awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Shift.Localization
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
L : Functor.{v₁, v₂, u₁, u₂} C D
W : MorphismProperty.{v₁, u₁} C
inst_3 : L.IsLocalization W
A : Type w
inst_4 : AddMonoid A
inst_5 : HasShift.{v₁, u₁, w} C A
F : Functor.{v₁, v₃, u₁, u₃} C E
F' : Functor.{v₂, v₃, u₂, u₃} D E
inst_6 : Lifting L W F F'
inst_7 : HasShift.{v₂, u₂, w} D A
inst_8 : HasShift.{v₃, u₃, w} E A
inst_9 : L.CommShift A
inst_10 : F.CommShift A
a b : A
X : C
erw :
  CategoryStruct.comp (Y :=
    F'.obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C b) (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))))
    (F'.map (NatTrans.app (Iso.inv (commShiftIso L b)) (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
    ((Lifting.iso L W F F').hom.app
        (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C b) (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
      CategoryStruct.comp (Y :=
        obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (F.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
        (NatTrans.app (Iso.hom (commShiftIso F b)) (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))
        (CategoryStruct.comp (Y :=
          obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
          (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
          (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
            NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X)))))
| F'.map (NatTrans.app (Iso.inv (commShiftIso L b)) (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
    F'.map (L.map (NatTrans.app (Iso.inv (shiftFunctorAdd C a b)) X)) ≫
      (Lifting.iso L W F F').hom.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C (a + b)) X) ≫
        F.map (NatTrans.app (Iso.hom (shiftFunctorAdd C a b)) X) ≫
          NatTrans.app (Iso.hom (commShiftIso F b)) (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X) ≫
            map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X) ≫
              NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F.obj X) ≫
                map (shiftFunctor E (a + b)) ((Lifting.iso L W F F').inv.app X)

Tactic:
erw [← NatTrans.naturality_assoc]
      dsimp
      rw [← Functor.map_comp_assoc, ← map_comp_assoc, Category.assoc,
        ← map_comp, Iso.inv_hom_id_app]
      dsimp
      rw [map_id, Category.comp_id, ← NatTrans.naturality]
      dsimp

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Category.comp_id, CategoryTheory.Functor.map_comp_assoc, CategoryTheory.Functor.map_id, CategoryTheory.NatTrans.naturality, CategoryTheory.Functor.map_comp, CategoryTheory.Iso.inv_hom_id_app, CategoryTheory.NatTrans.naturality_assoc
-- Module: Mathlib.CategoryTheory.Shift.Localization
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
L : Functor.{v₁, v₂, u₁, u₂} C D
W : MorphismProperty.{v₁, u₁} C
inst_3 : L.IsLocalization W
A : Type w
inst_4 : AddMonoid A
inst_5 : HasShift.{v₁, u₁, w} C A
F : Functor.{v₁, v₃, u₁, u₃} C E
F' : Functor.{v₂, v₃, u₂, u₃} D E
inst_6 : Lifting L W F F'
inst_7 : HasShift.{v₂, u₂, w} D A
inst_8 : HasShift.{v₃, u₃, w} E A
inst_9 : L.CommShift A
inst_10 : F.CommShift A
a b : A
X : C
erw :
  CategoryStruct.comp (Y := F'.obj (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D b) ((L ⋙ shiftFunctor D a).obj X)))
    (F'.map (map (shiftFunctor D b) (NatTrans.app (Iso.hom (commShiftIso L a)) X)))
    (CategoryStruct.comp (Y := (shiftFunctor D b ⋙ F').obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
      ((shiftFunctor D b ⋙ F').map (NatTrans.app (Iso.inv (commShiftIso L a)) X))
      (CategoryStruct.comp (Y := (F' ⋙ shiftFunctor E b).obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
        (NatTrans.app (Iso.hom (commShiftOfLocalization.iso L W F F' b))
          (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
        (map (shiftFunctor E b) ((Lifting.iso L W F F').hom.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
          CategoryStruct.comp (Y :=
            obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
            (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
            (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
              NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X))))))
| CategoryStruct.comp (Y := F'.obj (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D b) ((L ⋙ shiftFunctor D a).obj X)))
    (F'.map (map (shiftFunctor D b) (NatTrans.app (Iso.hom (commShiftIso L a)) X)))
    (CategoryStruct.comp (Y :=
      obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (F'.obj (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D a) (L.obj X))))
      (NatTrans.app (Iso.hom (commShiftOfLocalization.iso L W F F' b))
        (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D a) (L.obj X)))
      (CategoryStruct.comp (Y :=
        obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (F'.obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))))
        (map (shiftFunctor E b) (F'.map (NatTrans.app (Iso.inv (commShiftIso L a)) X)))
        (map (shiftFunctor E b) ((Lifting.iso L W F F').hom.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
          CategoryStruct.comp (Y :=
            obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
            (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
            (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
              NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X))))))

