Section Pplus.
Load "hCoefStructure".
Load "hOrderStructure".
Load "hEq".
Inductive plusP :
list (Term A n) -> list (Term A n) -> list (Term A n) -> Prop :=
| nillu1 : forall l1, plusP (pO A n) l1 l1
| nillu2 : forall l1, plusP l1 (pO A n) l1
| mainu1 :
forall a1 a2 l1 l2 l3,
ltT ltM a2 a1 ->
plusP l1 (pX a2 l2) l3 -> plusP (pX a1 l1) (pX a2 l2) (pX a1 l3)
| mainu2a :
forall a1 a2 l1 l2 l3,
plusP l1 l2 l3 ->
eqT a1 a2 ->
zeroP (A:=A) A0 eqA (n:=n) (plusTerm (A:=A) plusA (n:=n) a1 a2) ->
plusP (pX a1 l1) (pX a2 l2) l3
| mainu2b :
forall a1 a2 l1 l2 l3,
plusP l1 l2 l3 ->
eqT a1 a2 ->
~ zeroP (A:=A) A0 eqA (n:=n) (plusTerm (A:=A) plusA (n:=n) a1 a2) ->
plusP (pX a1 l1) (pX a2 l2)
(pX (plusTerm (A:=A) plusA (n:=n) a1 a2) l3)
| mainu3 :
forall a1 a2 l1 l2 l3,
ltT ltM a1 a2 ->
plusP (pX a1 l1) l2 l3 -> plusP (pX a1 l1) (pX a2 l2) (pX a2 l3).
Hint Resolve nillu1 nillu2 mainu1 mainu2a mainu2b mainu3.
Definition projsig1 (A : Set) (P : A -> Prop) (H : sig P) :=
let (a, _) return A := H in a.
Definition plusp : forall l, {a : _ | plusP (fst l) (snd l) a}.
intros l; pattern l in |- *.
apply well_founded_induction with (R := lessP A n); auto.
apply wf_lessP; auto.
intros x; case x; intros p q; simpl in |- *.
case p; clear p.
intros H'; exists q; auto.
intros a1 m1; case q; clear q.
intros H'; exists (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 ].
exists (pX a2 Orec); auto.
change (plusP (pX a1 m1) (pX a2 m2) (pX a2 Orec)) in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
lapply (H' (m1, pX a2 m2)); simpl in |- *;
[ intros Rec; case Rec; clear Rec; intros Orec Prec | idtac ].
exists (pX a1 Orec); auto.
change (plusP (pX a1 m1) (pX a2 m2) (pX a1 Orec)) in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto.
apply LetP with (h := plusTerm (A:=A) plusA (n:=n) a1 a2).
intros letA eqL0; case (zeroP_dec A A0 eqA eqA_dec n letA); intros H'0.
lapply (H' (m1, m2)); simpl in |- *;
[ intros Rec; case Rec; clear Rec; intros Orec Prec | idtac ].
exists Orec; auto.
change (plusP (pX a1 m1) (pX a2 m2) Orec) in |- *; auto.
rewrite eqL0 in H'0; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
lapply (H' (m1, m2)); simpl in |- *;
[ intros Rec; case Rec; clear Rec; intros Orec Prec | idtac ].
exists (pX letA Orec); auto.
change (plusP (pX a1 m1) (pX a2 m2) (pX letA Orec)) in |- *; auto.
rewrite eqL0; auto.
rewrite eqL0 in H'0; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
Defined.
Set Implicit Arguments.
Unset Strict Implicit.
Definition pluspf l1 l2 := projsig1 _ _ (plusp (l1, l2)).
Hint Unfold projsig1 pluspf.
Set Strict Implicit.
Unset Implicit Arguments.
intros p q l a b H'; simple inversion H'.
discriminate H.
discriminate H0.
rewrite <- H3; rewrite (pX_invl _ _ a2 b l2 q H2);
rewrite (pX_invr _ _ l2 q a2 b H2); rewrite (pX_invl _ _ a1 a l1 p H1);
rewrite (pX_invr _ _ l1 p a1 a H1).
intros H'0 H'1; exists l3; left; split;
[ idtac | split; [ try assumption | idtac ] ]; auto.
rewrite <- H3.
intros H'0 H'1 H'2; exists l3; right; right.
rewrite <- (pX_invl _ _ a2 b l2 q H3); rewrite <- (pX_invl _ _ a1 a l1 p H2);
auto.
rewrite <- (pX_invr _ _ l2 q a2 b H3); rewrite <- (pX_invr _ _ l1 p a1 a H2);
auto.
rewrite <- H4; auto.
intros H'0 H'1 H'2; exists l3; right; right.
rewrite <- (pX_invl _ _ a2 b l2 q H3); rewrite <- (pX_invl _ _ a1 a l1 p H2);
auto.
rewrite <- (pX_invr _ _ l2 q a2 b H3); rewrite <- (pX_invr _ _ l1 p a1 a H2);
auto.
intros H'0 H'1; exists l3; right; left; split;
[ idtac | split; [ try assumption | idtac ] ]; auto.
rewrite <- (pX_invl _ _ a2 b l2 q H2); rewrite <- (pX_invl _ _ a1 a l1 p H1);
auto.
rewrite <- (pX_invl _ _ a1 a l1 p H1); rewrite <- (pX_invr _ _ l2 q a2 b H2);
rewrite <- (pX_invr _ _ l1 p a1 a H1); auto.
rewrite <- H3; rewrite <- (pX_invl _ _ a2 b l2 q H2); auto.
