# Library Buchberger.Preduce

Require Export Pspminus.
Section Preduce.

Inductive inPolySet : list (Term A n) list (poly A0 eqA ltM) -> Prop :=
| incons :
(a : Term A n) (p : list (Term A n))
(H : canonical A0 eqA ltM (pX a p)) (P : list (poly A0 eqA ltM)),
inPolySet (pX a p)
(exist (fun l0 : list (Term A n) => canonical A0 eqA ltM l0)
(pX a p) H :: P)
| inskip :
(a : poly A0 eqA ltM) (p : list (Term A n))
(P : list (poly A0 eqA ltM)), inPolySet p PinPolySet p (a :: P).
Hint Resolve incons inskip.

Lemma inPolySet_imp_canonical :
(p : list (Term A n)) (L : list (poly A0 eqA ltM)),
inPolySet p L canonical A0 eqA ltM p.
intros p L H'; elim H'; auto.
Qed.
Hint Resolve inPolySet_imp_canonical.

Lemma not_nil_in_polySet_elm :
(Q : list (poly A0 eqA ltM)) (p : list (Term A n)),
inPolySet p Q p pO A n.
intros Q p H'; elim H'; auto.
intros a p0 H'0 H'1; red in |- ×; intros H'2; inversion H'2.
Qed.

Theorem not_nil_in_polySet :
(Q : list (poly A0 eqA ltM)) (p : list (Term A n)),
¬ inPolySet (pO A n) Q.
intros Q H'; red in |- *; intros H'0.
lapply (not_nil_in_polySet_elm Q (pO A n));
[ intros H'3; apply H'3 || elim H'3 | idtac ]; auto.
Qed.

Lemma inPolySet_is_pX :
(p : list (Term A n)) (L : list (poly A0 eqA ltM)),
inPolySet p L
a : Term A n, ( q : list (Term A n), p = pX a q).
intros p L H'; elim H'; auto.
intros a p0 H'0 H'1; a; p0; auto.
Qed.

Inductive reduce (Q : list (poly A0 eqA ltM)) :
list (Term A n) → list (Term A n) → Prop :=
| reducetop :
(a b : Term A n) (nZb : ¬ zeroP (A:=A) A0 eqA (n:=n) b)
(p q r : list (Term A n)),
inPolySet (pX b q) Q
divP A A0 eqA multA divA n a b
eqP A eqA n
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb p q) rreduce Q (pX a p) r
| reduceskip :
(a b : Term A n) (p q : list (Term A n)),
reduce Q p q
eqTerm (A:=A) eqA (n:=n) a breduce Q (pX a p) (pX b q).
Hint Resolve reduceskip.

Lemma pO_reduce :
(Q : list (poly A0 eqA ltM)) (p : list (Term A n)),
¬ reduce Q (pO A n) p.
intros Q p; red in |- *; intros H'; inversion H'; auto.
Qed.

Theorem reduce_in_pO :
(Q : list (poly A0 eqA ltM)) (p : list (Term A n)),
inPolySet p Qreduce Q p (pO A n).
intros Q p; case p; simpl in |- *; auto.
intros H'; elim (inPolySet_is_pX (pO A n) Q);
[ intros a E; elim E; intros q E0; inversion E0 | idtac ];
auto.
intros a l H'.
cut (canonical A0 eqA ltM (pX a l)); [ intros Op0 | idtac ].
cut (canonical A0 eqA ltM l);
[ intros Op1 | apply canonical_imp_canonical with (a := a) ];
auto.
cut (¬ zeroP (A:=A) A0 eqA (n:=n) a); [ intros Nza | idtac ].
cut
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a a Nza l
l = pO A n).
intros H'0; rewrite <- H'0.
change
(reduce Q (pX a l)
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a a
Nza l l)) in |- ×.
apply reducetop with (b := a) (nZb := Nza) (q := l); auto.
apply divP_on_eqT with (1 := cs); auto.
cut
(eqP A eqA n
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a a
Nza l l) (pO A n)).
intros H'0; inversion H'0; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec l
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=a) Nza)
l)); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec l
(mults (A:=A) multA (n:=n) (T1 A1 n) l));
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec l l);
auto.
apply minuspf_refl with (1 := cs); auto.
apply (canonical_nzeroP A A0 eqA n ltM) with (p := l); auto.
apply inPolySet_imp_canonical with (L := Q); auto.
Qed.

