(** ** Multiplication of polynomials

  - [pmult p1 p2] is the multiplication of the polynomials [p1] and [p2].

  - [pmultl pl] is the multiplication of the polynomials in the list [pl].
*)

Definition pmult_mon (m12:mon*mon) : mon :=
  (fst (fst m12) * fst (snd m12), snd (fst m12) ++  snd (snd m12)).Definition pmult (p1 p2:pol) : pol :=
  (max (fst p1) (fst p2), map pmult_mon (list_prod (snd p1) (snd p2))).

Lemma map_pmult_mon : forall m1 ml2 l, map (fun m2 => peval_mon (pmult_mon (m1, m2)) l) ml2 = map (fun m2 => peval_mon m1 l * peval_mon m2 l) ml2.