Qed.
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 (plusP_inv l1 l2 l4 n1 n2); auto.
intros x H'; elim H';
[ intros H'0; elim H'0; intros H'1 H'2; elim H'2; intros H'3 H'4;
rewrite H'4; clear H'2 H'0 H'
| intros H'0; clear H' ].
case (plusP_inv l1 l2 l3 n1 n2); auto.
intros x0 H'; elim H';
[ intros H'0; elim H'0; intros H'2 H'5; elim H'5; intros H'6 H'7;
rewrite H'7; clear H'5 H'0 H'
| intros H'0; clear H' ]; auto.
apply pX_inj; auto.
apply Induc with (y := (l1, n2 :: l2)); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto.
elim H'0;
[ intros H'; elim H'; intros H'2 H'5; clear H' H'0 | intros H'; clear H'0 ].
absurd (ltT ltM n1 n2); auto.
elim H'; intros H'0 H'2; try exact H'0; clear H'.
absurd (eqT n2 n1); auto.
apply (eqT_sym A n n1); auto.
elim H'0;
[ intros H'; elim H'; intros H'1 H'2; elim H'2; intros H'3 H'4; rewrite H'4;
clear H'2 H' H'0
| intros H'; clear H'0 ].
case (plusP_inv l1 l2 l3 n1 n2); auto.
intros x0 H'; elim H';
[ intros H'0; elim H'0; intros H'2 H'5; clear H'0 H'
| intros H'0; clear H' ].
absurd (ltT ltM n1 n2); auto.
elim H'0;
[ intros H'; elim H'; intros H'2 H'5; elim H'5; intros H'6 H'7; rewrite H'7;
clear H'5 H' H'0
| intros H'; clear H'0 ].
apply pX_inj; auto.
apply Induc with (y := (pX n1 l1, l2)); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
elim H'; intros H'0 H'2; try exact H'0; clear H'.
absurd (eqT n1 n2); auto.
elim H'; intros H'0 H'1; try exact H'0; clear H'.
case (plusP_inv l1 l2 l3 n1 n2); auto.
intros x0 H'; elim H';
[ intros H'2; elim H'2; intros H'3 H'4; try exact H'3; clear H'2 H'
| intros H'2; clear H' ].
absurd (eqT n2 n1); auto.
apply (eqT_sym A n n1); auto.
elim H'2;
[ intros H'; elim H'; intros H'3 H'4; try exact H'3; clear H' H'2
| intros H'; clear H'2 ].
absurd (eqT n1 n2); auto.
elim H'; intros H'2 H'3; elim H'3;
[ intros H'4; elim H'4; intros H'5 H'6; try exact H'5; clear H'4 H'3 H'
| intros H'4; clear H'3 H' ].
elim H'1;
[ intros H'; clear H'1
| intros H'; elim H'; intros H'3 H'4; apply H'3 || elim H'3; clear H' H'1 ];
auto.
elim H'; intros H'1 H'3; clear H'.
apply Induc with (y := (l1, l2)); auto; red in |- *; simpl in |- *; auto.
simpl in |- *; auto with arith.
elim H'4; intros H' H'3; elim H'3; intros H'5 H'6; rewrite H'6; clear H'3 H'4.
elim H'1;
[ intros H'3; clear H'1
| intros H'3; elim H'3; intros H'4 H'7; elim H'7; intros H'8 H'9;
rewrite H'9; clear H'7 H'3 H'1 ].
elim H'3; intros H'1 H'4; try exact H'1; clear H'3.
apply H' || elim H'; try assumption.
apply pX_inj; auto.
apply Induc with (y := (l1, l2)); auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
Qed.
intros l1 l2; try assumption.
unfold pluspf in |- *; case (plusp (l1, l2)); simpl in |- *; auto.
Qed.
Hint Resolve pluspf_is_plusP.
intros l1 l2 l3 a H'; generalize a; elim H'; auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 a0 H'3 H'4 H'5.
apply canonical_cons; auto.
apply (canonical_pX_order _ A0 eqA) with (l := l4); auto.
apply (canonical_nzeroP _ A0 eqA _ ltM) with (p := pX a1 l4); auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 H'3 a0 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 a0 H'4 H'5 H'6.
apply canonical_cons; auto.
apply ltT_eqTr with (a := a1); auto.
apply (eqT_sym A n (plusTerm (A:=A) plusA (n:=n) a1 a2)).
apply plusTerm_eqT1; auto.
apply (canonical_pX_order _ A0 eqA) with (l := l4); auto.
apply (canonical_nzeroP _ A0 eqA _ ltM) with (p := pX a1 l4); auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 a0 H'3 H'4 H'5.
apply canonical_cons; auto.
apply (canonical_pX_order _ A0 eqA) with (l := l5); auto.
apply (canonical_nzeroP _ A0 eqA _ ltM) with (p := pX a1 l4); auto.