Theorem ltP_reduce :
(Q : list (poly A0 eqA ltM)) (p q : list (Term A n)),
reduce Q p qcanonical A0 eqA ltM pltP (A:=A) (n:=n) ltM q p.
intros Q p q H'; elim H'; auto.
2: intros a b p0 q0 H'0 H'1 H'2 H'3; apply ltP_tl; auto.
2: apply (eqT_sym A n); apply (eqTerm_imp_eqT A eqA n); auto.
2: apply H'1; try apply canonical_imp_canonical with (a := a); auto.
intros a b nZb p0; case p0; auto.
intros q0 r H'0 H'1 H'2 H'3.
change (ltP (A:=A) (n:=n) ltM r (pX a (pO A n))) in |- ×.
apply (canonical_pX_ltP A A0 eqA); auto.
apply
eqp_imp_canonical
with
(1 := cs)
(p := pX a
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM
ltM_dec a b nZb (pO A n) q0)); auto.
apply canonical_spminusf_full_t with (1 := cs); auto.
apply inPolySet_imp_canonical with (L := Q); auto.
intros t l q0 r H'0 H'1 H'2 H'3; apply ltP_trans with (y := pX a (pO A n));
auto.
apply (canonical_pX_ltP A A0 eqA); auto.
apply
eqp_imp_canonical
with
(1 := cs)
(p := pX a
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM
ltM_dec a b nZb (pX t l) q0)); auto.
apply canonical_spminusf_full_t with (1 := cs); auto.
apply inPolySet_imp_canonical with (L := Q); auto.
change (ltP (A:=A) (n:=n) ltM (pX a (pO A n)) (pX a (pX t l))) in |- *; auto.
Qed.

Theorem canonical_reduce :
(Q : list (poly A0 eqA ltM)) (p q : list (Term A n)),
reduce Q p qcanonical A0 eqA ltM pcanonical A0 eqA ltM q.
intros Q p q H'; elim H'; auto.
intros a b nZb p0 q0 r H'0 H'1 H'2 H'3.
apply
eqp_imp_canonical
with
(1 := cs)
(p := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb p0 q0); auto.
apply canonical_spminusf with (1 := cs); auto.
apply canonical_imp_canonical with (a := a); auto.
apply canonical_imp_canonical with (a := b); auto.
apply inPolySet_imp_canonical with (L := Q); auto.
intros a b p0 q0 H'0 H'1 H'2 H'3.
apply eqp_imp_canonical with (1 := cs) (p := pX a q0); auto.
cut (canonical A0 eqA ltM p0);
[ intros C1 | apply canonical_imp_canonical with (a := a) ];
auto.
apply ltP_pX_canonical; auto.
apply canonical_nzeroP with (ltM := ltM) (p := p0); auto.
apply ltP_trans with (y := p0); auto.
apply ltP_reduce with (Q := Q); auto.
apply (canonical_pX_ltP A A0 eqA); auto.
Qed.