Tactic:
erw [← NatTrans.naturality_assoc]

Premises used: CategoryTheory.NatTrans.naturality_assoc
-- Module: Mathlib.CategoryTheory.Shift.Localization
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
L : Functor.{v₁, v₂, u₁, u₂} C D
W : MorphismProperty.{v₁, u₁} C
inst_3 : L.IsLocalization W
A : Type w
inst_4 : AddMonoid A
inst_5 : HasShift.{v₁, u₁, w} C A
F : Functor.{v₁, v₃, u₁, u₃} C E
F' : Functor.{v₂, v₃, u₂, u₃} D E
inst_6 : Lifting L W F F'
inst_7 : HasShift.{v₂, u₂, w} D A
inst_8 : HasShift.{v₃, u₃, w} E A
inst_9 : L.CommShift A
inst_10 : F.CommShift A
a b : A
X : C
dsimp :
  CategoryStruct.comp (Y :=
    F'.obj (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D b) (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D a) (L.obj X))))
    (F'.map (map (shiftFunctor D b) (NatTrans.app (Iso.hom (commShiftIso L a)) X)))
    (CategoryStruct.comp (Y :=
      F'.obj (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D b) (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))))
      (F'.map (map (shiftFunctor D b) (NatTrans.app (Iso.inv (commShiftIso L a)) X)))
      (CategoryStruct.comp (Y :=
        obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (F'.obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))))
        (NatTrans.app (Iso.hom (commShiftOfLocalization.iso L W F F' b))
          (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
        (map (shiftFunctor E b) ((Lifting.iso L W F F').hom.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
          CategoryStruct.comp (Y :=
            obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
            (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
            (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
              NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X))))))
| CategoryStruct.comp (Y := F'.obj (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D b) ((L ⋙ shiftFunctor D a).obj X)))
    (F'.map (map (shiftFunctor D b) (NatTrans.app (Iso.hom (commShiftIso L a)) X)))
    (CategoryStruct.comp (Y := (shiftFunctor D b ⋙ F').obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
      ((shiftFunctor D b ⋙ F').map (NatTrans.app (Iso.inv (commShiftIso L a)) X))
      (CategoryStruct.comp (Y := (F' ⋙ shiftFunctor E b).obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
        (NatTrans.app (Iso.hom (commShiftOfLocalization.iso L W F F' b))
          (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
        (map (shiftFunctor E b) ((Lifting.iso L W F F').hom.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
          CategoryStruct.comp (Y :=
            obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
            (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
            (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
              NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X))))))

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Shift.Localization
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
L : Functor.{v₁, v₂, u₁, u₂} C D
W : MorphismProperty.{v₁, u₁} C
inst_3 : L.IsLocalization W
A : Type w
inst_4 : AddMonoid A
inst_5 : HasShift.{v₁, u₁, w} C A
F : Functor.{v₁, v₃, u₁, u₃} C E
F' : Functor.{v₂, v₃, u₂, u₃} D E
inst_6 : Lifting L W F F'
inst_7 : HasShift.{v₂, u₂, w} D A
inst_8 : HasShift.{v₃, u₃, w} E A
inst_9 : L.CommShift A
inst_10 : F.CommShift A
a b : A
X : C
rw :
  F'.map (map (shiftFunctor D b) (𝟙 ((shiftFunctor C a ⋙ L).obj X))) ≫
    CategoryStruct.comp (Y :=
      obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (F'.obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))))
      (NatTrans.app (Iso.hom (commShiftOfLocalization.iso L W F F' b))
        (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
      (map (shiftFunctor E b) ((Lifting.iso L W F F').hom.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
        CategoryStruct.comp (Y :=
          obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
          (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
          (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
            NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X))))
| CategoryStruct.comp (Y :=
    F'.obj (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D b) (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D a) (L.obj X))))
    (F'.map (map (shiftFunctor D b) (NatTrans.app (Iso.hom (commShiftIso L a)) X)))
    (CategoryStruct.comp (Y :=
      F'.obj (obj.{v₂, v₂, u₂, u₂} (shiftFunctor D b) (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))))
      (F'.map (map (shiftFunctor D b) (NatTrans.app (Iso.inv (commShiftIso L a)) X)))
      (CategoryStruct.comp (Y :=
        obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (F'.obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))))
        (NatTrans.app (Iso.hom (commShiftOfLocalization.iso L W F F' b))
          (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
        (map (shiftFunctor E b) ((Lifting.iso L W F F').hom.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
          CategoryStruct.comp (Y :=
            obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
            (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
            (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
              NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X))))))

