-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Chebyshev (goal)
Goal:
R : Type u_1
inst : CommRing R
n : ℕ
ih1 :
  (derivative : (a : R[X]) → R[X]) (T R (-(↑n : ℤ))) =
    HMul.hMul (α := R[X]) (↑(-(↑n : ℤ)) : R[X]) (U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1))
ih2 :
  (derivative : (a : R[X]) → R[X]) (T R (HAdd.hAdd (α := ℤ) (-(↑n : ℤ)) 1)) =
    HMul.hMul (α := R[X]) (↑(HAdd.hAdd (α := ℤ) (-(↑n : ℤ)) 1) : R[X]) (U R (HAdd.hAdd (α := ℤ) (-(↑n : ℤ)) 1 - 1))
h₂ :
  U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 2) =
    HMul.hMul (α := R[X]) 2 X * U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1) - U R (-(↑n : ℤ))
h₃ : T R (-(↑n : ℤ)) = U R (-(↑n : ℤ)) - HMul.hMul (α := R[X]) X (U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1))
h₁ :
  (derivative : (a : R[X]) → R[X]) (T R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1)) =
    HSub.hSub (β := R[X])
      ((HMul.hMul (α := R[X]) 0 X + HMul.hMul (α := R[X]) 2 1) * T R (-(↑n : ℤ)) +
        HMul.hMul (β := R[X]) (HMul.hMul (α := R[X]) 2 X) ((derivative : (a : R[X]) → R[X]) (T R (-(↑n : ℤ))) : R[X]))
      ((derivative : (a : R[X]) → R[X]) (T R (HAdd.hAdd (α := ℤ) (-(↑n : ℤ)) 1)) : R[X])
⊢ HAdd.hAdd (α := R[X]) ((derivative : (a : R[X]) → R[X]) (T R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1)) : R[X])
        (HAdd.hAdd (α := R[X]) ((derivative : (a : R[X]) → R[X]) (T R (HAdd.hAdd (α := ℤ) (-(↑n : ℤ)) 1)) : R[X])
                (HMul.hMul (α := R[X]) 2 X *
                  HMul.hMul (α := R[X]) (↑(-(↑n : ℤ)) : R[X]) (U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1))) +
              HSub.hSub (β := R[X])
                ((HMul.hMul (α := R[X]) 0 X + HMul.hMul (α := R[X]) 2 1) * T R (-(↑n : ℤ)) +
                  HMul.hMul (β := R[X]) (HMul.hMul (α := R[X]) 2 X)
                    ((derivative : (a : R[X]) → R[X]) (T R (-(↑n : ℤ))) : R[X]))
                ((derivative : (a : R[X]) → R[X]) (T R (HAdd.hAdd (α := ℤ) (-(↑n : ℤ)) 1)) : R[X]) +
            2 * (U R (-(↑n : ℤ)) - HMul.hMul (α := R[X]) X (U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1))) +
          HAdd.hAdd (α := R[X]) (↑n : R[X]) 1 *
            (HMul.hMul (α := R[X]) 2 X * U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1) - U R (-(↑n : ℤ)))) -
      (HMul.hMul (α := R[X]) (↑(HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1) : R[X]) (U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1 - 1)) +
        (HAdd.hAdd (β := R[X])
              (HMul.hMul (α := R[X]) (↑(HAdd.hAdd (α := ℤ) (-(↑n : ℤ)) 1) : R[X])
                  (U R (HAdd.hAdd (α := ℤ) (-(↑n : ℤ)) 1 - 1)) +
                HMul.hMul (β := R[X]) (HMul.hMul (α := R[X]) 2 X)
                  ((derivative : (a : R[X]) → R[X]) (T R (-(↑n : ℤ))) : R[X]))
              ((derivative : (a : R[X]) → R[X]) (T R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1)) : R[X]) +
            HMul.hMul (α := R[X]) 2 (T R (-(↑n : ℤ))) +
          HAdd.hAdd (α := R[X]) (↑n : R[X]) 1 * U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 2))) =
    0
have a : HAdd.hAdd (α := R[X]) ((derivative : (a : R[X]) → R[X]) (T R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1)) : R[X])
      (HAdd.hAdd (α := R[X]) ((derivative : (a : R[X]) → R[X]) (T R (HAdd.hAdd (α := ℤ) (-(↑n : ℤ)) 1)) : R[X])
              (HMul.hMul (α := R[X]) 2 X *
                HMul.hMul (α := R[X]) (-(↑n : R[X])) (U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1))) +
            HSub.hSub (β := R[X])
              ((HMul.hMul (α := R[X]) 0 X + HMul.hMul (α := R[X]) 2 1) * T R (-(↑n : ℤ)) +
                HMul.hMul (β := R[X]) (HMul.hMul (α := R[X]) 2 X)
                  ((derivative : (a : R[X]) → R[X]) (T R (-(↑n : ℤ))) : R[X]))
              ((derivative : (a : R[X]) → R[X]) (T R (HAdd.hAdd (α := ℤ) (-(↑n : ℤ)) 1)) : R[X]) +
          2 * (U R (-(↑n : ℤ)) - HMul.hMul (α := R[X]) X (U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1))) +
        HAdd.hAdd (α := R[X]) (↑n : R[X]) 1 *
          (HMul.hMul (α := R[X]) 2 X * U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1) - U R (-(↑n : ℤ)))) -
    (HSub.hSub (α := R[X]) (-(↑n : R[X])) 1 * U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1 - 1) +
      (HAdd.hAdd (β := R[X])
            (HAdd.hAdd (α := R[X]) (-(↑n : R[X])) 1 * U R (HAdd.hAdd (α := ℤ) (-(↑n : ℤ)) 1 - 1) +
              HMul.hMul (β := R[X]) (HMul.hMul (α := R[X]) 2 X)
                ((derivative : (a : R[X]) → R[X]) (T R (-(↑n : ℤ))) : R[X]))
            ((derivative : (a : R[X]) → R[X]) (T R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 1)) : R[X]) +
          HMul.hMul (α := R[X]) 2 (T R (-(↑n : ℤ))) +
        HAdd.hAdd (α := R[X]) (↑n : R[X]) 1 * U R (HSub.hSub (α := ℤ) (-(↑n : ℤ)) 2))) =
  0
Tactic: push_cast