awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (goal)
Goal:
R : Type u
inst : CommRing R
A : ModuleCat R
inst_1 : MonObj A
x y z : (↑A : Type u)
⊢ x * (y + z) =
    (Hom.hom μ : (a : (↑(MonoidalCategoryStruct.tensorObj A A) : Type u)) → (↑A : Type u))
      (HAdd.hAdd (α := (↑(MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) A A) : Type u)) (x ⊗ₜ[R] y)
        (x ⊗ₜ[R] z))
have h : x * (y + z) = (Hom.hom μ : (a : (↑(MonoidalCategoryStruct.tensorObj A A) : Type u)) → (↑A : Type u)) (x ⊗ₜ[R] (y + z))
Tactic: rw [← TensorProduct.tmul_add]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (goal)
Goal:
R : Type u
inst : CommRing R
A : ModuleCat R
inst_1 : MonObj A
x y z : (↑A : Type u)
⊢ (x + y) * z = x * z + y * z
have h : (x + y) * z =
  (Hom.hom μ : (a : (↑(MonoidalCategoryStruct.tensorObj A A) : Type u)) → (↑A : Type u))
    (HAdd.hAdd (α := (↑(MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) A A) : Type u)) (x ⊗ₜ[R] z)
      (y ⊗ₜ[R] z))
Tactic: convert! μ[A].hom.map_add (x ⊗ₜ z) (y ⊗ₜ z)
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (goal)
Goal:
R : Type u
inst : CommRing R
A : ModuleCat R
inst_1 : MonObj A
x y z : (↑A : Type u)
⊢ (x + y) * z =
    (Hom.hom μ : (a : (↑(MonoidalCategoryStruct.tensorObj A A) : Type u)) → (↑A : Type u))
      (HAdd.hAdd (α := (↑(MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) A A) : Type u)) (x ⊗ₜ[R] z)
        (y ⊗ₜ[R] z))
have h : (x + y) * z = (Hom.hom μ : (a : (↑(MonoidalCategoryStruct.tensorObj A A) : Type u)) → (↑A : Type u)) ((x + y) ⊗ₜ[R] z)
Tactic: rw [← TensorProduct.add_tmul]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (hypothesis)
Goal:
R : Type u
inst : CommRing R
A : ModuleCat R
inst_1 : MonObj A
x y : R
⊢ AddHom.toFun (toAddHom (Hom.hom η)) (x * y) =
    HMul.hMul (α := (↑A : Type u)) (AddHom.toFun (toAddHom (Hom.hom η)) x) (AddHom.toFun (toAddHom (Hom.hom η)) y)
have h : Eq (α := (↑A : Type u))
  ((Hom.hom (Iso.hom (MonoidalCategoryStruct.leftUnitor A)) :
        (a : (↑(MonoidalCategoryStruct.tensorObj (MonoidalCategoryStruct.tensorUnit (ModuleCat R)) A) : Type u)) →
          (↑A : Type u))
      (x ⊗ₜ[R]
          (@ConcreteCategory.hom _ (moduleCategory R) (fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u))
                carrier _ _ (MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)) A η :
              (a : (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u)) → (↑A : Type u))
            y :
        (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)) : Type u) ⊗[R] (↑A : Type u)) :
    (↑A : Type u))
  ((Hom.hom (MonoidalCategoryStruct.whiskerRight η A ≫ μ) :
        (a : (↑(MonoidalCategoryStruct.tensorObj (MonoidalCategoryStruct.tensorUnit (ModuleCat R)) A) : Type u)) →
          (↑A : Type u))
      (x ⊗ₜ[R]
          (@ConcreteCategory.hom _ (moduleCategory R) (fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u))
                carrier _ _ (MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)) A η :
              (a : (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u)) → (↑A : Type u))
            y :
        (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)) : Type u) ⊗[R] (↑A : Type u)) :
    (↑A : Type u))
Tactic: have h := LinearMap.congr_fun (ModuleCat.hom_ext_iff.mp (one_mul A).symm) (x ⊗ₜ η[A] y)
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (goal)
Goal:
R : Type u
inst : CommRing R
A : ModuleCat R
inst_1 : MonObj A
r : R
a : (↑A : Type u)
⊢ HMul.hMul (α := (↑A : Type u))
      ((let __src :
              (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)) : Type u) →ₗ[R]
                (↑A : Type u) :=
              Hom.hom η;
            { toFun := __src.toFun, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ :
              R →+* (↑A : Type u) } :
            (a : R) → (↑A : Type u))
          r :