Tactic:
rw [← Functor.map_comp_assoc, ← map_comp, Iso.hom_inv_id_app]

Premises used: CategoryTheory.Iso.hom_inv_id_app, CategoryTheory.Functor.map_comp_assoc, CategoryTheory.Functor.map_comp
-- Module: Mathlib.CategoryTheory.Shift.Localization
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
L : Functor.{v₁, v₂, u₁, u₂} C D
W : MorphismProperty.{v₁, u₁} C
inst_3 : L.IsLocalization W
A : Type w
inst_4 : AddMonoid A
inst_5 : HasShift.{v₁, u₁, w} C A
F : Functor.{v₁, v₃, u₁, u₃} C E
F' : Functor.{v₂, v₃, u₂, u₃} D E
inst_6 : Lifting L W F F'
inst_7 : HasShift.{v₂, u₂, w} D A
inst_8 : HasShift.{v₃, u₃, w} E A
inst_9 : L.CommShift A
inst_10 : F.CommShift A
a b : A
X : C
dsimp :
  F'.map (map (shiftFunctor D b) (𝟙 (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))) ≫
    CategoryStruct.comp (Y :=
      obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (F'.obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))))
      (NatTrans.app (Iso.hom (commShiftOfLocalization.iso L W F F' b))
        (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
      (map (shiftFunctor E b) ((Lifting.iso L W F F').hom.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
        CategoryStruct.comp (Y :=
          obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
          (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
          (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
            NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X))))
| F'.map (map (shiftFunctor D b) (𝟙 ((shiftFunctor C a ⋙ L).obj X))) ≫
    CategoryStruct.comp (Y :=
      obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (F'.obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))))
      (NatTrans.app (Iso.hom (commShiftOfLocalization.iso L W F F' b))
        (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
      (map (shiftFunctor E b) ((Lifting.iso L W F F').hom.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
        CategoryStruct.comp (Y :=
          obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
          (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
          (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
            NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X))))

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Shift.Localization
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
L : Functor.{v₁, v₂, u₁, u₂} C D
W : MorphismProperty.{v₁, u₁} C
inst_3 : L.IsLocalization W
A : Type w
inst_4 : AddMonoid A
inst_5 : HasShift.{v₁, u₁, w} C A
F : Functor.{v₁, v₃, u₁, u₃} C E
F' : Functor.{v₂, v₃, u₂, u₃} D E
inst_6 : Lifting L W F F'
inst_7 : HasShift.{v₂, u₂, w} D A
inst_8 : HasShift.{v₃, u₃, w} E A
inst_9 : L.CommShift A
inst_10 : F.CommShift A
a b : A
X : C
rw :
  CategoryStruct.comp (Y := F'.obj ((shiftFunctor C b ⋙ L).obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
    (F'.map (NatTrans.app (Iso.inv (commShiftIso L b)) (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
    ((Lifting.iso L W F F').hom.app
        (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C b) (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
      CategoryStruct.comp (Y := (F ⋙ shiftFunctor E b).obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))
        (NatTrans.app (Iso.hom (commShiftIso F b)) (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))
        (CategoryStruct.comp (Y :=
          obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
          (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
          (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
            NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X)))))
| F'.map (map (shiftFunctor D b) (𝟙 (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))) ≫
    CategoryStruct.comp (Y :=
      obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (F'.obj (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X))))
      (NatTrans.app (Iso.hom (commShiftOfLocalization.iso L W F F' b))
        (L.obj (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)))
      (map (shiftFunctor E b) ((Lifting.iso L W F F').hom.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X)) ≫
        CategoryStruct.comp (Y :=
          obj.{v₃, v₃, u₃, u₃} (shiftFunctor E b) (obj.{v₃, v₃, u₃, u₃} (shiftFunctor E a) (F.obj X)))
          (map (shiftFunctor E b) (NatTrans.app (Iso.hom (commShiftIso F a)) X))
          (map (shiftFunctor E b) (map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)) ≫
            NatTrans.app (Iso.inv (shiftFunctorAdd E a b)) (F'.obj (L.obj X))))

