-- Module: Mathlib.RingTheory.Polynomial.Pochhammer
Goal:
R : Type u
inst : Ring R
r : R
k : ℕ
rw :
  eval (-r) (ascPochhammer R k * HAdd.hAdd (α := R[X]) X (↑k : R[X])) =
    HPow.hPow (α := R) (-1) (k + 1) * eval r (descPochhammer R (k + 1))
rw₁ :
  eval (-r) (HMul.hMul (α := R[X]) (ascPochhammer R k) X + HMul.hMul (α := R[X]) (ascPochhammer R k) (↑k : R[X])) =
    HPow.hPow (α := R) (-1) (k + 1) * eval r (descPochhammer R (k + 1))
rw₂ :
  eval (-r) (HMul.hMul (α := R[X]) (ascPochhammer R k) X) +
      eval (-r) (HMul.hMul (α := R[X]) (ascPochhammer R k) (↑k : R[X])) =
    HPow.hPow (α := R) (-1) (k + 1) * eval r (descPochhammer R (k + 1))
rw₃ :
  eval (-r) (ascPochhammer R k) * -r + eval (-r) (HMul.hMul (α := R[X]) (ascPochhammer R k) (↑k : R[X])) =
    HPow.hPow (α := R) (-1) (k + 1) * eval r (descPochhammer R (k + 1))
rw₄ :
  eval (-r) (ascPochhammer R k) * -r + eval (-r) (HMul.hMul (α := R[X]) (↑k : R[X]) (ascPochhammer R k)) =
    HPow.hPow (α := R) (-1) (k + 1) * eval r (descPochhammer R (k + 1))
rw₅ :
  eval (-r) (ascPochhammer R k) * -r + (↑k : R) * eval (-r) (ascPochhammer R k) =
    HPow.hPow (α := R) (-1) (k + 1) * eval r (descPochhammer R (k + 1))
rw₆ :
  eval (-r) (ascPochhammer R k) * -r + eval (-r) (ascPochhammer R k) * (↑k : R) =
    HPow.hPow (α := R) (-1) (k + 1) * eval r (descPochhammer R (k + 1))
rw₇ :
  eval (-r) (ascPochhammer R k) * (-r + (↑k : R)) = HPow.hPow (α := R) (-1) (k + 1) * eval r (descPochhammer R (k + 1))
rw₈ :
  HPow.hPow (α := R) (-1) k * eval r (descPochhammer R k) * (-r + (↑k : R)) =
    HPow.hPow (α := R) (-1) (k + 1) * eval r (descPochhammer R (k + 1))
rw₉ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) (k + 1) * eval r (descPochhammer R (k + 1))
rw₁₀ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) (k + 1) * eval r (descPochhammer R k * HSub.hSub (α := R[X]) X (↑k : R[X]))
rw₁₁ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) (k + 1) *
      eval r (HMul.hMul (α := R[X]) (descPochhammer R k) X - HMul.hMul (α := R[X]) (descPochhammer R k) (↑k : R[X]))
rw₁₂ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) (k + 1) *
      (eval r (HMul.hMul (α := R[X]) (descPochhammer R k) X) -
        eval r (HMul.hMul (α := R[X]) (descPochhammer R k) (↑k : R[X])))
rw₁₃ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) (k + 1) *
      (eval r (descPochhammer R k) * r - eval r (HMul.hMul (α := R[X]) (descPochhammer R k) (↑k : R[X])))
rw₁₄ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) (k + 1) *
      (eval r (descPochhammer R k) * r - eval r (HMul.hMul (α := R[X]) (↑k : R[X]) (descPochhammer R k)))
rw₁₅ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) (k + 1) * (eval r (descPochhammer R k) * r - (↑k : R) * eval r (descPochhammer R k))
rw₁₆ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) k * HPow.hPow (α := R) (β := ℕ) (-1) 1 *
      (eval r (descPochhammer R k) * r - (↑k : R) * eval r (descPochhammer R k))
rw₁₇ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) k * -1 * (eval r (descPochhammer R k) * r - (↑k : R) * eval r (descPochhammer R k))
rw₁₈ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) k * (-1 * (eval r (descPochhammer R k) * r - (↑k : R) * eval r (descPochhammer R k)))
rw₁₉ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) k * (-1 * (eval r (descPochhammer R k) * r) - -1 * ((↑k : R) * eval r (descPochhammer R k)))
rw₂₀ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) k * (-(eval r (descPochhammer R k) * r) - -1 * ((↑k : R) * eval r (descPochhammer R k)))
rw₂₁ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * -r - -1 * ((↑k : R) * eval r (descPochhammer R k)))
rw₂₂ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * -r - -1 * (eval r (descPochhammer R k) * (↑k : R)))
rw₂₃ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * -r + -(-1 * (eval r (descPochhammer R k) * (↑k : R))))
rw₂₄ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * -r + - -(eval r (descPochhammer R k) * (↑k : R)))
rw₂₅ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * -r + eval r (descPochhammer R k) * (↑k : R))
rw₂₆ rw₂₇ :
  HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R))) =
    HPow.hPow (α := R) (-1) k * (eval r (descPochhammer R k) * (-r + (↑k : R)))
⊢ eval (-r) (ascPochhammer R (k + 1)) = HPow.hPow (α := R) (-1) (k + 1) * eval r (descPochhammer R (k + 1))

Tactic:
rw [ascPochhammer_succ_right, mul_add, eval_add, eval_mul_X, ← Nat.cast_comm, eval_natCast_mul,
      Nat.cast_comm, ← mul_add, ascPochhammer_eval_neg_eq_descPochhammer r k, mul_assoc,
      descPochhammer_succ_right, mul_sub, eval_sub, eval_mul_X, ← Nat.cast_comm, eval_natCast_mul,
      pow_add, pow_one, mul_assoc ((-1) ^ k) (-1), mul_sub, neg_one_mul, neg_mul_eq_mul_neg,
      Nat.cast_comm, sub_eq_add_neg, neg_one_mul, neg_neg, ← mul_add]

Premises used: Nat.cast_comm, mul_sub, mul_assoc, sub_eq_add_neg, descPochhammer_succ_right, neg_one_mul, pow_add, neg_neg, neg_mul_eq_mul_neg, HPow.hPow, Polynomial.eval_natCast_mul, pow_one, Polynomial.eval_add, mul_add, Polynomial.eval_mul_X, OfNat.ofNat, Polynomial.eval_sub, Neg.neg, ascPochhammer_succ_right