Section Buch.
Load "hCoefStructure".
Load "hOrderStructure".
Load "hWfRO".
Inductive stable :
list (poly A0 eqA ltM) -> list (poly A0 eqA ltM) -> Prop :=
stable0 :
forall P Q : list (poly A0 eqA ltM),
(forall a : poly A0 eqA ltM,
Cb A A0 eqA plusA multA eqA_dec n ltM ltM_dec a P ->
Cb A A0 eqA plusA multA eqA_dec n ltM ltM_dec a Q) ->
(forall a : poly A0 eqA ltM,
Cb A A0 eqA plusA multA eqA_dec n ltM ltM_dec a Q ->
Cb A A0 eqA plusA multA eqA_dec n ltM ltM_dec a P) ->
stable P Q.
Hint Resolve stable0.
auto.
Qed.
intros Q y R H' H'0; inversion H'; inversion H'0; auto.
Qed.
intros Q R H'; elim H'; auto.
Qed.
Hint Resolve (Cb_in _ _ _ _ _ _ _ _ _ cs eqA_dec _ _ ltM_dec os).
intros a Q H'0; apply stable0; auto.
intros a0 H'1.
apply Cb_trans with (b := a) (1 := cs); auto.
Qed.
auto.
Qed.
Inductive reds :
poly A0 eqA ltM -> poly A0 eqA ltM -> list (poly A0 eqA ltM) -> Prop :=
| reds0 :
forall (P : list (poly A0 eqA ltM)) (a b : poly A0 eqA ltM),
red A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
(spolyp A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM
ltM_dec os a b) P -> reds a b P
| reds1 :
forall (P : list (poly A0 eqA ltM)) (a b c : poly A0 eqA ltM),
In c P ->
reds a c P ->
reds c b P ->
divp A A0 eqA multA divA n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM a b) c ->
reds a b P.
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.
intros a b P; elim P; simpl in |- *; auto.
unfold slicef in |- *; simpl in |- *; auto.
case (foreigner_dec A A0 A1 eqA multA n ltM a b); intros H; apply incl_refl;
auto with datatypes.
intros c; unfold slicef in |- *; simpl in |- *.
case
(divp_dec _ _ _ _ _ _ _ _ _ cs n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM a b) c);
auto with datatypes.
case
(divp_dec _ _ _ _ _ _ _ _ _ cs n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM a c) b);
auto with datatypes.
intros H' H'0 l; case (slice a b l); simpl in |- *; auto with datatypes.
Qed.
intros a b P; elim P; simpl in |- *; auto.
intros c H'; elim H'.
intros p aP1;
case
(divp_dec _ _ _ _ _ _ _ _ _ cs n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM a b) p);
simpl in |- *; auto.
case
(divp_dec _ _ _ _ _ _ _ _ _ cs n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM a p) b);
auto.
intros H' H'0 H'1 c H'2; elim H'2;
[ intros H'3; rewrite <- H'3; clear H'2 | intros H'3; clear H'2 ];
auto.
case (slice a b aP1); simpl in |- *; auto.
intros P0 H' H'0 H'1 c H'2; elim H'2;
[ intros H'3; rewrite <- H'3; clear H'2 | intros H'3; clear H'2 ];
auto.
elim (H'1 c); [ intros H'5; try exact H'5 | intros H'5 | idtac ]; auto.
intros P0 H' H'0 H'1 c H'2; elim H'2;
[ intros H'3; rewrite <- H'3; clear H'2 | intros H'3; clear H'2 ];
auto.
elim (H'1 c); [ intros H'5; try exact H'5 | intros H'5 | idtac ]; auto.
Qed.

Theorem slice_cons :
forall (i a : poly A0 eqA ltM) (aP Q : list (poly A0 eqA ltM)),
slice i a aP = DontKeep Q ->
(exists c : poly A0 eqA ltM,
In c Q /\
divp A A0 eqA multA divA n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM i a) c) \/
foreigner A A0 A1 eqA multA n ltM i a.