Fixpoint Cond0 (A:Type) (Interp:A->R) (l:list A) : Prop :=
  match l with
  | List.nil => True
  | a::l => Interp a == 0 /\ Cond0 A Interp l
  end.

(* Specifications: *)
Lemma mult_l_correct : forall l la lp, Cond0 PolZ (PhiR l) lp -> PhiR l (mult_l la lp) == 0.
Lemma compute_list_correct : forall l lla lp, Cond0 PolZ (PhiR l) lp -> Cond0 PolZ (PhiR l) (compute_list lla lp).
Lemma check_correct : forall l lpe qe certif, check lpe qe certif = true -> Cond0 PEZ (PEevalR l) lpe -> PEevalR l qe == 0.