awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
M : C
inst_2 : MonObj M
⊢ Eq.{v + 1} (α := tensorObj.{v, u_1} (𝒞 := inst) M M ⟶ M) μ μ
have H : Eq.{v + 1} (α := tensorObj.{v, u_1} (𝒞 := inst) M M ⟶ M)
  (CategoryStruct.comp (Y := tensorObj.{v, u_1} (𝒞 := inst) M M) (lift (fst M M) (snd M M)) μ) μ
Tactic: change lift (fst M M) (snd M M) ≫ μ = μ
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
M : C
inst_2 : MonObj M
⊢ Eq.{v + 1} (α := tensorObj.{v, u_1} (𝒞 := inst) M M ⟶ M)
    (CategoryStruct.comp (Y := tensorObj.{v, u_1} (𝒞 := inst) M M) (lift (fst M M) (snd M M)) μ) μ
have H : 𝟙 (tensorObj.{v, u_1} (𝒞 := inst) M M) ≫ μ = μ
Tactic: rw [lift_fst_snd, Category.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
M : C
inst_2 : MonObj M
H : 𝟙 (tensorObj.{v, u_1} (𝒞 := inst) M M) ≫ μ = μ
⊢ Eq.{v + 1} (α := tensorObj.{v, u_1} (𝒞 := inst) M M ⟶ M)
    (CategoryStruct.comp (Y := tensorObj.{v, u_1} (𝒞 := inst) M M) (lift (fst M M) (snd M M)) μ) μ
have H₁ : Eq.{v + 1} (α := tensorObj.{v, u_1} (𝒞 := inst) M M ⟶ M) μ μ
Tactic: rw [lift_fst_snd, Category.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
inst_2 : Category.{w, u_2} D
inst_3 : CartesianMonoidalCategory.{w, u_2} D
M_1 N_1 O X Y : C
inst_4 : MonObj M_1
inst_5 : MonObj N_1
inst_6 : MonObj O
M N : Mon.{v, u_1} C
α : Functor.obj yonedaMon M ⟶ Functor.obj yonedaMon N
⊢ Eq (α := tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ N.X)
    (CategoryStruct.comp (Y := M.X) η
        ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
              MonCat.carrier) (X := (Functor.obj yonedaMon M).obj (op M.X)) (α.app (op M.X)) :
            (a : (↑((yonedaMon.obj M).obj (op M.X)) : Type v)) → (↑((yonedaMon.obj N).obj (op M.X)) : Type v))
          (𝟙 M.X)) :
      tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ N.X)
    (η : tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ N.X)
have dsimp : Eq (α := tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ N.X)
  (CategoryStruct.comp (Y := M.X) η
      ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC := MonCat.carrier)
            (X := Functor.obj (yonedaMonObj M.X) (op M.X)) (α.app (op M.X)) :
          (a : (↑((yonedaMonObj M.X).obj (op M.X)) : Type v)) → (↑((yonedaMonObj N.X).obj (op M.X)) : Type v))
        (𝟙 M.X)) :
    tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ N.X)
  (η : tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ N.X)
Tactic: dsimp only [yonedaMon_obj] at α ⊢
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
inst_2 : Category.{w, u_2} D
inst_3 : CartesianMonoidalCategory.{w, u_2} D
M_1 N_1 O X Y : C
inst_4 : MonObj M_1
inst_5 : MonObj N_1
inst_6 : MonObj O
M N : Mon.{v, u_1} C
α : Quiver.Hom (V := Functor.{v, v, u_1, v + 1} Cᵒᵖ MonCat) (yonedaMonObj M.X) (yonedaMonObj N.X)
⊢ Eq (α := tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ N.X)
    (CategoryStruct.comp (Y := M.X) η
        ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
              MonCat.carrier) (X := Functor.obj (yonedaMonObj M.X) (op M.X)) (α.app (op M.X)) :
            (a : (↑((yonedaMonObj M.X).obj (op M.X)) : Type v)) → (↑((yonedaMonObj N.X).obj (op M.X)) : Type v))
          (𝟙 M.X)) :
      tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ N.X)
    (η : tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ N.X)
have rw : Eq (α := tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ N.X)
  ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC := MonCat.carrier)
          (X := Functor.obj (yonedaMonObj M.X) (op (tensorUnit.{v, u_1} C (𝒞 := inst))))
