Library Buchberger.Pminus
Require Export Pmults.
Require Import Arith.
Section Pminus.
Load "hCoefStructure".
Load "hOrderStructure".
Load "hMults".
Inductive minusP :
list (Term A n) -> list (Term A n) → list (Term A n) -> Prop :=
| mnillu1 :
∀ l1 : list (Term A n),
minusP (pO A n) l1
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) l1)
| mnillu2 : ∀ l1 : list (Term A n), minusP l1 (pO A n) l1
| mmainu1 :
∀ (a1 a2 : Term A n) (l1 l2 l3 : list (Term A n)),
ltT ltM a2 a1 →
minusP l1 (pX a2 l2) l3 -> minusP (pX a1 l1) (pX a2 l2) (pX a1 l3)
| mmainu2a :
∀ (a1 a2 : Term A n) (l1 l2 l3 : list (Term A n)),
minusP l1 l2 l3 →
eqT a1 a2 →
zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a1 a2) ->
minusP (pX a1 l1) (pX a2 l2) l3
| mmainu2b :
∀ (a1 a2 : Term A n) (l1 l2 l3 : list (Term A n)),
minusP l1 l2 l3 →
eqT a1 a2 →
¬ zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a1 a2) →
minusP (pX a1 l1) (pX a2 l2)
(pX (minusTerm (A:=A) minusA (n:=n) a1 a2) l3)
| mmainu3 :
∀ (a1 a2 : Term A n) (l1 l2 l3 : list (Term A n)),
ltT ltM a1 a2 →
minusP (pX a1 l1) l2 l3 →
minusP (pX a1 l1) (pX a2 l2) (pX (invTerm (A:=A) invA (n:=n) a2) l3).
Hint Resolve mnillu1 mnillu2 mmainu1 mmainu2a mmainu2b mmainu3.
Require Import LetP.
Definition minuspp :
∀ l : list (Term A n) × list (Term A n),
{a : list (Term A n) | minusP (fst l) (snd l) a}.
intros l; pattern l in |- ×.
apply
well_founded_induction
with (A := (list (Term A n) × list (Term A n))%type) (R := lessP A n);
auto.
apply wf_lessP; auto.
intros x; case x; intros l1 l2; simpl in |- ×.
case l1.
intros H';
∃ (mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) l2);
auto.
intros a1 m1; case l2.
intros H'; ∃ (pX a1 m1); auto.
intros a2 m2 H'; case (ltT_dec A n ltM ltM_dec a1 a2);
[ intros P; case P; clear P | idtac ]; intros H1.
lapply (H' (pX a1 m1, m2)); simpl in |- *;
[ intros Rec; case Rec; clear Rec; intros Orec Prec | idtac ].
∃ (pX (invTerm (A:=A) invA (n:=n) a2) Orec); auto.
change
(minusP (pX a1 m1) (pX a2 m2) (pX (invTerm (A:=A) invA (n:=n) a2) Orec))
in |- *; auto.
red in |- *; red in |- *; simpl in |- *; rewrite <- plus_n_Sm; auto.
lapply (H' (m1, pX a2 m2)); simpl in |- *;
[ intros Rec; case Rec; clear Rec; intros Orec Prec | idtac ].
∃ (pX a1 Orec); auto.
change (minusP (pX a1 m1) (pX a2 m2) (pX a1 Orec)) in |- *; auto.
red in |- *; red in |- *; simpl in |- *; rewrite <- plus_n_Sm; auto.
lapply (H' (m1, m2)); simpl in |- *;
[ intros Rec; case Rec; clear Rec; intros Orec Prec | idtac ].
apply LetP with (A := Term A n) (h := minusTerm (A:=A) minusA (n:=n) a1 a2).
intros u H'0; case (zeroP_dec A A0 eqA eqA_dec n u); intros Z0.
∃ Orec; auto.
rewrite H'0 in Z0; auto.
change (minusP (pX a1 m1) (pX a2 m2) Orec) in |- *; auto.
∃ (pX u Orec); auto.
rewrite H'0 in Z0; auto.
rewrite H'0; auto.
change
(minusP (pX a1 m1) (pX a2 m2)
(pX (minusTerm (A:=A) minusA (n:=n) a1 a2) Orec))
in |- *; auto.
red in |- *; red in |- *; simpl in |- *; rewrite <- plus_n_Sm; auto.
Defined.
Definition minuspf (l1 l2 : list (Term A n)) :=
projsig1 (list (Term A n)) _ (minuspp (l1, l2)).
Theorem zerop_is_eqTerm :
∀ a b : Term A n,
eqT a b →
zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) →
eqTerm (A:=A) eqA (n:=n) a b.
intros a b H' H'0.
cut (eqT a (invTerm (A:=A) invA (n:=n) b)); [ intros eq1 | idtac ].
cut (eqT a (plusTerm (A:=A) plusA (n:=n) b (invTerm (A:=A) invA (n:=n) b)));
[ intros eq2 | idtac ].
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n) a
(plusTerm (A:=A) plusA (n:=n) b (invTerm (A:=A) invA (n:=n) b)));
auto.
apply zeroP_plusTermr with (1 := cs); auto.
apply plusTerm_invTerm_zeroP with (1 := cs); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n) a
(plusTerm (A:=A) plusA (n:=n) (invTerm (A:=A) invA (n:=n) b) b));
auto.
apply plusTerm_comp_r with (1 := cs); auto.
apply (eqT_trans A n) with (plusTerm plusA b (invTerm invA b)); auto.
apply (eqTerm_imp_eqT A eqA); auto.
apply plusTerm_com with (1 := cs); auto.
apply plusTerm_com with (1 := cs); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n)
(plusTerm (A:=A) plusA (n:=n) a (invTerm (A:=A) invA (n:=n) b)) b);
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply plusTerm_assoc with (1 := cs); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply zeroP_plusTerml with (1 := cs); auto.
apply (eqT_trans A n) with (invTerm invA b); auto.
apply plusTerm_eqT2; auto.
apply (eqT_sym A n); apply invTerm_eqT; auto.
apply zeroP_comp_eqTerm with (1 := cs) (2 := H'0).
apply eqTerm_minusTerm_plusTerm_invTerm with (1 := cs); auto.
apply (eqT_trans A n) with (1 := eq1); auto.
apply (eqT_sym A n); apply plusTerm_eqT2; auto.
apply (eqT_trans A n) with (1 := H'); auto.