Tactic:
rw [map_id, map_id, Category.id_comp, commShiftOfLocalization.iso_hom_app,
        Category.assoc, Category.assoc, Category.assoc, ← map_comp_assoc,
        Iso.inv_hom_id_app, map_id, Category.id_comp]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Functor.map_comp_assoc, CategoryTheory.Functor.commShiftOfLocalization.iso_hom_app, CategoryTheory.Functor.map_id, CategoryTheory.Iso.inv_hom_id_app, CategoryTheory.Category.id_comp
-- Module: Mathlib.CategoryTheory.Shift.Localization
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
L : Functor.{v₁, v₂, u₁, u₂} C D
W : MorphismProperty.{v₁, u₁} C
inst_3 : L.IsLocalization W
A : Type w
inst_4 : AddMonoid A
inst_5 : HasShift.{v₁, u₁, w} C A
F : Functor.{v₁, v₃, u₁, u₃} C E
F' : Functor.{v₂, v₃, u₂, u₃} D E
inst_6 : Lifting L W F F'
inst_7 : HasShift.{v₂, u₂, w} D A
inst_8 : HasShift.{v₃, u₃, w} E A
inst_9 : L.CommShift A
inst_10 : F.CommShift A
a : A
X : C
⊢ Eq (α := (shiftFunctor D a ⋙ F').obj (L.obj X) ⟶ (F' ⋙ shiftFunctor E a).obj (L.obj X))
    (NatTrans.app (Iso.hom (commShiftIso F' a)) (L.obj X))
    (CategoryStruct.comp (Y := F'.obj ((shiftFunctor C a ⋙ L).obj X))
      (F'.map (NatTrans.app (Iso.inv (commShiftIso L a)) X))
      ((Lifting.iso L W F F').hom.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X) ≫
        NatTrans.app (Iso.hom (commShiftIso F a)) X ≫ map (shiftFunctor E a) ((Lifting.iso L W F F').inv.app X)))

Tactic:
apply commShiftOfLocalization.iso_hom_app

Premises used: CategoryTheory.Functor.commShiftOfLocalization.iso_hom_app
-- Module: Mathlib.CategoryTheory.Shift.Localization
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
L : Functor.{v₁, v₂, u₁, u₂} C D
W : MorphismProperty.{v₁, u₁} C
inst_3 : L.IsLocalization W
A : Type w
inst_4 : AddMonoid A
inst_5 : HasShift.{v₁, u₁, w} C A
F : Functor.{v₁, v₃, u₁, u₃} C E
F' : Functor.{v₂, v₃, u₂, u₃} D E
inst_6 : Lifting L W F F'
inst_7 : HasShift.{v₂, u₂, w} D A
inst_8 : HasShift.{v₃, u₃, w} E A
inst_9 : L.CommShift A
inst_10 : F.CommShift A
a : A
X : C
⊢ Eq (α := (F' ⋙ shiftFunctor E a).obj (L.obj X) ⟶ (shiftFunctor D a ⋙ F').obj (L.obj X))
    (NatTrans.app (Iso.inv (commShiftIso F' a)) (L.obj X))
    (map (shiftFunctor E a) ((Lifting.iso L W F F').hom.app X) ≫
      CategoryStruct.comp (Y := (shiftFunctor C a ⋙ F).obj X) (NatTrans.app (Iso.inv (commShiftIso F a)) X)
        ((Lifting.iso L W F F').inv.app (obj.{v₁, v₁, u₁, u₁} (shiftFunctor C a) X) ≫
          F'.map (NatTrans.app (Iso.hom (commShiftIso L a)) X)))

Tactic:
apply commShiftOfLocalization.iso_inv_app

Premises used: CategoryTheory.Functor.commShiftOfLocalization.iso_inv_app
-- Module: Mathlib.CategoryTheory.Shift.Localization
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
L : Functor.{v₁, v₂, u₁, u₂} C D
W : MorphismProperty.{v₁, u₁} C
inst_3 : L.IsLocalization W
A : Type w
inst_4 : AddMonoid A
inst_5 : HasShift.{v₁, u₁, w} C A
F : Functor.{v₁, v₃, u₁, u₃} C E
F' : Functor.{v₂, v₃, u₂, u₃} D E
inst_6 : Lifting L W F F'
inst_7 : HasShift.{v₂, u₂, w} D A
inst_8 : HasShift.{v₃, u₃, w} E A
inst_9 : L.CommShift A
inst_10 : F.CommShift A
this : F'.CommShift A := L.commShiftOfLocalization W A F F'
shift_comm :
  autoParam
    (∀ (a : A),
      Eq (α := shiftFunctor C a ⋙ L ⋙ F' ⟶ F ⋙ shiftFunctor E a)
        (Iso.hom (Functor.commShiftIso (L ⋙ F') a) ≫ Functor.whiskerRight (Lifting.iso L W F F').hom (shiftFunctor E a))
        (Functor.whiskerLeft (shiftFunctor C a) (Lifting.iso L W F F').hom ≫ Iso.hom (Functor.commShiftIso F a)))
    CommShift.shift_comm._autoParam
⊢ CommShift (Lifting.iso L W F F').hom A