Qed.
intros l1 l2 l3 H'; elim H'; auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 H'3 H'4; try assumption.
apply order_plusP with (l1 := l4) (l2 := pX a2 l5); auto.
apply canonical_cons; auto.
apply (canonical_nzeroP _ A0 eqA _ ltM) with (p := l4); auto.
apply H'2; auto.
apply (canonical_imp_canonical _ A0 eqA _ ltM) 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 _ A0 eqA _ ltM) with (a := a1); auto.
apply (canonical_imp_canonical _ A0 eqA _ ltM) with (a := a2); auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 H'3 H'4 H'5.
apply order_plusP with (l1 := l4) (l2 := l5); auto.
apply canonical_pX_eqT with (a := a1); auto.
apply (eqT_sym A n (plusTerm (A:=A) plusA (n:=n) a1 a2)).
apply plusTerm_eqT1; auto.
apply canonical_pX_eqT with (a := a2); auto; auto.
apply (eqT_sym A n (plusTerm (A:=A) plusA (n:=n) a1 a2)).
apply plusTerm_eqT2; auto.
apply H'1.
apply (canonical_imp_canonical _ A0 eqA _ ltM) with (a := a1); auto.
apply (canonical_imp_canonical _ A0 eqA _ ltM) with (a := a2); auto.
intros a1 a2 l4 l5 l6 H'0 H'1 H'2 H'3 H'4.
apply order_plusP with (l1 := pX a1 l4) (l2 := l5); auto.
apply canonical_cons; auto.
apply (canonical_nzeroP _ A0 eqA _ ltM) with (p := l5); auto.
apply H'2; auto.
apply canonical_imp_canonical with (a := a2); auto.
Qed.
intros l1 l2 H' H'0; generalize (pluspf_is_plusP l1 l2); intros u1.
apply canonical_plusP with (l1 := l1) (l2 := l2); auto.
Qed.
intros p; elim p.
intros q H'; inversion H'; auto.
intros a l H' q H'0; inversion H'0; auto.
Qed.
Hint Resolve eqp_refl.
intros p; elim p.
intros q H'; inversion H'; auto.
intros a l H' q H'0; inversion H'0; auto.
Qed.
Hint Resolve eqp_refl.
intros a p; case p; auto.
intros m l H'; try assumption.
change (plusP (pX a (pO A n)) (pX m l) (pX a (pX m l))) in |- *.
apply mainu1; auto.
apply (canonical_pX_order _ A0 eqA) with (l := l); auto.
Qed.
unfold pX in |- *; intros a b p q s H'; inversion_clear H'; intros.
change (pX a l3 = pX a (pluspf p (pX b q))) in |- *; apply pX_inj; auto.
apply uniq_plusp with (l := (p, pX b q)); simpl in |- *; auto.
absurd (eqT b a); auto.
apply eqT_sym; auto.
absurd (eqT b a); auto.
apply eqT_sym; auto.
absurd (ltT ltM b a); auto.
Qed.
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_plusp with (l := (pX a p, q)); simpl in |- *; auto.
Qed.
intros a b p q s Eqd Z0 H'; inversion_clear H'; intros.
absurd (eqT b a); auto.
apply eqT_sym; auto.
apply uniq_plusp with (l := (p, q)); auto.
elim H1; auto.
absurd (eqT a b); auto.
Qed.
unfold pX in |- *; intros a b p q s Eqd Z0 H'; inversion_clear H'; intros.
absurd (eqT b a); auto.
apply eqT_sym; auto.
elim Z0; try assumption.
change
(pX (plusTerm (A:=A) plusA (n:=n) a b) l3 =
pX (plusTerm (A:=A) plusA (n:=n) a b) (pluspf p q))
in |- *.
apply pX_inj; auto.
apply uniq_plusp with (l := (p, q)); auto.
absurd (eqT a b); auto.
Qed.
intros a b p q H'; try assumption.
rewrite (plusP_inv1 a b p q (pluspf (pX a p) (pX b q))); auto.
Qed.
intros a b p q H'; try assumption.
rewrite (plusP_inv2 a b p q (pluspf (pX a p) (pX b q))); auto.
Qed.
intros a b p q H' Z; try assumption.
rewrite (plusP_inv3a a b p q (pluspf (pX a p) (pX b q))); auto.
Qed.
intros a b p q H' Z; try assumption.
rewrite (plusP_inv3b a b p q (pluspf (pX a p) (pX b q))); auto.
Qed.
Hint Resolve pluspf_inv1 pluspf_inv2 pluspf_inv3a pluspf_inv3b.
intros p q r s H'; generalize s; elim H'; clear s H'; auto.
intros l1 s H'; try assumption.
rewrite (pO_plusP_inv2 l1 s); auto.
intros l1 s H'; rewrite (pO_plusP_inv1 l1 s); auto.
intros a1 a2 l1 l2 l3 H' H'0 H'1 s H'2.
rewrite (plusP_inv2 a2 a1 l2 l1 s); auto.
intros a1 a2 l1 l2 l3 H' H'0 H'1 H'2 s R.
rewrite (plusP_inv3a a2 a1 l2 l1 s); auto.
apply (eqT_sym A n a1); auto.
