Section Fcons_simpl.
(* Some general simpifications of the condition: eliminate duplicates, split multiplications *)
Fixpoint Fcons (e:PExpr C) (l:list (PExpr C)) {struct l} : list (PExpr C) :=
match l with
nil       => cons e nil
| cons a l1 => if PExpr_eq e a then l else cons a (Fcons e l1)
end.
(* equality of normal forms rather than syntactic equality *)
Fixpoint Fcons0 (e:PExpr C) (l:list (PExpr C)) {struct l} : list (PExpr C) :=
match l with
nil       => cons e nil
| cons a l1 =>
if Peq ceqb (Nnorm O nil e) (Nnorm O nil a) then l
else cons a (Fcons0 e l1)
end.
(* split factorized denominators *)
Fixpoint Fcons00 (e:PExpr C) (l:list (PExpr C)) {struct e} : list (PExpr C) :=
match e with
PEmul e1 e2 => Fcons00 e1 (Fcons00 e2 l)
| PEpow e1 _ => Fcons00 e1 l
| _ => Fcons0 e l
end.
Definition Pcond_simpl_gen :=
fcons_ok _ PFcons00_fcons_inv.
(* Specific case when the equality test of coefs is complete w.r.t. the field equality: non-zero coefs can be eliminated, and opposite can be simplified (if -1 <> 0) *)
Hypothesis ceqb_complete : forall c1 c2, [c1] == [c2] -> ceqb c1 c2 = true.
Fixpoint Fcons1 (e:PExpr C) (l:list (PExpr C)) {struct e} : list (PExpr C) :=
match e with
| PEmul e1 e2 => Fcons1 e1 (Fcons1 e2 l)
| PEpow e _ => Fcons1 e l
| PEopp e => if (-(1) =? 0)%coef then absurd_PCond else Fcons1 e l
| PEc c => if (c =? 0)%coef then absurd_PCond else l
| _ => Fcons0 e l
end.
Definition Fcons2 e l := Fcons1 (PEsimp e) l.

Theorem PFcons2_fcons_inv:
forall l a l1, PCond l (Fcons2 a l1) -> ~ a @ l == 0 /\ PCond l l1.