Section BuchRed.
Load "hCoefStructure".
Load "hOrderStructure".
Load "hBuch".
intros L p q H; apply Cb_incl with (1 := cs) (P := addEnd A A0 eqA n ltM p L);
auto.
elim L; simpl in |- *; auto.
intros a l H0 a0 H1; elim H1; clear H1; intros H1; auto.
case (H0 a0); auto.
Qed.
intros L p q H; apply Cb_incl with (1 := cs) (P := p :: L); auto.
elim L; simpl in |- *; auto.
intros a l H0 a0 H1; elim H1; clear H1; intros H1; auto.
case H1; auto.
Qed.
intros L p q H H0.
apply Cb_trans with (1 := cs) (b := p); auto.
apply Cb_cons_addEnd; auto.
Qed.
intros p L; unfold nf, LetP in |- *; auto.
case
(Reducef A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec
os L p); simpl in |- *; auto.
intros x0; case x0; simpl in |- *.
intros x c H'.
change
(CombLinear A A0 eqA plusA multA eqA_dec n ltM ltM_dec
(p :: L)
(mults (A:=A) multA (n:=n)
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM x c)) x))
in |- *.
apply CombLinear_mults1 with (1 := cs); auto.
apply unit_nZ with (1 := cs); auto.
apply reducestar_cb1 with (1 := cs); auto.
Qed.
intros L1 L2 H' q; case q; simpl in |- *.
intros x H'0 H'1.
apply CombLinear_compo with (1 := cs) (L1 := L1); auto.
intros q0 H'2.
case inPolySet_inv1 with (1 := H'2); auto.
intros q1 H; elim H; intros H0 H1; clear H.
lapply (H' q1); [ intros H'6 | idtac ]; auto.
generalize H'6 H1; case q1; simpl in |- *; auto.
intros x0 H'5 H'7 H'8; rewrite H'8; auto.
Defined.
intros p L; unfold nf in |- *.
case
(Reducef A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec
os L p); auto.
case p.
unfold LetP in |- *; intros x H' x0; case x0; simpl in |- *.
intros x1 c H'0.
change
(CombLinear A A0 eqA plusA multA eqA_dec n ltM ltM_dec
(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 c)) x1)
(canonical_mults A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec
n ltM os (unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM x1 c))
x1
(unit_nZ A A0 A1 eqA plusA invA minusA multA divA cs n ltM
(mks A A0 eqA n ltM x1 c)) c) :: L) x)
in |- *.
apply CombLinear_compo with (1 := cs) (L1 := mks A A0 eqA n ltM x1 c :: L);
auto.
change
(CombLinear A A0 eqA plusA multA eqA_dec n ltM ltM_dec
(mks A A0 eqA n ltM x1 c :: L)
(s2p A A0 eqA n ltM (mks A A0 eqA n ltM x H')))
in |- *.
apply reducestar_cb2 with (1 := cs); auto.
intros q H'1; inversion H'1; auto.
2: apply CombLinear_id with (1 := cs); auto.
2: apply inskip; auto.
generalize c H2; case x1; auto.
intros c0 H'2; inversion H'2.
intros a0 l c0 H'2.
cut
(~
zeroP (A:=A) A0 eqA (n:=n)
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM (a0 :: l) c0)));
[ intros nZu | idtac ]; auto.
apply
CombLinear_1
with
(a := divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(T1 A1 n)
(b:=unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM (pX a0 l) c0))
nZu)
(p := pO A n)
(q := mults (A:=A) multA (n:=n)
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM (pX a0 l) c0))
(pX a0 l)); auto.
simpl in |- *; auto.
change
(inPolySet A A0 eqA n ltM
(pX
(multTerm (A:=A) multA (n:=n)
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM (pX a0 l) c0)) a0)
(mults (A:=A) multA (n:=n)
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM (pX a0 l) c0)) l))
(exist (fun a1 => canonical A0 eqA ltM a1)
(pX
(multTerm (A:=A) multA (n:=n)
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM (a0 :: l) c0))
a0)
(mults (A:=A) multA (n:=n)
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM (a0 :: l) c0))
l))
(canonical_mults A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec
n ltM os
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM (pX a0 l) c0))
(pX a0 l)
(unit_nZ _ _ _ _ _ _ _ _ _ cs _ ltM
(mks A A0 eqA n ltM (pX a0 l) c0)) c0) :: L))
in |- *.
apply incons; auto.
apply CombLinear_0; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(T1 A1 n)
(b:=unit A A0 A1 eqA divA n ltM
(mks A A0 eqA n ltM (a0 :: l) c0)) nZu)
(mults (A:=A) multA (n:=n)
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM (a0 :: l) c0))
(a0 :: l))); auto.