Tactic:
constructor
-- Module: Mathlib.CategoryTheory.Shift.Localization
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
L : Functor.{v₁, v₂, u₁, u₂} C D
W : MorphismProperty.{v₁, u₁} C
inst_3 : L.IsLocalization W
A : Type w
inst_4 : AddMonoid A
inst_5 : HasShift.{v₁, u₁, w} C A
F : Functor.{v₁, v₃, u₁, u₃} C E
F' : Functor.{v₂, v₃, u₂, u₃} D E
inst_6 : Lifting L W F F'
inst_7 : HasShift.{v₂, u₂, w} D A
inst_8 : HasShift.{v₃, u₃, w} E A
inst_9 : L.CommShift A
inst_10 : F.CommShift A
this : F'.CommShift A := L.commShiftOfLocalization W A F F'
a : A
shift_comm :
  ∀ (X : C),
    Eq (α := (shiftFunctor C a ⋙ L ⋙ F').obj X ⟶ (F ⋙ shiftFunctor E a).obj X)
      (app
        (Iso.hom (Functor.commShiftIso (L ⋙ F') a) ≫ Functor.whiskerRight (Lifting.iso L W F F').hom (shiftFunctor E a))
        X)
      (app (Functor.whiskerLeft (shiftFunctor C a) (Lifting.iso L W F F').hom ≫ Iso.hom (Functor.commShiftIso F a)) X)
⊢ Eq (α := shiftFunctor C a ⋙ L ⋙ F' ⟶ F ⋙ shiftFunctor E a)
    (Iso.hom (Functor.commShiftIso (L ⋙ F') a) ≫ Functor.whiskerRight (Lifting.iso L W F F').hom (shiftFunctor E a))
    (Functor.whiskerLeft (shiftFunctor C a) (Lifting.iso L W F F').hom ≫ Iso.hom (Functor.commShiftIso F a))

Tactic:
ext X
-- Module: Mathlib.CategoryTheory.Shift.Localization
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
L : Functor.{v₁, v₂, u₁, u₂} C D
W : MorphismProperty.{v₁, u₁} C
inst_3 : L.IsLocalization W
A : Type w
inst_4 : AddMonoid A
inst_5 : HasShift.{v₁, u₁, w} C A
F : Functor.{v₁, v₃, u₁, u₃} C E
F' : Functor.{v₂, v₃, u₂, u₃} D E
inst_6 : Lifting L W F F'
inst_7 : HasShift.{v₂, u₂, w} D A
inst_8 : HasShift.{v₃, u₃, w} E A
inst_9 : L.CommShift A
inst_10 : F.CommShift A
this : F'.CommShift A := L.commShiftOfLocalization W A F F'
a : A
X : C
⊢ Eq (α := (shiftFunctor C a ⋙ L ⋙ F').obj X ⟶ (F ⋙ shiftFunctor E a).obj X)
    (app
      (Iso.hom (Functor.commShiftIso (L ⋙ F') a) ≫ Functor.whiskerRight (Lifting.iso L W F F').hom (shiftFunctor E a))
      X)
    (app (Functor.whiskerLeft (shiftFunctor C a) (Lifting.iso L W F F').hom ≫ Iso.hom (Functor.commShiftIso F a)) X)

Tactic:
simp only [comp_app, Functor.whiskerRight_app, Functor.whiskerLeft_app,
    Functor.commShiftIso_comp_hom_app,
    Functor.commShiftOfLocalization_iso_hom_app,
    Category.assoc, ← Functor.map_comp, ← Functor.map_comp_assoc,
    Iso.hom_inv_id_app, Functor.map_id, Iso.inv_hom_id_app,
    Category.comp_id, Category.id_comp, Functor.comp_obj]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.Category.assoc, CategoryTheory.Functor.commShiftIso_comp_hom_app, CategoryTheory.Functor.whiskerLeft_app, CategoryTheory.Iso.hom_inv_id_app, CategoryTheory.Category.comp_id, CategoryTheory.Functor.map_comp_assoc, CategoryTheory.Functor.map_id, CategoryTheory.Functor.whiskerRight_app, CategoryTheory.Functor.map_comp, CategoryTheory.Functor.commShiftOfLocalization_iso_hom_app, CategoryTheory.NatTrans.comp_app, CategoryTheory.Iso.inv_hom_id_app, CategoryTheory.Category.id_comp