# Library Buchberger.Pmult

Require Export Pmults.
Section Pmult.

Fixpoint multpf (p : list (Term A n)) : list (Term A n) list (Term A n) :=
fun q : list (Term A n)
match p with
| nil => pO A n
| a :: p'
pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a q) (multpf p' q)
end.

Theorem canonical_multpf :
l1 l2 : list (Term A n),
canonical A0 eqA ltM l1 ->
canonical A0 eqA ltM l2 canonical A0 eqA ltM (multpf l1 l2).
intros l1; elim l1; simpl in |- *; auto.
intros a l H' l2 H'0 H'1; try assumption.
apply canonical_pluspf; auto.
apply canonical_mults with (1 := cs); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l); auto.
apply H'; auto.
apply canonical_imp_canonical with (a := a); auto.
Qed.
Hint Resolve canonical_multpf.

Theorem pluspf_pX :
(p : list (Term A n)) (a : Term A n),
canonical A0 eqA ltM (pX a p)
eqP A eqA n (pX a p)
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a (pO A n)) p).
intros p; case p; clear p; auto.
intros a H;
change
(eqP A eqA n (pX a (pO A n))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a (pO A n)) (pO A n))) in |- ×.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros a1 p a0 H'.
cut (canonical A0 eqA ltM (pX a1 p));
[ intros Op0 | apply canonical_imp_canonical with (a := a0) ];
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 (pO A n) (pX a1 p))); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
change
(eqP A eqA n
(pX a0
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pO A n) (pX a1 p)))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a0 (pO A n)) (pX a1 p))) in |- *; auto.
apply pluspf_inv1 with (1 := cs); auto.
apply (canonical_pX_order _ A0 eqA) with (l := p); auto.
Qed.

Theorem multpf_disr_pX :
(p q : list (Term A n)) (a : Term A n),
canonical A0 eqA ltM p
canonical A0 eqA ltM (pX a q)
eqP A eqA n (multpf p (pX a q))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a p) (multpf p q)).
intros p; elim p; clear p; simpl in |- *; auto.
intros; apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros b p Rec q a Op0 Op1.
cut (canonical A0 eqA ltM q);
[ intros Op2 | apply canonical_imp_canonical with (a := a) ];
auto.
cut (canonical A0 eqA ltM p);
[ intros Op3 | apply canonical_imp_canonical with (a := b) ];
auto.
cut (¬ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros nZ0 | apply canonical_nzeroP with (ltM := ltM) (p := q) ];
auto.
cut (¬ zeroP (A:=A) A0 eqA (n:=n) b);
[ intros nZ1 | apply canonical_nzeroP with (ltM := ltM) (p := p) ];
auto.
cut (¬ zeroP (A:=A) A0 eqA (n:=n) (multTerm (A:=A) multA (n:=n) b a));
[ intros nZ2 | idtac ]; auto.
cut (¬ zeroP (A:=A) A0 eqA (n:=n) (multTerm (A:=A) multA (n:=n) a b));
[ intros nZ3 | idtac ]; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX (multTerm (A:=A) multA (n:=n) b a)
(mults (A:=A) multA (n:=n) b q))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) a p)
(multpf p q))); auto.
apply eqp_pluspf_com with (1 := cs); auto.
change (canonical A0 eqA ltM (mults (A:=A) multA (n:=n) b (pX a q))) in |- *;
auto.
change (canonical A0 eqA ltM (mults (A:=A) multA (n:=n) b (pX a q))) in |- *;
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 (pX (multTerm (A:=A) multA (n:=n) b a) (pO A n))
(mults (A:=A) multA (n:=n) b q))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) a p)
(multpf p q))); auto.
cut
(canonical A0 eqA ltM
(pX (multTerm (A:=A) multA (n:=n) b a) (mults (A:=A) multA (n:=n) b q)));
[ intros Op4 | idtac ]; auto.
apply eqp_pluspf_com with (1 := cs); auto.
apply pluspf_pX; auto.
change (canonical A0 eqA ltM (mults (A:=A) multA (n:=n) b (pX a q))) in |- *;
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX (multTerm (A:=A) multA (n:=n) b a) (pO A n))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) b q)
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) a p)
(multpf p q)))); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX (multTerm (A:=A) multA (n:=n) b a) (pO A n))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) b q)
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (multpf p q) (mults (A:=A) multA (n:=n) a p))));
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX (multTerm (A:=A) multA (n:=n) b a) (pO A n))
(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 (mults (A:=A) multA (n:=n) b q)
(multpf p q)) (mults (A:=A) multA (n:=n) a p)));
auto.
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 (multTerm (A:=A) multA (n:=n) b a) (pO A n))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) a p)
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) b q)
(multpf p q)))); 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 (pX (multTerm (A:=A) multA (n:=n) b a) (pO A n))
(mults (A:=A) multA (n:=n) a p))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) b q)
(multpf p q))); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply eqp_pluspf_com with (1 := cs); auto.
change (canonical A0 eqA ltM (mults (A:=A) multA (n:=n) a (pX b p))) in |- *;
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX (multTerm (A:=A) multA (n:=n) a b) (pO A n))
(mults (A:=A) multA (n:=n) a p)); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply pluspf_pX; auto.
change (canonical A0 eqA ltM (mults (A:=A) multA (n:=n) a (pX b p))) in |- *;
auto.
Qed.

