(** Definition of polynomial expressions *)
(* Inductive PExpr : Type := | 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. *)
(** Specification of the power function *)
Variable Cpow : Set.
Variable Cp_phi : N -> Cpow.
Variable rpow : R -> Cpow -> R.
Record power_theory : Prop := mkpow_th {
rpow_pow_N : forall r n,  (rpow r (Cp_phi n))== (pow_N r n)
}.
End POWER.
Variable Cpow : Set.
Variable Cp_phi : N -> Cpow.
Variable rpow : R -> Cpow -> R.
Variable pow_th : power_theory Cp_phi rpow.
(** evaluation of polynomial expressions towards R *)
Fixpoint PEeval (l:list R) (pe:PExpr C) {struct pe} : R :=
match pe with
| PEO => 0
| PEI => 1
| PEc c => [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].
Definition mk_X j := mkX j.
(** Correctness proofs *)

(* Specifications: *)
Lemma mkX_ok : forall p l, nth 0 p l == (mk_X p) @ l.
Ltac Esimpl3 :=
repeat match goal with
| |- context [(?P1 ++ ?P2)@?l] => rewrite (Padd_ok P1 P2 l)
| |- context [(?P1 -- ?P2)@?l] => rewrite (Psub_ok P1 P2 l)
end;try Esimpl2;try reflexivity;try apply ring_add_comm.