Section EVALUATION.
(* [mkpow x p] = x^p *)
Variable mkpow : R -> positive -> R.
(* [mkpow x p] = -(x^p) *)
Variable mkopp_pow : R -> positive -> R.
(* [mkmult_pow r x p] = r * x^p *)
Variable mkmult_pow : R -> R -> positive -> R.
Fixpoint mkmult_rec (r:R) (lm:list (R*positive)) {struct lm}: R :=
match lm with
| nil => r
| cons (x,p) t => mkmult_rec (mkmult_pow r x p) t
end.
Definition mkmult1 lm :=
match lm with
| nil => 1
| cons (x,p) t => mkmult_rec (mkpow x p) t
end.
Definition mkmultm1 lm :=
match lm with
| nil => ropp rI
| cons (x,p) t => mkmult_rec (mkopp_pow x p) t
end.
Definition mkmult_c_pos c lm :=
if c ?=! cI then mkmult1 (rev' lm)
else mkmult_rec [c] (rev' lm).
Definition mkmult_c c lm :=
match get_sign c with
| None => mkmult_c_pos c lm
| Some c' =>
if c' ?=! cI then mkmultm1 (rev' lm)
else mkmult_rec [c] (rev' lm)
end.
Definition mkadd_mult rP c lm :=
match get_sign c with
| None => rP + mkmult_c_pos c lm
| Some c' => rP - mkmult_c_pos c' lm
end.
Definition add_pow_list (r:R) n l :=
match n with
| N0 => l
| Npos p => (r,p)::l
end.
Fixpoint add_mult_dev
(rP:R) (P:Pol) (fv:list R) (n:N) (lm:list (R*positive)) {struct P} : R :=
match P with
| Pc c =>
let lm := add_pow_list (hd fv) n lm in
mkadd_mult rP c lm
| Pinj j Q =>
add_mult_dev rP Q (jump j fv) N0 (add_pow_list (hd fv) n lm)
| PX P i Q =>
let rP := add_mult_dev rP P fv (N.add (Npos i) n) lm in
if Q ?== P0 then rP
else add_mult_dev rP Q (tail fv) N0 (add_pow_list (hd fv) n lm)
end.
Fixpoint mult_dev (P:Pol) (fv : list R) (n:N)
(lm:list (R*positive)) {struct P} : R :=
(* P@l * (hd 0 l)^n * lm *)
match P with
| Pc c => mkmult_c c (add_pow_list (hd fv) n lm)
| Pinj j Q => mult_dev Q (jump j fv) N0 (add_pow_list (hd fv) n lm)
| PX P i Q =>
let rP := mult_dev P fv (N.add (Npos i) n) lm in
if Q ?== P0 then rP
else
let lmq := add_pow_list (hd fv) n lm in
add_mult_dev rP Q (tail fv) N0 lmq
end.
Definition Pphi_avoid fv P := mult_dev P fv N0 nil.
Fixpoint r_list_pow (l:list (R*positive)) : R :=
match l with
| nil => rI
| cons (r,p) l => pow_pos rmul r p * r_list_pow l
end.
Hypothesis mkpow_spec : forall r p, mkpow r p == pow_pos rmul r p.
Hypothesis mkopp_pow_spec : forall r p, mkopp_pow r p == - (pow_pos rmul r p).
Hypothesis mkmult_pow_spec : forall r x p, mkmult_pow r x p == r * pow_pos rmul x p.
End EVALUATION.
Definition Pphi_pow :=
let mkpow x p :=
match p with xH => x | _ => rpow x (Cp_phi (Npos p)) end in
let mkopp_pow x p := ropp (mkpow x p)  in
let mkmult_pow r x p := rmul r (mkpow x p) in
Pphi_avoid mkpow mkopp_pow mkmult_pow.
Fixpoint mkmult_pow (r x:R) (p: positive) {struct p} : R :=
match p with
| xH => r*x
| xO p => mkmult_pow (mkmult_pow r x p) x p
| xI p => mkmult_pow (mkmult_pow (r*x) x p) x p
end.
Definition mkpow x p :=
match p with
| xH => x
| xO p => mkmult_pow x x (Pos.pred_double p)
| xI p => mkmult_pow x x (xO p)
end.
Definition mkopp_pow x p :=
match p with
| xH => -x
| xO p => mkmult_pow (-x) x (Pos.pred_double p)
| xI p => mkmult_pow (-x) x (xO p)
end.
Definition Pphi_dev := Pphi_avoid mkpow mkopp_pow mkmult_pow.

Lemma ring_rw_correct : forall n lH l,
interp_PElist l lH ->
forall lmp, mk_monpol_list lH = lmp ->
forall pe npe, norm_subst n lmp pe = npe ->
PEeval l pe == Pphi_dev l npe.