(** ** 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)).

(* Specifications: *)
Lemma parity_pshift : forall p,
parity (pshift p) = S (parity p).
Lemma pWF_pshift : forall p, pWF p -> pWF (pshift p).
Lemma pshift_pow_correct : forall xn l,
peval_pow (pshift_pow xn) l = peval_pow xn (tl l).
Lemma pshift_mon_correct : forall m l,
peval_mon (pshift_mon m) l = peval_mon m (tl l).
Lemma pshift_correct : forall p l,
peval (pshift p) l = peval p (tl l).