awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
FD : D → D → Type u_1
CD : D → Type t
inst_2 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory.{t, w', w, u_1} D FD (CC := CD)
inst_3 : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D (FC := FD) (CC := CD))
inst_4 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_5 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : Functor.ReflectsIsomorphisms.{w', w, t, t + 1} (forget D (FC := FD) (CC := CD))
P : Functor.{v, w', u, w} Cᵒᵖ D
hsep :
  ∀ (X : C) (S : J.Cover X) (x y : ToType.{t, w', w, u_1} (FC := FD) (CC := CD) (P.obj (op X))),
    (∀ (I : S.Arrow),
        Eq (α := CD (P.obj (op I.Y)))
          ((ConcreteCategory.hom (FC := FD) (CC := CD) (P.map I.f.op) : (a : CD (P.obj (op X))) → CD (P.obj (op I.Y)))
              x :
            CD (P.obj (op I.Y)))
          ((ConcreteCategory.hom (FC := FD) (CC := CD) (P.map I.f.op) : (a : CD (P.obj (op X))) → CD (P.obj (op I.Y)))
              y :
            CD (P.obj (op I.Y)))) →
      x = y
X : C
S : J.Cover X
x :
  ToType.{t, w', w, u_1} (FC := FD) (CC := Functor.obj.{w', t, w, t + 1} (forget D (FC := FD) (CC := CD)))
    (multiequalizer (S.index (J.plusObj P)))
t : ToType.{t, w', w, u_1} (FC := FD) (CC := CD) ((J.plusObj P).obj (op X))
ht :
  Meq.mk (FD := FD) (CD := CD) S t =
    (Meq.equiv (FD := FD) (CD := CD) (J.plusObj P) S :
        (a : ToType (multiequalizer (S.index (J.plusObj P)))) → Meq (J.plusObj P) S)
      x
⊢ Eq (α := Meq (FD := FD) (CD := CD) (J.plusObj P) S)
    ((Meq.equiv (FD := FD) (CD := CD) (J.plusObj P) S :
          (a : ToType (multiequalizer (S.index (J.plusObj P)))) → Meq (J.plusObj P) S)
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (Functor.map (forget D (FC := FD) (CC := CD)) (S.toMultiequalizer (J.plusObj P))) :
            (a : (fun (X : Type t) ↦ X) ((forget D).obj ((J.plusObj P).obj (op X)))) →
              (fun (X : Type t) ↦ X) ((forget D).obj (multiequalizer (S.index (J.plusObj P)))))
          t) :
      Meq (FD := FD) (CD := CD) (J.plusObj P) S)
    ((Meq.equiv (FD := FD) (CD := CD) (J.plusObj P) S :
          (a : ToType (multiequalizer (S.index (J.plusObj P)))) → Meq (J.plusObj P) S)
        x :
      Meq (FD := FD) (CD := CD) (J.plusObj P) S)
have h : (Meq.equiv (FD := FD) (CD := CD) (J.plusObj P) S :
      (a : ToType (multiequalizer (S.index (J.plusObj P)))) → Meq (J.plusObj P) S)
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (Functor.map (forget D (FC := FD) (CC := CD)) (S.toMultiequalizer (J.plusObj P))) :
        (a : (fun (X : Type t) ↦ X) ((forget D).obj ((J.plusObj P).obj (op X)))) →
          (fun (X : Type t) ↦ X) ((forget D).obj (multiequalizer (S.index (J.plusObj P)))))
      t) =
  Meq.mk (FD := FD) (CD := CD) S t
Tactic: rw [← ht]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
FD : D → D → Type u_1
CD : D → Type t
inst_2 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory.{t, w', w, u_1} D FD (CC := CD)
inst_3 : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D (FC := FD) (CC := CD))