Qed.
Hint Unfold minuspf.
Theorem minusTerm_zeroP_r :
∀ a b : Term A n,
zeroP (A:=A) A0 eqA (n:=n) a →
eqT a b →
eqTerm (A:=A) eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b)
(invTerm (A:=A) invA (n:=n) b).
intros a b H' H0;
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := plusTerm (A:=A) plusA (n:=n) a (invTerm (A:=A) invA (n:=n) b));
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply zeroP_plusTerml with (1 := cs); auto.
apply (eqT_trans A n) with (y := b); auto.
Qed.
Theorem minusTerm_zeroP :
∀ a b : Term A n,
eqT a b →
zeroP (A:=A) A0 eqA (n:=n) b →
eqTerm (A:=A) eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) a.
intros a b H' H'0.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := plusTerm (A:=A) plusA (n:=n) a (invTerm (A:=A) invA (n:=n) b));
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply zeroP_plusTermr with (1 := cs); auto.
apply (eqT_trans A n) with (y := b); auto.
apply zeroP_invTerm_zeroP with (1 := cs); auto.
Qed.
Hint Resolve minusTerm_zeroP minusTerm_zeroP_r.
Theorem minusP_pO_is_eqP :
∀ p q r : list (Term A n),
minusP p q r → eqP A eqA n r (pO A n) → eqP A eqA n p q.
intros p q r H'; elim H'; auto.
intros l1; case l1; simpl in |- *; auto.
intros t l H; inversion H.
intros a1 a2 l1 l2 l3 H H0 H1 H2; inversion H2.
intros a1 a2 l1 l2 l3 H H0 H1 H2 H3.
apply eqpP1; auto.
apply zerop_is_eqTerm; auto.
intros a1 a2 l1 l2 l3 H H0 H1 H2 H3; inversion H3.
intros a1 a2 l1 l2 l3 H H0 H1 H2; inversion H2.
Qed.
Lemma minusP_inv :
∀ (p q l : list (Term A n)) (a b : Term A n),
minusP (pX a p) (pX b q) l →
∃ l1 : list (Term A n),
ltT ltM b a ∧ minusP p (pX b q) l1 ∧ l = pX a l1 ∨
ltT ltM a b ∧
minusP (pX a p) q l1 ∧ l = pX (invTerm (A:=A) invA (n:=n) b) l1 ∨
eqT a b ∧
(zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) ∧
minusP p q l ∨
¬ zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) ∧
minusP p q l1 ∧ l = pX (minusTerm (A:=A) minusA (n:=n) a b) l1).
intros p q l a b H'; simple inversion H'.
discriminate H.
discriminate H0.
rewrite <- H3; rewrite (pX_invl A n a2 b l2 q H2);
rewrite (pX_invr A n l2 q a2 b H2); rewrite (pX_invl A n a1 a l1 p H1);
rewrite (pX_invr A n l1 p a1 a H1).
intros H'0 H'1; ∃ l3; left; split; [ idtac | split ]; auto.
rewrite <- H4.
intros H'0 H'1 H'2; ∃ l3; right; right.
rewrite <- (pX_invl A n a2 b l2 q); try rewrite <- (pX_invl A n a1 a l1 p);
auto.
rewrite <- (pX_invr A n l2 q a2 b); try rewrite <- (pX_invr A n l1 p a1 a);
auto.
intros H'0 H'1 H'2; ∃ l3; right; right.
rewrite <- H4.
rewrite <- (pX_invl A n a2 b l2 q); try rewrite <- (pX_invl A n a1 a l1 p);
auto.
rewrite <- (pX_invr A n l2 q a2 b); try rewrite <- (pX_invr A n l1 p a1 a);
auto.
intros H'0 H'1; ∃ l3; right; left.
rewrite <- (pX_invl A n a2 b l2 q); try rewrite <- (pX_invl A n a1 a l1 p);
auto.
rewrite <- (pX_invr A n l2 q a2 b); try rewrite <- (pX_invr A n l1 p a1 a);
auto.
Qed.
Theorem uniq_minusp :
∀ (l : list (Term A n) × list (Term A n)) (l3 l4 : list (Term A n)),
minusP (fst l) (snd l) l3 → minusP (fst l) (snd l) l4 → l3 = l4.
unfold pX, pX in |- ×.
intros l; pattern l in |- ×.
apply well_founded_ind with (1 := wf_lessP A n).
intros (l1, l2); simpl in |- ×.
case l1; clear l1.
intros H' l3 l4 H'0; inversion_clear H'0; intros H'2; inversion_clear H'2;
auto.
case l2; clear l2.
simpl in |- *; intros n0 l0 H' l3 l4 H'0 H'2.
inversion H'2; inversion H'0; auto.
simpl in |- *; intros n2 l2 n1 l1 Induc l3 l4 Hl3 Hl4.
case (minusP_inv l1 l2 l4 n1 n2); auto.
intros x H'; elim H'; intros H'0; [ idtac | elim H'0; clear H'0; intros H'0 ];
clear H';
(case (minusP_inv l1 l2 l3 n1 n2); auto; intros x1 H'; elim H'; intros H'1;
[ idtac | elim H'1; clear H'1; intros H'1 ]; clear H').