Theorem reduce_eqp_com :
(Q : list (poly A0 eqA ltM)) (p q r s : list (Term A n)),
reduce Q p q
canonical A0 eqA ltM peqP A eqA n p reqP A eqA n q sreduce Q r s.
intros Q p q r s H'; generalize r s; clear r s; elim H'; auto.
intros a b nZb p0 q0 r H'0 H'1 H'2 r0 s H'3 H'4 H'5.
inversion_clear H'4.
apply reducetop with (b := b) (nZb := nZb) (q := q0); auto.
apply divP_eqTerm_comp with (1 := cs) (a := a); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb p0 q0); auto.
cut (canonical A0 eqA ltM p0);
[ intros Cp0 | apply canonical_imp_canonical with (a := a); auto ].
apply eqp_spminusf_com with (1 := cs); auto.
apply eqp_imp_canonical with (1 := cs) (p := p0); auto.
apply canonical_imp_canonical with (a := b); auto.
apply inPolySet_imp_canonical with (L := Q); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply divP_eqTerm_comp with (1 := cs) (a := a); auto.
apply (eqp_trans _ _ _ _ _ _ _ _ _ cs n) with (2 := H'5); auto.
intros a b p0 q0 H'0 H'1 H'2 r s H'3 H'4 H'5.
inversion_clear H'4.
inversion_clear H'5.
apply reduceskip; auto.
apply H'1; auto.
apply canonical_imp_canonical with (a := a); auto.
apply (eqTerm_trans _ _ _ _ _ _ _ _ _ cs n) with (y := a); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply (eqTerm_trans _ _ _ _ _ _ _ _ _ cs n) with (y := b); auto.
Qed.

Theorem reduce_inv :
(Q : list (poly A0 eqA ltM)) (a b : Term A n) (p q : list (Term A n)),
reduce Q (pX a p) (pX b q) →
eqTerm (A:=A) eqA (n:=n) a b
canonical A0 eqA ltM (pX a p) → reduce Q p q.
intros Q a b p q H'; inversion_clear H'; auto.
intros eq Cap.
elim (not_double_canonical A A0 eqA n ltM) with (a := b) (p := q); auto.
apply
eqp_imp_canonical
with
(p := pX b
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM
ltM_dec a b0 nZb p q0))
(1 := cs); auto.
apply (canonical_pX_eqT A A0 eqA n ltM) with (a := a); auto.
apply canonical_spminusf_full_t with (1 := cs); auto.
apply inPolySet_imp_canonical with (L := Q); auto.
apply (eqTerm_imp_eqT A eqA n); auto.
apply nzeroP_comp_eqTerm with (1 := cs) (a := a); auto.
apply canonical_nzeroP with (p := p) (ltM := ltM); auto.
Qed.

Theorem reducetop_sp :
(Q : list (poly A0 eqA ltM)) (a b : Term A n)
(nZb : ¬ zeroP (A:=A) A0 eqA (n:=n) b) (p q : list (Term A n)),
inPolySet (pX b q) Q
divP A A0 eqA multA divA n a b
reduce Q (pX a p)
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b nZb
p q).
intros Q a b nZb p q H' H'0.
apply reducetop with (b := b) (nZb := nZb) (q := q); auto.
Qed.
Hint Resolve reducetop_sp.

Theorem reduce_inv2 :
(Q : list (poly A0 eqA ltM)) (p q : list (Term A n)),
reduce Q p q
canonical A0 eqA ltM p
x : list (Term A n),
( a : Term A n,
¬ zeroP (A:=A) A0 eqA (n:=n) a
inPolySet x Q
canonical A0 eqA ltM x
eqP A eqA n q
(minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec p
(mults (A:=A) multA (n:=n) a x))).
intros Q p q H'; elim H'; auto.
intros a b nZb p0 q0 r H'0 H'1 H'2 H'3; cut (canonical A0 eqA ltM (pX b q0));
[ intros C1 | apply inPolySet_imp_canonical with (L := Q); auto ];
(pX b q0);
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b) nZb);
split; [ idtac | split; [ idtac | split; [ idtac | idtac ] ] ];
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb p0 q0); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb (pX a p0) (pX b q0)); auto.
apply spminusf_extend with (1 := cs); auto.
intros a b p0 q0 H'0 H'1 H'2 H'3.
cut (canonical A0 eqA ltM p0);
[ intros Op1 | apply canonical_imp_canonical with (a := a) ];
auto.
cut (canonical A0 eqA ltM (pX a q0));
[ intros Op2 | apply canonical_reduce with (Q := Q) (p := pX a p0) ];
auto.
cut (canonical A0 eqA ltM q0);
[ intros Op3 | apply canonical_imp_canonical with (a := a) ];
auto.
elim H'1;
[ intros x E; elim E; intros a0 E0; elim E0; intros H'4 H'5; elim H'5;
intros H'6 H'7; elim H'7; intros H'8 H'9; clear H'7 H'5 E0 E H'1
| clear H'1 ]; auto.
x; a0; split; [ idtac | split; [ idtac | split ] ]; auto.
apply (eqp_trans _ _ _ _ _ _ _ _ _ cs n) with (y := pX a q0);
[ auto | apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n) ].
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
(pX a p0)
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n) a0 x)));
[ auto | idtac ].
cut (¬ zeroP (A:=A) A0 eqA (n:=n) a); [ intros Z0 | 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 (pX a (pO A n)) p0)
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n) a0 x))).
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); apply eqp_pluspf_com with (1 := cs);
auto.
apply plusTerm_is_pX with (1 := cs); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a (pO A n))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec p0
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n) a0 x))));
[ auto | idtac ].
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a (pO A n))
(minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec p0
(mults (A:=A) multA (n:=n) a0 x))).
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
(pX a (pO A n)) q0); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply plusTerm_is_pX with (1 := cs); auto.
apply canonical_nzeroP with (ltM := ltM) (p := q0); auto.
Qed.

