(** Definition of polynomial expressions *)
Inductive PExpr : Type :=
| PEO : PExpr
| PEI : PExpr
| PEc : C -> PExpr
| PEX : positive -> PExpr
| PEadd : PExpr -> PExpr -> PExpr
| PEsub : PExpr -> PExpr -> PExpr
| PEmul : PExpr -> PExpr -> PExpr
| PEopp : PExpr -> PExpr
| PEpow : PExpr -> N -> PExpr.
(** evaluation of polynomial expressions towards R *)
Definition mk_X j := mkPinj_pred j mkX.
(** evaluation of polynomial expressions towards R *)
Fixpoint PEeval (l:list R) (pe:PExpr) {struct pe} : R :=
match pe with
| PEO => rO
| PEI => rI
| PEc c => phi c
| PEX j => nth 0 j l
| PEadd pe1 pe2 => (PEeval l pe1) + (PEeval l pe2)
| PEsub pe1 pe2 => (PEeval l pe1) - (PEeval l pe2)
| PEmul pe1 pe2 => (PEeval l pe1) * (PEeval l pe2)
| PEopp pe1 => - (PEeval l pe1)
| PEpow pe1 n => rpow (PEeval l pe1) (Cp_phi n)
end.
Strategy expand [PEeval].
(** Correctness proofs *)

Lemma mkX_ok p l : nth 0 p l == (mk_X p) @ l.
Hint Rewrite Padd_ok Psub_ok : Esimpl.