elim H'1; intros H' H'2; elim H'2; intros H'3 H'4; rewrite H'4; clear H'2 H'1.
elim H'0; intros H'1 H'2; elim H'2; intros H'5 H'6; rewrite H'6;
clear H'2 H'0.
apply pX_inj; auto.
apply Induc with (y := (l1, pX n2 l2)); auto.
red in |- *; red in |- *; simpl in |- *; auto.
elim H'1; intros H' H'2; clear H'1.
elim H'0; intros H'1 H'3; clear H'0.
absurd (ltT ltM n1 n2); auto.
elim H'1; intros H' H'2; clear H'1.
elim H'0; intros H'1 H'3; clear H'0.
absurd (eqT n2 n1); auto.
apply (eqT_sym A n); auto.
elim H'1; intros H' H'2; clear H'1.
elim H'0; intros H'1 H'3; clear H'0.
absurd (ltT ltM n1 n2); auto.
elim H'1; intros H' H'2; elim H'2; intros H'3 H'4; rewrite H'4; clear H'2 H'1.
elim H'0; intros H'1 H'2; elim H'2; intros H'5 H'6; rewrite H'6;
clear H'2 H'0.
apply pX_inj; auto.
apply Induc with (y := (pX n1 l1, l2)); auto; red in |- *; simpl in |- *;
auto.
rewrite <- plus_n_Sm; auto.
elim H'1; intros H' H'2; clear H'1.
elim H'0; intros H'1 H'3; clear H'0.
absurd (eqT n1 n2); auto.
elim H'1; intros H' H'2; clear H'1.
elim H'0; intros H'1 H'3; clear H'0.
absurd (eqT n2 n1); auto.
apply (eqT_sym A n); auto.
elim H'1; intros H' H'2; clear H'1.
elim H'0; intros H'1 H'3; clear H'0.
absurd (eqT n1 n2); auto.
elim H'1; intros H' H'2; elim H'2;
[ intros H'3; clear H'2 H'1
| intros H'3; elim H'3; intros H'4 H'5; elim H'5; intros H'6 H'7;
rewrite H'7; clear H'5 H'3 H'2 H'1 ].
elim H'0; intros H'1 H'2; elim H'2;
[ intros H'4; clear H'2 H'0
| intros H'4; elim H'4; intros H'5 H'6; apply H'5 || elim H'5;
try assumption; clear H'4 H'2 H'0 ].
elim H'4; intros H'0 H'2; clear H'4.
elim H'3; intros H'4 H'5; clear H'3.
apply Induc with (y := (l1, l2)); auto.
red in |- *; red in |- *; simpl in |- *; rewrite <- plus_n_Sm; auto.
elim H'3; intros H'0 H'2; clear H'3; auto.
elim H'0; intros H'1 H'2; elim H'2;
[ intros H'3; clear H'2 H'0
| intros H'3; elim H'3; intros H'5 H'8; elim H'8; intros H'9 H'10;
rewrite H'10; clear H'8 H'3 H'2 H'0 ].
elim H'3; intros H'0 H'2; clear H'3.
apply H'4 || elim H'4; auto.
apply pX_inj; auto.
apply Induc with (y := (l1, l2)); auto.
red in |- *; red in |- *; simpl in |- *; rewrite <- plus_n_Sm; auto.
Qed.
Theorem minuspf_is_minusP :
∀ l1 l2 : list (Term A n), minusP l1 l2 (minuspf l1 l2).
intros l1 l2; try assumption.
unfold minuspf in |- *; case (minuspp (pair l1 l2)); simpl in |- *; auto.
Qed.
Hint Resolve minuspf_is_minusP.
Theorem minuspf_pO_is_eqP :
∀ p q : list (Term A n),
eqP A eqA n (minuspf p q) (pO A n) → eqP A eqA n p q.
intros p q H'.
apply minusP_pO_is_eqP with (r := minuspf p q); auto.
Qed.
Theorem order_minusP :
∀ (l1 l2 l3 : list (Term A n)) (a : Term A n),
minusP l1 l2 l3 →
canonical A0 eqA ltM (pX a l1) →
canonical A0 eqA ltM (pX a l2) →
canonical A0 eqA ltM l3 → canonical A0 eqA ltM (pX a l3).
intros l1 l2 l3 a H'; elim H'; auto.
intros l4 H'0 H'1 H'2.
cut (canonical A0 eqA ltM l4);
try apply canonical_imp_canonical with (a := a); auto;
intros C0.
apply
canonical_pX_eqT
with
(a := multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
a); auto.
change
(canonical A0 eqA ltM
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(pX a l4))) in |- *; auto.
apply (eqT_sym A n);
apply (eqT_trans A n) with (y := multTerm (A:=A) multA (n:=n) (T1 A1 n) a);
auto.
apply (T1_eqT A A1 eqA); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l4); auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 H'3 H'4 H'5.
apply canonical_cons; auto.
apply (canonical_pX_order A A0 eqA) with (ltM := ltM) (l := l4); auto.
apply canonical_nzeroP with (ltM := ltM) (p := pX a1 l4); auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 H'3 H'4 H'5 H'6.
apply H'1; auto.
apply canonical_skip_fst with (b := a1); auto.
apply canonical_skip_fst with (b := a2); auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 H'3 H'4 H'5 H'6.
apply canonical_cons; auto.
apply minusTerm_ltT_l; auto.
apply (canonical_pX_order A A0 eqA) with (l := l4); auto.
apply canonical_nzeroP with (ltM := ltM) (p := pX a1 l4); auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 H'3 H'4 H'5.
apply canonical_cons; auto.
apply invTerm_ltT_l; auto.
apply (canonical_pX_order A A0 eqA) with (l := l5); auto.
apply canonical_nzeroP with (ltM := ltM) (p := pX a1 l4); auto.
