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); unfold Tl in |- *;
simpl in |- *; auto.
intros c l.
unfold slicef in |- *; simpl in |- *; auto.
case
(divp_dec _ _ _ _ _ _ _ _ _ cs n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM a b) c);
auto.
unfold Tl in |- *; simpl in |- *; auto.
case
(divp_dec _ _ _ _ _ _ _ _ _ cs n ltM
(ppcp A A0 A1 eqA plusA invA minusA multA divA cs n ltM a c) b);
auto.
intros H' H'0; case (slice a b l); simpl in |- *; auto.
unfold Tl in |- *; simpl in |- *; auto.
unfold Tl in |- *; simpl in |- *; auto.
unfold Tl in |- *; intros H' H'0; case (slice a b l); simpl in |- *;
auto with arith.
Qed.
Inductive genPcP :
poly A0 eqA ltM ->
list (poly A0 eqA ltM) ->
list (poly A0 eqA ltM) -> list (poly A0 eqA ltM) -> Prop :=
| genPcP0 :
forall (i : poly A0 eqA ltM) (L : list (poly A0 eqA ltM)),
genPcP i nil L L
| genPcP1 :
forall (L L1 L2 L3 : list _) (a i : poly A0 eqA ltM),
slice i a L1 = Keep L2 ->
genPcP i L2 L L3 ->
genPcP i (a :: L1) L
(addEnd A A0 eqA n ltM
(spolyp A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM
ltM_dec os i a) L3)
| genPcP2 :
forall (L L1 L2 L3 : list _) (a i : poly A0 eqA ltM),
slice i a L1 = DontKeep L2 ->
genPcP i L2 L L3 -> genPcP i (a :: L1) L L3.
Hint Resolve genPcP0.

Theorem genPcP_spolyp1 :
forall (i : poly A0 eqA ltM) (L L1 L2 : list _),
genPcP i L1 L L2 ->
forall a : poly A0 eqA ltM,
In a L2 ->
(exists b : poly A0 eqA ltM,
In b L1 /\
a =
spolyp A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec
os i b) \/ In a L.