awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Unitor (goal)
Goal:
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Zero I
inst_3 : DecidableEq I
inst_4 : HasInitial.{v_1, u_1} C
F : D ⥤ Functor.{v_1, v_2, u_1, u_2} C D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst_5 : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j' : J
s : Cofan ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j')
h : (j', 0) ∈ p ⁻¹' {j'}
pos :
  Eq (α := (F.obj (X j')).obj (Functor.obj (single₀ I) Y 0) ⟶ s.pt)
    (((F.obj (X j')).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j')) ≫
      e.inv.app (X j') ≫
        CategoryStruct.comp (Y := (F.obj (X j')).obj (Functor.obj (single 0) Y 0))
          ((F.obj (X j')).map (Iso.inv (singleObjApplyIso 0 Y)))
          (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j'}) (j', 0) ⋯)))
    (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j'}) (j', 0) h))
pos₁ :
  Eq (α := (F.obj (X j')).obj (Functor.obj (single₀ I) Y 0) ⟶ s.pt)
    ((F.obj (X j')).map (Iso.hom (singleObjApplyIso 0 Y)) ≫
      e.hom.app (X j') ≫
        e.inv.app (X j') ≫
          CategoryStruct.comp (Y := (F.obj (X j')).obj (Functor.obj (single 0) Y 0))
            ((F.obj (X j')).map (Iso.inv (singleObjApplyIso 0 Y)))
            (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j'}) (j', 0) ⋯)))
    (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j'}) (j', 0) h))
⊢ Eq (α := (F.obj (X j')).obj (Functor.obj (single₀ I) Y 0) ⟶ s.pt)
    (Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j')
          (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j'}) (j', 0) h) ≫
        e.inv.app (X j') ≫
          CategoryStruct.comp (Y := (F.obj (X j')).obj (Functor.obj (single 0) Y 0))
            ((F.obj (X j')).map (Iso.inv (singleObjApplyIso 0 Y)))
            (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j'}) (j', 0) ⋯)) :
      (F.obj (X j')).obj (Functor.obj (single₀ I) Y 0) ⟶ s.pt)
    (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j'}) (j', 0) h) :
      (((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j'
          (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j'}) (j', 0) h) ⟶
        s.pt)
have pos₂ : Eq (α := (F.obj (X j')).obj (Functor.obj (single₀ I) Y 0) ⟶ s.pt)
  ((F.obj (X j')).map (Iso.hom (singleObjApplyIso 0 Y)) ≫
    CategoryStruct.comp (Y := (F.obj (X j')).obj (Functor.obj (single 0) Y 0))
      ((F.obj (X j')).map (Iso.inv (singleObjApplyIso 0 Y)))
      (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j'}) (j', 0) sorry)))
  (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j'}) (j', 0) h))
Tactic: rw [mapBifunctorRightUnitorCofan_inj, assoc, Iso.hom_inv_id_app_assoc,
          ← Functor.map_comp_assoc, Iso.hom_inv_id, Functor.map_id, id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Unitor (goal)
Goal:
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Zero I
inst_3 : DecidableEq I
inst_4 : HasInitial.{v_1, u_1} C
F : D ⥤ Functor.{v_1, v_2, u_1, u_2} C D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst_5 : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j)
j' : J
i : I
h : (j', i) ∈ p ⁻¹' {j}
hi : ¬i = 0
⊢ Eq (α :=
    (((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j
        (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j', i) h) ⟶
      s.pt)
    (CategoryStruct.comp (Y :=
      Cocone.pt (F := Discrete.functor ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j))
        (mapBifunctorRightUnitorCofan F Y e p hp X j))
      (Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j)
        (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j', i) h))
      ((fun (s : Cofan ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j)) ↦
          (e.inv.app (X j) ≫
              CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
                ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
                (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯)) :
            Cocone.pt (F :=
                Discrete.functor ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j))