          (α.app (op (tensorUnit.{v, u_1} C (𝒞 := inst)))) :
        (a : (↑((yonedaMonObj M.X).obj (op (𝟙_ C))) : Type v)) → (↑((yonedaMonObj N.X).obj (op (𝟙_ C))) : Type v))
      (CategoryStruct.comp (Y := tensorUnit.{v, u_1} C (𝒞 := inst)) (toUnit (tensorUnit.{v, u_1} C (𝒞 := inst))) η) :
    (↑(Functor.obj (yonedaMonObj N.X) (op (tensorUnit.{v, u_1} C (𝒞 := inst)))) : Type v))
  (CategoryStruct.comp (Y := tensorUnit.{v, u_1} C (𝒞 := inst)) (toUnit (tensorUnit.{v, u_1} C (𝒞 := inst))) η :
    tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ N.X)
Tactic: rw [← yonedaMon_naturality, Category.comp_id,
            ← Category.id_comp η[M.X], toUnit_unique (𝟙 _) (toUnit _),
            ← Category.id_comp η[N.X], toUnit_unique (𝟙 _) (toUnit _)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
inst_2 : Category.{w, u_2} D
inst_3 : CartesianMonoidalCategory.{w, u_2} D
M_1 N_1 O X Y : C
inst_4 : MonObj M_1
inst_5 : MonObj N_1
inst_6 : MonObj O
M N : Mon.{v, u_1} C
α : Functor.obj yonedaMon M ⟶ Functor.obj yonedaMon N
⊢ Eq (α := tensorObj.{v, u_1} (𝒞 := inst) M.X M.X ⟶ N.X)
    (μ ≫
        (ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC := MonCat.carrier)
              (X := (Functor.obj yonedaMon M).obj (op M.X)) (α.app (op M.X)) :
            (a : (↑((yonedaMon.obj M).obj (op M.X)) : Type v)) → (↑((yonedaMon.obj N).obj (op M.X)) : Type v))
          (𝟙 M.X) :
      tensorObj.{v, u_1} (𝒞 := inst) M.X M.X ⟶ N.X)
    (CategoryStruct.comp (Y := tensorObj.{v, u_1} (𝒞 := inst) N.X N.X)
        ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
                MonCat.carrier) (X := (Functor.obj yonedaMon M).obj (op M.X)) (α.app (op M.X)) :
              (a : (↑((yonedaMon.obj M).obj (op M.X)) : Type v)) → (↑((yonedaMon.obj N).obj (op M.X)) : Type v))
            (𝟙 M.X) ⊗ₘ
          (ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
                MonCat.carrier) (X := (Functor.obj yonedaMon M).obj (op M.X)) (α.app (op M.X)) :
              (a : (↑((yonedaMon.obj M).obj (op M.X)) : Type v)) → (↑((yonedaMon.obj N).obj (op M.X)) : Type v))
            (𝟙 M.X))
        μ :
      tensorObj.{v, u_1} (𝒞 := inst) M.X M.X ⟶ N.X)
have dsimp : Eq (α := tensorObj.{v, u_1} (𝒞 := inst) M.X M.X ⟶ N.X)
  (μ ≫
      (ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC := MonCat.carrier)
            (X := Functor.obj (yonedaMonObj M.X) (op M.X)) (α.app (op M.X)) :
          (a : (↑((yonedaMonObj M.X).obj (op M.X)) : Type v)) → (↑((yonedaMonObj N.X).obj (op M.X)) : Type v))
        (𝟙 M.X) :
    tensorObj.{v, u_1} (𝒞 := inst) M.X M.X ⟶ N.X)
  (CategoryStruct.comp (Y := tensorObj.{v, u_1} (𝒞 := inst) N.X N.X)
      ((ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC := MonCat.carrier)
              (X := Functor.obj (yonedaMonObj M.X) (op M.X)) (α.app (op M.X)) :
            (a : (↑((yonedaMonObj M.X).obj (op M.X)) : Type v)) → (↑((yonedaMonObj N.X).obj (op M.X)) : Type v))
          (𝟙 M.X) ⊗ₘ
        (ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC := MonCat.carrier)
              (X := Functor.obj (yonedaMonObj M.X) (op M.X)) (α.app (op M.X)) :
            (a : (↑((yonedaMonObj M.X).obj (op M.X)) : Type v)) → (↑((yonedaMonObj N.X).obj (op M.X)) : Type v))
          (𝟙 M.X))
      μ :
    tensorObj.{v, u_1} (𝒞 := inst) M.X M.X ⟶ N.X)
Tactic: dsimp only [yonedaMon_obj] at α ⊢
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
inst_2 : Category.{w, u_2} D
inst_3 : CartesianMonoidalCategory.{w, u_2} D
M_1 N_1 O X Y : C
inst_4 : MonObj M_1
inst_5 : MonObj N_1
inst_6 : MonObj O
M N : Mon.{v, u_1} C
α : Functor.obj yonedaMon M ⟶ Functor.obj yonedaMon N
