ⟶ Q
hQ : Presheaf.IsSheaf J Q
h : J.toPlus P ≫ η = J.toPlus P ≫ γ
⊢ η = γ
have this : γ = J.plusLift (J.toPlus P ≫ γ) hQ
Tactic: have : γ = J.plusLift (J.toPlus P ≫ γ) hQ := by
    apply plusLift_unique
    rfl
-- 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)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Functor.{v, w', u, w} Cᵒᵖ D
η γ : J.plusObj P ⟶ Q
hQ : Presheaf.IsSheaf J Q
h : J.toPlus P ≫ η = J.toPlus P ≫ γ
this : γ = J.plusLift (J.toPlus P ≫ γ) hQ
⊢ η = γ
have rw : η = J.plusLift (J.toPlus P ≫ γ) hQ
Tactic: rw [this]
-- 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
hP : Presheaf.IsSheaf J P
⊢ Eq (α := J.plusObj P ⟶ P) (Iso.inv (J.isoToPlus P hP)) (J.plusLift (𝟙 P) hP)
have hγ : CategoryStruct.comp (Y := J.plusObj P) (J.toPlus P) (Iso.inv (J.isoToPlus P hP)) = 𝟙 P
Tactic: apply J.plusLift_unique
-- 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
hP : Presheaf.IsSheaf J P
⊢ CategoryStruct.comp (Y := J.plusObj P) (J.toPlus P) (Iso.inv (J.isoToPlus P hP)) = 𝟙 P
have hγ : Eq (α := P ⟶ J.plusObj P) (J.toPlus P) (Iso.hom (J.isoToPlus P hP))
Tactic: rw [Iso.comp_inv_eq, Category.id_comp]
-- 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)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q R : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
γ : Q ⟶ R
hR : Presheaf.IsSheaf J R
⊢ Eq (α := J.plusObj P ⟶ R) (J.plusMap η ≫ J.plusLift γ hR) (J.plusLift (η ≫ γ) hR)
have hγ : CategoryStruct.comp (Y := J.plusObj P) (J.toPlus P) (J.plusMap η ≫ J.plusLift γ hR) = η ≫ γ
Tactic: apply J.plusLift_unique
-- 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)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q R : Functor.{v, w', u, w} Cᵒᵖ D
η : P ⟶ Q
γ : Q ⟶ R
hR : Presheaf.IsSheaf J R
⊢ CategoryStruct.comp (Y := J.plusObj P) (J.toPlus P) (J.plusMap η ≫ J.plusLift γ hR) = η ≫ γ
have hγ : CategoryStruct.comp (Y := J.plusObj Q) (J.toPlus P ≫ J.plusMap η) (J.plusLift γ hR) = η ≫ γ
Tactic: rw [← Category.assoc, ← J.toPlus_naturality, Category.assoc, J.toPlus_plusLift]