                (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟶
              s.pt))
        s))
    (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j', i) h))
have neg : IsInitial.{v_2, u_2}
  ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j
    (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j', i) h))
Tactic: apply IsInitial.hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Unitor (goal)
Goal:
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Zero I
inst_3 : DecidableEq I
inst_4 : HasInitial.{v_1, u_1} C
F : D ⥤ Functor.{v_1, v_2, u_1, u_2} C D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst_5 : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j)
m :
  Cocone.pt (F := Discrete.functor ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j))
      (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟶
    s.pt
hm :
  ∀ (j_1 : (↑(p ⁻¹' {j}) : Type (max u_3 u_4))),
    Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) j_1 ≫ m = s.inj j_1
⊢ m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯))
have rw : m =
  e.inv.app (X j) ≫
    CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
      ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
      (Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j)
          (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) sorry) ≫
        m)
Tactic: rw [← hm ⟨⟨j, 0⟩, hp j⟩, mapBifunctorRightUnitorCofan_inj, assoc, ← Functor.map_comp_assoc,
        Iso.inv_hom_id, Functor.map_id, id_comp, Iso.inv_hom_id_app_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Unitor (goal)
Goal:
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Zero I
inst_3 : DecidableEq I
inst_4 : HasInitial.{v_1, u_1} C
F : D ⥤ Functor.{v_1, v_2, u_1, u_2} C D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst_5 : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j)
m :
  Cocone.pt (F := Discrete.functor ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j))
      (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟶
    s.pt
hm :
  ∀ (j_1 : (↑(p ⁻¹' {j}) : Type (max u_3 u_4))),
    Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) j_1 ≫ m = s.inj j_1
rw :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j)
            (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯) ≫
          m)
⊢ m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯))
have rw₁ : m =
  e.inv.app (X j) ≫
    CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
      ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
      (((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j)) ≫ m)
Tactic: rw [← hm ⟨⟨j, 0⟩, hp j⟩, mapBifunctorRightUnitorCofan_inj, assoc, ← Functor.map_comp_assoc,
        Iso.inv_hom_id, Functor.map_id, id_comp, Iso.inv_hom_id_app_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Unitor (goal)
Goal:
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Zero I
inst_3 : DecidableEq I
inst_4 : HasInitial.{v_1, u_1} C
F : D ⥤ Functor.{v_1, v_2, u_1, u_2} C D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst_5 : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j)
m :
  Cocone.pt (F := Discrete.functor ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j))
      (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟶
    s.pt
hm :
  ∀ (j_1 : (↑(p ⁻¹' {j}) : Type (max u_3 u_4))),
    Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) j_1 ≫ m = s.inj j_1
rw :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j)
            (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯) ≫
          m)
rw₁ :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j)) ≫ m)
⊢ m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯))
have rw₂ : m =
  e.inv.app (X j) ≫
    CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
      ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
      ((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j) ≫ m)
Tactic: rw [← hm ⟨⟨j, 0⟩, hp j⟩, mapBifunctorRightUnitorCofan_inj, assoc, ← Functor.map_comp_assoc,
