-- Module: Mathlib.RingTheory.Polynomial.Chebyshev
Goal:
R : Type u_1
inst : CommRing R
n : ℤ
k : ℕ
h :
  (⇑derivative : (a : R[X]) → R[X])^[k + 2] (T R n) -
      ((⇑derivative : (a : R[X]) → R[X])^[k + 2] (T R n) * HPow.hPow (α := R[X]) (β := ℕ) X 2 +
          HSMul.hSMul (β := R[X]) (2 * k) ((⇑derivative : (a : R[X]) → R[X])^[k + 1] (T R n)) * X +
        HSMul.hSMul (β := R[X]) (k * (k - 1)) ((⇑derivative : (a : R[X]) → R[X])^[k] (T R n))) =
    HMul.hMul (α := R[X]) ((⇑derivative : (a : R[X]) → R[X])^[k + 1] (T R n)) X +
        HSMul.hSMul (β := R[X]) k ((⇑derivative : (a : R[X]) → R[X])^[k] (T R n)) -
      HMul.hMul (α := R[X]) (↑(HPow.hPow (β := ℕ) n 2) : R[X]) ((⇑derivative : (a : R[X]) → R[X])^[k] (T R n))
a :
  ∀
    (h :
      (⇑derivative : (a : R[X]) → R[X])^[HAdd.hAdd (α := ℕ) 0 2] (T R n) -
          ((⇑derivative : (a : R[X]) → R[X])^[HAdd.hAdd (α := ℕ) 0 2] (T R n) * HPow.hPow (α := R[X]) (β := ℕ) X 2 +
              HSMul.hSMul (β := R[X]) (HMul.hMul (α := ℕ) 2 0)
                  ((⇑derivative : (a : R[X]) → R[X])^[HAdd.hAdd (α := ℕ) 0 1] (T R n)) *
                X +
            HSMul.hSMul (β := R[X]) (0 * HSub.hSub (α := ℕ) 0 1) ((⇑derivative : (a : R[X]) → R[X])^[0] (T R n))) =
        HMul.hMul (α := R[X]) ((⇑derivative : (a : R[X]) → R[X])^[HAdd.hAdd (α := ℕ) 0 1] (T R n)) X +
            HSMul.hSMul (α := ℕ) (β := R[X]) 0 ((⇑derivative : (a : R[X]) → R[X])^[0] (T R n)) -
          HMul.hMul (α := R[X]) (↑(HPow.hPow (β := ℕ) n 2) : R[X]) ((⇑derivative : (a : R[X]) → R[X])^[0] (T R n))),
    HMul.hMul (α := R[X]) ((⇑derivative : (a : R[X]) → R[X])^[0] (T R n)) (↑0 : R[X]) +
          HMul.hMul (α := R[X]) ((⇑derivative : (a : R[X]) → R[X])^[0] (T R n)) (↑0 : R[X]) *
            (↑(HSub.hSub (α := ℕ) 0 1) : R[X]) -
        (⇑derivative : (a : R[X]) → R[X])^[0] (T R n) * HPow.hPow (α := R[X]) (β := ℕ) (↑0 : R[X]) 2 =
      0
a₁ :
  ∀ (n_1 : ℕ)
    (h :
      (⇑derivative : (a : R[X]) → R[X])^[n_1 + 1 + 2] (T R n) -
          ((⇑derivative : (a : R[X]) → R[X])^[n_1 + 1 + 2] (T R n) * HPow.hPow (α := R[X]) (β := ℕ) X 2 +
              HSMul.hSMul (β := R[X]) (2 * (n_1 + 1)) ((⇑derivative : (a : R[X]) → R[X])^[n_1 + 1 + 1] (T R n)) * X +
            HSMul.hSMul (β := R[X]) ((n_1 + 1) * (n_1 + 1 - 1)) ((⇑derivative : (a : R[X]) → R[X])^[n_1 + 1] (T R n))) =
        HMul.hMul (α := R[X]) ((⇑derivative : (a : R[X]) → R[X])^[n_1 + 1 + 1] (T R n)) X +
            HSMul.hSMul (β := R[X]) (n_1 + 1) ((⇑derivative : (a : R[X]) → R[X])^[n_1 + 1] (T R n)) -
          HMul.hMul (α := R[X]) (↑(HPow.hPow (β := ℕ) n 2) : R[X])
            ((⇑derivative : (a : R[X]) → R[X])^[n_1 + 1] (T R n))),
    HMul.hMul (α := R[X]) ((⇑derivative : (a : R[X]) → R[X])^[n_1 + 1] (T R n)) (↑(n_1 + 1) : R[X]) +
          HMul.hMul (α := R[X]) ((⇑derivative : (a : R[X]) → R[X])^[n_1 + 1] (T R n)) (↑(n_1 + 1) : R[X]) *
            (↑(n_1 + 1 - 1) : R[X]) -
        (⇑derivative : (a : R[X]) → R[X])^[n_1 + 1] (T R n) * HPow.hPow (α := R[X]) (β := ℕ) (↑(n_1 + 1) : R[X]) 2 =
      0
⊢ HMul.hMul (α := R[X]) ((⇑derivative : (a : R[X]) → R[X])^[k] (T R n)) (↑k : R[X]) +
        HMul.hMul (α := R[X]) ((⇑derivative : (a : R[X]) → R[X])^[k] (T R n)) (↑k : R[X]) * (↑(k - 1) : R[X]) -
      (⇑derivative : (a : R[X]) → R[X])^[k] (T R n) * HPow.hPow (α := R[X]) (β := ℕ) (↑k : R[X]) 2 =
    0

Tactic:
cases k