inst_4 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_5 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : Functor.ReflectsIsomorphisms.{w', w, t, t + 1} (forget D (FC := FD) (CC := CD))
P : Functor.{v, w', u, w} Cᵒᵖ D
hsep :
  ∀ (X : C) (S : J.Cover X) (x y : ToType.{t, w', w, u_1} (FC := FD) (CC := CD) (P.obj (op X))),
    (∀ (I : S.Arrow),
        Eq (α := CD (P.obj (op I.Y)))
          ((ConcreteCategory.hom (FC := FD) (CC := CD) (P.map I.f.op) : (a : CD (P.obj (op X))) → CD (P.obj (op I.Y)))
              x :
            CD (P.obj (op I.Y)))
          ((ConcreteCategory.hom (FC := FD) (CC := CD) (P.map I.f.op) : (a : CD (P.obj (op X))) → CD (P.obj (op I.Y)))
              y :
            CD (P.obj (op I.Y)))) →
      x = y
X : C
S : J.Cover X
x :
  ToType.{t, w', w, u_1} (FC := FD) (CC := Functor.obj.{w', t, w, t + 1} (forget D (FC := FD) (CC := CD)))
    (multiequalizer (S.index (J.plusObj P)))
t : ToType.{t, w', w, u_1} (FC := FD) (CC := CD) ((J.plusObj P).obj (op X))
ht :
  Meq.mk (FD := FD) (CD := CD) S t =
    (Meq.equiv (FD := FD) (CD := CD) (J.plusObj P) S :
        (a : ToType (multiequalizer (S.index (J.plusObj P)))) → Meq (J.plusObj P) S)
      x
⊢ (Meq.equiv (FD := FD) (CD := CD) (J.plusObj P) S :
        (a : ToType (multiequalizer (S.index (J.plusObj P)))) → Meq (J.plusObj P) S)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (Functor.map (forget D (FC := FD) (CC := CD)) (S.toMultiequalizer (J.plusObj P))) :
          (a : (fun (X : Type t) ↦ X) ((forget D).obj ((J.plusObj P).obj (op X)))) →
            (fun (X : Type t) ↦ X) ((forget D).obj (multiequalizer (S.index (J.plusObj P)))))
        t) =
    Meq.mk (FD := FD) (CD := CD) S t
have h : ∀ (i : S.Arrow),
  (↑((Meq.equiv (FD := FD) (CD := CD) (J.plusObj P) S :
              (a : ToType (multiequalizer (S.index (J.plusObj P)))) → Meq (J.plusObj P) S)
            ((ConcreteCategory.hom (FC := TypeCat.Fun)
                  (Functor.map (forget D (FC := FD) (CC := CD)) (S.toMultiequalizer (J.plusObj P))) :
                (a : (fun (X : Type t) ↦ X) ((forget D).obj ((J.plusObj P).obj (op X)))) →
                  (fun (X : Type t) ↦ X) ((forget D).obj (multiequalizer (S.index (J.plusObj P)))))
              t)) :
        (I : S.Arrow) → ToType ((J.plusObj P).obj (op I.Y)))
      i =
    (↑(Meq.mk (FD := FD) (CD := CD) S t) : (I : S.Arrow) → ToType ((J.plusObj P).obj (op I.Y))) i
Tactic: ext i
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
FD : D → D → Type u_1
CD : D → Type t
inst_2 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory.{t, w', w, u_1} D FD (CC := CD)
inst_3 : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D (FC := FD) (CC := CD))
inst_4 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_5 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : Functor.ReflectsIsomorphisms.{w', w, t, t + 1} (forget D (FC := FD) (CC := CD))
P : Functor.{v, w', u, w} Cᵒᵖ D
hsep :
  ∀ (X : C) (S : J.Cover X) (x y : ToType.{t, w', w, u_1} (FC := FD) (CC := CD) (P.obj (op X))),
