Section BuchAux.
Load "hCoefStructure".
Load "hOrderStructure".
Load "hComb".
Definition red (a : poly A0 eqA ltM) (P : list (poly A0 eqA ltM)) :=
reducestar A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec P
(s2p A A0 eqA n ltM a) (pO A n).
Definition addEnd :
poly A0 eqA ltM -> list (poly A0 eqA ltM) -> list (poly A0 eqA ltM).
intros a H'0; elim H'0.
exact (a :: nil).
intros b L1 Rec; exact (b :: Rec).
Defined.
intros a b aL; elim aL; simpl in |- *; auto.
intros H'; case H'; [ intros H'0; rewrite <- H'0 | intros H'0; clear H' ];
auto.
intros a0 l H' H'0; case H'0;
[ intros H'1; rewrite <- H'1; clear H'0 | intros H'1; clear H'0 ];
auto.
case (H' H'1); auto.
Qed.
intros a aL; elim aL; simpl in |- *; auto.
Qed.
intros a b aL; elim aL; simpl in |- *; auto.
intros a0 l H' H'0; case H'0; auto.
Qed.
intros a P; elim P; simpl in |- *; auto.
intros a0 l H'; elim H'; auto.
Qed.
Definition spolyp : poly A0 eqA ltM -> poly A0 eqA ltM -> poly A0 eqA ltM.
intros p q; case p; case q.
intros x Cpx x0 Cpx0;
exists
(spolyf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec x x0 Cpx
Cpx0); auto.
apply spolyf_canonical with (1 := cs); auto.
Defined.
intros a b; case a; case b; simpl in |- *.
unfold red in |- *; simpl in |- *.
intros x Cx x0 Cx0 aL H'1; inversion H'1.
cut
(canonical A0 eqA ltM
(spolyf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec x x0 Cx
Cx0)); [ intros Op1 | apply spolyf_canonical with (1 := cs) ];
auto.
cut
(canonical A0 eqA ltM
(spolyf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec x0 x Cx0
Cx)); [ intros Op2 | apply spolyf_canonical with (1 := cs) ];
auto.
apply reducestar0; auto.
apply
reduceplus_eqp_com
with
(1 := cs)
(p := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(spolyf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
x x0 Cx Cx0))
(q := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(pO A n)); auto.
apply reduceplus_mults with (1 := cs); auto.
inversion H; auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply spolyf_com with (1 := cs); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply spolyf_com with (1 := cs); auto.
Qed.
intros Q p H' H'0.
elim
reduce0_reducestar
with
(1 := cs)
(eqA_dec := eqA_dec)
(ltM_dec := ltM_dec)
(Q := Q)
(p := pO A n); auto.
intros t E; apply reducestar0; auto.
apply pO_irreducible; auto.
Qed.
Definition spO : poly A0 eqA ltM.
exists (pO A n); simpl in |- *; auto.
Defined.
Definition sp1 : poly A0 eqA ltM.
exists (pX (A1, M1 n) nil); auto.
Defined.
Definition sgen : nat -> poly A0 eqA ltM.
intros m; exists (pX (A1, gen_mon n m) (pO A n)).
apply canonicalp1; auto.
Defined.
Definition sscal : A -> poly A0 eqA ltM -> poly A0 eqA ltM.
intros a p; case p.
intros x H'1; exists (tmults A0 multA eqA_dec (a, M1 n) x); auto.
unfold tmults in |- *; case (zeroP_dec A A0 eqA eqA_dec n (a, M1 n));
simpl in |- *; auto.
Qed.
intros a; case (seqp_dec A A0 eqA eqA_dec n ltM a spO).
case a; unfold red, spO, seqP in |- *; simpl in |- *.
intros x c H' p H'0; inversion H'; auto.
apply reducestar_pO_is_pO with (p := x); auto.
case a; unfold red, spO, seqP in |- *; simpl in |- *.
intros x c H' p H'0.
apply rstar_rtopO; auto.
apply Rstar_n with (y := pO A n); auto.
apply reduce_in_pO with (1 := cs); auto.
change
(inPolySet A A0 eqA n ltM
(s2p A A0 eqA n ltM
(exist (fun l : list (Term A n) => canonical A0 eqA ltM l) x c)) p)
in |- *; apply In_inp_inPolySet; auto.
red in |- *; intros H'2; apply H'; apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n);
auto.
apply Rstar_0; auto.