Theorem mults_eqp_inv :
(a : Term A n) (p q : list (Term A n)),
eqP A eqA n (mults (A:=A) multA (n:=n) a p) (mults (A:=A) multA (n:=n) a q) →
¬ zeroP (A:=A) A0 eqA (n:=n) aeqP A eqA n p q.
intros a p; elim p; auto.
intros q; case q; simpl in |- *; auto.
intros t l H'; inversion_clear H'; auto.
intros a0 l H' q; case q; simpl in |- *; auto.
intros H'0; inversion_clear H'0; auto.
intros t l0 H'0; inversion_clear H'0; auto.
intros H'1.
change (eqP A eqA n (pX a0 l) (pX t l0)) in |- ×.
apply eqpP1; auto.
apply multTerm_eqTerm_inv with (1 := cs) (a := a); auto.
Qed.

Theorem reduce_mults_invf :
(a : Term A n) (nZa : ¬ zeroP (A:=A) A0 eqA (n:=n) a),
eqT a (T1 A1 n) →
(Q : list (poly A0 eqA ltM)) (p q : list (Term A n)),
canonical A0 eqA ltM p
reduce Q (mults (A:=A) multA (n:=n) a p) q
reduce Q p
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) (T1 A1 n) (b:=a) nZa) q).
intros a H' eq0 Q p; elim p; auto.
simpl in |- *; intros q H'0 H'1; inversion_clear H'1; auto.
intros a0 l H'0 q H'1 H'2.
cut (canonical A0 eqA ltM l);
[ intros Op0 | apply canonical_imp_canonical with (a := a0) ];
auto.
cut (¬ zeroP (A:=A) A0 eqA (n:=n) a0);
[ intros Z0 | apply canonical_nzeroP with (ltM := ltM) (p := l) ];
auto.
generalize H'0 H'1; inversion_clear H'2; clear H'0 H'1; auto.
intros H'0 H'1.
cut (canonical A0 eqA ltM (pX b q0));
[ intros Op1 | apply inPolySet_imp_canonical with (L := Q) ];
auto.
cut (canonical A0 eqA ltM q0);
[ intros Op2 | apply canonical_imp_canonical with (a := b) ];
auto.
cut (divP A A0 eqA multA divA n a0 b); [ intros d0 | idtac ].
apply
reduce_eqp_com
with
(p := pX a0 l)
(q := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
a0 b nZb l q0); auto.
apply mults_eqp_inv with (a := a); 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')) q); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := mults (A:=A) multA (n:=n) (T1 A1 n) q);
auto.
apply (eqp_trans _ _ _ _ _ _ _ _ _ cs n) with (y := q); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
(multTerm (A:=A) multA (n:=n) a a0) b nZb
(mults (A:=A) multA (n:=n) a l) q0); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply spminusf_multTerm with (1 := cs); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply mults_comp with (1 := cs); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(multTerm (A:=A) multA (n:=n) a (T1 A1 n)) (b:=a) H');
auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=a) H');
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply divTerm_multTerm_l with (1 := cs); auto.
apply divTerm_on_eqT with (1 := cs); auto.
apply (eqT_sym A n); auto.
apply
eqT_nzero_eqT_divP
with (1 := cs) (c := multTerm (A:=A) multA (n:=n) a a0) (nZb := nZb);
auto.
apply (eqTerm_imp_eqT A eqA n); auto.
apply (eqT_trans A n) with (y := multTerm (A:=A) multA (n:=n) (T1 A1 n) a0);
auto.
apply (eqTerm_imp_eqT A eqA n); auto.
apply (eqT_sym A n); auto.
intros H'0 H'1.
change
(reduce Q (pX a0 l)
(pX
(multTerm (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(T1 A1 n) (b:=a) H') b)
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(T1 A1 n) (b:=a) H') q0))) in |- ×.
apply reduceskip; auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := multTerm (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(T1 A1 n) (b:=a) H') (multTerm (A:=A) multA (n:=n) a a0));
auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := multTerm (A:=A) multA (n:=n)
(multTerm (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(T1 A1 n) (b:=a) H') a) a0); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := multTerm (A:=A) multA (n:=n) (T1 A1 n) a0);
auto.
apply eqTerm_multTerm_comp with (1 := cs); auto.
apply divTerm_on_eqT with (1 := cs); auto.
apply (eqT_sym A n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply multTerm_assoc with (1 := cs); auto.
Qed.