    (∀ (I : S.Arrow),
        Eq (α := CD (P.obj (op I.Y)))
          ((ConcreteCategory.hom (FC := FD) (CC := CD) (P.map I.f.op) : (a : CD (P.obj (op X))) → CD (P.obj (op I.Y)))
              x :
            CD (P.obj (op I.Y)))
          ((ConcreteCategory.hom (FC := FD) (CC := CD) (P.map I.f.op) : (a : CD (P.obj (op X))) → CD (P.obj (op I.Y)))
              y :
            CD (P.obj (op I.Y)))) →
      x = y
X : C
S : J.Cover X
x :
  ToType.{t, w', w, u_1} (FC := FD) (CC := Functor.obj.{w', t, w, t + 1} (forget D (FC := FD) (CC := CD)))
    (multiequalizer (S.index (J.plusObj P)))
t : ToType.{t, w', w, u_1} (FC := FD) (CC := CD) ((J.plusObj P).obj (op X))
ht :
  Meq.mk (FD := FD) (CD := CD) S t =
    (Meq.equiv (FD := FD) (CD := CD) (J.plusObj P) S :
        (a : ToType (multiequalizer (S.index (J.plusObj P)))) → Meq (J.plusObj P) S)
      x
i : S.Arrow
⊢ (↑((Meq.equiv (FD := FD) (CD := CD) (J.plusObj P) S :
              (a : ToType (multiequalizer (S.index (J.plusObj P)))) → Meq (J.plusObj P) S)
            ((ConcreteCategory.hom (FC := TypeCat.Fun)
                  (Functor.map (forget D (FC := FD) (CC := CD)) (S.toMultiequalizer (J.plusObj P))) :
                (a : (fun (X : Type t) ↦ X) ((forget D).obj ((J.plusObj P).obj (op X)))) →
                  (fun (X : Type t) ↦ X) ((forget D).obj (multiequalizer (S.index (J.plusObj P)))))
              t)) :
        (I : S.Arrow) → ToType ((J.plusObj P).obj (op I.Y)))
      i =
    (↑(Meq.mk (FD := FD) (CD := CD) S t) : (I : S.Arrow) → ToType ((J.plusObj P).obj (op I.Y))) i
have h : (ConcreteCategory.hom (FC := FD) (CC := CD) (Multiequalizer.ι (S.index (J.plusObj P)) i) :
      (a : CD (multiequalizer (S.index (J.plusObj P)))) → CD ((J.plusObj P).obj (op i.Y)))
    ((ConcreteCategory.hom (FC := FD) (CC := CD) (S.toMultiequalizer (J.plusObj P)) :
          (a : CD ((J.plusObj P).obj (op X))) → CD (multiequalizer (S.index (J.plusObj P))))
        t :
      CD (multiequalizer (S.index (J.plusObj P)))) =
  (↑(Meq.mk (FD := FD) (CD := CD) S t) : (I : S.Arrow) → ToType ((J.plusObj P).obj (op I.Y))) i
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
FD : D → D → Type u_1
CD : D → Type t
inst_2 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory.{t, w', w, u_1} D FD (CC := CD)
inst_3 : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D (FC := FD) (CC := CD))
inst_4 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_5 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : Functor.ReflectsIsomorphisms.{w', w, t, t + 1} (forget D (FC := FD) (CC := CD))
P : Functor.{v, w', u, w} Cᵒᵖ D
hsep :
  ∀ (X : C) (S : J.Cover X) (x y : ToType.{t, w', w, u_1} (FC := FD) (CC := CD) (P.obj (op X))),
    (∀ (I : S.Arrow),
        Eq (α := CD (P.obj (op I.Y)))
          ((ConcreteCategory.hom (FC := FD) (CC := CD) (P.map I.f.op) : (a : CD (P.obj (op X))) → CD (P.obj (op I.Y)))
              x :
            CD (P.obj (op I.Y)))
          ((ConcreteCategory.hom (FC := FD) (CC := CD) (P.map I.f.op) : (a : CD (P.obj (op X))) → CD (P.obj (op I.Y)))
