(** ** Composition of polynomials

  - [pcomp p pl] is the polynomial [p] where each variable [i] is replaced by the [i]th polynomial in 
    the list [pl].
*)

Definition pcomp_pow' (xn:pow)(pl:list pol) : pol :=
  ppower (nth (fst xn) pl (pcst 0 0)) (snd xn).Definition pcomp_pow (xn:pow)(pl:list pol) : pol :=
  (maxl (map parity pl), snd (pcomp_pow' xn pl)).Definition pcomp_mon' (m:mon)(pl:list pol) : pol :=
  pscalar (fst m) (pmultl (map (fun xn => pcomp_pow xn pl) (snd m))).Definition pcomp_mon (m:mon)(pl:list pol) : pol :=
  (maxl (map parity pl), snd (pcomp_mon' m pl)).Definition pcomp' (p:pol)(pl:list pol) : pol :=
  pplusl (map (fun m => pcomp_mon m pl) (snd p)).Definition pcomp (p:pol)(pl:list pol) : pol :=
  (maxl (map parity pl), snd (pcomp' p pl)).

Lemma parity_pcomp_mon' : forall m pl, parity (pcomp_mon' m pl) <= maxl (map parity pl).