Qed.
Theorem canonical_minusP :
∀ l1 l2 l3 : list (Term A n),
minusP l1 l2 l3 →
canonical A0 eqA ltM l1 →
canonical A0 eqA ltM l2 → canonical A0 eqA ltM l3.
intros l1 l2 l3 H'; elim H'; auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 H'3 H'4.
apply order_minusP with (l1 := l4) (l2 := pX a2 l5); auto.
apply canonical_cons; auto.
apply canonical_nzeroP with (ltM := ltM) (p := l4); auto.
apply H'2; auto.
apply canonical_imp_canonical with (a := a1); auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 H'3 H'4 H'5.
apply H'1; auto.
apply canonical_imp_canonical with (a := a1); auto.
apply canonical_imp_canonical with (a := a2); auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 H'3 H'4 H'5.
apply order_minusP with (l1 := l4) (l2 := l5); auto.
apply canonical_pX_eqT with (a := a1); auto.
apply (eqT_sym A n); apply minusTerm_eqT; auto.
apply canonical_pX_eqT with (a := a2); auto.
apply (eqT_trans A n) with (y := a1); apply (eqT_sym A n); auto.
apply minusTerm_eqT; auto.
apply H'1.
apply canonical_imp_canonical with (a := a1); auto.
apply canonical_imp_canonical with (a := a2); auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 H'3 H'4.
apply order_minusP with (l1 := pX a1 l4) (l2 := l5); auto.
apply canonical_pX_eqT with (a := a2); auto.
apply canonical_cons; auto.
apply canonical_nzeroP with (ltM := ltM) (p := l5); auto.
apply nZero_invTerm_nZero with (1 := cs); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l5); auto.
apply canonical_pX_eqT with (a := a2); auto.
apply nZero_invTerm_nZero with (1 := cs); auto.
apply canonical_nzeroP with (ltM := ltM) (p := l5); auto.
apply H'2; auto.
apply canonical_imp_canonical with (a := a2); auto.
Qed.
Theorem canonical_minuspf :
∀ l1 l2 : list (Term A n),
canonical A0 eqA ltM l1 →
canonical A0 eqA ltM l2 → canonical A0 eqA ltM (minuspf l1 l2).
intros l1 l2 H' H'0; generalize (minuspf_is_minusP l1 l2); intros u1.
apply canonical_minusP with (l1 := l1) (l2 := l2); auto.
Qed.
Lemma invTerm_eqT_comp :
∀ a b : Term A n, eqT a b → eqT a (invTerm (A:=A) invA (n:=n) b).
intros a b H'.
apply (eqT_trans A n) with (y := b); auto.
Qed.
Lemma invTerm_T1_eqT_comp :
∀ a b : Term A n,
eqT a b →
eqT a
(invTerm (A:=A) invA (n:=n) (multTerm (A:=A) multA (n:=n) (T1 A1 n) b)).
intros a b H'.
apply (eqT_trans A n) with (y := b); auto.
apply (eqT_trans A n) with (y := multTerm (A:=A) multA (n:=n) (T1 A1 n) b);
auto.
apply (T1_eqT A A1 eqA); auto.
Qed.
Lemma multTerm_invTerm_T1_eqT_comp :
∀ a b : Term A n,
eqT a b →
eqT a
(multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) b).
intros a b H'.
apply (eqT_trans A n) with (y := b); auto.
apply (eqT_trans A n) with (y := multTerm (A:=A) multA (n:=n) (T1 A1 n) b);
auto.
apply (T1_eqT A A1 eqA); auto.
Qed.
Hint Resolve invTerm_eqT_comp invTerm_T1_eqT_comp
multTerm_invTerm_T1_eqT_comp.
Lemma minusP_is_plusP_mults :
∀ p q r : list (Term A n),
minusP p q r →
eqP A eqA n r
(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)) q)).
intros p q r H'; elim H'; auto.
intros; apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
simpl in |- *; intros; apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros a1 a2 l1 l2 l3 H'0 H'1 H'2.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pX a1
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec l1
(pX
(multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) a2)
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) l2))));
auto.
simpl in |- *; apply pluspf_inv1 with (1 := cs); auto.
change
(ltT ltM
(multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) a2)
a1) in |- ×.
apply eqT_compat_ltTl with (b := a2); auto.
intros a1 a2 l1 l2 l3 H'0 H'1 H'2 H'3.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
l1
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
l2)); auto.
simpl in |- *; apply pluspf_inv3a with (1 := cs); auto.
change
(eqT a1
(multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) a2))
in |- ×.
apply (eqT_trans A n) with (y := a2); auto.
change
(zeroP (A:=A) A0 eqA (n:=n)
(plusTerm (A:=A) plusA (n:=n) a1
(multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
a2))) in |- *; auto.
apply
zeroP_comp_eqTerm with (1 := cs) (a := minusTerm (A:=A) minusA (n:=n) a1 a2);
auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := plusTerm (A:=A) plusA (n:=n) a1 (invTerm (A:=A) invA (n:=n) a2));
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply plusTerm_comp_r with (1 := cs); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := invTerm (A:=A) invA (n:=n)
(multTerm (A:=A) multA (n:=n) (T1 A1 n) a2));
auto.
apply eqTerm_invTerm_comp with (1 := cs); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros a1 a2 l1 l2 l3 H'0 H'1 H'2 H'3.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pX
(plusTerm (A:=A) plusA (n:=n) a1
(multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) a2))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec l1
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) l2)));
auto.
