(** ** Shifting of a polynomial

  - [pshift p] is the polynomial [p] with one more variable and where each variable [i] is replaced by [i+1].
*)

Definition pshift_pow (xn:pow) : pow :=
  (S (fst xn), snd xn).Definition pshift_mon (m:mon) : mon :=
  (fst m, map pshift_pow (snd m)).Definition pshift (p:pol) : pol :=
  (S (fst p), map pshift_mon (snd p)).

Lemma parity_pshift : forall p, parity (pshift p) = S (parity p).