intros p L; case p; unfold nf in |- *; auto.
unfold nf in |- *; auto.
intros x c;
case
(Reducef A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec
os L (exist (fun l => canonical A0 eqA ltM l) x c)).
unfold LetP in |- *.
intros x0; case x0; auto.
intros x1 c0 H' H'0 H'1.
lapply
(reducestar_divp (mks A A0 eqA n ltM x c) (mks A A0 eqA n ltM x1 c0) L);
[ intros H'5; lapply H'5; [ intros H'6; clear H'5 | clear H'5 ] | idtac ];
auto.
case H'6; intros r E; case E; intros H'2 H'3; case H'3; intros H'4 H'5;
clear H'3 E H'6.
case H'2; [ intros H'3; clear H'2 | intros H'3; clear H'2 ].
exists
(mks A A0 eqA n ltM
(mults (A:=A) multA (n:=n)
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM x1 c0)) x1)
(canonical_mults _ _ _ _ _ _ _ _ _ cs eqA_dec _ _ os
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM x1 c0)) x1
(unit_nZ _ _ _ _ _ _ _ _ _ cs n ltM (mks A A0 eqA n ltM x1 c0)) c0));
split; [ idtac | split ]; auto.
simpl in |- *; auto.
apply (divp_trans _ _ _ _ _ _ _ _ _ cs n ltM) with (y := r); auto.
generalize H'5; rewrite <- H'3.
generalize c0; case x1; auto.
simpl in |- *; auto.
intros a l c1 H1.
cut (~ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros nZa | apply canonical_nzeroP with (ltM := ltM) (p := l) ];
auto.
cut
(~
zeroP (A:=A) A0 eqA (n:=n)
(multTerm (A:=A) multA (n:=n)
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM (pX a l) c1)) a));
[ intros nZu | idtac ]; auto.
simpl in |- *; apply divTerm_def with (nZb := nZu); auto.
apply divTerm_on_eqT with (1 := cs) (a := a); auto.
apply (eqT_trans A n) with (y := multTerm (A:=A) multA (n:=n) (T1 A1 n) a);
auto.
apply (eqTerm_imp_eqT A eqA n); auto.
apply multTerm_eqT; auto.
apply (eqT_sym A n); auto.
apply unit_T1; auto.
apply nzeroP_multTerm with (1 := cs); auto.
apply unit_nZ with (1 := cs); auto.
exists r; split; [ idtac | split ]; auto.
simpl in |- *; auto.
Qed.

Theorem divp_reduce1 :
forall (p : poly A0 eqA ltM) (L1 L2 : list (poly A0 eqA ltM)),
(forall r1 : poly A0 eqA ltM,
In r1 L1 ->
~ zerop A A0 eqA n ltM r1 ->
exists r2 : poly A0 eqA ltM,
In r2 L2 /\ divp A A0 eqA multA divA n ltM r1 r2) ->
forall q : poly A0 eqA ltM,
reducep L1 p q -> exists r : poly A0 eqA ltM, reducep L2 p r.