-- Module: Mathlib.Algebra.MvPolynomial.Derivation
Goal:
σ : Type u_1
R : Type u_2
A : Type u_3
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : Module R A
inst_3 : Module (MvPolynomial σ R) A
inst_4 : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : Eq.{u_3 + 1} (α := A) ((D : (a : MvPolynomial σ R) → A) 1 : A) (0 : A)
H :
  ∀ (s : σ →₀ ℕ) (i : σ),
    (D : (a : MvPolynomial σ R) → A)
        (HMul.hMul (α := MvPolynomial σ R) ((monomial s : (a : R) → MvPolynomial σ R) 1) (X i)) =
      HSMul.hSMul (α := MvPolynomial σ R) (β := A) ((monomial s : (a : R) → MvPolynomial σ R) 1)
          ((D : (a : MvPolynomial σ R) → A) (X i) : A) +
        HSMul.hSMul (α := MvPolynomial σ R) (β := A) (X i)
          ((D : (a : MvPolynomial σ R) → A) ((monomial s : (a : R) → MvPolynomial σ R) 1) : A)
hC :
  ∀ (r : R),
    Eq.{u_3 + 1} (α := A)
      ((D : (a : MvPolynomial σ R) → A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) r : MvPolynomial σ R) : A)
      (0 : A)
this :
  ∀ (p : MvPolynomial σ R) (i : σ),
    (D : (a : MvPolynomial σ R) → A) (p * X i) =
      HSMul.hSMul (β := A) p ((D : (a : MvPolynomial σ R) → A) (X i) : A) +
        HSMul.hSMul (α := MvPolynomial σ R) (β := A) (X i) ((D : (a : MvPolynomial σ R) → A) p : A)
p q : MvPolynomial σ R
C :
  ∀ (c : R),
    (D : (a : MvPolynomial σ R) → A)
        (HMul.hMul (β := MvPolynomial σ R) p ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R)) =
      HSMul.hSMul (β := A) p
          ((D : (a : MvPolynomial σ R) → A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R) : A) +
        HSMul.hSMul (α := MvPolynomial σ R) (β := A)
          ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R) ((D : (a : MvPolynomial σ R) → A) p : A)
add :
  ∀ (q₁ q₂ : MvPolynomial σ R)
    (h₁ :
      (D : (a : MvPolynomial σ R) → A) (p * q₁) =
        HSMul.hSMul (β := A) p ((D : (a : MvPolynomial σ R) → A) q₁ : A) +
          HSMul.hSMul (β := A) q₁ ((D : (a : MvPolynomial σ R) → A) p : A))
    (h₂ :
      (D : (a : MvPolynomial σ R) → A) (p * q₂) =
        HSMul.hSMul (β := A) p ((D : (a : MvPolynomial σ R) → A) q₂ : A) +
          HSMul.hSMul (β := A) q₂ ((D : (a : MvPolynomial σ R) → A) p : A)),
    (D : (a : MvPolynomial σ R) → A) (p * (q₁ + q₂)) =
      HSMul.hSMul (β := A) p ((D : (a : MvPolynomial σ R) → A) (q₁ + q₂) : A) +
        HSMul.hSMul (β := A) (q₁ + q₂) ((D : (a : MvPolynomial σ R) → A) p : A)
mul_X :
  ∀ (q : MvPolynomial σ R) (i : σ)
    (hq :
      (D : (a : MvPolynomial σ R) → A) (p * q) =
        HSMul.hSMul (β := A) p ((D : (a : MvPolynomial σ R) → A) q : A) +
          HSMul.hSMul (β := A) q ((D : (a : MvPolynomial σ R) → A) p : A)),
    (D : (a : MvPolynomial σ R) → A) (p * (q * X i)) =
      HSMul.hSMul (β := A) p ((D : (a : MvPolynomial σ R) → A) (q * X i) : A) +
        HSMul.hSMul (β := A) (q * X i) ((D : (a : MvPolynomial σ R) → A) p : A)
⊢ (D : (a : MvPolynomial σ R) → A) (p * q) =
    HSMul.hSMul (β := A) p ((D : (a : MvPolynomial σ R) → A) q : A) +
      HSMul.hSMul (β := A) q ((D : (a : MvPolynomial σ R) → A) p : A)

Tactic:
induction q using MvPolynomial.induction_on with
  | C c =>
    rw [mul_comm, C_mul', hC, smul_zero, zero_add, D.map_smul, C_eq_smul_one, smul_one_smul]
  | add q₁ q₂ h₁ h₂ => simp only [mul_add, map_add, h₁, h₂, smul_add, add_smul]; abel
  | mul_X q i hq =>
    simp only [this, ← mul_assoc, hq, mul_smul, smul_add, add_assoc]
    rw [smul_comm (X i), smul_comm (X i)]

Premises used: add_assoc, mul_assoc, smul_zero, map_add, MvPolynomial.induction_on, zero_add, MvPolynomial.C_mul', add_smul, smul_one_smul, LinearMap.map_smul, mul_comm, mul_add, MvPolynomial.C, SemigroupAction.mul_smul, smul_add, MvPolynomial.C_eq_smul_one