        (↑A : Type u))
      a =
    HMul.hMul (β := (↑A : Type u)) a
      ((let __src :
              (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)) : Type u) →ₗ[R]
                (↑A : Type u) :=
              Hom.hom η;
            { toFun := __src.toFun, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ } :
            (a : R) → (↑A : Type u))
          r :
        (↑A : Type u))
have dsimp : HMul.hMul (α := (↑A : Type u))
    ((Hom.hom η : (a : (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u)) → (↑A : Type u)) r :
      (↑A : Type u))
    a =
  HMul.hMul (β := (↑A : Type u)) a
    ((Hom.hom η : (a : (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u)) → (↑A : Type u)) r :
      (↑A : Type u))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (hypothesis)
Goal:
R : Type u
inst : CommRing R
A : ModuleCat R
inst_1 : MonObj A
r : R
a : (↑A : Type u)
⊢ HMul.hMul (α := (↑A : Type u))
      ((Hom.hom η : (a : (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u)) → (↑A : Type u)) r :
        (↑A : Type u))
      a =
    HMul.hMul (β := (↑A : Type u)) a
      ((Hom.hom η : (a : (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u)) → (↑A : Type u)) r :
        (↑A : Type u))
have h₁ : Eq (α := (↑A : Type u))
  ((Hom.hom (MonoidalCategoryStruct.whiskerRight η A ≫ μ) :
        (a : (↑(MonoidalCategoryStruct.tensorObj (MonoidalCategoryStruct.tensorUnit (ModuleCat R)) A) : Type u)) →
          (↑A : Type u))
      (r ⊗ₜ[R] a) :
    (↑A : Type u))
  ((Hom.hom (Iso.hom (MonoidalCategoryStruct.leftUnitor A)) :
        (a : (↑(MonoidalCategoryStruct.tensorObj (MonoidalCategoryStruct.tensorUnit (ModuleCat R)) A) : Type u)) →
          (↑A : Type u))
      (r ⊗ₜ[R] a) :
    (↑A : Type u))
Tactic: have h₁ := LinearMap.congr_fun (ModuleCat.hom_ext_iff.mp (one_mul A)) (r ⊗ₜ a)
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (hypothesis)
Goal:
R : Type u
inst : CommRing R
A : ModuleCat R
inst_1 : MonObj A
r : R
a : (↑A : Type u)
h₁ :
  Eq (α := (↑A : Type u))
    ((Hom.hom (MonoidalCategoryStruct.whiskerRight η A ≫ μ) :
          (a : (↑(MonoidalCategoryStruct.tensorObj (MonoidalCategoryStruct.tensorUnit (ModuleCat R)) A) : Type u)) →
            (↑A : Type u))
        (r ⊗ₜ[R] a) :
      (↑A : Type u))
    ((Hom.hom (Iso.hom (MonoidalCategoryStruct.leftUnitor A)) :
          (a : (↑(MonoidalCategoryStruct.tensorObj (MonoidalCategoryStruct.tensorUnit (ModuleCat R)) A) : Type u)) →
            (↑A : Type u))
        (r ⊗ₜ[R] a) :
      (↑A : Type u))
⊢ HMul.hMul (α := (↑A : Type u))
      ((Hom.hom η : (a : (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u)) → (↑A : Type u)) r :
        (↑A : Type u))
      a =
    HMul.hMul (β := (↑A : Type u)) a
      ((Hom.hom η : (a : (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u)) → (↑A : Type u)) r :
        (↑A : Type u))
have h₂ : Eq (α := (↑A : Type u))
  ((Hom.hom (MonoidalCategoryStruct.whiskerLeft A η ≫ μ) :
        (a : (↑(MonoidalCategoryStruct.tensorObj A (MonoidalCategoryStruct.tensorUnit (ModuleCat R))) : Type u)) →
          (↑A : Type u))
      (a ⊗ₜ[R] r) :
    (↑A : Type u))
  ((Hom.hom (Iso.hom (MonoidalCategoryStruct.rightUnitor A)) :
        (a : (↑(MonoidalCategoryStruct.tensorObj A (MonoidalCategoryStruct.tensorUnit (ModuleCat R))) : Type u)) →
          (↑A : Type u))
      (a ⊗ₜ[R] r) :
    (↑A : Type u))
Tactic: have h₂ := LinearMap.congr_fun (ModuleCat.hom_ext_iff.mp (mul_one A)) (a ⊗ₜ r)
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (goal)
Goal:
R : Type u
inst : CommRing R
A : AlgCat R
⊢ Eq (α :=
    Quiver.Hom (V := ModuleCat R)
      (MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R)
        (MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)) (of R (↑A : Type u)))
      (of R (↑A : Type u)))
    (CategoryStruct.comp (Y :=
      MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u)) (of R (↑A : Type u)))
      (MonoidalCategoryStruct.whiskerRight (↟(Algebra.linearMap R (↑A : Type u))) (of R (↑A : Type u)))
      (↟(mul' R (↑A : Type u))))
    (Iso.hom (MonoidalCategoryStruct.leftUnitor (of R (↑A : Type u))))
have hf : Eq (α :=
  (↑(MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R)
          (MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)) (of R (↑A : Type u))) :
      Type u) →ₗ[R]
    (↑(of R (↑A : Type u)) : Type u))
  (Hom.hom
    (CategoryStruct.comp (Y :=
      MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u)) (of R (↑A : Type u)))
      (MonoidalCategoryStruct.whiskerRight (↟(Algebra.linearMap R (↑A : Type u))) (of R (↑A : Type u)))
      (↟(mul' R (↑A : Type u)))))
  (Hom.hom (Iso.hom (MonoidalCategoryStruct.leftUnitor (of R (↑A : Type u)))))
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (goal)
Goal:
R : Type u
inst : CommRing R
A : AlgCat R
⊢ Eq (α :=
    Quiver.Hom (V := ModuleCat R)
      (MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u))
        (MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)))
      (of R (↑A : Type u)))
    (CategoryStruct.comp (Y :=
      MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u)) (of R (↑A : Type u)))
      (MonoidalCategoryStruct.whiskerLeft (of R (↑A : Type u)) (↟(Algebra.linearMap R (↑A : Type u))))
      (↟(mul' R (↑A : Type u))))
    (Iso.hom (MonoidalCategoryStruct.rightUnitor (of R (↑A : Type u))))
have hf : Eq (α :=
  (↑(MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u))
          (MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R))) :
      Type u) →ₗ[R]
    (↑(of R (↑A : Type u)) : Type u))
  (Hom.hom
    (CategoryStruct.comp (Y :=
      MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u)) (of R (↑A : Type u)))
      (MonoidalCategoryStruct.whiskerLeft (of R (↑A : Type u)) (↟(Algebra.linearMap R (↑A : Type u))))
      (↟(mul' R (↑A : Type u)))))
  (Hom.hom (Iso.hom (MonoidalCategoryStruct.rightUnitor (of R (↑A : Type u)))))
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (goal)
Goal:
R : Type u
inst : CommRing R
A : AlgCat R
x : (↑(of R (↑A : Type u)) : Type u)
⊢ Eq (α := (↑A : Type u))
    (((mul' R (↑A : Type u) : (↑A : Type u) ⊗[R] (↑A : Type u) →ₗ[R] (↑A : Type u)) :
          (a : (↑(MonoidalCategoryStruct.tensorObj (of R (↑A : Type u)) (of R (↑A : Type u))) : Type u)) →
            (↑A : Type u))
        (((Hom.hom (MonoidalCategoryStruct.whiskerLeft (of R (↑A : Type u)) (↟(Algebra.linearMap R (↑A : Type u)))) :
                (↑(MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u))
                        (MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R))) :
                    Type u) →ₗ[R]
                  (↑(MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u))
                        (of R (↑A : Type u))) :
                    Type u)) :
              (a : (↑A : Type u) ⊗[R] (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u)) →
                (↑(MonoidalCategoryStruct.tensorObj (of R (↑A : Type u)) (of R (↑A : Type u))) : Type u))
            ((((TensorProduct.mk R (↑A : Type u)
                        (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)) : Type u) :
                      (a : (↑A : Type u)) →
                        R →ₗ[R] (↑A : Type u) ⊗[R] (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u))
                    x :
                  R →ₗ[R]
                    (↑A : Type u) ⊗[R]
                      (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)) : Type u)) :
                (a : R) → (↑A : Type u) ⊗[R] (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u))
              1) :
          (↑(MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u)) (of R (↑A : Type u))) :
            Type u)) :
      (↑A : Type u))
