/-- If `M` is a representation of two different `R`-algebras `A` and `B` whose actions commute,
then it is a representation the `R`-algebra `A ⊗[R] B`.

An important example arises from a semiring `S`; allowing `S` to act on itself via left and right
multiplication, the roles of `R`, `A`, `B`, `M` are played by `ℕ`, `S`, `Sᵐᵒᵖ`, `S`. This example
is important because a submodule of `S` as a `Module` over `S ⊗[ℕ] Sᵐᵒᵖ` is a two-sided ideal.

NB: This is not an instance because in the case `B = A` and `M = A ⊗[R] A` we would have a diamond
of `smul` actions. Furthermore, this would not be a mere definitional diamond but a true
mathematical diamond in which `A ⊗[R] A` had two distinct scalar actions on itself: one from its
multiplication, and one from this would-be instance. Arguably we could live with this but in any
case the real fix is to address the ambiguity in notation, probably along the lines outlined here:
https://leanprover.zulipchat.com/#narrow/stream/144837-PR-reviews/topic/.234773.20base.20change/near/240929258
-/
protected def module : Module (A ⊗[R] B) M where
  smul x m := moduleAux x m
  zero_smul m := by simp only [(· • ·), map_zero, LinearMap.zero_apply]
  smul_zero x := by simp only [(· • ·), map_zero]
  smul_add x m₁ m₂ := by simp only [(· • ·), map_add]
  add_smul x y m := by simp only [(· • ·), map_add, LinearMap.add_apply]
  one_smul m := by
    -- porting note: was one `simp only` not two in lean 3
    simp only [(· • ·), Algebra.TensorProduct.one_def]
    simp only [moduleAux_apply, one_smul]
  mul_smul x y m := by
    refine TensorProduct.induction_on x ?_ ?_ ?_ <;> refine TensorProduct.induction_on y ?_ ?_ ?_
    · simp only [(· • ·), mul_zero, map_zero, LinearMap.zero_apply]
    · intro a b
      simp only [(· • ·), zero_mul, map_zero, LinearMap.zero_apply]
    · intro z w _ _
      simp only [(· • ·), zero_mul, map_zero, LinearMap.zero_apply]
    · intro a b
      simp only [(· • ·), mul_zero, map_zero, LinearMap.zero_apply]
    · intro a₁ b₁ a₂ b₂
      -- porting note; was one `simp only` not two and a `rw` in mathlib3
      simp only [(· • ·), Algebra.TensorProduct.tmul_mul_tmul]
      simp only [moduleAux_apply, mul_smul]
      rw [smul_comm a₁ b₂]
    · intro z w hz hw a b
      --porting note: was one `simp only` but random stuff doesn't work
      simp only [(· • ·)] at hz hw ⊢
      simp only [moduleAux_apply]
      rw [mul_add]  -- simp only doesn't work
      

Proof state:
case refine_2.refine_3
R : Type u_1
A : Type u_2
B : Type u_3
M : Type u_4
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Module A M
inst✝⁵ : Module B M
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
inst✝² : IsScalarTower R A M
inst✝¹ : IsScalarTower R B M
inst✝ : SMulCommClass A B M
x y : A ⊗[R] B
m : M
z w : A ⊗[R] B
hz : ∀ (x : A) (y : B), (moduleAux (x ⊗ₜ[R] y * z)) m = (moduleAux (x ⊗ₜ[R] y)) ((moduleAux z) m)
hw : ∀ (x : A) (y : B), (moduleAux (x ⊗ₜ[R] y * w)) m = (moduleAux (x ⊗ₜ[R] y)) ((moduleAux w) m)
a : A
b : B
⊢ (moduleAux (a ⊗ₜ[R] b * z + a ⊗ₜ[R] b * w)) m = a • b • (moduleAux (z + w)) m

Next tactic:
simp only [LinearMap.map_add, LinearMap.add_apply, moduleAux_apply, hz, hw, smul_add]