Section Preduceplus.
Load "hCoefStructure".
Load "hOrderStructure".
Load "hReduce".
Inductive reduceplus (Q : list (poly A0 eqA ltM)) :
list (Term A n) -> list (Term A n) -> Prop :=
| Rstar_0 :
forall x y : list (Term A n), eqP A eqA n x y -> reduceplus Q x y
| Rstar_n :
forall x y z : list (Term A n),
reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q x y ->
reduceplus Q y z -> reduceplus Q x z.
Hint Resolve Rstar_0.
intros Q p q r s H'; generalize r s; clear r s; elim H'; auto.
intros x y H'0 r s H'1 H'2 H'3.
apply Rstar_0; auto.
apply (eqp_trans _ _ _ _ _ _ _ _ _ cs n) with (y := y); auto.
apply (eqp_trans _ _ _ _ _ _ _ _ _ cs n) with (y := x); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros x y z H'0 H'1 H'2 r s H'3 H'4 H'5.
apply Rstar_n with (y := y); auto.
apply reduce_eqp_com with (1 := cs) (p := x) (q := y); auto.
apply H'2; auto.
apply canonical_reduce with (1 := cs) (3 := H'0); auto.
Qed.
intros Q x y z H'; elim H'; auto.
intros x0 y0 H'0 H'1 H'2.
apply reduceplus_eqp_com with (p := y0) (q := z); auto.
apply eqp_imp_canonical with (1 := cs) (p := x0); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros x0 y0 z0 H'0 H'1 H'2 H'3 H'4.
apply Rstar_n with (y := y0); auto.
apply H'2; auto.
apply canonical_reduce with (1 := cs) (3 := H'0); auto.
Qed.
intros Q p q H'; elim H'; auto.
intros x y H H0.
apply eqp_imp_canonical with (1 := cs) (p := x); auto.
intros x y z H'0 H'1 H'2 H'3.
apply H'2; auto.
apply canonical_reduce with (1 := cs) (3 := H'0); auto.
Qed.
intros Q p q H'; elim H'; auto.
intros x y z H'0 H'1 H'2 H'3; left; try assumption.
elim H'2;
[ intros H'5; try exact H'5; clear H'2 | intros H'5; clear H'2 | clear H'2 ].
apply ltP_trans with (y := y); auto.
apply ltP_reduce with (1 := cs) (3 := H'0); auto.
apply (ltp_eqp_comp A A0 eqA) with (p := y) (q := x); auto.
apply ltP_reduce with (1 := cs) (3 := H'0); auto.
apply canonical_reduce with (1 := cs) (3 := H'0); auto.
apply canonical_reduce with (1 := cs) (3 := H'0); auto.
Qed.
intros Q a b p q H'; generalize a b; clear a b; elim H'; auto.
intros x y z H'0 H'1 H'2 a b H'3 H'4.
apply Rstar_n with (y := pX b y); auto.
apply H'2; auto.
apply ltP_pX_canonical; auto.
apply canonical_reduce with (1 := cs) (3 := H'0); auto.
apply canonical_imp_canonical with (a := a); auto.
apply canonical_nzeroP with (ltM := ltM) (p := x); auto.
apply eqp_imp_canonical with (1 := cs) (p := pX a x); auto.
apply ltP_trans with (y := x); auto.
apply ltP_reduce with (1 := cs) (3 := H'0); auto.
apply canonical_imp_canonical with (a := a); auto.
apply (canonical_pX_ltP A A0 eqA); auto.
apply eqp_imp_canonical with (1 := cs) (p := pX a x); auto.
Qed.
Hint Resolve reduceplus_skip.
intros Q p q H'.
apply Rstar_n with (y := q); auto.
Qed.
Hint Resolve reduce_imp_reduceplus.
intros Q p q H'; inversion H'; auto.
inversion H; auto.
case (pO_reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q y);
auto.