Qed.
intros a P; case a.
unfold red in |- *; simpl in |- *; auto.
intros x H'.
apply rstar_rtopO; auto.
apply spolyf_canonical with (1 := cs); auto.
apply Rstar_0; auto.
apply spolyf_pO with (1 := cs); auto.
Qed.
intros P Q H' a b H'0; elim H'0; auto.
intros a0 b0 nZb p q r H'1 H'2 H'3; auto.
apply reducetop with (b := b0) (nZb := nZb) (q := q); auto.
Qed.
intros P Q H' a b H'0; elim H'0; auto.
intros; apply Rstar_0; auto.
intros x y z H'1 H'2 H'3.
apply Rstar_n with (y := y); auto.
apply inP_reduce with (Q := Q); auto.
Qed.
intros P Q H' a b H'0; inversion H'0; auto.
apply inP_reduceplus with (Q := Q); auto.
Qed.
unfold red in |- *.
intros a p q H' H'0.
inversion H'0.
apply rstar_rtopO; auto.
case a; auto.
apply inP_reducestar with (Q := p); auto.
intros a0 H'1.
apply Incl_inp_inPolySet with (P := p); auto.
Qed.
Definition zerop : poly A0 eqA ltM -> Prop.
intros H'; case H'.
intros x; case x.
intros H'0; exact True.
intros a l H'0; exact False.
Defined.
intros H'; case H'.
intros x; case x.
intros c; left; simpl in |- *; auto.
intros a l c; right; red in |- *; intros H'0; inversion H'0.
Qed.
Definition divp : poly A0 eqA ltM -> poly A0 eqA ltM -> Prop.
intros H'; case H'.
intros x; case x.
intros H'0 H'1; exact False.
intros a l H'0 H'1; case H'1.
intros x0; case x0.
intros H'2; exact False.
intros a0 l0 H'2.
exact (divP A A0 eqA multA divA n a a0).
Defined.
red in |- *.
intros x y z; case x; case y; case z.
intros x0 c x1 c0 x2 c1; generalize c c0 c1; case x0; case x1; case x2;
simpl in |- *; auto.
intros a l a0 l0 H' H'0 H'1 H'2; elim H'2.
intros a l a0 l0 a1 l1 H' H'0 H'1 H'2 H'3.
apply (divP_trans _ _ _ _ _ _ _ _ _ cs n) with (y := a0); auto.
Qed.
intros a b; case a; case b.
intros x c x0 c0; generalize c c0; case x; case x0; simpl in |- *.
intros H' H'0; right; red in |- *; intros H'1; elim H'1.
intros a0 l H' H'0; right; red in |- *; intros H'1; elim H'1.
intros a0 l H' H'0; right; red in |- *; intros H'1; elim H'1.
intros a0 l a1 l0 H' H'0.
apply divP_dec with (1 := cs); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l0); auto.
Qed.
Definition ppcp : poly A0 eqA ltM -> poly A0 eqA ltM -> poly A0 eqA ltM.
intros H'; case H'.
intros x; case x.
intros H'0 H'1; exists (pO A n); auto.
intros a l H'0 H'1; case H'1.
intros x0; case x0.
intros H'2; exists (pO A n); auto.
intros a0 l0 H'2; exists (ppc (A:=A) A1 (n:=n) a a0 :: pO A n).
change (canonical A0 eqA ltM (pX (ppc (A:=A) A1 (n:=n) a a0) (pO A n)))
in |- *; apply canonicalp1; auto.
apply ppc_nZ with (1 := cs); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l0); auto.
Defined.
intros a b c; (case a; case b; case c).
intros x c0 x0 c1 x1 c2; generalize c0 c1 c2; case x; case x0; case x1;
simpl in |- *; auto.
intros a0 l a1 l0 a2 l1 H' H'0 H'1 H'2.
apply divP_eqTerm_comp with (1 := cs) (a := ppc (A:=A) A1 (n:=n) a0 a1); auto.
Qed.
intros a b; (case a; case b).
intros x c0 x0 c1; generalize c0 c1; case x; case x0; simpl in |- *; auto.
intros a0 l a1 l0 H' H'0 H'1 H'2.
apply divP_ppcr with (1 := cs); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l0); auto.
Qed.
intros a b; (case a; case b).
intros x c0 x0 c1; generalize c0 c1; case x; case x0; simpl in |- *; auto.