apply eqpP1; auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := plusTerm (A:=A) plusA (n:=n) a1 (invTerm (A:=A) invA (n:=n) a2));
auto.
apply plusTerm_comp_r with (1 := cs); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := invTerm (A:=A) invA (n:=n)
(multTerm (A:=A) multA (n:=n) (T1 A1 n) a2));
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
simpl in |- *; apply pluspf_inv3b with (1 := cs); auto.
change
(eqT a1
(multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) a2))
in |- ×.
apply (eqT_trans A n) with (y := a2); auto.
change
(¬
zeroP (A:=A) A0 eqA (n:=n)
(plusTerm (A:=A) plusA (n:=n) a1
(multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
a2))) in |- *; auto.
apply
nzeroP_comp_eqTerm
with (1 := cs) (a := minusTerm (A:=A) minusA (n:=n) a1 a2);
auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := plusTerm (A:=A) plusA (n:=n) a1 (invTerm (A:=A) invA (n:=n) a2));
auto.
apply plusTerm_comp_r with (1 := cs); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := invTerm (A:=A) invA (n:=n)
(multTerm (A:=A) multA (n:=n) (T1 A1 n) a2));
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros a1 a2 l1 l2 l3 H'0 H'1 H'2.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pX
(multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) a2)
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (pX a1 l1)
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) l2)));
auto.
apply eqpP1; auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := invTerm (A:=A) invA (n:=n)
(multTerm (A:=A) multA (n:=n) (T1 A1 n) a2));
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
simpl in |- *; apply pluspf_inv2 with (1 := cs); auto.
change
(ltT ltM a1
(multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) a2))
in |- *; auto.
apply eqT_compat_ltTr with (b := a2); auto.
Qed.
Theorem minuspf_is_pluspf_mults :
∀ p q : list (Term A n),
eqP A eqA n (minuspf p q)
(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)) q)).
intros p q; try assumption.
apply minusP_is_plusP_mults with (p := p) (q := q); auto.
Qed.
Hint Resolve minuspf_is_pluspf_mults.
Theorem pO_minusP_inv1 :
∀ p q : list (Term A n),
minusP (pO A n) p q →
q = mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) p.
intros p; elim p.
intros q H'; inversion H'; auto.
intros a l H' q H'0; inversion H'0; auto.
Qed.
Theorem pO_minusP_inv2 :
∀ p q : list (Term A n), minusP p (pO A n) q → p = q.
intros p; elim p.
intros q H'; inversion H'; auto.
intros a l H' q H'0; inversion H'0; auto.
Qed.
Theorem minusP_inv1 :
∀ (a b : Term A n) (p q s : list (Term A n)),
minusP (pX a p) (pX b q) s → ltT ltM b a → s = pX a (minuspf p (pX b q)).
intros a b p q s H'; inversion_clear H'; intros.
apply pX_inj; auto.
apply uniq_minusp with (l := (p, pX b q)); simpl in |- *; auto.
absurd (ltT ltM b a); auto.
apply ltT_not_eqT; auto; apply eqT_sym; auto.
absurd (ltT ltM b a); auto.
apply ltT_not_eqT; auto; apply eqT_sym; auto.
absurd (ltT ltM b a); auto.
Qed.
Theorem minusP_inv2 :
∀ (a b : Term A n) (p q s : list (Term A n)),
minusP (pX a p) (pX b q) s →
ltT ltM a b → s = pX (invTerm (A:=A) invA (n:=n) b) (minuspf (pX a p) q).
intros a b p q s H'; inversion_clear H'; intros.
absurd (ltT ltM a b); auto.
absurd (ltT ltM a b); auto.
absurd (ltT ltM a b); auto.
apply pX_inj; auto.
apply uniq_minusp with (l := (pX a p, q)); simpl in |- *; auto.
Qed.
Theorem minusP_inv3a :
∀ (a b : Term A n) (p q s : list (Term A n)),
eqT a b →
zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) →
minusP (pX a p) (pX b q) s → s = minuspf p q.
intros a b p q s Eqd Z0 H'; inversion_clear H'.
absurd (ltT ltM b a); auto.
apply ltT_not_eqT; auto; apply eqT_sym; auto.
apply uniq_minusp with (l := (p, q)); simpl in |- *; auto.
case H1; auto.
absurd (ltT ltM a b); auto.
Qed.
Theorem minusP_inv3b :
∀ (a b : Term A n) (p q s : list (Term A n)),
eqT a b →
¬ zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) →
minusP (pX a p) (pX b q) s →
s = pX (minusTerm (A:=A) minusA (n:=n) a b) (minuspf p q).
intros a b p q s Eqd Z0 H'; inversion_clear H'.
absurd (ltT ltM b a); auto.
apply ltT_not_eqT; auto; apply eqT_sym; auto.
case Z0; auto.
apply pX_inj; auto.
apply uniq_minusp with (l := (p, q)); auto.
absurd (ltT ltM a b); auto.
Qed.
Theorem minuspf_inv1_eq :
∀ (a b : Term A n) (p q : list (Term A n)),
ltT ltM b a → pX a (minuspf p (pX b q)) = minuspf (pX a p) (pX b q).
intros a b p q H'; try assumption.
rewrite (minusP_inv1 a b p q (minuspf (pX a p) (pX b q))); auto.
Qed.
Theorem minuspf_inv2_eq :
∀ (a b : Term A n) (p q : list (Term A n)),
ltT ltM a b →
pX (invTerm (A:=A) invA (n:=n) b) (minuspf (pX a p) q) =
minuspf (pX a p) (pX b q).
intros a b p q H'; try assumption.
rewrite (minusP_inv2 a b p q (minuspf (pX a p) (pX b q))); auto.
Qed.