apply
zeroP_comp_eqTerm with (1 := cs) (a := plusTerm (A:=A) plusA (n:=n) a1 a2);
auto.
apply plusTerm_com with (1 := cs); auto.
intros a1 a2 l1 l2 l3 H' H'0 H'1 H'2 s H'3.
rewrite (plusP_inv3b a2 a1 l2 l1 s); auto.
apply eqpP1; auto; apply plusTerm_com with (1 := cs); auto.
apply (eqT_sym A n a1); auto.
red in |- *; intros H'4; apply H'2.
apply
zeroP_comp_eqTerm with (1 := cs) (a := plusTerm (A:=A) plusA (n:=n) a2 a1);
auto.
apply plusTerm_com with (1 := cs); auto.
apply (eqT_sym A n); auto.
intros a1 a2 l1 l2 l3 H' H'0 H'1 s H'2.
rewrite (plusP_inv1 a2 a1 l2 l1 s); auto.
Qed.
intros p q; apply (plusP_com p q (pluspf p q) (pluspf q p)); auto.
Qed.
intros p q H'; inversion H'; auto.
Qed.
intros p q H'; inversion H'; auto.
Qed.
Hint Resolve plusP_zero_pOl plusP_zero_pOr.
Hint Resolve eqp_trans.
intros p q r s H' H'0; rewrite (uniq_plusp (p, q) r s); auto.
Qed.
Hint Resolve plusP_uniq_eqP.
intros p.
apply uniq_plusp with (l := (pO A n, p)); auto.
Qed.
intros p.
apply uniq_plusp with (l := (p, pO A n)); auto.
Qed.
intros a b p q H'; rewrite (plusP_inv1 a b p q (pluspf (pX a p) (pX b q)));
auto.
Qed.
intros a b p q H'; rewrite (plusP_inv2 a b p q (pluspf (pX a p) (pX b q)));
auto.
Qed.
intros a b p q H' H1';
rewrite (plusP_inv3a a b p q (pluspf (pX a p) (pX b q)));
auto.
Qed.
intros a b p q H' H1';
rewrite (plusP_inv3b a b p q (pluspf (pX a p) (pX b q)));
auto.
Qed.
intros l1 l2 a H' H'0.
apply order_plusP with (l1 := l1) (l2 := l2); auto.
apply canonical_pluspf; auto.
apply canonical_imp_canonical with (a := a); auto.
apply canonical_imp_canonical with (a := a); auto.
Qed.
intros a p q; case q; auto.
rewrite <- pO_pluspf_inv2; auto.
rewrite <- pO_pluspf_inv2; auto.
intros a0 l H'0 H'1.
change (pX a (pluspf p (pX a0 l)) = pluspf (pX a p) (pX a0 l)) in |- *.
apply pluspf_inv1_eq; auto.
apply (canonical_pX_order _ A0 eqA) with (l := l); auto.
Qed.
intros a p; case p; auto.
intros q H'0 H'1.
rewrite <- pO_pluspf_inv1; auto.
rewrite <- pO_pluspf_inv1; auto.
intros a0 l q H'0 H'1.
change (pX a (pluspf (pX a0 l) q) = pluspf (pX a0 l) (pX a q)) in |- *.
apply pluspf_inv2_eq; auto.
apply (canonical_pX_order A A0 eqA) with (l := l); auto.
Qed.
intros p; rewrite <- pO_pluspf_inv1; auto.
Qed.
intros p; rewrite <- pO_pluspf_inv2; auto.
Qed.
Hint Resolve p0_pluspf_l p0_pluspf_r.
intros a p; case p; auto.
intros a0 l H'.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := pX a (pluspf (pO A n) (a0 :: l))); auto.
change
(eqP A eqA n (pluspf (pX a (pO A n)) (pX a0 l))
(pX a (pluspf (pO A n) (pX a0 l)))) in |- *.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); apply pluspf_inv1; auto.
apply (canonical_pX_order A A0 eqA) with (l := l); auto.
Qed.
Hint Resolve plusTerm_is_pX.
intros l; pattern l in |- *.
apply (well_founded_ind (wf_lessP3 A n)).
intros x; case x; intros p z; case z; intros q r; simpl in |- *; clear x z;
auto.
case p.
rewrite <- pO_pluspf_inv1; auto.
case q.
intros a l0; rewrite <- pO_pluspf_inv1; rewrite <- pO_pluspf_inv2; auto.
case r; simpl in |- *; auto.
intros a l0 a0 l1; rewrite <- pO_pluspf_inv2; rewrite <- pO_pluspf_inv2; auto.
intros a l0 a0 l1 a1 l2 H' H'0 H'1 H'2.
cut (canonical A0 eqA ltM (pX a l0)); [ clear H'2; intros H'2 | auto ].
cut (canonical A0 eqA ltM (pX a0 l1)); [ clear H'1; intros H'1 | auto ].
cut (canonical A0 eqA ltM (pX a1 l2)); [ clear H'0; intros H'0 | auto ].
cut (canonical A0 eqA ltM l0);
[ intros C0 | apply canonical_imp_canonical with (a := a) ];
auto.
cut (canonical A0 eqA ltM l1);
[ intros C1 | apply canonical_imp_canonical with (a := a0) ];
auto.