        Iso.inv_hom_id, Functor.map_id, id_comp, Iso.inv_hom_id_app_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Unitor (goal)
Goal:
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Zero I
inst_3 : DecidableEq I
inst_4 : HasInitial.{v_1, u_1} C
F : D ⥤ Functor.{v_1, v_2, u_1, u_2} C D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst_5 : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j)
m :
  Cocone.pt (F := Discrete.functor ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j))
      (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟶
    s.pt
hm :
  ∀ (j_1 : (↑(p ⁻¹' {j}) : Type (max u_3 u_4))),
    Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) j_1 ≫ m = s.inj j_1
rw :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j)
            (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯) ≫
          m)
rw₁ :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j)) ≫ m)
rw₂ :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        ((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j) ≫ m)
⊢ m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯))
have rw₃ : m =
  e.inv.app (X j) ≫
    (F.obj (X j)).map
        (CategoryStruct.comp (Y := Functor.obj (single 0) Y 0) (Iso.inv (singleObjApplyIso 0 Y))
          (Iso.hom (singleObjApplyIso 0 Y))) ≫
      e.hom.app (X j) ≫ m
Tactic: rw [← hm ⟨⟨j, 0⟩, hp j⟩, mapBifunctorRightUnitorCofan_inj, assoc, ← Functor.map_comp_assoc,
        Iso.inv_hom_id, Functor.map_id, id_comp, Iso.inv_hom_id_app_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Unitor (goal)
Goal:
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Zero I
inst_3 : DecidableEq I
inst_4 : HasInitial.{v_1, u_1} C
F : D ⥤ Functor.{v_1, v_2, u_1, u_2} C D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst_5 : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j)
m :
  Cocone.pt (F := Discrete.functor ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j))
      (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟶
    s.pt
hm :
  ∀ (j_1 : (↑(p ⁻¹' {j}) : Type (max u_3 u_4))),
    Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) j_1 ≫ m = s.inj j_1
rw :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j)
            (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯) ≫
          m)
rw₁ :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j)) ≫ m)
rw₂ :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        ((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j) ≫ m)
rw₃ :
  m =
    e.inv.app (X j) ≫
      (F.obj (X j)).map
          (CategoryStruct.comp (Y := Functor.obj (single 0) Y 0) (Iso.inv (singleObjApplyIso 0 Y))
            (Iso.hom (singleObjApplyIso 0 Y))) ≫
        e.hom.app (X j) ≫ m
⊢ m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯))
have rw₄ : m = e.inv.app (X j) ≫ (F.obj (X j)).map (𝟙 Y) ≫ e.hom.app (X j) ≫ m
Tactic: rw [← hm ⟨⟨j, 0⟩, hp j⟩, mapBifunctorRightUnitorCofan_inj, assoc, ← Functor.map_comp_assoc,
        Iso.inv_hom_id, Functor.map_id, id_comp, Iso.inv_hom_id_app_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Unitor (goal)
Goal:
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Zero I
inst_3 : DecidableEq I
inst_4 : HasInitial.{v_1, u_1} C
F : D ⥤ Functor.{v_1, v_2, u_1, u_2} C D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst_5 : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j)
m :
  Cocone.pt (F := Discrete.functor ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j))
      (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟶
    s.pt
hm :
  ∀ (j_1 : (↑(p ⁻¹' {j}) : Type (max u_3 u_4))),
    Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) j_1 ≫ m = s.inj j_1
rw :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j)
            (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯) ≫
          m)
rw₁ :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j)) ≫ m)
rw₂ :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        ((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j) ≫ m)
rw₃ :
  m =
    e.inv.app (X j) ≫
      (F.obj (X j)).map