Qed.
intros Q p q; pattern p, q in |- *.
apply (Opm_ind A n ltM ltM_dec); intros; auto.
exists (pO A n); exists s; split; [ idtac | split ]; auto.
apply reduce_imp_reduceplus.
apply
reduce_eqp_com
with
(1 := cs)
(p := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pO A n) p0)
(q := s); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
exists s; exists (pO A n); split; [ idtac | split ]; auto.
apply reduce_imp_reduceplus.
apply
reduce_eqp_com
with
(1 := cs)
(p := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p0 (pO A n))
(q := s); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
cut (canonical A0 eqA ltM p0);
[ intros C0 | apply canonical_imp_canonical with (a := a); auto ].
cut (canonical A0 eqA ltM q0);
[ intros C1 | apply canonical_imp_canonical with (a := b); auto ].
cut (~ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros nZ1 | apply canonical_nzeroP with (ltM := ltM) (p := p0); auto ].
cut (~ zeroP (A:=A) A0 eqA (n:=n) b);
[ intros nZ2 | apply canonical_nzeroP with (ltM := ltM) (p := q0); auto ].
inversion H1.
cut (canonical A0 eqA ltM (pX b0 q1));
[ intros C2 | apply inPolySet_imp_canonical with (L := Q); auto ].
cut (canonical A0 eqA ltM q1);
[ intros C3 | apply canonical_imp_canonical with (a := b0); auto ].
exists (pX a p0);
exists
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec b b0 nZb
q0 q1); split; [ idtac | split ]; auto.
apply reduce_imp_reduceplus.
apply reducetop with (b := b0) (nZb := nZb) (q := q1); auto.
cut (eqTerm (A:=A) eqA (n:=n) a0 b); [ intros eqb | idtac ].
apply divP_eqT with (1 := cs) (a := a0); auto.
rewrite <-
pluspf_inv2_eq with (1 := os) (a := a) (b := b) (p := p0) (q := q0) in H4;
auto.
rewrite
(pX_invl _ _ a0 b p1
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p0) q0)); auto.
rewrite <-
pluspf_inv2_eq with (1 := os) (a := a) (b := b) (p := p0) (q := q0) in H4;
auto.
rewrite
(pX_invl _ _ a0 b p1
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p0) q0)); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
a0 b0 nZb p1 q1); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
rewrite <-
pluspf_inv2_eq with (1 := os) (a := a) (b := b) (p := p0) (q := q0) in H4;
auto.
rewrite
(pX_invl _ _ a0 b p1
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p0) q0)); auto.
rewrite
(pX_invr _ _ p1
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p0) q0) a0 b); auto.
apply spminusf_plusTerm_r; auto.
rewrite <-
(pX_invl _ _ a0 b p1
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p0) q0)); auto.
elim (H q1);
[ intros p2 E; elim E; intros q2 E0; elim E0; intros H'4 H'5; elim H'5;
intros H'6 H'7; clear H'5 E0 E
| idtac
| idtac
| idtac ]; auto.
exists p2; exists (pX b q2); split; [ idtac | split ]; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pX a0
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec p2 q2)); auto.
apply eqpP1; auto; (apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto).
rewrite <-
pluspf_inv2_eq with (1 := os) (a := a) (b := b) (p := p0) (q := q0) in H4;
auto.
rewrite
(pX_invl _ _ a0 b p1
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p0) q0)); auto.
rewrite pluspf_inv2_eqa; auto.
apply ltP_pX_canonical; auto.
apply canonical_reduceplus with (Q := Q) (p := pX a p0); auto.
elim (order_reduceplus Q (pX a p0) p2);
[ intros H'5 | intros H'5 | idtac | idtac ]; auto.
apply ltP_trans with (y := pX a p0); auto.
apply (ltp_eqp_comp A A0 eqA) with (p := pX a p0) (q := pX b (pO A n)); auto.
apply ltP_pX_canonical; auto.
apply canonical_reduceplus with (Q := Q) (p := q0); auto.