Qed.
intros a b; (case a; case b).
intros x c0 x0 c1; generalize c0 c1; case x; case x0; simpl in |- *; auto.
Qed.
Hint Resolve pO_irreducible.
intros Q; apply reducestar0; auto.
apply Rstar_0; auto.
Qed.
Hint Resolve reducetopO_pO.
intros a b; (case a; case b).
intros x c0 x0 c1; generalize c0 c1; case x; case x0; simpl in |- *; auto.
intros c2 c3 H' Q; unfold red in |- *; simpl in |- *; auto.
intros a0 l c2 c3 H'; elim H'.
intros a0 l c2 c3 H' Q; unfold red in |- *; simpl in |- *; auto.
intros a0 l a1 l0 c2 c3 H'; elim H'.
Qed.
intros a b; (case a; case b).
intros x c0 x0 c1; generalize c0 c1; case x; case x0; simpl in |- *; auto.
intros c2 c3 H' Q; unfold red in |- *; simpl in |- *; auto.
intros a0 l c2 c3 H' Q; unfold red in |- *; simpl in |- *; auto.
intros a0 l c2 c3 H'; elim H'.
intros a0 l a1 l0 c2 c3 H'; elim H'.
Qed.
intros a b c; (case a; case b; case c).
intros x c0 x0 c1 x1 c2; generalize c0 c1 c2; case x; case x0; case x1;
simpl in |- *; auto.
intros a0 l a1 l0 a2 l1 H' H'0 H'1 H'2 H'3.
elim ppc_is_ppcm with (1 := cs) (a := a1) (b := a2); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l0); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l1); auto.
Qed.
Definition Cb : poly A0 eqA ltM -> list (poly A0 eqA ltM) -> Prop.
intros H'; case H'.
intros x H'0 Q;
exact (CombLinear A A0 eqA plusA multA eqA_dec n ltM ltM_dec Q x).
Defined.
intros a; case a; simpl in |- *.
intros x; case x; auto.
intros c Q H'.
replace (nil (A:=Term A n)) with (pO A n); auto; apply CombLinear_0; auto.
intros a0 l c Q H'.
apply CombLinear_id 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) c)) Q)
in |- *; apply in_inPolySet; auto.
red in |- *; intros H'4; inversion H'4.
Qed.
intros a p l; elim l; simpl in |- *; auto.
intros H'; inversion H'; auto.
intros a0 l0 H' H'0; inversion H'0; auto.
right.
exact (incons A A0 eqA n ltM a1 p0 H l0).
case H'; auto.
intros H3; right; try assumption.
apply inskip; auto.
Qed.
Remark CombLinear_trans1 :
forall (a : list (Term A n)) (R : list (poly A0 eqA ltM)),
CombLinear A A0 eqA plusA multA eqA_dec n ltM ltM_dec R a ->
forall (b : poly A0 eqA ltM) (Q : list (poly A0 eqA ltM)),
R = addEnd b Q ->
CombLinear A A0 eqA plusA multA eqA_dec n ltM ltM_dec Q
(s2p A A0 eqA n ltM b) ->
CombLinear A A0 eqA plusA multA eqA_dec n ltM ltM_dec Q a.
intros a R H'; elim H'; auto.
intros b Q H'0 H'1.
apply CombLinear_0; auto.
intros a0 p q s H'0 H'1 H'2 H'3 H'4 b Q H'5 H'6.
apply
CombLinear_comp
with
(1 := cs)
(p := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a0 q) p); auto.
2: apply
eqp_imp_canonical
with
(1 := cs)
(p := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) a0 q) p);
auto.
2: apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
2: apply canonical_pluspf with (1 := os); auto.
2: apply canonical_mults with (1 := cs); auto.
2: apply inPolySet_imp_canonical with (L := R); auto.
2: apply
CombLinear_canonical
with (eqA_dec := eqA_dec) (ltM_dec := ltM_dec) (1 := cs) (Q := R);
auto.
2: apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply CombLinear_pluspf with (1 := cs); auto.
apply CombLinear_mults1 with (1 := cs); auto.
2: apply H'3 with (b := b); auto.
case (inPolySet_addEnd q b Q); auto.
rewrite <- H'5; auto.
intros H'7; rewrite H'7; auto.
intros; (apply CombLinear_id with (1 := cs); auto).