(p q : list (Term A n)) (a b : Term A n),
canonical A0 eqA ltM (pX a p)
canonical A0 eqA ltM (pX b q)
c : list (Term A n),
multpf (pX a p) (pX b q) = pX (multTerm (A:=A) multA (n:=n) a b) c.
intros p; elim p; clear p; auto.
simpl in |- *; intros q a b H' H'0.
cut (canonical A0 eqA ltM q);
[ intros Op0 | apply canonical_imp_canonical with (a := b) ];
auto.
(mults (A:=A) multA (n:=n) a q); simpl in |- *; auto.
rewrite <- pO_pluspf_inv2; auto.
intros a1 p H' q a0 b H'0 H'1; auto.
elim (H' q a1 b); simpl in |- *; [ intros c E | idtac | idtac ]; auto.
2: apply canonical_imp_canonical with (a := a0); auto.
rewrite E.

(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a0 q)
(pX (multTerm (A:=A) multA (n:=n) a1 b) c)); auto.
rewrite <- pluspf_inv1_eq; auto.
apply multTerm_ltT_r; auto.
apply (canonical_pX_order _ A0 eqA) with (l := p); auto.
Qed.

(p q : list (Term A n)) (a b : Term A n),
canonical A0 eqA ltM (pX a p)
canonical A0 eqA ltM (pX b q)
In (multTerm (A:=A) multA (n:=n) a b) (multpf (pX a p) (pX b q)).
intros p q a b H' H'0.
elim (multpf_head p q a b); [ intros c E; rewrite E | idtac | idtac ];
simpl in |- *; auto.
Qed.

Theorem multpf_comp :
p r : list (Term A n),
eqP A eqA n p r
q s : list (Term A n),
canonical A0 eqA ltM p
canonical A0 eqA ltM q
canonical A0 eqA ltM r
canonical A0 eqA ltM s
eqP A eqA n q s eqP A eqA n (multpf p q) (multpf r s).
intros p r H'; elim H'; simpl in |- *; auto.
intros ma mb p0 q H'0 H'1 H'2 q0 s H'3 H'4 H'5 H'6 H'7.
cut (canonical A0 eqA ltM p0);
[ intros Op0 | apply canonical_imp_canonical with (a := ma); auto ].
cut (canonical A0 eqA ltM q);
[ intros Op1 | apply canonical_imp_canonical with (a := mb); auto ].
cut (¬ zeroP (A:=A) A0 eqA (n:=n) ma);
[ intros Z0 | apply canonical_nzeroP with (ltM := ltM) (p := p0); auto ].
cut (¬ zeroP (A:=A) A0 eqA (n:=n) mb);
[ intros Z1 | apply canonical_nzeroP with (ltM := ltM) (p := q) ];
auto.
Qed.

Theorem multpf_com :
p q : list (Term A n),
canonical A0 eqA ltM p
canonical A0 eqA ltM q eqP A eqA n (multpf p q) (multpf q p).
intros p; elim p; simpl in |- *; auto.
intros q; elim q; simpl in |- *; auto.
intros a l H' H'0 H'1.
cut (canonical A0 eqA ltM l);
[ intros Op0 | apply canonical_imp_canonical with (a := a) ];
auto.
cut (¬ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros Z0 | apply canonical_nzeroP with (ltM := ltM) (p := l) ];
auto.
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) (pO A n)); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros a l H' q H'0 H'1.
cut (canonical A0 eqA ltM l);
[ intros Op0 | apply canonical_imp_canonical with (a := a) ];
auto.
cut (¬ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros Z0 | apply canonical_nzeroP with (ltM := ltM) (p := l) ];
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a q) (multpf q l));
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
change
(eqP A eqA n (multpf q (pX a l))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a q) (multpf q l)))
in |- ×.
apply multpf_disr_pX; auto.
Qed.