cut (canonical A0 eqA ltM l2);
[ intros C2 | apply canonical_imp_canonical with (a := a1) ];
auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros nZ0 | apply (canonical_nzeroP _ A0 eqA _ ltM) with (p := l0) ];
auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) a0);
[ intros nZ1 | apply (canonical_nzeroP _ A0 eqA _ ltM) with (p := l1) ];
auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) a1);
[ intros nZ2 | apply (canonical_nzeroP _ A0 eqA _ ltM) with (p := l2) ];
auto.
change
(eqP A eqA n (pluspf (pluspf (pX a1 l2) (pX a0 l1)) (pX a l0))
(pluspf (pX a1 l2) (pluspf (pX a0 l1) (pX a l0))))
in |- *.
case (ltT_dec A n ltM ltM_dec a1 a0);
[ intros H0; case H0; clear H0 | idtac ]; intros H0.
case (ltT_dec A n ltM ltM_dec a0 a); [ intros H1; case H1; clear H1 | idtac ];
intros H1.
cut (ltT ltM a1 a);
[ intros Ord0 | apply (ltT_trans A _ _ os) with (y := a0) ];
auto.
rewrite <- (pluspf_inv2_eq a1 a0 l2 l1); auto.
rewrite <- (pluspf_inv2_eq a0 a (pluspf (pX a1 l2) l1) l0); auto.
rewrite <- (pluspf_inv2_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv2_eq a1 a l2 (pluspf (pX a0 l1) l0)); auto.
apply eqpP1; auto.
rewrite (pluspf_inv2_eq a1 a0 l2 l1); auto.
apply H' with (y := (pX a1 l2, (pX a0 l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
repeat rewrite (fun a b : nat => plus_comm a (S b)); simpl in |- *.
repeat rewrite (plus_comm (size A n l1)); auto.
rewrite <- (pluspf_inv1_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv2_eq a1 a0 l2 l1); auto.
rewrite <- (pluspf_inv2_eq a1 a0 l2 (pluspf l1 (pX a l0))); auto.
rewrite <- (pluspf_inv1_eq a0 a (pluspf (pX a1 l2) l1) l0); auto.
apply eqpP1; auto.
apply H' with (y := (pX a1 l2, (l1, pX a l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
cut (ltT ltM a1 a); [ intros Ord0 | apply eqT_compat_ltTr with (b := a0) ];
auto.
case (zeroP_dec A A0 eqA eqA_dec n (plusTerm (A:=A) plusA (n:=n) a0 a));
intros Z0.
rewrite <- (pluspf_inv3a_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv2_eq a1 a0 l2 l1); auto.
rewrite <- (pluspf_inv3a_eq a0 a (pluspf (pX a1 l2) l1) l0); auto.
apply H' with (y := (pX a1 l2, (l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
repeat rewrite (fun a b : nat => plus_comm a (S b)); simpl in |- *.
repeat rewrite (plus_comm (size A n l2)); auto.
repeat rewrite (plus_comm (size A n l1)); auto.
cut (ltT ltM a1 (plusTerm (A:=A) plusA (n:=n) a0 a));
[ intros Ord1
| apply eqT_compat_ltTr with (b := a0); auto; apply (eqT_sym A n);
apply (plusTerm_eqT1 A plusA n) ]; auto.
rewrite <- (pluspf_inv2_eq a1 a0 l2 l1); auto.
rewrite <- (pluspf_inv3b_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv3b_eq a0 a (pluspf (pX a1 l2) l1) l0); auto.
rewrite <-
(pluspf_inv2_eq a1 (plusTerm (A:=A) plusA (n:=n) a0 a) l2 (pluspf l1 l0))
; auto.
apply eqpP1; auto.
apply H' with (y := (pX a1 l2, (l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
repeat rewrite (fun a b : nat => plus_comm a (S b)); simpl in |- *.
repeat rewrite (plus_comm (size A n l2)); auto.
repeat rewrite (plus_comm (size A n l1)); auto.
rewrite <- (pluspf_inv1_eq a1 a0 l2 l1); auto.
case (ltT_dec A n ltM ltM_dec a0 a); [ intros H1; case H1; clear H1 | idtac ];
intros H1.
rewrite <- (pluspf_inv2_eq a0 a l1 l0); auto.
case (ltT_dec A n ltM ltM_dec a1 a); [ intros H2; case H2; clear H2 | idtac ];
intros H2.
rewrite <- (pluspf_inv2_eq a1 a (pluspf l2 (pX a0 l1)) l0); auto.
rewrite <- (pluspf_inv2_eq a1 a l2 (pluspf (pX a0 l1) l0)); auto.
apply eqpP1; auto.
rewrite (pluspf_inv1_eq a1 a0 l2 l1); auto.
apply H' with (y := (pX a1 l2, (pX a0 l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
repeat rewrite (fun a b : nat => plus_comm a (S b)); simpl in |- *.
repeat rewrite (plus_comm (size A n l1)); auto.
rewrite <- (pluspf_inv1_eq a1 a l2 (pluspf (pX a0 l1) l0)); auto.
rewrite <- (pluspf_inv1_eq a1 a (pluspf l2 (pX a0 l1)) l0); auto.