elim (order_reduceplus Q q0 q2); [ intros H'5 | intros H'5 | idtac | idtac ];
auto.
apply ltP_trans with (y := q0); auto.
apply (canonical_pX_ltP A A0 eqA); auto.
apply (ltp_eqp_comp A A0 eqA) with (p := q0) (q := pX b (pO A n)); auto.
apply (canonical_pX_ltP A A0 eqA); auto.
apply reduce_eqp_com with (1 := cs) (p := p1) (q := q1); auto.
apply canonical_imp_canonical with (a := a0); auto.
rewrite H4; auto.
rewrite <-
pluspf_inv2_eq with (1 := os) (a := a) (b := b) (p := p0) (q := q0) in H4;
auto.
rewrite
(pX_invr _ _ p1
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p0) q0) a0 b); auto.
rewrite <- pluspf_inv1_eq with (a := a) (b := b) (p := p0) (q := q0) in H1;
auto.
cut (canonical A0 eqA ltM p0);
[ intros C0 | apply canonical_imp_canonical with (a := a); auto ].
cut (canonical A0 eqA ltM q0);
[ intros C1 | apply canonical_imp_canonical with (a := b); auto ].
cut (~ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros nZ1 | apply canonical_nzeroP with (ltM := ltM) (p := p0); auto ].
cut (~ zeroP (A:=A) A0 eqA (n:=n) b);
[ intros nZ2 | apply canonical_nzeroP with (ltM := ltM) (p := q0); auto ].
inversion_clear H1.
cut (canonical A0 eqA ltM (pX b0 q1));
[ intros C2 | apply inPolySet_imp_canonical with (L := Q); auto ].
cut (canonical A0 eqA ltM q1);
[ intros C3 | apply canonical_imp_canonical with (a := b0); auto ].
cut (~ zeroP (A:=A) A0 eqA (n:=n) b0);
[ intros nZ3 | apply canonical_nzeroP with (ltM := ltM) (p := q1); auto ].
exists
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b0 nZb
p0 q1); exists (pX b q0); split; [ idtac | split ];
auto.
apply reduce_imp_reduceplus; auto.
apply reducetop with (b := b0) (nZb := nZb) (q := q1); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b0 nZb
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec p0 (pX b q0)) q1).
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply spminusf_pluspf; auto.
elim (H q1);
[ intros p2 E; elim E; intros q2 E0; elim E0; intros H'4 H'5; elim H'5;
intros H'6 H'7; clear H'5 E0 E
| idtac
| idtac
| idtac ]; auto.
exists (pX a p2); exists q2; split; [ idtac | split ]; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pX a
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec p2 q2)); auto.
apply eqpP1; auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
rewrite pluspf_inv1_eqa; auto.
apply ltP_pX_canonical; auto.
apply canonical_reduceplus with (Q := Q) (p := p0); auto.
elim (order_reduceplus Q p0 p2);
[ intros H'14 | intros H'14 | idtac | idtac ]; auto.
apply ltP_trans with (y := p0); auto.
apply (canonical_pX_ltP A A0 eqA); auto.
apply (ltp_eqp_comp A A0 eqA) with (p := p0) (q := pX a (pO A n)); auto.
apply (canonical_pX_ltP A A0 eqA); auto.
apply ltP_pX_canonical; auto.
apply canonical_reduceplus with (Q := Q) (p := pX b q0); auto.
elim (order_reduceplus Q (pX b q0) q2);
[ intros H'14 | intros H'14 | idtac | idtac ]; auto.
apply ltP_trans with (y := pX b q0); auto.
apply (ltp_eqp_comp A A0 eqA) with (p := pX b q0) (q := pX a (pO A n)); auto.
cut (canonical A0 eqA ltM p0);
[ intros C0 | apply canonical_imp_canonical with (a := a); auto ].
cut (canonical A0 eqA ltM q0);
[ intros C1 | apply canonical_imp_canonical with (a := b); auto ].