          (CategoryStruct.comp (Y := Functor.obj (single 0) Y 0) (Iso.inv (singleObjApplyIso 0 Y))
            (Iso.hom (singleObjApplyIso 0 Y))) ≫
        e.hom.app (X j) ≫ m
rw₄ : m = e.inv.app (X j) ≫ (F.obj (X j)).map (𝟙 Y) ≫ e.hom.app (X j) ≫ m
⊢ m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯))
have rw₅ : m = e.inv.app (X j) ≫ 𝟙 ((F.obj (X j)).obj Y) ≫ e.hom.app (X j) ≫ m
Tactic: rw [← hm ⟨⟨j, 0⟩, hp j⟩, mapBifunctorRightUnitorCofan_inj, assoc, ← Functor.map_comp_assoc,
        Iso.inv_hom_id, Functor.map_id, id_comp, Iso.inv_hom_id_app_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Unitor (goal)
Goal:
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Zero I
inst_3 : DecidableEq I
inst_4 : HasInitial.{v_1, u_1} C
F : D ⥤ Functor.{v_1, v_2, u_1, u_2} C D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst_5 : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j)
m :
  Cocone.pt (F := Discrete.functor ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j))
      (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟶
    s.pt
hm :
  ∀ (j_1 : (↑(p ⁻¹' {j}) : Type (max u_3 u_4))),
    Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) j_1 ≫ m = s.inj j_1
rw :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j)
            (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯) ≫
          m)
rw₁ :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j)) ≫ m)
rw₂ :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        ((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j) ≫ m)
rw₃ :
  m =
    e.inv.app (X j) ≫
      (F.obj (X j)).map
          (CategoryStruct.comp (Y := Functor.obj (single 0) Y 0) (Iso.inv (singleObjApplyIso 0 Y))
            (Iso.hom (singleObjApplyIso 0 Y))) ≫
        e.hom.app (X j) ≫ m
rw₄ : m = e.inv.app (X j) ≫ (F.obj (X j)).map (𝟙 Y) ≫ e.hom.app (X j) ≫ m
rw₅ : m = e.inv.app (X j) ≫ 𝟙 ((F.obj (X j)).obj Y) ≫ e.hom.app (X j) ≫ m
⊢ m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯))
have rw₆ : m = e.inv.app (X j) ≫ e.hom.app (X j) ≫ m
Tactic: rw [← hm ⟨⟨j, 0⟩, hp j⟩, mapBifunctorRightUnitorCofan_inj, assoc, ← Functor.map_comp_assoc,
        Iso.inv_hom_id, Functor.map_id, id_comp, Iso.inv_hom_id_app_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Unitor (goal)
Goal:
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Zero I
inst_3 : DecidableEq I
inst_4 : HasInitial.{v_1, u_1} C
F : D ⥤ Functor.{v_1, v_2, u_1, u_2} C D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst_5 : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j)
m :
  Cocone.pt (F := Discrete.functor ((((mapBifunctor F J I).obj X).obj (Functor.obj (single₀ I) Y)).mapObjFun p j))
      (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟶
    s.pt
hm :
  ∀ (j_1 : (↑(p ⁻¹' {j}) : Type (max u_3 u_4))),
    Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) j_1 ≫ m = s.inj j_1
rw :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j)
            (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯) ≫
          m)
rw₁ :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j)) ≫ m)
rw₂ :
  m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        ((F.obj (X j)).map (Iso.hom (singleObjApplyIso 0 Y)) ≫ e.hom.app (X j) ≫ m)
rw₃ :
  m =
    e.inv.app (X j) ≫
      (F.obj (X j)).map
          (CategoryStruct.comp (Y := Functor.obj (single 0) Y 0) (Iso.inv (singleObjApplyIso 0 Y))
            (Iso.hom (singleObjApplyIso 0 Y))) ≫
        e.hom.app (X j) ≫ m
rw₄ : m = e.inv.app (X j) ≫ (F.obj (X j)).map (𝟙 Y) ≫ e.hom.app (X j) ≫ m
rw₅ : m = e.inv.app (X j) ≫ 𝟙 ((F.obj (X j)).obj Y) ≫ e.hom.app (X j) ≫ m
rw₆ : m = e.inv.app (X j) ≫ e.hom.app (X j) ≫ m
⊢ m =
    e.inv.app (X j) ≫
      CategoryStruct.comp (Y := (F.obj (X j)).obj (Functor.obj (single 0) Y 0))
        ((F.obj (X j)).map (Iso.inv (singleObjApplyIso 0 Y)))
        (s.inj (Subtype.mk (p := fun (x : J × I) ↦ x ∈ p ⁻¹' {j}) (j, 0) ⋯))
have rw₇ : m = m
Tactic: rw [← hm ⟨⟨j, 0⟩, hp j⟩, mapBifunctorRightUnitorCofan_inj, assoc, ← Functor.map_comp_assoc,
        Iso.inv_hom_id, Functor.map_id, id_comp, Iso.inv_hom_id_app_assoc]