Theorem minuspf_inv3a_eq :
∀ (a b : Term A n) (p q : list (Term A n)),
eqT a b →
zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) →
minuspf p q = minuspf (pX a p) (pX b q).
intros a b p q H' Z; try assumption.
rewrite (minusP_inv3a a b p q (minuspf (pX a p) (pX b q))); auto.
Qed.
Theorem minuspf_inv3b_eq :
∀ (a b : Term A n) (p q : list (Term A n)),
eqT a b →
¬ zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) →
pX (minusTerm (A:=A) minusA (n:=n) a b) (minuspf p q) =
minuspf (pX a p) (pX b q).
intros a b p q H' Z; try assumption.
rewrite (minusP_inv3b a b p q (minuspf (pX a p) (pX b q))); auto.
Qed.
Theorem minuspf_inv1 :
∀ (a b : Term A n) (p q : list (Term A n)),
ltT ltM b a →
eqP A eqA n (pX a (minuspf p (pX b q))) (minuspf (pX a p) (pX b q)).
intros a b p q H'; try assumption.
rewrite (minusP_inv1 a b p q (minuspf (pX a p) (pX b q))); auto.
Qed.
Theorem minuspf_inv2 :
∀ (a b : Term A n) (p q : list (Term A n)),
ltT ltM a b →
eqP A eqA n (pX (invTerm (A:=A) invA (n:=n) b) (minuspf (pX a p) q))
(minuspf (pX a p) (pX b q)).
intros a b p q H'; try assumption.
rewrite (minusP_inv2 a b p q (minuspf (pX a p) (pX b q))); auto.
Qed.
Theorem minuspf_inv3a :
∀ (a b : Term A n) (p q : list (Term A n)),
eqT a b →
zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) →
eqP A eqA n (minuspf p q) (minuspf (pX a p) (pX b q)).
intros a b p q H' Z; try assumption.
rewrite (minusP_inv3a a b p q (minuspf (pX a p) (pX b q))); auto.
Qed.
Theorem minuspf_inv3b :
∀ (a b : Term A n) (p q : list (Term A n)),
eqT a b →
¬ zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) →
eqP A eqA n (pX (minusTerm (A:=A) minusA (n:=n) a b) (minuspf p q))
(minuspf (pX a p) (pX b q)).
intros a b p q H' Z; try assumption.
rewrite (minusP_inv3b a b p q (minuspf (pX a p) (pX b q))); auto.
Qed.
Hint Resolve pluspf_inv1 pluspf_inv2 pluspf_inv3a pluspf_inv3b.
Hint Resolve minuspf_inv1 minuspf_inv2 minuspf_inv3a minuspf_inv3b.
Theorem minuspf_comp :
∀ p q r 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 p r →
eqP A eqA n q s → eqP A eqA n (minuspf p q) (minuspf r s).
intros p q r s H' H'0 H'1 H'2 H'3 H'4;
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := 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)) q)).
apply minuspf_is_pluspf_mults; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
r
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
s)); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
Qed.
Theorem mults_dist_minuspf :
∀ (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 (minuspf p q))
(minuspf (mults (A:=A) multA (n:=n) a p) (mults (A:=A) multA (n:=n) a q)).
intros p q a H' H'0 H'1.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n) a
(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)) q)));
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)
(mults (A:=A) multA (n:=n) a
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) q)));
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)
(mults (A:=A) multA (n:=n)
(multTerm (A:=A) multA (n:=n) a
(invTerm (A:=A) invA (n:=n) (T1 A1 n))) q));
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)
(mults (A:=A) multA (n:=n)
(multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) a) q));
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)
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n) a q))); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
Qed.
Hint Resolve mults_dist_minuspf.
Theorem minuspf_pO_refl :
∀ p : list (Term A n), eqP A eqA n (minuspf p (pO A n)) p.
intros p;
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := 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))
(pO A n))); auto; simpl in |- *; auto.
Qed.
Hint Resolve minuspf_pO_refl.
Theorem minuspf_pOmults :
∀ p : list (Term A n),
eqP A eqA n (minuspf (pO A n) p)
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) p).
intros p;
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)
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) p));
auto; simpl in |- *; auto.
Qed.
Hint Resolve minuspf_pOmults.
Theorem mults_pO :
∀ (p : list (Term A n)) (a b : Term A n),
eqT a b →
zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) →
eqP A eqA n (pO A n)
(minuspf (mults (A:=A) multA (n:=n) a p) (mults (A:=A) multA (n:=n) b p)).
intros p; elim p.
simpl in |- *; auto.
intros; apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros a l H' a0 b H'0 H'1; simpl in |- ×.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := minuspf (mults (A:=A) multA (n:=n) a0 l)
(mults (A:=A) multA (n:=n) b l)); auto.
apply minuspf_inv3a; auto.
apply
zeroP_comp_eqTerm
with
(1 := cs)
(a := multTerm (A:=A) multA (n:=n) (minusTerm (A:=A) minusA (n:=n) a0 b)
a); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply multTerm_minusTerm_dist_l with (1 := cs); auto.
Qed.