    ((Hom.hom (Iso.hom (MonoidalCategoryStruct.rightUnitor (of R (↑A : Type u)))) :
          (a : (↑A : Type u) ⊗[R] (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u)) → (↑A : Type u))
        ((((TensorProduct.mk R (↑A : Type u)
                    (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)) : Type u) :
                  (a : (↑A : Type u)) →
                    R →ₗ[R] (↑A : Type u) ⊗[R] (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u))
                x :
              R →ₗ[R]
                (↑A : Type u) ⊗[R]
                  (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R) (𝒞 := moduleCategory R)) : Type u)) :
            (a : R) → (↑A : Type u) ⊗[R] (↑(MonoidalCategoryStruct.tensorUnit (ModuleCat R)) : Type u))
          1) :
      (↑A : Type u))
have hf : HSMul.hSMul (α := R) (β := (↑A : Type u)) (x, 1).2
    ((LinearMap.id : (a : (↑(of R (↑A : Type u)) : Type u)) → (↑(of R (↑A : Type u)) : Type u)) (x, 1).1 :
      (↑(of R (↑A : Type u)) : Type u)) =
  HSMul.hSMul (α := R) 1 x
Tactic: erw [LinearMap.mul'_apply, ModuleCat.MonoidalCategory.rightUnitor_hom_apply, ← Algebra.commutes,
      ← Algebra.smul_def]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (goal)
Goal:
R : Type u
inst : CommRing R
A : AlgCat R
⊢ Eq (α :=
    Quiver.Hom (V := ModuleCat R)
      (MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R)
        (MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u)) (of R (↑A : Type u)))
        (of R (↑A : Type u)))
      (of R (↑A : Type u)))
    (CategoryStruct.comp (Y :=
      MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u)) (of R (↑A : Type u)))
      (MonoidalCategoryStruct.whiskerRight (↟(mul' R (↑A : Type u))) (of R (↑A : Type u))) (↟(mul' R (↑A : Type u))))
    (Iso.hom (MonoidalCategoryStruct.associator (of R (↑A : Type u)) (of R (↑A : Type u)) (of R (↑A : Type u))) ≫
      CategoryStruct.comp (Y :=
        MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u)) (of R (↑A : Type u)))
        (MonoidalCategoryStruct.whiskerLeft (of R (↑A : Type u)) (↟(mul' R (↑A : Type u)))) (↟(mul' R (↑A : Type u))))
have hf : Eq (α :=
  (↑(MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R)
          (MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u)) (of R (↑A : Type u)))
          (of R (↑A : Type u))) :
      Type u) →ₗ[R]
    (↑(of R (↑A : Type u)) : Type u))
  (Hom.hom
    (CategoryStruct.comp (Y :=
      MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u)) (of R (↑A : Type u)))
      (MonoidalCategoryStruct.whiskerRight (↟(mul' R (↑A : Type u))) (of R (↑A : Type u))) (↟(mul' R (↑A : Type u)))))
  (Hom.hom
    (Iso.hom (MonoidalCategoryStruct.associator (of R (↑A : Type u)) (of R (↑A : Type u)) (of R (↑A : Type u))) ≫
      CategoryStruct.comp (Y :=
        MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (of R (↑A : Type u)) (of R (↑A : Type u)))
        (MonoidalCategoryStruct.whiskerLeft (of R (↑A : Type u)) (↟(mul' R (↑A : Type u)))) (↟(mul' R (↑A : Type u)))))
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (goal)
Goal:
R : Type u
inst : CommRing R
A : Mon (ModuleCat R)
⊢ Eq.{u + 1} (α :=
    MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (Functor.obj (𝟭 (Mon (ModuleCat R))) A).X
        (Functor.obj (𝟭 (Mon (ModuleCat R))) A).X ⟶
      (Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X)
    (μ ≫ ↟{ toFun := _root_.id, map_add' := ⋯, map_smul' := ⋯ })
    (MonoidalCategoryStruct.tensorHom (↟{ toFun := _root_.id, map_add' := ⋯, map_smul' := ⋯ })
        (↟{ toFun := _root_.id, map_add' := ⋯, map_smul' := ⋯ }) ≫
      μ)
have hf : Eq.{u + 1} (α :=
  (↑(MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R) (Functor.obj (𝟭 (Mon (ModuleCat R))) A).X
          (Functor.obj (𝟭 (Mon (ModuleCat R))) A).X) :
      Type u) →ₗ[R]
    (↑(Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X : Type u))
