intros Q p; elim p; auto.
simpl in |- *; auto.
intros; apply Rstar_0; auto.
intros a1 p1 H' q H'0 H'1.
cut (canonical A0 eqA ltM p1);
[ intros Op0 | apply canonical_imp_canonical with (a := a1) ];
auto.
cut (canonical A0 eqA ltM q);
[ intros Op1 | apply inPolySet_imp_canonical with (L := Q) ];
auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) a1);
[ intros Z0 | apply canonical_nzeroP with (ltM := ltM) (p := p1) ];
auto.
apply
reduceplus_trans
with
(1 := cs)
(y := multpf A A0 eqA plusA multA eqA_dec n ltM ltM_dec p1 q);
auto.
generalize Op1 H'0; case q; clear Op1 H'0 q.
intros H'0 H'2; elim (not_nil_in_polySet A A0 eqA n ltM Q); auto.
intros b1 q1 H'0 H'2.
cut (canonical A0 eqA ltM q1);
[ intros Op2 | apply canonical_imp_canonical with (a := b1) ];
auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) b1);
[ intros nZb1 | apply canonical_nzeroP with (ltM := ltM) (p := q1) ];
auto.
apply
reduceplus_eqp_com
with
(1 := cs)
(p := multpf A A0 eqA plusA multA eqA_dec n ltM ltM_dec
(pX a1 p1) (pX b1 q1))
(q := minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec
(multpf A A0 eqA plusA multA eqA_dec n ltM ltM_dec
(pX a1 p1) (pX b1 q1))
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(multTerm (A:=A) multA (n:=n) a1 b1) (b:=b1) nZb1)
(pX b1 q1))); auto.
apply reduce_imp_reduceplus with (1 := cs); auto.
apply minus_is_reduce; auto.
apply in_multpf_head; auto.
cut
(eqTerm (A:=A) eqA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(multTerm (A:=A) multA (n:=n) a1 b1) (b:=b1) nZb1) a1);
[ intros Eq0 | auto ].
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec
(multpf A A0 eqA plusA multA eqA_dec n ltM ltM_dec
(pX a1 p1) (pX b1 q1))
(mults (A:=A) multA (n:=n) a1 (pX b1 q1)));
[ auto | idtac ].
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) a1 (pX b1 q1))
(multpf A A0 eqA plusA multA eqA_dec n ltM ltM_dec p1
(pX b1 q1)))
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n) a1 (pX b1 q1))));
[ auto | idtac ].
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec
(multpf A A0 eqA plusA multA eqA_dec n ltM ltM_dec p1
(pX b1 q1)) (mults (A:=A) multA (n:=n) a1 (pX b1 q1)))
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n) a1 (pX b1 q1))));
[ auto | idtac ].
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(multpf A A0 eqA plusA multA eqA_dec n ltM ltM_dec p1 (pX b1 q1))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) a1 (pX b1 q1))
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n) a1 (pX b1 q1)))));
[ auto | idtac ].
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(multpf A A0 eqA plusA multA eqA_dec n ltM ltM_dec p1 (pX b1 q1))
(minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec
(mults (A:=A) multA (n:=n) a1 (pX b1 q1))
(mults (A:=A) multA (n:=n) a1 (pX b1 q1)))).
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(multpf A A0 eqA plusA multA eqA_dec n ltM ltM_dec p1 (pX b1 q1))
(pO A n)); auto.
apply eqp_pluspf_com with (1 := cs); auto.
apply minuspf_refl with (1 := cs); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := multTerm (A:=A) multA (n:=n) a1
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) b1 (b:=b1) nZb1));
auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := multTerm (A:=A) multA (n:=n) a1 (T1 A1 n));
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
Qed.

Theorem fspoly_Reduceplus_pO :
forall (Q : list (poly A0 eqA ltM)) (a b : Term A n),
eqT (ppc (A:=A) A1 (n:=n) a b) (multTerm (A:=A) multA (n:=n) a b) ->
forall p q : list (Term A n),
inPolySet A A0 eqA n ltM (pX b q) Q ->
inPolySet A A0 eqA n ltM (pX a p) Q ->
reduceplus A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q
(minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec
(mults (A:=A) multA (n:=n) b p) (mults (A:=A) multA (n:=n) a q))
(pO A n).