/-- This is not an instance for the same reasons as `Polynomial.hasSMulPi'`. -/
noncomputable def Polynomial.algebraPi : Algebra R'[X] (S' → T') :=
  { Polynomial.hasSMulPi' R' S' T' with
    toFun := fun p z => algebraMap S' T' (aeval z p)
    map_one' := by
      funext z
      simp only [Polynomial.aeval_one, Pi.one_apply, map_one]
    map_mul' := fun f g => by
      funext z
      simp only [Pi.mul_apply, map_mul]
    map_zero' := by
      funext z
      simp only [Polynomial.aeval_zero, Pi.zero_apply, map_zero]
    map_add' := fun f g => by
      funext z
      simp only [Polynomial.aeval_add, Pi.add_apply, map_add]
    commutes' := fun p f => by
      funext z
      exact mul_comm _ _
    smul_def' := fun p f => by
      

Proof state:
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : IsDomain R
inst✝⁵ : CommRing S
R' : Type u
S' : Type v
T' : Type w
inst✝⁴ : CommSemiring R'
inst✝³ : CommSemiring S'
inst✝² : CommSemiring T'
inst✝¹ : Algebra R' S'
inst✝ : Algebra S' T'
src✝ : SMul R'[X] (S' → T') := hasSMulPi' R' S' T'
p : R'[X]
f : S' → T'
⊢ p • f =
    {
          toMonoidHom :=
            {
              toOneHom :=
                { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                  map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) },
              map_mul' :=
                (_ :
                  ∀ (f g : R'[X]),
                    OneHom.toFun
                        { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                          map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                        (f * g) =
                      OneHom.toFun
                          { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                            map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                          f *
                        OneHom.toFun
                          { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                            map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                          g) },
          map_zero' :=
            (_ :
              OneHom.toFun
                  (↑{
                      toOneHom :=
                        { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                          map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) },
                      map_mul' :=
                        (_ :
                          ∀ (f g : R'[X]),
                            OneHom.toFun
                                { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                                  map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                                (f * g) =
                              OneHom.toFun
                                  { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                                    map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                                  f *
                                OneHom.toFun
                                  { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                                    map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                                  g) })
                  0 =
                0),
          map_add' :=
            (_ :
              ∀ (f g : R'[X]),
                OneHom.toFun
                    (↑{
                        toOneHom :=
                          { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                            map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) },
                        map_mul' :=
                          (_ :
                            ∀ (f g : R'[X]),
                              OneHom.toFun
                                  { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                                    map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                                  (f * g) =
                                OneHom.toFun
                                    { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                                      map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                                    f *
                                  OneHom.toFun
                                    { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                                      map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                                    g) })
                    (f + g) =
                  OneHom.toFun
                      (↑{
                          toOneHom :=
                            { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                              map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) },
                          map_mul' :=
                            (_ :
                              ∀ (f g : R'[X]),
                                OneHom.toFun
                                    { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                                      map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                                    (f * g) =
                                  OneHom.toFun
                                      { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                                        map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                                      f *
                                    OneHom.toFun
                                      { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                                        map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                                      g) })
                      f +
                    OneHom.toFun
                      (↑{
                          toOneHom :=
                            { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                              map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) },
                          map_mul' :=
                            (_ :
                              ∀ (f g : R'[X]),
                                OneHom.toFun
                                    { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                                      map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                                    (f * g) =
                                  OneHom.toFun
                                      { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                                        map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                                      f *
                                    OneHom.toFun
                                      { toFun := fun p z => (algebraMap S' T') ((aeval z) p),
                                        map_one' := (_ : (fun p z => (algebraMap S' T') ((aeval z) p)) 1 = 1) }
                                      g) })
                      g) }
        p *
      f

Next tactic:
funext z