English
For any semiring S, and natural numbers n,k, the evaluation at k of descPochhammer S (n+1) equals the product of the evaluation of descPochhammer S n at k with (k−n).
Русский
Для любой полупремененной полупрет у S и натуральных n,k, eval(descPochhammer S (n+1)) в k равно eval(descPochhammer S n) в k умножить на (k−n).
LaTeX
$$$\operatorname{eval}_k(\operatorname{descPochhammer}_S(n+1)) = (\operatorname{eval}_k(\operatorname{descPochhammer}_S(n))) \cdot (k - n)$$$
Lean4
theorem descPochhammer_succ_eval {S : Type*} [Ring S] (n : ℕ) (k : S) :
(descPochhammer S (n + 1)).eval k = (descPochhammer S n).eval k * (k - n) := by
rw [descPochhammer_succ_right, mul_sub, eval_sub, eval_mul_X, ← Nat.cast_comm, ← C_eq_natCast, eval_C_mul,
Nat.cast_comm, ← mul_sub]
