intros Q s t H'; elim H'.
intros x y eqxy Op p q H'0 H'1 H'2 H'3; exists p; split; auto.
apply Rstar_0; auto.
cut (canonical A0 eqA ltM y); [ intros Cx | idtac ].
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pO A n)
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
p)); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
y
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
p)); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
x
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
p)); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n);
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 p q)
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) p));
auto.
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 q p)
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
p)); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec p
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) p)));
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n);
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec q
(minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec p p));
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n);
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec q (pO A n)); auto.
apply eqp_pluspf_com with (1 := cs); auto.
apply minuspf_refl with (1 := cs); auto.
apply eqp_imp_canonical with (1 := cs) (p := x); auto.
intros x y z H'0 H'1 H'2 H'3 p q H'4 H'5 H'6 H'7.
elim (rep_plus_reduce Q p q y);
[ intros p1 E; elim E; intros q1 E0; elim E0; intros H'15 H'16; elim H'16;
intros H'17 H'18; try exact H'17; clear H'16 E0 E
| idtac
| idtac
| idtac ]; auto.
lapply H'2;
[ intros H'8; elim (H'8 p1 q1);
[ intros r1 E; elim E; intros H'16 H'19; clear E H'2
| clear H'2
| clear H'2
| clear H'2
| clear H'2 ]
| clear H'2 ]; auto.
exists r1; split; auto.
apply reduceplus_trans with (y := p1); auto.
apply reduceplus_trans with (y := q1); auto.
apply canonical_reduceplus with (Q := Q) (p := p); auto.
apply canonical_reduceplus with (Q := Q) (p := q); auto.
apply
eqp_imp_canonical
with
(1 := cs)
(p := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p1 q1); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply canonical_pluspf; auto.
apply canonical_reduceplus with (Q := Q) (p := p); auto.
apply canonical_reduceplus with (Q := Q) (p := q); auto.
apply reduce_eqp_com with (p := x) (q := y) (1 := cs); auto.
Qed.

Theorem red_plus_zero_reduce :
forall (Q : list (poly A0 eqA ltM)) (p q : list (Term A n)),
canonical A0 eqA ltM p ->
canonical A0 eqA ltM q ->
reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p q)
(pO A n) ->
exists r1 : list (Term A n),
reduceplus Q p r1 /\
reduceplus Q q
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) r1).