2: apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (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:=unit A A0 A1 eqA divA n ltM
(mks A A0 eqA n ltM (a0 :: l) c0)) nZu)
(unit A A0 A1 eqA divA n ltM (mks A A0 eqA n ltM (a0 :: l) c0)))
(a0 :: l)); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := mults (A:=A) multA (n:=n) (T1 A1 n) (a0 :: l));
auto.
rewrite H'2; auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply mults_comp with (1 := cs); auto.
apply divTerm_on_eqT with (1 := cs); auto.
apply (eqT_sym A n); auto.
apply unit_T1; auto.
apply unit_nZ with (1 := cs); auto.
Qed.
intros L p q H' H'0.
apply Cb_comp with (L1 := p :: L); auto.
simpl in |- *; auto.
intros p0 H'1; case H'1;
[ intros H'2; rewrite <- H'2; clear H'1 | intros H'2; clear H'1 ];
auto.
generalize H'; case p; simpl in |- *; auto.
intros x; case x; simpl in |- *; auto.
intros H'1 H'3; try assumption.
change (CombLinear A A0 eqA plusA multA eqA_dec n ltM ltM_dec L (pO A n))
in |- *.
apply CombLinear_0; auto.
intros a l H'1 H'3; elim H'3; auto.
apply Cb_id with (1 := cs); auto.
Qed.
intros p L1 H' L2 H'0.
apply Cb_comp with (L1 := L1); auto.
Qed.
Definition reducep (L : list (poly A0 eqA ltM)) (p q : poly A0 eqA ltM) :
Prop :=
reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec L
(s2p A A0 eqA n ltM p) (s2p A A0 eqA n ltM q).
intros L H'; inversion H'; auto.
intros p H'0.
case
(Reducef A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec
os L p).
intros x H'1; inversion H'.
lapply (H0 (s2p A A0 eqA n ltM p) (s2p A A0 eqA n ltM x));
[ intros H'4; lapply H'4; [ clear H'4 | clear H'4 ] | idtac ];
auto.
inversion H'1; auto.
inversion H1; auto.
intros H'2; left.
cut (eqP A eqA n (s2p A A0 eqA n ltM p) (pO A n)); auto.
case p; simpl in |- *; auto.
intros x1; case x1; auto.
intros a l H'3 H'4; inversion H'4.
apply (eqp_trans _ _ _ _ _ _ _ _ _ cs n) with (y := s2p A A0 eqA n ltM x);
auto.
intros H'2; right; cut (canonical A0 eqA ltM y); auto.
intros H'3; exists (mks A A0 eqA n ltM y H'3); generalize H5; case p;
simpl in |- *; auto.
apply canonical_reduce with (1 := cs) (3 := H5); auto.
generalize H'0; case p; simpl in |- *; auto.
generalize H'0; case p; simpl in |- *; auto.
Qed.
intros L H'.
apply Grobner0.
intros p q H'0 H'1.
cut (canonical A0 eqA ltM p).
intros H'2.
cut (canonical A0 eqA ltM q).
intros H'3.
elim (H' (mks A A0 eqA n ltM q H'3)); [ intros H'6 | intros H'6 | idtac ];
auto.
generalize H'3 H'6; case q; simpl in |- *; auto.
intros a l H'4 H'5; elim H'5; auto.
case H'6; intros q0 E; clear H'6.
inversion H'1.
generalize H0 E; case q0; auto.
simpl in |- *; auto.
intros x H'4 H'5 H'6;
absurd
(reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec L q x);
auto.
simpl in |- *; auto.
apply reducestar_cb with (a := p) (1 := cs); auto.
inversion H'1.
apply canonical_reduceplus with (1 := cs) (3 := H); auto.
apply CombLinear_canonical with (1 := cs) (3 := H'0); auto.