⊢ Functor.map yonedaMon
      {
        hom :=
          (ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
                MonCat.carrier) (X := (Functor.obj yonedaMon M).obj (op M.X)) (α.app (op M.X)) :
              (a : (↑((yonedaMon.obj M).obj (op M.X)) : Type v)) → (↑((yonedaMon.obj N).obj (op M.X)) : Type v))
            (𝟙 M.X),
        isMonHom_hom := ⋯ } =
    α
have w : ∀ (Y : Cᵒᵖ) (f : (↑((Functor.obj yonedaMon M).obj Y) : Type v)),
  Eq (α := (↑((Functor.obj yonedaMon N).obj Y) : Type v))
    (((MonCat.Hom.hom
              (NatTrans.app
                (Functor.map yonedaMon
                  {
                    hom :=
                      (ConcreteCategory.hom (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
                            MonCat.carrier) (X := (Functor.obj yonedaMon M).obj (op M.X)) (α.app (op M.X)) :
                          (a : (↑((yonedaMon.obj M).obj (op M.X)) : Type v)) →
                            (↑((yonedaMon.obj N).obj (op M.X)) : Type v))
                        (𝟙 M.X),
                    isMonHom_hom := sorry })
                Y) :
            (↑((Functor.obj yonedaMon M).obj Y) : Type v) →* (↑((Functor.obj yonedaMon N).obj Y) : Type v)) :
          (a : (↑((yonedaMon.obj M).obj Y) : Type v)) → (↑((yonedaMon.obj N).obj Y) : Type v))
        f :
      (↑((Functor.obj yonedaMon N).obj Y) : Type v))
    ((MonCat.Hom.hom (α.app Y) : (a : (↑((yonedaMon.obj M).obj Y) : Type v)) → (↑((yonedaMon.obj N).obj Y) : Type v))
        f :
      (↑((Functor.obj yonedaMon N).obj Y) : Type v))
Tactic: ext Y f
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
⊢ Eq (α := ObjectProperty (Functor.{v, v, u_1, v + 1} Cᵒᵖ MonCat)) (Functor.essImage (C := Mon.{v, u_1} C) yonedaMon)
    fun (F : Functor.{v, v, u_1, v + 1} Cᵒᵖ MonCat) ↦
    (F ⋙
        forget MonCat (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
          MonCat.carrier)).IsRepresentable
have h : ∀ (F : Functor.{v, v, u_1, v + 1} Cᵒᵖ MonCat),
  Functor.essImage (C := Mon.{v, u_1} C) yonedaMon F ↔
    (F ⋙
        forget MonCat (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
          MonCat.carrier)).IsRepresentable
Tactic: ext F
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
F : Functor.{v, v, u_1, v + 1} Cᵒᵖ MonCat
⊢ Functor.essImage (C := Mon.{v, u_1} C) yonedaMon F ↔
    (F ⋙
        forget MonCat (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
          MonCat.carrier)).IsRepresentable
have h : Functor.essImage (C := Mon.{v, u_1} C) yonedaMon F →
  (F ⋙
      forget MonCat (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
        MonCat.carrier)).IsRepresentable
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
F : Functor.{v, v, u_1, v + 1} Cᵒᵖ MonCat
h :
  Functor.essImage (C := Mon.{v, u_1} C) yonedaMon F →
    (F ⋙
        forget MonCat (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
          MonCat.carrier)).IsRepresentable
⊢ Functor.essImage (C := Mon.{v, u_1} C) yonedaMon F ↔
    (F ⋙
        forget MonCat (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
          MonCat.carrier)).IsRepresentable
have h₁ : (F ⋙
      forget MonCat (FC := fun (x1 x2 : MonCat.{v}) ↦ (↑x1 : Type v) →* (↑x2 : Type v)) (CC :=
        MonCat.carrier)).IsRepresentable →
  Functor.essImage (C := Mon.{v, u_1} C) yonedaMon F
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
M N X : C
inst_2 : MonObj M
inst_3 : MonObj N
f : X ⟶ M
n : ℕ
g : M ⟶ N
inst_4 : IsMonHom g
⊢ (f ^ n) ≫ g = (f ≫ g) ^ n
have zero : HPow.hPow (β := ℕ) f 0 ≫ g = HPow.hPow (β := ℕ) (f ≫ g) 0