  (Hom.hom (μ ≫ ↟{ toFun := _root_.id, map_add' := sorry, map_smul' := sorry }))
  (Hom.hom
    (MonoidalCategoryStruct.tensorHom (↟{ toFun := _root_.id, map_add' := sorry, map_smul' := sorry })
        (↟{ toFun := _root_.id, map_add' := sorry, map_smul' := sorry }) ≫
      μ))
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Module (goal)
Goal:
R : Type u
inst : CommRing R
A : Mon (ModuleCat R)
⊢ Eq.{u + 1} (α :=
    MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R)
        (Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X
        (Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X ⟶
      (Functor.obj (𝟭 (Mon (ModuleCat R))) A).X)
    (μ ≫ ↟{ toFun := _root_.id, map_add' := ⋯, map_smul' := ⋯ })
    (MonoidalCategoryStruct.tensorHom (↟{ toFun := _root_.id, map_add' := ⋯, map_smul' := ⋯ })
        (↟{ toFun := _root_.id, map_add' := ⋯, map_smul' := ⋯ }) ≫
      μ)
have hf : Eq.{u + 1} (α :=
  (↑(MonoidalCategoryStruct.tensorObj (𝒞 := moduleCategory R)
          (Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X
          (Functor.obj (functor ⋙ MonModuleEquivalenceAlgebra.inverse) A).X) :
      Type u) →ₗ[R]
    (↑(Functor.obj (𝟭 (Mon (ModuleCat R))) A).X : Type u))
  (Hom.hom (μ ≫ ↟{ toFun := _root_.id, map_add' := sorry, map_smul' := sorry }))
  (Hom.hom
    (MonoidalCategoryStruct.tensorHom (↟{ toFun := _root_.id, map_add' := sorry, map_smul' := sorry })
        (↟{ toFun := _root_.id, map_add' := sorry, map_smul' := sorry }) ≫
      μ))
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Types.Basic (goal)
Goal:
A : MonCat
⊢ Eq (α :=
    MonoidalCategoryStruct.tensorObj (𝒞 := types)
        (MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u)) (↑A : Type u) ⟶
      (↑A : Type u))
    (CategoryStruct.comp (Y := MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u))
      (MonoidalCategoryStruct.whiskerRight
        (↾fun (p : MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u)) ↦ p.1 * p.2)
        (↑A : Type u))
      (↾fun (p : MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u)) ↦ p.1 * p.2))
    (Iso.hom (MonoidalCategoryStruct.associator (↑A : Type u) (↑A : Type u) (↑A : Type u)) ≫
      CategoryStruct.comp (Y := MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u))
        (MonoidalCategoryStruct.whiskerLeft (↑A : Type u)
          (↾fun (p : MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u)) ↦ p.1 * p.2))
        (↾fun (p : MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u)) ↦ p.1 * p.2))
have h : ∀ (z x y : (↑A : Type u)),
  Eq (α := (↑A : Type u))
    (TypeCat.Fun.toFun
      (hom (FC := TypeCat.Fun)
        (CategoryStruct.comp (Y := MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u))
          (MonoidalCategoryStruct.whiskerRight
            (↾fun (p : MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u)) ↦ p.1 * p.2)
            (↑A : Type u))
          (↾fun (p : MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u)) ↦ p.1 * p.2)))
      ((x, y), z))
    (TypeCat.Fun.toFun
      (hom (FC := TypeCat.Fun)
        (Iso.hom (MonoidalCategoryStruct.associator (↑A : Type u) (↑A : Type u) (↑A : Type u)) ≫
          CategoryStruct.comp (Y := MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u))
            (MonoidalCategoryStruct.whiskerLeft (↑A : Type u)
              (↾fun (p : MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u)) ↦ p.1 * p.2))
            (↾fun (p : MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑A : Type u) (↑A : Type u)) ↦ p.1 * p.2)))
      ((x, y), z))
Tactic: ext ⟨⟨x, y⟩, z⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Types.Basic (goal)
Goal:
X Y : MonCat
f : X ⟶ Y
⊢ Eq (α :=
    MonoidalCategoryStruct.tensorUnit (Type u) (𝒞 := types) ⟶
      Mon.X
        { X := (↑Y : Type u),
          mon :=
