Section Pspminus.
Load "hCoefStructure".
Load "hOrderStructure".
Load "hMinus".
intros a b H'; inversion H'.
case
(ltT_dec A n ltM ltM_dec (T1 A1 n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b) nZb));
[ intros tmp; case tmp; clear tmp | idtac ]; intros H'2;
auto.
apply ltT_not_ltT; auto.
apply
ltT_eqTl
with
(a := multTerm (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b) nZb) b);
auto.
apply (eqT_sym A n); apply (eqTerm_imp_eqT A eqA); auto.
apply ltT_eqTr with (a := multTerm (A:=A) multA (n:=n) (T1 A1 n) b); auto.
apply (eqT_sym A n); apply (eqTerm_imp_eqT A eqA);
apply T1_multTerm_l with (1 := cs); auto.
apply multTerm_ltT_r with (1 := os); auto.
elim
(T1_is_min_ltT A A1)
with
(a := divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b) nZb)
(1 := os); auto.
apply ltT_not_eqT; auto.
apply
(eqT_trans A n)
with
(y := multTerm (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b) nZb) b);
auto.
apply (eqTerm_imp_eqT A eqA); auto.
apply (eqT_trans A n) with (y := multTerm (A:=A) multA (n:=n) (T1 A1 n) b);
auto.
apply (eqT_sym A n); auto.
apply (eqT_sym A n); apply (eqTerm_imp_eqT A eqA); auto.
Qed.
Hint Resolve divP_is_not_order.
intros a b p; case p; auto.
intros H' H'0.
change (canonical A0 eqA ltM (pX a (pO A n))) in |- *; apply canonicalp1;
auto.
apply divP_inv1 with (1 := H'0); auto.
intros a0 l H' H'0; change (canonical A0 eqA ltM (pX a (pX a0 l))) in |- *;
apply canonical_cons; auto.
case (ltT_dec A n ltM ltM_dec a b);
[ intros tmp; case tmp; clear tmp | idtac ]; intros H'2;
auto.
elim (divP_is_not_order a b); auto.
apply (ltT_trans A _ _ os) with (y := b); auto.
apply (canonical_pX_order _ A0 eqA) with (l := l); auto.
apply eqT_compat_ltTr with (b := b); auto.
apply (eqT_sym A n); auto.
apply (canonical_pX_order _ A0 eqA) with (l := l); auto.
apply divP_inv1 with (1 := H'0); auto.
apply canonical_imp_canonical with (a := b); auto.
Qed.
Definition spminusf (a b : Term A n) (nZb : ~ zeroP (A:=A) A0 eqA (n:=n) b)
(p q : list (Term A n)) :=
minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec p
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b) nZb) q).
auto; auto.
intros a b nZ1 nZ2 p q; unfold spminusf in |- *.
change
(minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec p
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b) nZ1) q) =
minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec p
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b) nZ2) q))
in |- *.
rewrite
divTerm_rew with (1 := cs) (a := a) (b := b) (nZ1 := nZ1) (nZ2 := nZ2);
auto.
Qed.
simpl in |- *; auto.
Qed.
unfold spminusf in |- *.
intros a b nZb p q H' H'0 H'1; apply canonical_minuspf with (1 := cs); auto.
Qed.
Hint Resolve canonical_spminusf.
intros a b nZb p q H'; unfold spminusf in |- *; simpl in |- *; inversion H'.
intros H'0 H'1;
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec
(pX a p)
(pX a
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b)
nZb) q))); auto.
apply minuspf_inv3a with (1 := cs); auto.
apply minuspf_comp with (1 := cs); auto.
apply
canonical_pX_eqT
with
(a := multTerm (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b) nZb) b);
auto.
change
(canonical A0 eqA ltM
(mults (A:=A) multA (n:=n) (divTerm divA a nZb) (pX b q)))
in |- *; auto.
apply (eqT_sym A n); apply (eqTerm_imp_eqT A eqA n); auto.
change
(canonical A0 eqA ltM
(mults (A:=A) multA (n:=n) (divTerm divA a nZb) (pX b q)))
in |- *; auto.
Qed.
intros a b nZb p q H' H'0 H'1; apply canonical_spminusf; auto.
apply canonical_imp_canonical with (a := a); auto.
apply canonical_imp_canonical with (a := b); auto.
Qed.
Hint Resolve canonical_spminusf_full.
unfold spminusf in |- *.
intros a b nZb p q H' H'0 H'1; try assumption.
inversion H'1.
apply order_minuspf with (1 := cs); auto.
apply
canonical_pX_eqT
with
(a := multTerm (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b) nZb) b);
auto.
change
(canonical A0 eqA ltM
(mults (A:=A) multA (n:=n) (divTerm divA a nZb) (pX b q)))
in |- *; auto.
apply (eqT_sym A n); apply (eqTerm_imp_eqT A eqA n); auto.
Qed.
Hint Resolve canonical_spminusf_full_t.
intros a b nZb p q r H' H'0 H'1 H'2; unfold spminusf in |- *.
inversion H'2.
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))
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b)
nZb) r))); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec q
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b)
nZb) r)))); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b)
nZb) r)) q)); auto.
apply eqp_pluspf_com 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
(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))
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a (b:=b)
nZb) r))) q); apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n);
auto.
apply eqp_pluspf_com with (1 := cs); auto.
Qed.
Hint Resolve spminusf_pluspf.
unfold spminusf in |- *; auto.
Qed.
intros a b c nZc p q H' H'0 H'1 H'2.
cut (divP A A0 eqA multA divA n b c); [ intros H'3 | auto ].
unfold spminusf in |- *; auto.
apply divP_eqTerm_comp with (1 := cs) (a := a); auto.
Qed.
intros a b c nZc p q r H' H'0 H'1 H'2 H'3 H'4.
apply (eqp_trans _ _ _ _ _ _ _ _ _ cs n) with (y := spminusf b c nZc p r);
auto.
apply eqTerm_spminusf_com; auto.
apply eqptail_spminusf_com; auto.
apply divP_eqTerm_comp with (1 := cs) (a := a); auto.
Qed.
Hint Resolve eqTerm_spminusf_com eqp_spminusf_com eqp_spminusf_com.

Theorem spminusf_minuspf :
forall (a b : Term A n) (nZb : ~ zeroP (A:=A) A0 eqA (n:=n) b)
(p q r : list (Term A n)),
canonical A0 eqA ltM p ->
canonical A0 eqA ltM q ->
canonical A0 eqA ltM r ->
divP A A0 eqA multA divA n a b ->
eqP A eqA n
(spminusf a b nZb
(minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec p q) r)
(minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec
(spminusf a b nZb p r) q).