Theorem multpf_dist_plusr :
(p q r : list (Term A n)) (a : Term A n),
canonical A0 eqA ltM p
canonical A0 eqA ltM q
canonical A0 eqA ltM r
eqP A eqA n
(multpf
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p
q) r)
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(multpf p r) (multpf q r)).
intros p q r; elim r.
intros H' H'0 H'1 H'2.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := multpf (pO A n)
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec p q)); auto.
apply multpf_com; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(multpf (pO A n) p) (multpf (pO A n) q));
auto.
simpl in |- *; apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply eqp_pluspf_com with (1 := cs); auto; apply multpf_com; auto.
intros a l H' H'0 H'1 H'2 H'3.
cut (¬ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros nZ0 | apply canonical_nzeroP with (ltM := ltM) (p := l) ];
auto.
cut (canonical A0 eqA ltM l);
[ intros Op0 | apply canonical_imp_canonical with (a := a) ];
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec p q))
(multpf
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec p q) l)); auto.
change
(eqP A eqA n
(multpf
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p q) (pX a l))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec p q))
(multpf
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec p q) l))) in |- ×.
apply multpf_disr_pX; 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 (mults (A:=A) multA (n:=n) a p)
(mults (A:=A) multA (n:=n) a q))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (multpf p l) (multpf q l)));
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a p)
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) a q)
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (multpf p l) (multpf q l))));
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a p)
(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 (mults (A:=A) multA (n:=n) a q)
(multpf p l)) (multpf q l))); auto.
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
(mults (A:=A) multA (n:=n) a p)
(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 (multpf p l) (mults (A:=A) multA (n:=n) a q))
(multpf q l))); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a p)
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (multpf p l)
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) a q)
(multpf q l)))); 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 (mults (A:=A) multA (n:=n) a p)
(multpf p l))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) a q)
(multpf q l))); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply eqp_pluspf_com with (1 := cs); auto.
change
(eqP A eqA n
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a p) (multpf p l))
(multpf p (pX a l))) in |- ×.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); apply multpf_disr_pX; auto.
change
(eqP A eqA n
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a q) (multpf q l))
(multpf q (pX a l))) in |- ×.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); apply multpf_disr_pX; auto.
Qed.

Theorem multpf_dist_plusl :
(p q r : list (Term A n)) (a : Term A n),
canonical A0 eqA ltM p
canonical A0 eqA ltM q
canonical A0 eqA ltM r
eqP A eqA n
(multpf p
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec q
r))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(multpf p q) (multpf p r)).
intros p q r H' H'0 H'1 H'2.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := multpf
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec q r) p); auto.
apply multpf_com; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(multpf q p) (multpf r p)); auto.
apply multpf_dist_plusr; auto.
apply eqp_pluspf_com with (1 := cs); auto.
apply multpf_com; auto.
apply multpf_com; auto.
Qed.

Theorem multpf_smultm_assoc :
(p q : list (Term A n)) (a : Term A n),
canonical A0 eqA ltM p
canonical A0 eqA ltM q
¬ zeroP (A:=A) A0 eqA (n:=n) a
eqP A eqA n (mults (A:=A) multA (n:=n) a (multpf p q))
(multpf (mults (A:=A) multA (n:=n) a p) q).
intros p q a; elim p; simpl in |- *; auto.
intros a0 l H' H'0 H'1 H'2.
cut (¬ zeroP (A:=A) A0 eqA (n:=n) a0);
[ intros nZ0 | apply canonical_nzeroP with (ltM := ltM) (p := l) ];
auto.
cut (canonical A0 eqA ltM l);
[ intros Op0 | apply canonical_imp_canonical with (a := a0) ];
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a (mults (A:=A) multA (n:=n) a0 q))
(mults (A:=A) multA (n:=n) a (multpf l q)));
auto.
apply eqp_pluspf_com with (1 := cs); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
Qed.

Theorem multpf_assoc :
p q r : list (Term A n),
canonical A0 eqA ltM p
canonical A0 eqA ltM q
canonical A0 eqA ltM r
eqP A eqA n (multpf p (multpf q r)) (multpf (multpf p q) r).
intros p q r; elim p; simpl in |- *; auto.
intros a l H' H'0 H'1 H'2.
cut (¬ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros nZ0 | apply canonical_nzeroP with (ltM := ltM) (p := l) ];
auto.
cut (canonical A0 eqA ltM l);
[ intros Op0 | apply canonical_imp_canonical with (a := a) ];
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(multpf (mults (A:=A) multA (n:=n) a q) r)
(multpf (multpf l q) r)); auto.
apply eqp_pluspf_com with (1 := cs); auto.
apply multpf_smultm_assoc; auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply multpf_dist_plusr; auto.
Qed.

Definition smult : poly A0 eqA ltM poly A0 eqA ltM poly A0 eqA ltM.
intros sp1 sp2.
case sp1; case sp2.
intros p1 H'1 p2 H'2; (multpf p1 p2); auto.
Defined.
End Pmult.