Theorem mults_minusTerm :
∀ (p : list (Term A n)) (a b : Term A n),
eqT a b →
¬ zeroP (A:=A) A0 eqA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) →
canonical A0 eqA ltM p →
eqP A eqA n
(mults (A:=A) multA (n:=n) (minusTerm (A:=A) minusA (n:=n) a b) p)
(minuspf (mults (A:=A) multA (n:=n) a p) (mults (A:=A) multA (n:=n) b p)).
intros p; elim p.
simpl in |- *; auto.
intros a b H H0 H1; apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros a l H' a0 b H'0 H'1 H'2; simpl in |- *; auto.
cut (canonical A0 eqA ltM l); try apply canonical_imp_canonical with (a := a);
auto; intros Z0; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pX
(minusTerm (A:=A) minusA (n:=n)
(multTerm (A:=A) multA (n:=n) a0 a)
(multTerm (A:=A) multA (n:=n) b a))
(minuspf (mults (A:=A) multA (n:=n) a0 l)
(mults (A:=A) multA (n:=n) b l))); auto.
apply (eqpP1 A eqA); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply multTerm_minusTerm_dist_l with (1 := cs); auto.
apply minuspf_inv3b; auto.
apply
nzeroP_comp_eqTerm
with
(1 := cs)
(a := multTerm (A:=A) multA (n:=n) (minusTerm (A:=A) minusA (n:=n) a0 b)
a); auto.
apply nzeroP_multTerm with (1 := cs); auto.
apply (canonical_nzeroP A A0 eqA n ltM) with (p := l); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply multTerm_minusTerm_dist_l with (1 := cs); auto.
Qed.
Theorem order_pluspf :
∀ (l1 l2 : list (Term A n)) (a : Term A n),
canonical A0 eqA ltM (pX a l1) →
canonical A0 eqA ltM (pX a l2) →
canonical A0 eqA ltM
(pX a
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec l1
l2)).
intros l1 l2 a H' H'0.
apply order_plusP with (1 := os) (plusA := plusA) (l1 := l1) (l2 := l2); auto.
apply pluspf_is_plusP; auto.
apply canonical_pluspf; auto.
apply canonical_imp_canonical with (a := a); auto.
apply canonical_imp_canonical with (a := a); auto.
Qed.
Hint Resolve order_pluspf.
Theorem order_minuspf :
∀ (l1 l2 : list (Term A n)) (a : Term A n),
canonical A0 eqA ltM (pX a l1) →
canonical A0 eqA ltM (pX a l2) →
canonical A0 eqA ltM (pX a (minuspf l1 l2)).
intros l1 l2 a H' H'0.
apply
eqp_imp_canonical
with
(1 := cs)
(p := pX a
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec l1
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) l2)));
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply order_pluspf; auto.
apply
canonical_pX_eqT
with
(a := multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
a); auto.
change
(canonical A0 eqA ltM
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(pX a l2))) in |- *; auto.
apply (eqT_sym A n); auto.
apply (canonical_nzeroP A A0 eqA n ltM) with (p := l1); auto.
Qed.
Theorem minusP_refl :
∀ p q r : list (Term A n), minusP p q r → p = q → r = pO A n.
intros p q r H'; elim H'; auto.
intros l1 H'0; rewrite <- H'0; simpl in |- *; auto.
intros a1 a2 l1 l2 l3 H'0 H'1 H'2 H'3; generalize H'0.
rewrite (pX_invl A n a1 a2 l1 l2); auto; intros Ord.
absurd (ltT ltM a2 a2); auto.
intros a1 a2 l1 l2 l3 H'0 H'1 H'2 H'3 H'4.
apply H'1; auto.
apply pX_invr with (a := a1) (b := a2); auto.
intros a1 a2 l1 l2 l3 H'0 H'1 H'2 H'3 H'4; elim H'3; auto.
rewrite (pX_invl A n a1 a2 l1 l2); auto.
intros a1 a2 l1 l2 l3 H'0 H'1 H'2 H'3; generalize H'0.
rewrite (pX_invl A n a1 a2 l1 l2); auto; intros Ord.
absurd (ltT ltM a2 a2); auto.
Qed.
Theorem minuspf_refl_eq : ∀ p : list (Term A n), minuspf p p = pO A n.
intros p; rewrite (minusP_refl p p (minuspf p p)); auto.
Qed.
Theorem minuspf_refl :
∀ p : list (Term A n), eqP A eqA n (minuspf p p) (pO A n).
intros p.
rewrite (minusP_refl p p (minuspf p p)); auto.
Qed.
Theorem mults_comp_minuspf :
∀ (a : Term A n) (p q : list (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
(minuspf (mults (A:=A) multA (n:=n) a p) (mults (A:=A) multA (n:=n) a q))
(mults (A:=A) multA (n:=n) a (minuspf p q)).
intros a p q H' H'0 H'1.
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)
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
(mults (A:=A) multA (n:=n) a q))); 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)
(mults (A:=A) multA (n:=n)
(multTerm (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) a) q));
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)
(mults (A:=A) multA (n:=n)
(multTerm (A:=A) multA (n:=n) a
(invTerm (A:=A) invA (n:=n) (T1 A1 n))) q));
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)
(mults (A:=A) multA (n:=n) a
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) q)));
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n) a
(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)) q)));
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
Qed.
Theorem minuspf_zero :
∀ (a : Term A n) (p q : list (Term A n)),
eqP A eqA n (minuspf (pX a p) (pX a q)) (minuspf p q).
intros a p q; try assumption.
apply (eqp_trans _ _ _ _ _ _ _ _ _ cs n) with (y := minuspf p q); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n).
apply minuspf_inv3a; auto.
Qed.
Hint Resolve canonical_minuspf.
Theorem pluspf_minuspf_id :
∀ p q : list (Term A n),
canonical A0 eqA ltM p →
canonical A0 eqA ltM q →
eqP A eqA n
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(minuspf p q) q) p.
intros p q H' H'0.
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)) q)) q);
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)) q) q));
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)) q)));
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p (pO A n)); auto.
Qed.
Theorem minusP_pO_refl_eq :
∀ p q : list (Term A n), minusP p (pO A n) q → p = q.
unfold pO in |- *; intros p q H'; inversion H'; simpl in |- *; auto.
Qed.
Theorem minuspf_pO_refl_eq :
∀ p : list (Term A n), minuspf p (pO A n) = p.
intros p.
rewrite <- (minusP_pO_refl_eq p (minuspf p (pO A n))); auto.