Qed.
intros p q; case p; case q; simpl in |- *.
intros x H' x0 c Q H'0; inversion H'0.
generalize c; rewrite <- H2; simpl in |- *; auto.
case inPolySet_inv1 with (1 := H); auto.
intros q1 E; case E; intros H'1 H'2; clear E.
intros c0; exists q1; split; [ right | split ]; auto.
generalize H'2; case q1; simpl in |- *; auto.
intros x1; case x1; auto.
intros H'4 H'5; discriminate H'5; auto.
intros a0 l H'4 H'5; injection H'5; intros.
rewrite <- H5; trivial.
generalize H'1 H'2 c0; case q1; simpl in |- *; auto.
intros x1; case x1; auto.
intros c1 H'3 H'4; discriminate H'4.
exists (exist (fun l : list (Term A n) => canonical A0 eqA ltM l) x H');
split; [ left | split ]; simpl in |- *; auto.
generalize c; rewrite <- H1; unfold pX in |- *; auto.
generalize H'; rewrite <- H2; unfold pX in |- *; auto.
intros H'1 H'2.
apply divP_eqTerm_comp with (a := b) (1 := cs); auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) b); [ intros Z | idtac ]; auto.
apply divTerm_def with (nZb := Z); auto.
apply canonical_nzeroP with (ltM := ltM) (p := q0); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
generalize c H'; rewrite <- H2; auto.
Qed.
intros a b Q H'; elim H'; auto.
intros x y H'0 H'1 x0 y0 H'2 H'3 H'4; exists y0; split; [ idtac | split ];
auto with datatypes.
generalize H'0 H'4; clear H'0 H'4; rewrite <- H'3; rewrite <- H'2; case x0;
case y0; simpl in |- *; auto.
intros x1; case x1; simpl in |- *; auto.
intros H2 x2; case x2; auto.
intros t l H'5 H'6; inversion_clear H'6; auto.
intros t l H'0 x2; case x2; simpl in |- *; auto.
intros t0 l0 H'5 H'6; inversion_clear H'6; auto.
intros H5; apply divP_eqTerm_comp with (a := t) (1 := cs); auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) t); [ intros Z | idtac ]; auto.
apply divTerm_def with (nZb := Z); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
generalize H'0 H'4; clear H'0 H'4; rewrite <- H'3; rewrite <- H'2; case x0;
case y0; simpl in |- *; auto.
intros x1; case x1; simpl in |- *; auto.
intros H'0 x2; case x2; simpl in |- *; auto.
intros t l H'4 H'5; inversion_clear H'5.
intros x y z H'0 H'1 H'2 H'3 x0 y0 H'4 H'5 H'6.
cut (canonical A0 eqA ltM y); [ intros Z | idtac ].
2: apply canonical_reduce with (1 := cs) (3 := H'0); auto.
lapply (reduce_divp x0 (mks A A0 eqA n ltM y Z) Q); [ intros H'9 | idtac ];
auto.
2: rewrite H'4; simpl in |- *; auto.
case H'9; intros r E; case E; simpl in |- *; intros H'7 H'8; case H'8;
intros H'10 H'11; clear H'8 E H'9.
case H'7; [ intros H'8; clear H'7 | intros H'8; clear H'7 ].
2: exists r; split; [ right | idtac ]; auto.
lapply H'2;
[ intros H'7; lapply (H'7 (mks A A0 eqA n ltM y Z) y0); simpl in |- *;
[ intros H'13; lapply H'13;
[ intros H'14; lapply H'14;
[ intros H'15; clear H'14 H'13 H'2 | clear H'14 H'13 H'2 ]
| clear H'13 H'2 ]
| clear H'2 ]
| clear H'2 ]; auto.
case H'15; intros r0 E; case E; intros H'2 H'9; case H'9; intros H'12 H'13;
clear H'9 E H'15.
exists r0; split; [ idtac | split ]; auto.
apply (divp_trans _ _ _ _ _ _ _ _ _ cs n ltM) with (y := r); auto.
rewrite <- H'8; auto.
generalize H'11; rewrite <- H'8.
generalize Z; case y; auto.