Theorem reduce_mults :
(Q : list (poly A0 eqA ltM)) (a : Term A n) (p q : list (Term A n)),
reduce Q p q
canonical A0 eqA ltM p
¬ zeroP (A:=A) A0 eqA (n:=n) a
reduce Q (mults (A:=A) multA (n:=n) a p) (mults (A:=A) multA (n:=n) a q).
intros Q a p q H'; generalize a; elim H'; clear a H'; auto.
simpl in |- *; auto.
intros a b nZb p0 q0 r H' H'0 H'1 a0 H'2 H'3.
cut (¬ zeroP (A:=A) A0 eqA (n:=n) (multTerm (A:=A) multA (n:=n) a0 a));
[ intros nZ0 | idtac ]; auto.
apply reducetop with (b := b) (nZb := nZb) (q := q0); auto.
apply divTerm_def with (nZb := nZb); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := multTerm (A:=A) multA (n:=n)
(multTerm (A:=A) multA (n:=n) a0
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b) nZb))
b); [ auto | idtac ].
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := multTerm (A:=A) multA (n:=n) a0
(multTerm (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b) nZb)
b)); [ auto | idtac ].
apply multTerm_assoc with (1 := cs); auto.
apply eqTerm_multTerm_comp with (1 := cs); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply divTerm_multTerm_l with (1 := cs); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n) a0
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM
ltM_dec a b nZb p0 q0)); auto.
apply spminusf_multTerm with (1 := cs); auto.
apply canonical_imp_canonical with (a := a); auto.
apply canonical_imp_canonical with (a := b); auto.
apply inPolySet_imp_canonical with (L := Q); auto.
apply nzeroP_multTerm with (1 := cs); auto.
apply canonical_nzeroP with (ltM := ltM) (p := p0); auto.
intros a b p0 q0 H' H'0 H'1 a0 H'2 H'3.
simpl in |- *; apply reduceskip; auto.
apply H'0; auto.
apply canonical_imp_canonical with (a := a); auto.
Qed.
Hint Resolve reduce_mults.