apply eqpP1; auto.
rewrite (pluspf_inv2_eq a0 a l1 l0); auto.
apply H' with (y := (l2, (pX a0 l1, pX a l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto.
cut (ltT ltM a0 (plusTerm (A:=A) plusA (n:=n) a1 a));
[ intros Ord1
| apply eqT_compat_ltTr with (b := a1); auto; apply (eqT_sym A n);
apply (plusTerm_eqT1 A plusA n) ]; auto.
case (zeroP_dec A A0 eqA eqA_dec n (plusTerm (A:=A) plusA (n:=n) a1 a));
intros Z0.
rewrite <- (pluspf_inv3a_eq a1 a (pluspf l2 (pX a0 l1)) l0); auto.
rewrite <- (pluspf_inv3a_eq a1 a l2 (pluspf (pX a0 l1) l0)); auto.
apply H' with (y := (l2, (pX a0 l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
rewrite <- (pluspf_inv3b_eq a1 a (pluspf l2 (pX a0 l1)) l0); auto.
rewrite <- (pluspf_inv3b_eq a1 a l2 (pluspf (pX a0 l1) l0)); auto.
apply eqpP1; auto.
apply H' with (y := (l2, (pX a0 l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
cut (ltT ltM a a1);
[ intros Ord0 | apply (ltT_trans A n ltM os) with (y := a0) ];
auto with arith.
rewrite <- (pluspf_inv1_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv1_eq a1 a0 l2 (pluspf l1 (pX a l0))); auto.
rewrite <- (pluspf_inv1_eq a1 a (pluspf l2 (pX a0 l1)) l0); auto.
apply eqpP1; auto.
rewrite (pluspf_inv1_eq a0 a l1 l0); auto.
apply H' with (y := (l2, (pX a0 l1, pX a l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto.
cut (ltT ltM a a1);
[ intros Ord0 | apply eqT_compat_ltTl with (b := a0); auto ];
auto.
case (zeroP_dec A A0 eqA eqA_dec n (plusTerm (A:=A) plusA (n:=n) a0 a));
intros Z0.
rewrite <- (pluspf_inv3a_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv1_eqa a1 l2 (pluspf l1 l0)); auto.
rewrite <- (pluspf_inv1_eq a1 a (pluspf l2 (pX a0 l1)) l0); auto.
apply eqpP1; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := pluspf l2 (pluspf (pX a0 l1) (pX a l0)));
auto.
apply H' with (y := (l2, (pX a0 l1, pX a l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto.
rewrite <- (pluspf_inv3a_eq a0 a l1 l0); auto.
apply order_pluspf; auto.
apply canonical_skip_fst with (b := a0); auto.
apply canonical_skip_fst with (b := a); auto.
cut (ltT ltM (plusTerm (A:=A) plusA (n:=n) a0 a) a1);
[ intros Ord1
| apply eqT_compat_ltTl with (b := a0); auto; apply (eqT_sym A n); auto;
apply (plusTerm_eqT1 A plusA n) ]; auto.
rewrite <- (pluspf_inv3b_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv1_eq a1 a (pluspf l2 (pX a0 l1)) l0); auto.
rewrite <-
(pluspf_inv1_eq a1 (plusTerm (A:=A) plusA (n:=n) a0 a) l2 (pluspf l1 l0))
; auto.
apply eqpP1; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := pluspf l2 (pluspf (pX a0 l1) (pX a l0)));
auto.
apply H' with (y := (l2, (pX a0 l1, pX a l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto.
rewrite <- (pluspf_inv3b_eq a0 a l1 l0); auto.
case (zeroP_dec A A0 eqA eqA_dec n (plusTerm (A:=A) plusA (n:=n) a1 a0));
intros Z0.
rewrite <- (pluspf_inv3a_eq a1 a0 l2 l1); auto.
case (ltT_dec A n ltM ltM_dec a1 a); [ intros H2; case H2; clear H2 | idtac ];
intros H2.
cut (ltT ltM a0 a); [ intros Ord0 | apply eqT_compat_ltTl with (b := a1) ];
auto.
rewrite <- (pluspf_inv2_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv2_eq a1 a l2 (pluspf (pX a0 l1) l0)); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := pX a (pluspf (pluspf (pX a1 l2) (pX a0 l1)) l0));
auto.
rewrite <- (pluspf_inv3a_eq a1 a0 l2 l1); auto.
rewrite <- (pluspf_inv2_eqa a (pluspf l2 l1) l0); auto.
apply order_pluspf; auto.
apply canonical_skip_fst with (b := a1); auto.
apply canonical_skip_fst with (b := a0); auto.
apply eqpP1; auto.
apply H' with (y := (pX a1 l2, (pX a0 l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
repeat rewrite (fun a b : nat => plus_comm a (S b)); simpl in |- *.
repeat rewrite (plus_comm (size A n l1)); auto.
cut (ltT ltM a a0); [ intros Ord0 | apply eqT_compat_ltTr with (b := a1) ];
auto.
rewrite <- (pluspf_inv1_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv3a_eq a1 a0 l2 (pluspf l1 (pX a l0))); auto.