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intros x c x0 H' Q H'0 H'1.
apply
CombLinear_trans1
with
(R := addEnd
(exist (fun l : list (Term A n) => canonical A0 eqA ltM l) x c) Q)
(b := exist (fun l : list (Term A n) => canonical A0 eqA ltM l) x c);
auto.
Qed.
intros a; case a; simpl in |- *; auto.
intros x H' P Q H'0 H'1.
apply CombLinear_incl with (1 := cs) (P := P); auto.
intros a0 H'2.
apply Incl_inp_inPolySet with (P := P); auto.
Qed.
intros a b Q H'.
apply Cb_incl with (P := Q); simpl in |- *; auto.
Qed.
intros a b Q H'.
apply Cb_incl with (P := Q); simpl in |- *; auto.
intros a0 H'0; apply addEnd_id2; auto.
Qed.
intros a b; case a; case b; simpl in |- *; auto.
intros x; case x; auto.
intros a0 l H' x0; case x0; auto.
intros a1 l0 H'0 Q H'1 H'2.
cut (~ zeroP (A:=A) A0 eqA (n:=n) a0);
[ intros Z0 | apply canonical_nzeroP with (ltM := ltM) (p := l); auto ].
cut (~ zeroP (A:=A) A0 eqA (n:=n) a1);
[ intros Z1 | apply canonical_nzeroP with (ltM := ltM) (p := l0); auto ].
apply
CombLinear_comp
with
(1 := cs)
(p := minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(ppc (A:=A) A1 (n:=n) a0 a1) (b:=a0) Z0)
(pX a0 l))
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(ppc (A:=A) A1 (n:=n) a0 a1) (b:=a1) Z1)
(pX a1 l0))); auto.
apply CombLinear_minuspf with (1 := cs); auto.
apply CombLinear_mults1 with (1 := cs); auto.
apply CombLinear_mults1 with (1 := cs); auto.
apply spolyf_canonical with (1 := cs); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
change
(eqP A eqA n
(spolyf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec
(pX a0 l) (pX a1 l0) H' H'0)
(minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(ppc (A:=A) A1 (n:=n) a0 a1) (b:=a0) Z0)
(pX a0 l))
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(ppc (A:=A) A1 (n:=n) a0 a1) (b:=a1) Z1)
(pX a1 l0)))) in |- *.
apply spoly_is_minus with (1 := cs); auto.
Qed.
Definition unit : poly A0 eqA ltM -> Term A n.
intros p; case p.
intros x; case x.
intros H'; exact (T1 A1 n).
intros a l; case a.
intros co m H'; cut (~ eqA co A0).
intros H'0; exact (divA A1 co H'0, M1 n).
inversion H'; auto.
simpl in H0.
intuition.
Defined.
unfold eqT in |- *; intros p; case p.
intros x; case x; simpl in |- *; auto.
intros a l; case a; simpl in |- *; auto.
Qed.
intros a b H' nZa; red in |- *; intros H'1; auto.
case H';
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := multA (divA b a nZa) a);
auto.
apply divA_is_multA with (1 := cs); auto.
apply (eqA_trans _ _ _ _ _ _ _ _ _ cs) with (y := multA A0 a); auto.
apply multA_eqA_comp with (1 := cs); auto.
apply (eqA_ref _ _ _ _ _ _ _ _ _ cs); auto.
apply multA_A0_l with (1 := cs); auto.
Qed.
Hint Resolve divA_nZ.
intros p; case p.
intros x; case x; simpl in |- *; auto.
intros a l; case a; simpl in |- *; auto.
Qed.
Definition nf : poly A0 eqA ltM -> list (poly A0 eqA ltM) -> poly A0 eqA ltM.
intros p L.
case
(Reducef A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec
os L p).
intros x H'.
apply LetP with (A := poly A0 eqA ltM) (h := x).
intros u H'1; case u.
intros x0 H'2;
exists (mults (A:=A) multA (n:=n) (unit (mks A A0 eqA n ltM x0 H'2)) x0);
auto.
apply canonical_mults with (1 := cs); auto.
apply unit_nZ; auto.
Defined.

Theorem nf_irreducible :
forall (p : poly A0 eqA ltM) (aP : list (poly A0 eqA ltM)),
irreducible A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec aP
(s2p A A0 eqA n ltM (nf p aP)).