cut (~ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros nZ1 | apply canonical_nzeroP with (ltM := ltM) (p := p0); auto ].
cut (~ zeroP (A:=A) A0 eqA (n:=n) b);
[ intros nZ2 | apply canonical_nzeroP with (ltM := ltM) (p := q0); auto ].
case (zeroP_dec A A0 eqA eqA_dec n (plusTerm (A:=A) plusA (n:=n) a b));
intros Z0.
elim (H s);
[ intros p1 E; elim E; intros q1 E0; elim E0; intros H'3 H'5; elim H'5;
intros H'6 H'7; clear H'5 E0 E
| idtac
| idtac
| idtac ]; auto.
exists (pX a p1); exists (pX b q1); split; [ idtac | split ]; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p1 q1); auto.
apply pluspf_inv3a with (1 := cs); auto.
rewrite pluspf_inv3a_eq with (1 := os) (a := a) (b := b) (p := p0) (q := q0);
auto.
inversion H1.
cut (canonical A0 eqA ltM (pX b0 q1));
[ intros C2 | apply inPolySet_imp_canonical with (L := Q); auto ].
cut (canonical A0 eqA ltM q1);
[ intros C3 | apply canonical_imp_canonical with (a := b0); auto ].
cut (divP A A0 eqA multA divA n a b0); [ intros div0 | idtac ].
cut (divP A A0 eqA multA divA n b b0); [ intros div1 | idtac ].
exists
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b0 nZb
p0 q1);
exists
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec b b0 nZb
q0 q1); split; [ idtac | split ]; auto.
apply reduce_imp_reduceplus; auto.
apply reducetop with (b := b0) (nZb := nZb) (q := q1); auto.
apply reduce_imp_reduceplus; auto.
apply reducetop with (b := b0) (nZb := nZb) (q := q1); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
a0 b0 nZb p1 q1).
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
rewrite <-
pluspf_inv3b_eq with (1 := os) (a := a) (b := b) (p := p0) (q := q0) in H4;
auto.
rewrite pX_invr with (1 := H4); auto.
rewrite pX_invl with (1 := H4); auto.
apply spminusf_plusTerm; auto.
apply divP_eqT with (1 := cs) (a := plusTerm (A:=A) plusA (n:=n) a b); auto.
apply plusTerm_eqT2; auto.
rewrite <-
pluspf_inv3b_eq with (1 := os) (a := a) (b := b) (p := p0) (q := q0) in H4;
auto.
rewrite <- pX_invl with (1 := H4); auto.
apply divP_eqT with (1 := cs) (a := plusTerm (A:=A) plusA (n:=n) a b); auto.
apply plusTerm_eqT1; auto.
rewrite <-
pluspf_inv3b_eq with (1 := os) (a := a) (b := b) (p := p0) (q := q0) in H4;
auto.
rewrite <- pX_invl with (1 := H4); auto.
rewrite <-
pluspf_inv3b_eq with (1 := os) (a := a) (b := b) (p := p0) (q := q0) in H4;
auto.
elim (H q1);
[ intros p2 E; elim E; intros q2 E0; elim E0; intros H'3 H'5; elim H'5;
intros H'6 H'7; clear H'5 E0 E
| idtac
| idtac
| idtac ]; auto.
exists (pX a p2); exists (pX b q2); split; [ idtac | split ]; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pX a0
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec p2 q2)); auto.
apply eqpP1; auto; apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
rewrite pX_invl with (1 := H4); auto.
apply pluspf_inv3b with (1 := cs); auto.
rewrite <- pX_invr with (1 := H4); auto.
Qed.
intros Q a p q H'; elim H'; auto.
intros x y z H'0 H'1 H'2 H'3 H'4.
apply Rstar_n with (y := mults (A:=A) multA (n:=n) a y); auto.
apply reduce_mults with (1 := cs); auto.
apply H'2; auto.
apply canonical_reduce with (1 := cs) (3 := H'0); auto.