Tactic: induction n
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
M N X : C
inst_2 : MonObj M
inst_3 : MonObj N
f : X ⟶ M
n : ℕ
g : M ⟶ N
inst_4 : IsMonHom g
zero : HPow.hPow (β := ℕ) f 0 ≫ g = HPow.hPow (β := ℕ) (f ≫ g) 0
⊢ (f ^ n) ≫ g = (f ≫ g) ^ n
have succ : ∀ (n : ℕ), (f ^ n) ≫ g = (f ≫ g) ^ n → (f ^ (n + 1)) ≫ g = (f ≫ g) ^ (n + 1)
Tactic: induction n
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
M X Y : C
inst_2 : MonObj M
f : X ⟶ M
n : ℕ
h : Y ⟶ X
⊢ h ≫ f ^ n = (h ≫ f) ^ n
have zero : h ≫ HPow.hPow (β := ℕ) f 0 = HPow.hPow (β := ℕ) (h ≫ f) 0
Tactic: induction n
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
M X Y : C
inst_2 : MonObj M
f : X ⟶ M
n : ℕ
h : Y ⟶ X
zero : h ≫ HPow.hPow (β := ℕ) f 0 = HPow.hPow (β := ℕ) (h ≫ f) 0
⊢ h ≫ f ^ n = (h ≫ f) ^ n
have succ : ∀ (n : ℕ), h ≫ f ^ n = (h ≫ f) ^ n → h ≫ f ^ (n + 1) = (h ≫ f) ^ (n + 1)
Tactic: induction n
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
M : C
inst_2 : MonObj M
⊢ Eq (α := tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ M) η
    (CategoryStruct.comp (Y := tensorUnit.{v, u_1} C (𝒞 := inst)) (toUnit (tensorUnit.{v, u_1} C (𝒞 := inst))) η)
have rw : Eq (α := tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ M) η (𝟙 (tensorUnit.{v, u_1} C (𝒞 := inst)) ≫ η)
Tactic: rw [toUnit_unique (toUnit _) (𝟙 _), Category.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
M : C
inst_2 : MonObj M
rw : Eq (α := tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ M) η (𝟙 (tensorUnit.{v, u_1} C (𝒞 := inst)) ≫ η)
⊢ Eq (α := tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ M) η
    (CategoryStruct.comp (Y := tensorUnit.{v, u_1} C (𝒞 := inst)) (toUnit (tensorUnit.{v, u_1} C (𝒞 := inst))) η)
have rw₁ : Eq.{v + 1} (α := tensorUnit.{v, u_1} C (𝒞 := inst) ⟶ M) η η
Tactic: rw [toUnit_unique (toUnit _) (𝟙 _), Category.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
M : C
inst_2 : MonObj M
⊢ Eq.{v + 1} (α := tensorObj.{v, u_1} (𝒞 := inst) M M ⟶ M) μ
    (CategoryStruct.comp (Y := tensorObj.{v, u_1} (𝒞 := inst) M M) (lift (fst M M) (snd M M)) μ)
have rw : μ = 𝟙 (tensorObj.{v, u_1} (𝒞 := inst) M M) ≫ μ
Tactic: rw [lift_fst_snd, Category.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Mon (goal)
Goal:
C : Type u_1
inst : Category.{v, u_1} C
inst_1 : CartesianMonoidalCategory.{v, u_1} C
M : C
inst_2 : MonObj M
rw : μ = 𝟙 (tensorObj.{v, u_1} (𝒞 := inst) M M) ≫ μ
⊢ Eq.{v + 1} (α := tensorObj.{v, u_1} (𝒞 := inst) M M ⟶ M) μ
    (CategoryStruct.comp (Y := tensorObj.{v, u_1} (𝒞 := inst) M M) (lift (fst M M) (snd M M)) μ)
have rw₁ : Eq.{v + 1} (α := tensorObj.{v, u_1} (𝒞 := inst) M M ⟶ M) μ μ
Tactic: rw [lift_fst_snd, Category.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H K : C
inst_2 : GrpObj G
inst_3 : GrpObj H
inst_4 : GrpObj K
φ : H ⟶ G
⊢ ∃ (ψ :
    tensorObj.{v_1, u_1} (𝒞 := inst) G (tensorUnit.{v_1, u_1} C (𝒞 := inst)) ⟶ tensorUnit.{v_1, u_1} C (𝒞 := inst)),
    ψ ≫ η = G ◁ η ≫ conj G
have h : CategoryStruct.comp (Y := tensorUnit.{v_1, u_1} C (𝒞 := inst))
    (toUnit (tensorObj.{v_1, u_1} (𝒞 := inst) G (tensorUnit.{v_1, u_1} C (𝒞 := inst)))) η =
  G ◁ η ≫ conj G
Tactic: use toUnit _
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
inst_5 : Mono φ
⊢ Normal φ ↔ ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
have rw : (∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G) ↔
  ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
Tactic: rw [isNormalHom_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
inst_5 : Mono φ
⊢ (∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G) ↔
    ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
