End POWER.
(** Normalization and rewriting *)
Section NORM_SUBST_REC.
Variable n : nat.
Variable lmp:list (C*Mon*Pol).
Let subst_l P := PNSubstL P lmp n n.
Let Pmul_subst P1 P2 := subst_l (P1 ** P2).
Let Ppow_subst := Ppow_N subst_l.
Fixpoint norm_aux (pe:PExpr) : Pol :=
match pe with
| PEO => Pc cO
| PEI => Pc cI
| PEc c => Pc c
| PEX j => mk_X j
| PEadd (PEopp pe1) pe2 => (norm_aux pe2) -- (norm_aux pe1)
| PEadd pe1 (PEopp pe2) => (norm_aux pe1) -- (norm_aux pe2)
| PEadd pe1 pe2 => (norm_aux pe1) ++ (norm_aux pe2)
| PEsub pe1 pe2 => (norm_aux pe1) -- (norm_aux pe2)
| PEmul pe1 pe2 => (norm_aux pe1) ** (norm_aux pe2)
| PEopp pe1 => -- (norm_aux pe1)
| PEpow pe1 n => Ppow_N (fun p => p) (norm_aux pe1) n
end.
Definition norm_subst pe := subst_l (norm_aux pe).
(** Internally, [norm_aux] is expanded in a large number of cases. To speed-up proofs, we use an alternative definition. *)
Definition get_PEopp pe :=
match pe with
| PEopp pe' => Some pe'
| _ => None
end.

(* Specifications: *)
Lemma norm_aux_PEadd pe1 pe2 :
norm_aux (PEadd pe1 pe2) =
match get_PEopp pe1, get_PEopp pe2 with
| Some pe1', _ => (norm_aux pe2) -- (norm_aux pe1')
| None, Some pe2' => (norm_aux pe1) -- (norm_aux pe2')
| None, None => (norm_aux pe1) ++ (norm_aux pe2)
end.
Lemma norm_aux_PEopp pe :
match get_PEopp pe with
| Some pe' => norm_aux pe = -- (norm_aux pe')
| None => True
end.