Theorem reduce_mults_inv_lem :
(Q : list (poly A0 eqA ltM)) (p q : list (Term A n)),
reduce Q p q
r : list (Term A n),
canonical A0 eqA ltM r
p = mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) r
reduce Q r
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) q).
intros Q p q H'; elim H'; clear H'; auto.
2: intros a b p0 q0 H' H'0 H'1 r; case r; auto.
2: intros H'2 H'3; inversion_clear H'3; auto.
2: intros a0 l H'2 H'3.
2: change
(reduce Q (pX a0 l)
(pX
(multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) b)
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
q0))) in |- ×.
2: apply reduceskip; auto.
2: apply H'0; auto.
2: apply canonical_imp_canonical with (a := a0); auto.
2: inversion_clear H'3; auto.
2: apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) a);
auto.
2: replace a with
(multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) a0);
[ idtac | inversion H'3 ]; auto.
2: apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := multTerm (A:=A) multA (n:=n) (T1 A1 n) a0);
auto.
2: apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := multTerm (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))) a0);
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros a b nZb p0 q0 r H' H'0 H'1 r0; case r0; auto.
intros; discriminate.
intros a0 l H'2 H'3.
change
(pX a p0 =
pX
(multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) a0)
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) l))
in H'3.
cut (canonical A0 eqA ltM l);
try apply canonical_imp_canonical with (a := a0);
auto; intros C0.
change (reduce Q (pX a0 l) (mults multA (invTerm invA (T1 A1 n)) r)) in |- ×.
apply reducetop with (b := b) (nZb := nZb) (q := q0); auto.
apply divP_eqT with (1 := cs) (a := a); auto.
rewrite pX_invl with (1 := H'3); auto.
apply (eqT_sym A n);
apply (eqT_trans A n) with (y := multTerm (A:=A) multA (n:=n) (T1 A1 n) a0);
auto.
apply (eqTerm_imp_eqT A eqA n); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l); auto.
cut (canonical A0 eqA ltM (pX b q0));
try apply inPolySet_imp_canonical with (L := Q); auto;
intros C1.
cut (canonical A0 eqA ltM q0);
try apply canonical_imp_canonical with (a := b); auto;
intros C2.
cut (canonical A0 eqA ltM (pX a p0)); [ intros C3 | idtac ]; auto.
cut (canonical A0 eqA ltM p0);
try apply canonical_imp_canonical with (a := a); auto;
intros C4.
cut (divP A A0 eqA multA divA n a0 b); [ intros divP0 | idtac ].
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM
ltM_dec a b nZb p0 q0)).
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n);
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
(multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) a) b nZb
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) p0) q0);
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n).
apply spminusf_multTerm with (1 := cs); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
a0 b nZb
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
p0) q0); auto.
apply eqTerm_spminusf_com with (1 := cs); auto.
rewrite pX_invl with (1 := H'3); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(multTerm (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))) a0);
auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (multTerm (A:=A) multA (n:=n) (T1 A1 n) a0);
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply divP_eqT with (1 := cs) (a := a); auto.
apply (eqT_trans A n) with (multTerm (A:=A) multA (n:=n) (T1 A1 n) a); auto.
apply (eqTerm_imp_eqT A eqA n); auto.
apply nzeroP_comp_eqTerm with (1 := cs) (a := invTerm (A:=A) invA (n:=n) a);
auto.
apply nZero_invTerm_nZero with (1 := cs); auto.
apply canonical_nzeroP with (ltM := ltM) (p := p0); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(invTerm (A:=A) invA (n:=n) (multTerm (A:=A) multA (n:=n) (T1 A1 n) a));
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply eqp_spminusf_com with (1 := cs); auto.
rewrite pX_invr with (1 := H'3); 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))) l);
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply mults_comp with (1 := cs); auto.
apply divP_eqT with (1 := cs) (a := a); auto.
rewrite pX_invl with (1 := H'3); auto.
apply (eqT_sym A n);
apply (eqT_trans A n) with (y := multTerm (A:=A) multA (n:=n) (T1 A1 n) a0);
auto.
apply (eqTerm_imp_eqT A eqA n); auto.
apply nzeroP_comp_eqTerm with (1 := cs) (a := invTerm (A:=A) invA (n:=n) a);
auto.
apply nZero_invTerm_nZero with (1 := cs); auto.
apply canonical_nzeroP with (ltM := ltM) (p := p0); auto.
rewrite pX_invl with (1 := H'3); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))) a0);
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply mult_invTerm_com with (1 := cs); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (multTerm (A:=A) multA (n:=n) (T1 A1 n) a0);
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
rewrite H'3;
change
(canonical A0 eqA ltM
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(pX a0 l))) in |- *; auto.
Qed.