apply H' with (y := (l2, (l1, pX a l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
case (zeroP_dec A A0 eqA eqA_dec n (plusTerm (A:=A) plusA (n:=n) a0 a));
intros Z1.
cut (eqT a0 a);
[ intros eqTerm0
| apply (eqT_trans A n) with (y := a1); auto; apply (eqT_sym A n) ];
auto.
cut (eqT a (plusTerm (A:=A) plusA (n:=n) a1 a0));
[ intros eqTerm1
| apply (eqT_trans A n) with (y := a0); auto; apply (eqT_sym A n); auto;
apply (plusTerm_eqT2 A plusA n) ]; auto.
cut (eqT (plusTerm (A:=A) plusA (n:=n) a a1) a0);
[ intros eqTerm2
| apply (eqT_trans A n) with (y := a);
[ apply (plusTerm_eqT1 A plusA n)
| apply (eqT_trans A n) with (y := a1) ]; auto;
apply (eqT_sym A n) ]; auto.
cut (eqT (plusTerm (A:=A) plusA (n:=n) a1 a) a0);
[ intros eqTerm3
| apply (eqT_trans A n) with (y := a); auto;
[ apply (plusTerm_eqT2 A plusA n) | apply (eqT_sym A n) ] ];
auto.
cut (eqT a1 (plusTerm (A:=A) plusA (n:=n) a a0));
[ intros eqTerm4
| apply (eqT_trans A n) with (y := a0); auto; apply (eqT_sym A n);
apply (plusTerm_eqT2 A plusA n); auto; apply (eqT_sym A n) ];
auto.
cut (eqT a1 (plusTerm (A:=A) plusA (n:=n) a0 a));
[ intros eqTerm5
| apply (eqT_trans A n) with (y := a0); auto; apply (eqT_sym A n);
apply (plusTerm_eqT1 A plusA n) ]; auto.
rewrite <- (pluspf_inv3a_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv2_eqa a (pluspf l2 l1) l0); auto.
rewrite <- (pluspf_inv1_eqa a1 l2 (pluspf l1 l0)); auto.
apply eqpP1; auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n) a (plusTerm (A:=A) plusA (n:=n) a1 a0));
auto.
apply zeroP_plusTermr with (1 := cs); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n) a1 (plusTerm (A:=A) plusA (n:=n) a0 a));
auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n) a1 (plusTerm (A:=A) plusA (n:=n) a a0));
auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n) (plusTerm (A:=A) plusA (n:=n) a a1) a0);
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply plusTerm_assoc with (1 := cs); auto.
apply (eqT_sym A n); auto.
apply (eqT_sym A n); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n) (plusTerm (A:=A) plusA (n:=n) a1 a) a0);
auto.
apply plusTerm_comp_l with (1 := cs); auto.
apply plusTerm_com with (1 := cs); auto.
apply (eqT_sym A n); auto.
apply plusTerm_assoc with (1 := cs); auto.
apply plusTerm_comp_r with (1 := cs); auto.
apply plusTerm_com with (1 := cs); auto.
apply (eqT_sym A n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply zeroP_plusTermr with (1 := cs); auto.
apply H' with (y := (l2, (l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
apply order_pluspf; auto.
apply canonical_pX_eqT with (a := a0); auto.
apply (eqT_sym A n); auto.
apply canonical_pX_eqT with (a := a); auto.
apply (eqT_sym A n); auto.
apply order_pluspf; auto.
apply canonical_pX_eqT with (a := a1); auto.
apply canonical_pX_eqT with (a := a0); auto.
cut (eqT a0 a);
[ intros Ord0
| apply (eqT_trans A n) with (y := a1); auto; apply (eqT_sym A n) ];
auto.
cut (eqT a (plusTerm (A:=A) plusA (n:=n) a1 a0));
[ intros eqTerm1
| apply (eqT_trans A n) with (y := a0); auto; apply (eqT_sym A n); auto;
apply (plusTerm_eqT2 A plusA n) ]; auto.
rewrite <- (pluspf_inv3b_eq a0 a l1 l0); auto.
rewrite <-
(pluspf_inv3b_eq a1 (plusTerm (A:=A) plusA (n:=n) a0 a) l2 (pluspf l1 l0))
; auto.
rewrite <- (pluspf_inv2_eqa a (pluspf l2 l1) l0); auto.
apply eqpP1; auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n) a (plusTerm (A:=A) plusA (n:=n) a1 a0));
auto.
apply zeroP_plusTermr with (1 := cs); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n) (plusTerm (A:=A) plusA (n:=n) a1 a0) a);
auto.
apply plusTerm_com with (1 := cs); auto.
apply plusTerm_assoc with (1 := cs); auto.
apply (eqT_sym A n); auto.
apply H' with (y := (l2, (l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
apply order_pluspf; auto.
apply canonical_pX_eqT with (a := a1); auto.
apply canonical_pX_eqT with (a := a0); auto.
apply (eqT_trans A n) with (y := a); auto; apply (eqT_sym A n); auto.
apply (plusTerm_eqT2 A plusA n); auto.
red in |- *; intros H'3; absurd (zeroP (A:=A) A0 eqA (n:=n) a); auto.