              y :
            CD (P.obj (op I.Y)))) →
      x = y
X : C
S : J.Cover X
x :
  ToType.{t, w', w, u_1} (FC := FD) (CC := Functor.obj.{w', t, w, t + 1} (forget D (FC := FD) (CC := CD)))
    (multiequalizer (S.index (J.plusObj P)))
t : ToType.{t, w', w, u_1} (FC := FD) (CC := CD) ((J.plusObj P).obj (op X))
ht :
  Meq.mk (FD := FD) (CD := CD) S t =
    (Meq.equiv (FD := FD) (CD := CD) (J.plusObj P) S :
        (a : ToType (multiequalizer (S.index (J.plusObj P)))) → Meq (J.plusObj P) S)
      x
i : S.Arrow
⊢ (ConcreteCategory.hom (FC := FD) (CC := CD) (Multiequalizer.ι (S.index (J.plusObj P)) i) :
        (a : CD (multiequalizer (S.index (J.plusObj P)))) → CD ((J.plusObj P).obj (op i.Y)))
      ((ConcreteCategory.hom (FC := FD) (CC := CD) (S.toMultiequalizer (J.plusObj P)) :
            (a : CD ((J.plusObj P).obj (op X))) → CD (multiequalizer (S.index (J.plusObj P))))
          t :
        CD (multiequalizer (S.index (J.plusObj P)))) =
    (↑(Meq.mk (FD := FD) (CD := CD) S t) : (I : S.Arrow) → ToType ((J.plusObj P).obj (op I.Y))) i
have h : (ConcreteCategory.hom (FC := FD) (CC := CD) ((J.plusObj P).map i.f.op) :
      (a : CD ((J.plusObj P).obj (op X))) → CD ((J.plusObj P).obj (op i.Y)))
    t =
  (↑(Meq.mk (FD := FD) (CD := CD) S t) : (I : S.Arrow) → ToType ((J.plusObj P).obj (op I.Y))) i
Tactic: rw [← ConcreteCategory.comp_apply, Multiequalizer.lift_ι]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
FD : D → D → Type u_1
CD : D → Type t
inst_2 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory.{t, w', w, u_1} D FD (CC := CD)
inst_3 : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D (FC := FD) (CC := CD))
inst_4 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_5 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : Functor.ReflectsIsomorphisms.{w', w, t, t + 1} (forget D (FC := FD) (CC := CD))
P : Functor.{v, w', u, w} Cᵒᵖ D
⊢ Presheaf.IsSheaf J (J.plusObj (J.plusObj P))
have hsep : ∀ (X : C) (S : J.Cover X) (x y : ToType.{t, w', w, u_1} (FC := FD) (CC := CD) ((J.plusObj P).obj (op X))),
  (∀ (I : S.Arrow),
      Eq (α := CD ((J.plusObj P).obj (op I.Y)))
        ((ConcreteCategory.hom (FC := FD) (CC := CD) ((J.plusObj P).map I.f.op) :
              (a : CD ((J.plusObj P).obj (op X))) → CD ((J.plusObj P).obj (op I.Y)))
            x :
          CD ((J.plusObj P).obj (op I.Y)))
        ((ConcreteCategory.hom (FC := FD) (CC := CD) ((J.plusObj P).map I.f.op) :
              (a : CD ((J.plusObj P).obj (op X))) → CD ((J.plusObj P).obj (op I.Y)))
            y :
          CD ((J.plusObj P).obj (op I.Y)))) →
    x = y
Tactic: apply isSheaf_of_sep
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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 : Functor.{v, w', u, w} Cᵒᵖ D
hP : Presheaf.IsSheaf J P
⊢ IsIso (Y := J.sheafify P) (J.toPlus P ≫ J.plusMap (J.toPlus P))
have this : IsIso (Y := J.plusObj P) (J.toPlus P)
Tactic: haveI := isIso_toPlus_of_isSheaf J P hP