Qed.
intros a b Q H' H'0.
apply
reduceplus_divp_lem
with (a := s2p A A0 eqA n ltM a) (b := s2p A A0 eqA n ltM b);
auto.
apply canonical_s2p; auto.
Qed.
intros a b Q H' H'0; apply reduceplus_divp; auto.
inversion H'; auto.
Qed.
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.
intros p L1 L2 H' q; case p; case q; simpl in |- *; auto.
intros x H'0 x0 c H'1; generalize c; unfold reducep in H'1; simpl in H'1;
elim H'1; auto.
intros a b nZb p0 q0 r H'2 H'3 H'4 c0.
case inPolySet_inv1 with (1 := H'2).
intros q1 E; case E; intros H4 H5; clear E.
elim (H' q1);
[ intros r2 E; elim E; intros H'10 H'11; clear E | idtac | idtac ];
auto.
generalize H'10 H'11 H5 c0 H4; case q1; simpl in |- *; auto.
intros x1 c1; case r2; simpl in |- *; auto.
intros x2; case x2; simpl in |- *; auto.
generalize c1; case x1; auto.
intros c2 c3 H'6 H'8 H'9 c4 H'12; discriminate.
intros t l c2 c3 H'6 H'8; elim H'8.
intros a0 l c2; generalize c1; case x1; auto.
intros c3 H'6 H'8; elim H'8.
intros t l0 c3 H'6 H'8 H'9 c4 H'12.
cut (~ zeroP (A:=A) A0 eqA (n:=n) a0); [ intros nZa0 | idtac ].
cut
(canonical A0 eqA ltM
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a a0
nZa0 p0 l)); [ intros Z | idtac ].
exists
(mks A A0 eqA n ltM
(spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a a0
nZa0 p0 l) Z); simpl in |- *; auto.
red in |- *; simpl in |- *; apply reducetop_sp with (1 := cs); auto.
change
(inPolySet A A0 eqA n ltM
(s2p A A0 eqA n ltM (mks A A0 eqA n ltM (pX a0 l) c2)) L2)
in |- *.
apply in_inPolySet; auto.
simpl in |- *.
red in |- *; intros H'13; inversion H'13.
injection H'9; intros.
apply (divP_trans _ _ _ _ _ _ _ _ _ cs n) with (y := b); auto.
rewrite H0; auto.
apply canonical_spminusf_full with (1 := cs); auto.
injection H'9; intros.
apply (divP_trans _ _ _ _ _ _ _ _ _ cs n) with (y := b); auto.
rewrite H0; auto.
apply canonical_nzeroP with (ltM := ltM) (p := l); auto.
generalize H5; case q1; simpl in |- *; auto.
intros x1; case x1; simpl in |- *; auto.
intros c1 H'6; discriminate.
intros a b p0 q0 H'2 H'3 H'4 c0.
cut (canonical A0 eqA ltM p0);
[ intro | apply canonical_imp_canonical with (a := a); auto ].
case (H'3 H); intros r; case r.
intros x1 H'6 H'7.
cut (canonical A0 eqA ltM (pX b x1)); [ intros Z | idtac ].
exists (mks A A0 eqA n ltM (pX b x1) Z); simpl in |- *; auto.
red in |- *; simpl in |- *; red in H'7; simpl in H'7; auto.
apply eqp_imp_canonical with (1 := cs) (p := pX a x1); 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 (1 := cs) (3 := H'7); auto.
apply (canonical_pX_ltP A A0 eqA); auto.
Qed.

Theorem nf_divp_zero :
forall (p : poly A0 eqA ltM) (L : list (poly A0 eqA ltM)),
~ zerop A A0 eqA n ltM p ->
zerop A A0 eqA n ltM
(nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os p
L) ->
exists q : poly A0 eqA ltM,
In q L /\ divp A A0 eqA multA divA n ltM p q /\ ~ zerop A A0 eqA n ltM q.