Theorem reduce_mults_invr :
(Q : list (poly A0 eqA ltM)) (p q : list (Term A n)),
reduce Q
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) p) q
canonical A0 eqA ltM p
reduce Q p
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) q).
intros Q p q H' H'0.
apply
reduce_mults_inv_lem
with
(p := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) p);
auto.
Qed.

Definition irreducible (Q : list (poly A0 eqA ltM))
(p : list (Term A n)) := q : list (Term A n), ¬ reduce Q p q.

Lemma irreducible_inv_px_l :
(Q : list (poly A0 eqA ltM)) (a : Term A n) (p : list (Term A n)),
canonical A0 eqA ltM (pX a p) →
irreducible Q (pX a p) → irreducible Q (pX a (pO A n)).
unfold irreducible in |- ×.
intros Q a p H' H'0 q; red in |- *; intros H'1.
inversion_clear H'1.
apply
H'0
with
(q := spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
a b nZb p q0); auto.
inversion_clear H; auto.
Qed.

Lemma pO_irreducible :
Q : list (poly A0 eqA ltM), irreducible Q (pO A n).
unfold irreducible in |- *; auto.
intros Q q; red in |- *; intros H'; inversion H'.
Qed.
Hint Resolve pO_irreducible.

Theorem irreducible_eqp_com :
(Q : list (poly A0 eqA ltM)) (p q : list (Term A n)),
irreducible Q p
canonical A0 eqA ltM peqP A eqA n p qirreducible Q q.
unfold irreducible in |- ×.
intros Q p q H' H'0 H'1 q0; red in |- *; intros H'2.
apply H' with (q := q0); auto.
apply reduce_eqp_com with (p := q) (q := q0); auto.
apply eqp_imp_canonical with (1 := cs) (p := p); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
Qed.

Inductive pickinSetp :
Term A nlist (Term A n) → list (poly A0 eqA ltM) → Prop :=
| pickinSeteqp :
(a b : Term A n) (p : list (Term A n))
(H : canonical A0 eqA ltM (pX b p)) (P : list (poly A0 eqA ltM)),
divP A A0 eqA multA divA n a b
pickinSetp a (pX b p)
(exist (fun l0 : list (Term A n) ⇒ canonical A0 eqA ltM l0)
(pX b p) H :: P)
| pickinSetskip :
(a b : Term A n) (p : list (Term A n))
(q : poly A0 eqA ltM) (P : list (poly A0 eqA ltM)),
pickinSetp a p PpickinSetp a p (q :: P).
Hint Resolve pickinSeteqp pickinSetskip.

Lemma pickin_is_pX :
(a : Term A n) (p : list (Term A n)) (Q : list (poly A0 eqA ltM)),
pickinSetp a p Q
b : Term A n, ( q : list (Term A n), p = pX b q).
intros a p Q H'; elim H'; auto.
intros a0 b p0 H'0 H'1 H'2; b; p0; auto.
Qed.

Inductive reducehead (Q : list (poly A0 eqA ltM)) :
list (Term A n) → list (Term A n) → Prop :=
(a b : Term A n) (nZb : ¬ zeroP (A:=A) A0 eqA (n:=n) b)
(p q : list (Term A n)),
pickinSetp a (pX b q) Q
reducehead Q (pX a p)
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a
b nZb p q).

Lemma pick_inv_in :
(Q : list (poly A0 eqA ltM)) (a : Term A n) (p : list (Term A n)),
pickinSetp a p QinPolySet p Q.
intros Q a p H'; elim H'; auto.
Qed.

Lemma pick_inv_eqT_lem :
(Q : list (poly A0 eqA ltM)) (a : Term A n) (p : list (Term A n)),
pickinSetp a p Q
(b : Term A n) (q : list (Term A n)),
p = pX b qdivP A A0 eqA multA divA n a b.
intros Q a p H'; elim H'; auto.
intros a0 b p0 H'0 H'1 H'2 b0 q H'3; injection H'3; auto.
intros H'4 H'5; rewrite <- H'5; auto.
Qed.