Qed.
intros Q p q H'; elim H'; auto.
intros x y H'0 r H'1 H'2.
apply Rstar_0; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) x);
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
r)); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n)
(multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n))
(invTerm (A:=A) invA (n:=n) (T1 A1 n))) r);
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n);
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := mults (A:=A) multA (n:=n) (T1 A1 n) r);
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros x y z H'0 H'1 H'2 r H'3 H'4.
apply
Rstar_n
with
(y := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) y);
auto.
apply reduce_mults_invr with (1 := cs); auto.
apply reduce_eqp_com with (1 := cs) (p := x) (q := y); auto.
apply
eqp_imp_canonical
with
(p := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) r)
(1 := cs); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply reduceplus_mults; auto.
apply canonical_reduce with (1 := cs) (3 := H'0); auto.
apply
eqp_imp_canonical
with
(p := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) r)
(1 := cs); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
Qed.
intros Q p q H' H'0.
apply
reduceplus_mults_invr_lem
with
(p := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) p);
auto.
Qed.
intros a H' H'0 Q p q H'1; elim H'1; auto.
intros x y H'2 H'3 r s H'4 H'5 H'6.
apply Rstar_0; auto.
apply mults_eqp_inv with (a := a) (1 := cs); auto.
apply (eqp_trans _ _ _ _ _ _ _ _ _ cs n) with (y := y); auto.
apply (eqp_trans _ _ _ _ _ _ _ _ _ cs n) with (y := x); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros x y z H'2 H'3 H'4 H'5 r s H'6 H'7 H'8.
apply
Rstar_n
with
(y := mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(T1 A1 n) (b:=a) H') y); auto.
apply reduce_mults_invf with (1 := cs); auto.
apply reduce_eqp_com with (1 := cs) (p := x) (q := y); auto.
apply H'4; auto.
apply canonical_reduce with (1 := cs) (3 := H'2); auto.
apply canonical_mults with (1 := cs); auto.
apply canonical_reduce with (1 := cs) (3 := H'2); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n)
(multTerm (A:=A) multA (n:=n) a
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(T1 A1 n) (b:=a) H')) y); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(multTerm (A:=A) multA (n:=n) a (T1 A1 n)) (b:=a) H') y);
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=a) H') y);
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply mults_comp with (1 := cs); auto.
apply divTerm_multTerm_l with (1 := cs); auto.
apply divTerm_on_eqT with (1 := cs); auto.
apply (eqT_sym A n); auto.
Qed.
intros a H' H'0 Q p q H'1 H'2.
apply
reduceplus_mults_invf0
with
(a := a)
(p := mults (A:=A) multA (n:=n) a p)
(q := mults (A:=A) multA (n:=n) a q); auto.
Qed.
intros Q p q s H' H'0 H'1.
elim
(rep_plus_reduce Q p
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) q) s);
[ intros p1 E; elim E; intros q1 E0; elim E0; intros H'9 H'10; elim H'10;
intros H'11 H'12; clear H'10 E0 E
| idtac
| idtac
| idtac ]; auto.
exists p1;
exists (mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) q1);
split; [ idtac | split ]; auto.
apply reduceplus_mults_invr; auto.
cut (canonical A0 eqA ltM p1); [ intros Cp1 | idtac ].
cut (canonical A0 eqA ltM q1); [ intros Cq1 | idtac ].
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p1 q1); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p1
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) q1)));
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p1
(mults (A:=A) multA (n:=n)
(multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n))
(invTerm (A:=A) invA (n:=n) (T1 A1 n))) q1));
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 p1 (mults (A:=A) multA (n:=n) (T1 A1 n) q1));
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply
canonical_reduceplus
with
(Q := Q)
(p := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) q);
auto.
apply canonical_reduceplus with (Q := Q) (p := p); auto.
apply
reduce_eqp_com
with
(1 := cs)
(p := minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec p q)
(q := s); auto.