have refine_1 : (∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G) →
  ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
inst_5 : Mono φ
refine_1 :
  (∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G) →
    ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
⊢ (∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G) ↔
    ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
have refine_2 : (∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))) →
  ∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
inst_5 : Mono φ
ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H
hψ : ψ ≫ φ = G ◁ φ ≫ conj G
X : C
⊢ Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
have refine_1 : ∀ (n : X ⟶ G),
  Membership.mem (γ := Subgroup (X ⟶ G)) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X)) n →
    ∀ (g : X ⟶ G), Membership.mem (γ := Subgroup (X ⟶ G)) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X)) (g * n * g⁻¹)
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
inst_5 : Mono φ
hnormal : ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
⊢ ∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G
have this : ∀ (X : C) (g : X ⟶ G) (h : X ⟶ H), ∃ (h' : X ⟶ H), h' ≫ φ = g * h ≫ φ * g⁻¹
Tactic: have (X : C) (g : X ⟶ G) (h : X ⟶ H) : ∃ (h' : X ⟶ H), h' ≫ φ = g * h ≫ φ * g⁻¹ :=
      (hnormal X).conj_mem (h ≫ φ) (by simp) g
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
inst_5 : Mono φ
hnormal : ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
this : ∀ (X : C) (g : X ⟶ G) (h : X ⟶ H), ∃ (h' : X ⟶ H), h' ≫ φ = g * h ≫ φ * g⁻¹
⊢ ∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G
have refine_2 : ∀ (h' : (X : C) → (g : X ⟶ G) → (h : X ⟶ H) → X ⟶ H)
  (hh' : ∀ (X : C) (g : X ⟶ G) (h : X ⟶ H), h' X g h ≫ φ = g * h ≫ φ * g⁻¹),
  ∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G
Tactic: choose h' hh' using this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
inst_5 : Mono φ
hnormal : ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
h' : (X : C) → (g : X ⟶ G) → (h : X ⟶ H) → X ⟶ H
hh' : ∀ (X : C) (g : X ⟶ G) (h : X ⟶ H), h' X g h ≫ φ = g * h ≫ φ * g⁻¹
⊢ ∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G
have refine_2 : Quiver.Hom (V := Functor.{v_1, v_1, u_1, v_1 + 1} Cᵒᵖ (Type v_1))
  (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) (Functor.obj yoneda H)
Tactic: refine ⟨Yoneda.fullyFaithful.preimage ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Cartesian.Normal (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : CartesianMonoidalCategory.{v_1, u_1} C
G H : C
inst_2 : GrpObj G
inst_3 : GrpObj H
φ : H ⟶ G
inst_4 : IsMonHom φ
inst_5 : Mono φ
hnormal : ∀ (X : C), Subgroup.Normal.{v_1} (G := X ⟶ G) (MonoidHom.range (G := X ⟶ H) (monoidHom φ X))
h' : (X : C) → (g : X ⟶ G) → (h : X ⟶ H) → X ⟶ H
hh' : ∀ (X : C) (g : X ⟶ G) (h : X ⟶ H), h' X g h ≫ φ = g * h ≫ φ * g⁻¹
refine_2 :
  Quiver.Hom (V := Functor.{v_1, v_1, u_1, v_1 + 1} Cᵒᵖ (Type v_1))
    (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) (Functor.obj yoneda H)
⊢ ∃ (ψ : tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ H), ψ ≫ φ = G ◁ φ ≫ conj G
have refine_2₁ : Eq (α := tensorObj.{v_1, u_1} (𝒞 := inst) G H ⟶ G)
  (Functor.FullyFaithful.preimage Yoneda.fullyFaithful
      {
        app := fun (X : Cᵒᵖ) ↦
          ↾fun (x : Functor.obj (Functor.obj yoneda (tensorObj.{v_1, u_1} (𝒞 := inst) G H)) X) ↦
            h' (Opposite.unop X) (x ≫ fst G H) (x ≫ snd G H),
        naturality := sorry } ≫
    φ)
  (G ◁ φ ≫ conj G)
Tactic: refine ⟨Yoneda.fullyFaithful.preimage ?_, ?_⟩