intros P a b H'; elim H'; simpl in |- *; auto.
intros P0 a0 b0 H'0.
apply reds0; auto.
apply red_com; auto.
intros P0 a0 b0 c H'0 H'1 H'2 H'3 H'4 H'5.
apply reds1 with (c := c); auto.
apply divp_ppc; auto.
Qed.
(* Now we are ready!! We start with the definition of genCpC *)
Inductive cpRes : Set :=
| Keep : forall P : list (poly A0 eqA ltM), cpRes
| DontKeep : forall P : list (poly A0 eqA ltM), cpRes.
Definition getRes : cpRes -> list (poly A0 eqA ltM).
intros H'; case H'; auto.
Defined.
Definition addRes : poly A0 eqA ltM -> cpRes -> cpRes.
intros i H'; case H'.
intros H'0; exact (Keep (i :: H'0)).
intros H'0; exact (DontKeep (i :: H'0)).
Defined.
Definition slice :
poly A0 eqA ltM -> poly A0 eqA ltM -> list (poly A0 eqA ltM) -> cpRes.
intros i a q; elim q; clear q.
case (foreigner_dec A A0 A1 eqA multA n ltM i a).
intros H; exact (DontKeep nil).
intros H; exact (Keep nil).
intros b q1 Rec.
case
(divp_dec _ _ _ _ _ _ _ _ _ cs n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM i a) b).
intros divp10; exact (DontKeep (b :: q1)).
intros divp10.
case
(divp_dec _ _ _ _ _ _ _ _ _ cs n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM i b) a).
intros divp11; exact Rec.
intros divp11; exact (addRes b Rec).
Defined.
Definition slicef :
poly A0 eqA ltM ->
poly A0 eqA ltM -> list (poly A0 eqA ltM) -> list (poly A0 eqA ltM).
intros i a q; case (slice i a q); auto.
Defined.

Theorem slicef_incl :
forall (a b : poly A0 eqA ltM) (P : list (poly A0 eqA ltM)),
incl (slicef a b P) P.