(** evaluation of polynomial expressions towards R *)
 Definition mk_X j := mkPinj_pred j mkX.(** evaluation of polynomial expressions towards R *)

 Fixpoint PEeval (l:Env R) (pe:PExpr) : R :=
   match pe with
   | PEc c => phi c
   | PEX j => nth 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.

Lemma mkX_ok p l : nth p l == (mk_X p) @ l.