Qed.
Theorem Opm_ind :
∀ (P : list (Term A n) → list (Term A n) → Prop)
(p q : list (Term A n)),
(∀ p : list (Term A n), P (pO A n) p) →
(∀ p : list (Term A n), P p (pO A n)) →
(∀ (a b : Term A n) (p q : list (Term A n)),
P (pX a p) q → ltT ltM a b → P (pX a p) (pX b q)) →
(∀ (a b : Term A n) (p q : list (Term A n)),
P p (pX b q) → ltT ltM b a → P (pX a p) (pX b q)) →
(∀ (a b : Term A n) (p q : list (Term A n)),
P p q → eqT a b → P (pX a p) (pX b q)) →
∀ p q : list (Term A n), P p q.
intros P H' H'0 H'1 H'2 H'3 H'4 H'5 p; elim p; auto.
intros a l H'6 q; elim q; auto.
intros a0 l0 H'7;
(case (ltT_dec A n ltM ltM_dec a a0);
[ intros temp; case temp; clear temp | idtac ];
intros test); change (P (pX a l) (pX a0 l0)) in |- *;
auto.
Qed.
Theorem minuspf_eq_inv1 :
∀ (a : Term A n) (p q : list (Term A n)),
canonical A0 eqA ltM (pX a p) →
canonical A0 eqA ltM (pX a q) →
eqP A eqA n (pX a (minuspf p q)) (minuspf (pX a p) q).
intros a p q; case q; simpl in |- *; auto.
intros H' H'0; rewrite minuspf_pO_refl_eq; rewrite minuspf_pO_refl_eq; auto.
intros a0 l H' H'0.
change
(eqP A eqA n (pX a (minuspf p (pX a0 l))) (minuspf (pX a p) (pX a0 l)))
in |- *; apply minuspf_inv1; auto.
apply (canonical_pX_order A A0 eqA) with (l := l); auto.
Qed.
Theorem minuspf_pOmults_eq :
∀ p : list (Term A n),
minuspf (pO A n) p =
mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) p.
intros p; apply uniq_minusp with (l := (pO A n, p)); auto.
Qed.
Theorem minuspf_eq_inv2 :
∀ (a : Term A n) (p q : list (Term A n)),
canonical A0 eqA ltM (pX a p) →
canonical A0 eqA ltM (pX a q) →
eqP A eqA n (pX (invTerm (A:=A) invA (n:=n) a) (minuspf p q))
(minuspf p (pX a q)).
intros a p; elim p; auto.
intros q H' H'0; rewrite minuspf_pOmults_eq.
rewrite minuspf_pOmults_eq; simpl in |- ×.
apply eqpP1; auto.
change
(eqTerm (A:=A) eqA (n:=n) (invTerm (A:=A) invA (n:=n) a)
(multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) a))
in |- ×.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := invTerm (A:=A) invA (n:=n)
(multTerm (A:=A) multA (n:=n) (T1 A1 n) a));
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
intros a0 l H' q H'0 H'1.
change
(eqP A eqA n (pX (invTerm (A:=A) invA (n:=n) a) (minuspf (pX a0 l) q))
(minuspf (pX a0 l) (pX a q))) in |- ×.
rewrite <- (minuspf_inv2_eq a0 a l q); auto.
apply (canonical_pX_order A A0 eqA) with (l := l); auto.
Qed.
Definition inv : list (Term A n) → Term A n → Term A n.
intros p; case p.
intros a;
exact
(multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) a).
intros a1 p1 a; exact (invTerm (A:=A) invA (n:=n) a).
Defined.
Theorem inv_prop :
∀ (a : Term A n) (p q : list (Term A n)),
canonical A0 eqA ltM (pX a p) →
minuspf p (pX a q) = pX (inv p a) (minuspf p q).
intros a p q; case p; simpl in |- *; auto.
intros H'.
change
(minuspf (pO A n) (pX a q) =
pX (multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) a)
(minuspf (pO A n) q)) in |- ×.
rewrite (minuspf_pOmults_eq (pX a q)); simpl in |- *; auto.
rewrite (minuspf_pOmults_eq q); simpl in |- *; auto.
intros a0 l H'.
change
(minuspf (pX a0 l) (pX a q) =
pX (invTerm (A:=A) invA (n:=n) a) (minuspf (pX a0 l) q))
in |- ×.
rewrite <- (minuspf_inv2_eq a0 a l q); auto.
apply (canonical_pX_order A A0 eqA) with (l := l); auto.
Qed.
Hint Resolve inv_prop.
Theorem invTerm_T1_multTerm_T1 :
eqTerm (A:=A) eqA (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))) (T1 A1 n).
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := invTerm (A:=A) invA (n:=n)
(multTerm (A:=A) multA (n:=n) (T1 A1 n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n))));
[ auto | idtac ].
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := invTerm (A:=A) invA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)));
auto.
apply eqTerm_invTerm_comp with (1 := cs); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); apply T1_multTerm_l with (1 := cs);
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); apply invTerm_invol with (1 := cs);
auto.
Qed.
Hint Resolve invTerm_T1_multTerm_T1.
Theorem pluspf_is_minuspf :
∀ p q : list (Term A n),
canonical A0 eqA ltM p →
canonical A0 eqA ltM q →
eqP A eqA n
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p q)
(minuspf p
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) q)).
intros p q Opp Opq;
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := 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)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) q)));
auto.
apply eqp_pluspf_com with (1 := cs); 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))) q);
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n);
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := mults (A:=A) multA (n:=n) (T1 A1 n) q);
auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
Qed.
Definition sminus : 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; ∃ (minuspf p1 p2); auto.
Defined.
End Pminus.