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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.sheafify P ⟶ Q
h : J.toSheafify P ≫ γ = η
⊢ γ = J.sheafifyLift η hQ
have hγ : J.toPlus (J.plusObj P) ≫ γ = J.plusLift η hQ
Tactic: apply plusLift_unique
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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.sheafify P ⟶ Q
h : J.toSheafify P ≫ γ = η
⊢ J.toPlus (J.plusObj P) ≫ γ = J.plusLift η hQ
have hγ : J.toPlus P ≫ J.toPlus (J.plusObj P) ≫ γ = η
Tactic: apply plusLift_unique
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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.sheafify P ⟶ Q
h : J.toSheafify P ≫ γ = η
⊢ J.toPlus P ≫ J.toPlus (J.plusObj P) ≫ γ = η
have hγ : (J.toPlus P ≫ J.plusMap (J.toPlus P)) ≫ γ = η
Tactic: rw [← Category.assoc, ← plusMap_toPlus]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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 : Functor.{v, w', u, w} Cᵒᵖ D
hP : Presheaf.IsSheaf J P
⊢ Eq (α := J.sheafify P ⟶ P) (Iso.inv (J.isoSheafify hP)) (J.sheafifyLift (𝟙 P) hP)
have a : CategoryStruct.comp (Y := J.sheafify P) (J.toSheafify P) (Iso.inv (J.isoSheafify hP)) = 𝟙 P
Tactic: apply J.sheafifyLift_unique
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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.sheafify P ⟶ Q
hQ : Presheaf.IsSheaf J Q
h : J.toSheafify P ≫ η = J.toSheafify P ≫ γ
⊢ η = γ
have apply : J.toPlus (J.plusObj P) ≫ η = J.toPlus (J.plusObj P) ≫ γ
Tactic: apply J.plus_hom_ext _ _ hQ
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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.sheafify P ⟶ Q
hQ : Presheaf.IsSheaf J Q
h : J.toSheafify P ≫ η = J.toSheafify P ≫ γ
⊢ J.toPlus (J.plusObj P) ≫ η = J.toPlus (J.plusObj P) ≫ γ
have apply : J.toPlus P ≫ J.toPlus (J.plusObj P) ≫ η = J.toPlus P ≫ J.toPlus (J.plusObj P) ≫ γ
Tactic: apply J.plus_hom_ext _ _ hQ
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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.sheafify P ⟶ Q
hQ : Presheaf.IsSheaf J Q
h : J.toSheafify P ≫ η = J.toSheafify P ≫ γ
⊢ J.toPlus P ≫ J.toPlus (J.plusObj P) ≫ η = J.toPlus P ≫ J.toPlus (J.plusObj P) ≫ γ
have rw : (J.toPlus P ≫ J.plusMap (J.toPlus P)) ≫ η = (J.toPlus P ≫ J.plusMap (J.toPlus P)) ≫ γ
Tactic: rw [← Category.assoc, ← Category.assoc, ← plusMap_toPlus]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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.sheafify P ⟶ R) (J.sheafifyMap η ≫ J.sheafifyLift γ hR) (J.sheafifyLift (η ≫ γ) hR)
have a : CategoryStruct.comp (Y := J.sheafify P) (J.toSheafify P) (J.sheafifyMap η ≫ J.sheafifyLift γ hR) = η ≫ γ
Tactic: apply J.sheafifyLift_unique
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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.sheafify P) (J.toSheafify P) (J.sheafifyMap η ≫ J.sheafifyLift γ hR) = η ≫ γ
have a : CategoryStruct.comp (Y := J.sheafify Q) (J.toSheafify P ≫ J.sheafifyMap η) (J.sheafifyLift γ hR) = η ≫ γ
