-- Module: Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
Goal:
R_1 : Type u₁
S_1 : Type u₂
inst : CommRing R_1
inst_1 : CommRing S_1
f_1 : R_1 →+* S_1
R : Type u₁
S : Type u₂
inst_2 : CommRing R
inst_3 : CommRing S
f : R →+* S
X : ModuleCat R
Y : ModuleCat S
g : X ⟶ (restrictScalars f).obj Y
m1 : Module R S := Module.compHom S f
m2 : Module R (↑Y : Type (max u₂ u_1)) := Module.compHom (↑Y : Type (max u₂ u_1)) f
s : S
z : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))
refine_3 :
  (lift
          { toFun := fun (s : (↑((restrictScalars f).obj (of S S)) : Type u₂)) ↦ evalAt f s g, map_add' := ⋯,
            map_smul' := ⋯ } :
        (a : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) → (↑Y : Type (max u₂ u_1)))
      (HSMul.hSMul (β := (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) s 0) =
    HSMul.hSMul (β := (↑Y : Type (max u₂ u_1))) s
      ((lift
            { toFun := fun (s : (↑((restrictScalars f).obj (of S S)) : Type u₂)) ↦ evalAt f s g, map_add' := ⋯,
              map_smul' := ⋯ :
              (↑((restrictScalars f).obj (of S S)) : Type u₂) →ₗ[R]
                (↑X : Type (max u_1 u₂)) →ₗ[R] (↑Y : Type (max u₂ u_1)) } :
          (a : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) →
            (↑Y : Type (max u₂ u_1)))
        0)
refine_3₁ :
  ∀ (s' : (↑((restrictScalars f).obj (of S S)) : Type u₂)) (x : (↑X : Type (max u_1 u₂))),
    (lift
            { toFun := fun (s : (↑((restrictScalars f).obj (of S S)) : Type u₂)) ↦ evalAt f s g, map_add' := ⋯,
              map_smul' := ⋯ } :
          (a : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) →
            (↑Y : Type (max u₂ u_1)))
        (HSMul.hSMul (β := (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) s
          (s' ⊗ₜ[R] x)) =
      HSMul.hSMul (β := (↑Y : Type (max u₂ u_1))) s
        ((lift
                { toFun := fun (s : (↑((restrictScalars f).obj (of S S)) : Type u₂)) ↦ evalAt f s g, map_add' := ⋯,
                  map_smul' := ⋯ :
                  (↑((restrictScalars f).obj (of S S)) : Type u₂) →ₗ[R]
                    (↑X : Type (max u_1 u₂)) →ₗ[R] (↑Y : Type (max u₂ u_1)) } :
              (a : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) →
                (↑Y : Type (max u₂ u_1)))
            (s' ⊗ₜ[R] x) :
          (↑Y : Type (max u₂ u_1)))
refine_3₂ :
  ∀ (x y : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂)))
    (ih1 :
      (lift
              { toFun := fun (s : (↑((restrictScalars f).obj (of S S)) : Type u₂)) ↦ evalAt f s g, map_add' := ⋯,
                map_smul' := ⋯ } :
            (a : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) →
              (↑Y : Type (max u₂ u_1)))
          (s • x) =
        HSMul.hSMul (β := (↑Y : Type (max u₂ u_1))) s
          ((lift
                  { toFun := fun (s : (↑((restrictScalars f).obj (of S S)) : Type u₂)) ↦ evalAt f s g, map_add' := ⋯,
                    map_smul' := ⋯ :
                    (↑((restrictScalars f).obj (of S S)) : Type u₂) →ₗ[R]
                      (↑X : Type (max u_1 u₂)) →ₗ[R] (↑Y : Type (max u₂ u_1)) } :
                (a : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) →
                  (↑Y : Type (max u₂ u_1)))
              x :
            (↑Y : Type (max u₂ u_1))))
    (ih2 :
      (lift
              { toFun := fun (s : (↑((restrictScalars f).obj (of S S)) : Type u₂)) ↦ evalAt f s g, map_add' := ⋯,
                map_smul' := ⋯ } :
            (a : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) →
              (↑Y : Type (max u₂ u_1)))
          (s • y) =
        HSMul.hSMul (β := (↑Y : Type (max u₂ u_1))) s
          ((lift
                  { toFun := fun (s : (↑((restrictScalars f).obj (of S S)) : Type u₂)) ↦ evalAt f s g, map_add' := ⋯,
                    map_smul' := ⋯ :
                    (↑((restrictScalars f).obj (of S S)) : Type u₂) →ₗ[R]
                      (↑X : Type (max u_1 u₂)) →ₗ[R] (↑Y : Type (max u₂ u_1)) } :
                (a : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) →
                  (↑Y : Type (max u₂ u_1)))
              y :
            (↑Y : Type (max u₂ u_1)))),
    (lift
            { toFun := fun (s : (↑((restrictScalars f).obj (of S S)) : Type u₂)) ↦ evalAt f s g, map_add' := ⋯,
              map_smul' := ⋯ } :
          (a : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) →
            (↑Y : Type (max u₂ u_1)))
        (s • (x + y)) =
      HSMul.hSMul (β := (↑Y : Type (max u₂ u_1))) s
        ((lift
                { toFun := fun (s : (↑((restrictScalars f).obj (of S S)) : Type u₂)) ↦ evalAt f s g, map_add' := ⋯,
                  map_smul' := ⋯ :
                  (↑((restrictScalars f).obj (of S S)) : Type u₂) →ₗ[R]
                    (↑X : Type (max u_1 u₂)) →ₗ[R] (↑Y : Type (max u₂ u_1)) } :
              (a : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) →
                (↑Y : Type (max u₂ u_1)))
            (x + y) :
          (↑Y : Type (max u₂ u_1)))
⊢ (lift
          { toFun := fun (s : (↑((restrictScalars f).obj (of S S)) : Type u₂)) ↦ evalAt f s g, map_add' := ⋯,
            map_smul' := ⋯ } :
        (a : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) → (↑Y : Type (max u₂ u_1)))
      (s • z) =
    HSMul.hSMul (β := (↑Y : Type (max u₂ u_1))) s
      ((lift
              { toFun := fun (s : (↑((restrictScalars f).obj (of S S)) : Type u₂)) ↦ evalAt f s g, map_add' := ⋯,
                map_smul' := ⋯ :
                (↑((restrictScalars f).obj (of S S)) : Type u₂) →ₗ[R]
                  (↑X : Type (max u_1 u₂)) →ₗ[R] (↑Y : Type (max u₂ u_1)) } :
            (a : (↑((restrictScalars f).obj (of S S)) : Type u₂) ⊗[R] (↑X : Type (max u_1 u₂))) →
              (↑Y : Type (max u₂ u_1)))
          z :
        (↑Y : Type (max u₂ u_1)))

Tactic:
induction z using TensorProduct.induction_on with
    | zero => rw [smul_zero, map_zero, smul_zero]
    | tmul s' x => simp [mul_smul]
    | add _ _ ih1 ih2 => rw [smul_add, map_add, ih1, ih2, map_add, smul_add]

Premises used: smul_zero, map_zero, map_add, TensorProduct.tmul, SemigroupAction.mul_smul, smul_add