Qed.
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.
intros Q p q H' H'0 H'1.
apply
rep_plus_zero_reduce
with
(s := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p q)
(t := pO A n); auto.
Qed.
intros Q p q H' H'0 H'1.
elim
(red_plus_zero_reduce Q p
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) q));
[ intros r1 E; elim E; intros H'8 H'9; clear E | idtac | idtac | idtac ];
auto.
exists r1; split; auto.
apply
reduceplus_eqp_com
with
(p := q)
(q := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
r1)); auto.
apply reduceplus_mults_invr; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n)
(multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n))
(invTerm (A:=A) invA (n:=n) (T1 A1 n))) r1);
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply
reduceplus_eqp_com
with
(p := minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec p q)
(q := pO A n); auto.
Qed.
intros Q r; elim r; auto.
intros H' a b nZb p q H'0 H'1 H'2;
exists
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b nZb
p q); split; auto.
apply
reduceplus_eqp_com
with
(p := pX a p)
(q := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb p q); auto.
apply reduce_imp_reduceplus; auto.
apply reducetop with (b := b) (nZb := nZb) (q := q); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
change
(eqP A eqA n
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p) (pO A n)) (pX a p)) in |- *; auto.
change
(reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb p q) (pO A n))
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b
nZb p q)) in |- *; auto.
clear r; intros c r H' H'0 a b nZb p q H'1 H'2 H'3.
cut (canonical A0 eqA ltM p);
[ intros C0 | apply canonical_imp_canonical with (a := a); auto ].
cut (canonical A0 eqA ltM (pX b q));
[ intros C1 | apply inPolySet_imp_canonical with (L := Q); auto ].
cut (canonical A0 eqA ltM q);
[ intros C2 | apply canonical_imp_canonical with (a := b); auto ].
cut (canonical A0 eqA ltM r);
[ intros C3 | apply canonical_imp_canonical with (a := c); auto ].
cut (~ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros nZ1 | apply canonical_nzeroP with (ltM := ltM) (p := p); auto ].
cut (~ zeroP (A:=A) A0 eqA (n:=n) c);
[ intros nZ3 | apply canonical_nzeroP with (ltM := ltM) (p := r); auto ].
case (ltT_dec A n ltM ltM_dec a c);
[ intros temp; case temp; clear temp | idtac ]; intros test.
lapply H';
[ intros H'4; elim (H'4 a b nZb p q);
[ intros s E; elim E; intros H'13 H'14; clear E H'
| clear H'
| clear H'
| clear H' ]
| clear H' ]; auto.
exists (pX c s); split; auto.
change
(reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p) (pX c r)) (pX c s)) in |- *.
rewrite <- pluspf_inv2_eq with (1 := os) (a := a) (b := c) (p := p) (q := r);
auto.
apply reduceplus_skip; auto.
rewrite pluspf_inv2_eq with (1 := os) (a := a) (b := c) (p := p) (q := r);
auto.
change
(reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb p q) (pX c r)) (pX c s)) in |- *.
cut
(canonical A0 eqA ltM
(pX a
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b
nZb p q))); auto.
generalize H'14;
case
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b nZb
p q); simpl in |- *; auto.
intros H' H'5.
apply
reduceplus_eqp_com
with
(p := pX c
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (pO A n) r))
(q := pX c s); auto.
apply reduceplus_skip; auto.
apply eqp_imp_canonical with (1 := cs) (p := pX c r); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply eqp_imp_canonical with (1 := cs) (p := pX c r); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply (eqp_trans _ _ _ _ _ _ _ _ _ cs n) with (y := pX c r); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
change
(eqP A eqA n
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pO A n) (pX c r)) (pX c r)) in |- *; auto.
intros t l H' H'5.
change
(reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX t l) (pX c r)) (pX c s)) in |- *.
rewrite <- pluspf_inv2_eq with (1 := os) (a := t) (b := c) (p := l) (q := r);
auto.