Tactic: rw [← Category.assoc, ← J.toSheafify_naturality, Category.assoc, toSheafify_sheafifyLift]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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
a : CategoryStruct.comp (Y := J.sheafify Q) (J.toSheafify P ≫ J.sheafifyMap η) (J.sheafifyLift γ hR) = η ≫ γ
⊢ CategoryStruct.comp (Y := J.sheafify P) (J.toSheafify P) (J.sheafifyMap η ≫ J.sheafifyLift γ hR) = η ≫ γ
have a₁ : CategoryStruct.comp (Y := J.sheafify Q) (η ≫ J.toSheafify Q) (J.sheafifyLift γ hR) = η ≫ γ
Tactic: rw [← Category.assoc, ← J.toSheafify_naturality, Category.assoc, toSheafify_sheafifyLift]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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
a : CategoryStruct.comp (Y := J.sheafify Q) (J.toSheafify P ≫ J.sheafifyMap η) (J.sheafifyLift γ hR) = η ≫ γ
a₁ : CategoryStruct.comp (Y := J.sheafify Q) (η ≫ J.toSheafify Q) (J.sheafifyLift γ hR) = η ≫ γ
⊢ CategoryStruct.comp (Y := J.sheafify P) (J.toSheafify P) (J.sheafifyMap η ≫ J.sheafifyLift γ hR) = η ≫ γ
have a₂ : η ≫ CategoryStruct.comp (Y := J.sheafify Q) (J.toSheafify Q) (J.sheafifyLift γ hR) = η ≫ γ
Tactic: rw [← Category.assoc, ← J.toSheafify_naturality, Category.assoc, toSheafify_sheafifyLift]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (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
a : CategoryStruct.comp (Y := J.sheafify Q) (J.toSheafify P ≫ J.sheafifyMap η) (J.sheafifyLift γ hR) = η ≫ γ
a₁ : CategoryStruct.comp (Y := J.sheafify Q) (η ≫ J.toSheafify Q) (J.sheafifyLift γ hR) = η ≫ γ
a₂ : η ≫ CategoryStruct.comp (Y := J.sheafify Q) (J.toSheafify Q) (J.sheafifyLift γ hR) = η ≫ γ
⊢ CategoryStruct.comp (Y := J.sheafify P) (J.toSheafify P) (J.sheafifyMap η ≫ J.sheafifyLift γ hR) = η ≫ γ
have a₃ : η ≫ γ = η ≫ γ
Tactic: rw [← Category.assoc, ← J.toSheafify_naturality, Category.assoc, toSheafify_sheafifyLift]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.ConcreteSheafification (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
FD : D → D → Type u_1
CD : D → Type t
inst_2 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory.{t, w', w, u_1} D FD (CC := CD)
inst_3 : ∀ {X : C} (S : J.Cover X), PreservesLimitsOfShape (WalkingMulticospan S.shape) (forget D (FC := FD) (CC := CD))
inst_4 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_5 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : Functor.ReflectsIsomorphisms.{w', w, t, t + 1} (forget D (FC := FD) (CC := CD))
inst_8 : Preadditive.{w', w} D
F G : Functor.{v, w', u, w} Cᵒᵖ D
⊢ Eq (α :=
    Functor.obj (plusPlusSheaf J D (FD := FD) (CD := CD)) F ⟶ Functor.obj (plusPlusSheaf J D (FD := FD) (CD := CD)) G)
    (Functor.map (plusPlusSheaf J D (FD := FD) (CD := CD)) 0) 0
have h : ∀ (x : Cᵒᵖ),
  Eq.{w' + 1} (α :=
    (Functor.obj (plusPlusSheaf J D (FD := FD) (CD := CD)) F).obj (P := Presheaf.IsSheaf J).obj x ⟶
      (Functor.obj (plusPlusSheaf J D (FD := FD) (CD := CD)) G).obj (P := Presheaf.IsSheaf J).obj x)
    (NatTrans.app (InducedCategory.Hom.hom (Functor.map (plusPlusSheaf J D (FD := FD) (CD := CD)) 0)) x)
    (NatTrans.app (InducedCategory.Hom.hom 0) x)
Tactic: ext : 3