Lemma pick_inv_eqT :
(Q : list (poly A0 eqA ltM)) (a b : Term A n) (p q : list (Term A n)),
pickinSetp a (pX b q) QdivP A A0 eqA multA divA n a b.
intros Q a b H' q H'0.
apply pick_inv_eqT_lem with (Q := Q) (p := pX b q) (q := q); auto.
Qed.

(Q : list (poly A0 eqA ltM)) (p q : list (Term A n)),
reducehead Q p qreduce Q p q.
intros Q p q H'; elim H'; auto.
intros a b nZb p0 q0 H'0.
apply reducetop with (b := b) (nZb := nZb) (q := q0); auto.
apply pick_inv_in with (a := a); auto.
apply pick_inv_eqT with (Q := Q) (q := q0); auto.
Qed.

Definition s2p : poly A0 eqA ltMlist (Term A n).
intros H'; elim H'.
intros x H'0; exact x.
Defined.

Theorem In_inp_inPolySet :
(Q : list (poly A0 eqA ltM)) (p : poly A0 eqA ltM),
In p Q¬ eqP A eqA n (s2p p) (pO A n) → inPolySet (s2p p) Q.
intros Q; elim Q; simpl in |- *; auto.
intros p H'; elim H'.
intros a l H' p H'0; elim H'0;
[ intros H'1; rewrite H'1; clear H'0 | intros H'1; clear H'0 ];
auto.
case p; simpl in |- *; auto.
intros x; case x; simpl in |- *; auto.
intros c H'0; elim H'0; auto.
intros t l0 c H;
change (inPolySet (pX t l0) (exist (canonical A0 eqA ltM) (pX t l0) c :: l))
in |- *; apply incons.
Qed.

Theorem in_inPolySet :
(P : list (poly A0 eqA ltM)) (p : poly A0 eqA ltM),
In p P¬ eqP A eqA n (s2p p) (pO A n) → inPolySet (s2p p) P.
intros P; elim P; auto.
intros p H'; inversion H'.
simpl in |- ×.
intros a l H' p H'0; elim H'0;
[ intros H'1; rewrite <- H'1; clear H'0 | intros H'1; clear H'0 ];
auto.
case a; simpl in |- ×.
intros x; case x; auto.
intros c H'0; elim H'0; auto.
intros t l0 c H;
change (inPolySet (pX t l0) (exist (canonical A0 eqA ltM) (pX t l0) c :: l))
in |- *; apply incons.
Qed.

Theorem inPolySet_inv0 :
(P : list (poly A0 eqA ltM)) (p : list (Term A n)),
inPolySet p P¬ eqP A eqA n p (pO A n).
intros P p H'; elim H'; auto.
intros a p0 H'0 H'1; red in |- *; intros H'2; inversion H'2.
Qed.

Theorem inPolySet_inv1 :
(P : list (poly A0 eqA ltM)) (p : list (Term A n)),
inPolySet p P q : poly A0 eqA ltM, In q P p = s2p q.
intros P p H'; elim H'; auto.
intros a p0 H P0.
(exist (canonical A0 eqA ltM) (a :: p0) H); split; simpl in |- *; auto.
intros a p0 P0 H'0 H'1; elim H'1; intros q E; elim E; intros H'2 H'3;
clear E H'1.
q; split; simpl in |- *; auto.
Qed.

Theorem Incl_inp_inPolySet :
P Q : list (poly A0 eqA ltM),
incl P Q p : list (Term A n), inPolySet p PinPolySet p Q.
intros P Q H' p H'0; auto.
elim (inPolySet_inv1 P p);
[ intros q E; elim E; intros H'4 H'5; clear E | idtac ];
auto.
rewrite H'5.
apply in_inPolySet; auto.
rewrite <- H'5; auto.
apply inPolySet_inv0 with (P := P); auto.
Qed.
End Preduce.