apply reduceplus_skip; auto.
rewrite pluspf_inv2_eq with (1 := os) (a := t) (b := c) (p := l) (q := r);
auto.
apply canonical_pluspf; auto.
apply canonical_imp_canonical with (a := a); auto.
apply (ltT_trans A _ _ os) with (y := a); auto.
apply (canonical_pX_order A A0 eqA) with (l := l); auto.
apply (ltT_trans A _ _ os) with (y := a); auto.
apply (canonical_pX_order A A0 eqA) with (l := l); auto.
apply canonical_spminusf_full_t with (1 := cs); auto.
exists
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b
nZb p q) (pX c r)); split; auto.
apply reduce_imp_reduceplus.
change
(reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p) (pX c r))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb p q) (pX c r))) in |- *.
rewrite <- pluspf_inv1_eq with (1 := os) (a := a) (b := c) (p := p) (q := r);
auto.
apply reducetop with (b := b) (nZb := nZb) (q := q); auto.
apply spminusf_pluspf with (1 := cs); auto.
cut (divP A A0 eqA multA divA n c b);
[ intros divc | apply divP_eqT with (1 := cs) (a := a) ];
auto.
cut
(canonical A0 eqA ltM
(pX c
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b
nZb p q))); [ intros Ca | apply canonical_pX_eqT with (a := a) ];
auto.
case (zeroP_dec A A0 eqA eqA_dec n (plusTerm (A:=A) plusA (n:=n) a c));
intros Z0.
exists
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p r);
split; auto.
change
(reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p) (pX c r))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p r))
in |- *.
rewrite <- pluspf_inv3a_eq with (1 := os) (a := a) (b := c) (p := p) (q := r);
auto.
change
(reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb p q) (pX c r))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p r))
in |- *.
rewrite <- pluspf_inv2_eqa; auto.
apply reduce_imp_reduceplus.
apply reducetop with (b := b) (nZb := nZb) (q := q); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM
ltM_dec a b nZb p q)
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM
ltM_dec c b nZb r q)); auto.
apply spminusf_plusTerm_r with (1 := cs); auto.
apply canonical_spminusf with (1 := cs); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply spminusf_plusTerm_z; auto.
exists
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
(plusTerm (A:=A) plusA (n:=n) a c) b nZb
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p r)
q); split; auto.
change
(reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a p) (pX c r))
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
(plusTerm (A:=A) plusA (n:=n) a c) b nZb
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p r) q)) in |- *.
rewrite <- pluspf_inv3b_eq with (1 := os) (a := a) (b := c) (p := p) (q := r);
auto.
apply reduce_imp_reduceplus.
apply reducetop with (b := b) (nZb := nZb) (q := q); auto.
change
(reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb p q) (pX c r))
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
(plusTerm (A:=A) plusA (n:=n) a c) b nZb
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p r) q)) in |- *.
rewrite <-
pluspf_inv2_eqa
with
(1 := os)
(a := c)
(p :=
spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM
ltM_dec a b nZb p q)
(q := r); auto.
apply reduce_imp_reduceplus.
apply reducetop with (b := b) (nZb := nZb) (q := q); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM
ltM_dec a b nZb p q)
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM
ltM_dec c b nZb r q)); auto.
apply spminusf_plusTerm_r with (1 := cs); auto.
apply canonical_spminusf with (1 := cs); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply spminusf_plusTerm with (1 := cs); auto.
apply canonical_spminusf_full_t with (1 := cs); auto.
Qed.

Theorem one_plus_reduceplus :
forall (Q : list (poly A0 eqA ltM)) (p1 p2 : list (Term A n)),
reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q p1 p2 ->
forall r : list (Term A n),
canonical A0 eqA ltM p1 ->
canonical A0 eqA ltM r ->
exists s : list (Term A n),
reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p1
r) s /\
reduceplus Q
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p2
r) s.