            { one := ↾fun (x : MonoidalCategoryStruct.tensorUnit (Type u)) ↦ 1,
              mul := ↾fun (p : MonoidalCategoryStruct.tensorObj (↑Y : Type u) (↑Y : Type u)) ↦ p.1 * p.2, one_mul := ⋯,
              mul_one := ⋯, mul_assoc := ⋯ } })
    (CategoryStruct.comp (Y :=
      Mon.X
        { X := (↑X : Type u),
          mon :=
            { one := ↾fun (x : MonoidalCategoryStruct.tensorUnit (Type u)) ↦ 1,
              mul := ↾fun (p : MonoidalCategoryStruct.tensorObj (↑X : Type u) (↑X : Type u)) ↦ p.1 * p.2, one_mul := ⋯,
              mul_one := ⋯, mul_assoc := ⋯ } })
      η
      (↾(⇑(hom (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := MonCat.carrier) f) :
          (a : (↑X : Type u)) → (↑Y : Type u))))
    η
have simp : Eq (α := MonoidalCategoryStruct.tensorUnit (Type u) (𝒞 := types) ⟶ (↑Y : Type u))
  (CategoryStruct.comp (Y := (↑X : Type u)) (↾fun (x : MonoidalCategoryStruct.tensorUnit (Type u) (𝒞 := types)) ↦ 1)
    (↾(⇑(hom (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := MonCat.carrier) f) :
        (a : (↑X : Type u)) → (↑Y : Type u))))
  (↾fun (x : MonoidalCategoryStruct.tensorUnit (Type u) (𝒞 := types)) ↦ 1)
Tactic: simp +instances only
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Internal.Types.Basic (goal)
Goal:
X Y : MonCat
f : X ⟶ Y
⊢ Eq (α :=
    MonoidalCategoryStruct.tensorObj (𝒞 := types)
        (Mon.X
          { X := (↑X : Type u),
            mon :=
              { one := ↾fun (x : MonoidalCategoryStruct.tensorUnit (Type u)) ↦ 1,
                mul := ↾fun (p : MonoidalCategoryStruct.tensorObj (↑X : Type u) (↑X : Type u)) ↦ p.1 * p.2,
                one_mul := ⋯, mul_one := ⋯, mul_assoc := ⋯ } :
            Mon (Type u) })
        (Mon.X
          { X := (↑X : Type u),
            mon :=
              { one := ↾fun (x : MonoidalCategoryStruct.tensorUnit (Type u)) ↦ 1,
                mul := ↾fun (p : MonoidalCategoryStruct.tensorObj (↑X : Type u) (↑X : Type u)) ↦ p.1 * p.2,
                one_mul := ⋯, mul_one := ⋯, mul_assoc := ⋯ } :
            Mon (Type u) }) ⟶
      Mon.X
        { X := (↑Y : Type u),
          mon :=
            { one := ↾fun (x : MonoidalCategoryStruct.tensorUnit (Type u)) ↦ 1,
              mul := ↾fun (p : MonoidalCategoryStruct.tensorObj (↑Y : Type u) (↑Y : Type u)) ↦ p.1 * p.2, one_mul := ⋯,
              mul_one := ⋯, mul_assoc := ⋯ } })
    (μ ≫
      ↾(⇑(hom (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := MonCat.carrier) f) :
          (a : (↑X : Type u)) → (↑Y : Type u)))
    (MonoidalCategoryStruct.tensorHom
        (↾(⇑(hom (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := MonCat.carrier) f) :
            (a : (↑X : Type u)) → (↑Y : Type u)))
        (↾(⇑(hom (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := MonCat.carrier) f) :
            (a : (↑X : Type u)) → (↑Y : Type u))) ≫
      μ)
have simp : Eq (α := MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑X : Type u) (↑X : Type u) ⟶ (↑Y : Type u))
  (CategoryStruct.comp (Y := (↑X : Type u))
    (↾fun (p : MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑X : Type u) (↑X : Type u)) ↦ p.1 * p.2)
    (↾(⇑(hom (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := MonCat.carrier) f) :
        (a : (↑X : Type u)) → (↑Y : Type u))))
  (CategoryStruct.comp (Y := MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑Y : Type u) (↑Y : Type u))
    (MonoidalCategoryStruct.tensorHom
      (↾(⇑(hom (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := MonCat.carrier) f) :
          (a : (↑X : Type u)) → (↑Y : Type u)))
      (↾(⇑(hom (FC := fun (x1 x2 : MonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CC := MonCat.carrier) f) :
          (a : (↑X : Type u)) → (↑Y : Type u))))
    (↾fun (p : MonoidalCategoryStruct.tensorObj (𝒞 := types) (↑Y : Type u) (↑Y : Type u)) ↦ p.1 * p.2))
Tactic: simp +instances only