apply
zeroP_comp_eqTerm
with
(1 := cs)
(a := plusTerm (A:=A) plusA (n:=n) a1 (plusTerm (A:=A) plusA (n:=n) a0 a));
auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n) (plusTerm (A:=A) plusA (n:=n) a1 a0) a);
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply plusTerm_assoc with (1 := cs); auto.
apply (eqT_sym A n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply zeroP_plusTerml with (1 := cs); auto.
apply (eqT_sym A n); auto.
rewrite <- (pluspf_inv3b_eq a1 a0 l2 l1); auto.
case (ltT_dec A n ltM ltM_dec a0 a); [ intros H1; case H1; clear H1 | idtac ];
intros H1.
cut (ltT ltM a1 a);
[ intros Ord0
| apply eqT_compat_ltTl with (b := a0); auto; apply (eqT_sym A n) ];
auto.
cut (ltT ltM (plusTerm (A:=A) plusA (n:=n) a1 a0) a);
[ intros Ord1
| apply eqT_compat_ltTl with (b := a1); auto; apply (eqT_sym A n);
apply (plusTerm_eqT1 A plusA n) ]; auto.
rewrite <- (pluspf_inv2_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv2_eq a1 a l2 (pluspf (pX a0 l1) l0)); auto.
rewrite <-
(pluspf_inv2_eq (plusTerm (A:=A) plusA (n:=n) a1 a0) a (pluspf l2 l1) l0)
; auto.
apply eqpP1; auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := pluspf (pluspf (pX a1 l2) (pX a0 l1)) l0);
auto.
rewrite <- (pluspf_inv3b_eq a1 a0 l2 l1); auto.
apply H' with (y := (pX a1 l2, (pX a0 l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
repeat rewrite (fun a b : nat => plus_comm a (S b)); simpl in |- *.
repeat rewrite (plus_comm (size A n l1)); auto.
cut (ltT ltM a a1);
[ intros Ord0
| apply eqT_compat_ltTr with (b := a0); auto; apply (eqT_sym A n) ];
auto.
cut (ltT ltM a (plusTerm (A:=A) plusA (n:=n) a1 a0));
[ intros Ord1
| apply eqT_compat_ltTr with (b := a0); auto; apply (eqT_sym A n);
apply (plusTerm_eqT2 A plusA n) ]; auto.
rewrite <- (pluspf_inv1_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv3b_eq a1 a0 l2 (pluspf l1 (pX a l0))); auto.
rewrite <-
(pluspf_inv1_eq (plusTerm (A:=A) plusA (n:=n) a1 a0) a (pluspf l2 l1) l0)
; auto.
apply eqpP1; auto.
apply H' with (y := (l2, (l1, pX a l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
cut (eqT a1 (plusTerm (A:=A) plusA (n:=n) a0 a));
[ intros eqTerm1
| apply (eqT_trans A n) with (y := a0); auto; apply (eqT_sym A n);
apply (plusTerm_eqT1 A plusA n) ]; auto.
cut (eqT a1 a);
[ intros eqT0
| apply (eqT_trans A n) with (y := a0); auto; apply (eqT_sym A n) ];
auto.
case (zeroP_dec A A0 eqA eqA_dec n (plusTerm (A:=A) plusA (n:=n) a0 a));
intros Z1.
rewrite <- (pluspf_inv3a_eq a0 a l1 l0); auto.
rewrite <- (pluspf_inv1_eqa a1 l2 (pluspf l1 l0)); auto.
cut
(~
zeroP (A:=A) A0 eqA (n:=n)
(plusTerm (A:=A) plusA (n:=n) (plusTerm (A:=A) plusA (n:=n) a1 a0) a));
[ intros Z2 | idtac ].
rewrite <-
(pluspf_inv3b_eq (plusTerm (A:=A) plusA (n:=n) a1 a0) a (pluspf l2 l1) l0)
; auto.
apply eqpP1; auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n) a1 (plusTerm (A:=A) plusA (n:=n) a0 a));
auto.
apply plusTerm_assoc with (1 := cs); auto.
apply (eqT_sym A n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply zeroP_plusTermr with (1 := cs); auto.
apply H' with (y := (l2, (l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
apply (eqT_trans A n) with (y := a1); auto.
apply (plusTerm_eqT1 A plusA n); auto.
red in |- *; intros H'3; absurd (zeroP (A:=A) A0 eqA (n:=n) a1); auto.
apply
zeroP_comp_eqTerm
with
(1 := cs)
(a := plusTerm (A:=A) plusA (n:=n) (plusTerm (A:=A) plusA (n:=n) a1 a0) a);
auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := plusTerm (A:=A) plusA (n:=n) a1 (plusTerm (A:=A) plusA (n:=n) a0 a));
auto.
apply plusTerm_assoc with (1 := cs); auto.
apply (eqT_sym A n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply zeroP_plusTermr with (1 := cs); auto.
apply order_pluspf; auto.
apply canonical_pX_eqT with (a := a0); auto.
apply (eqT_trans A n) with a; auto.
apply (eqT_sym A n); auto.
apply canonical_pX_eqT with (a := a); auto.
apply (eqT_sym A n); auto.
rewrite <- (pluspf_inv3b_eq a0 a l1 l0); auto.
cut
(eqTerm eqA
(plusTerm (A:=A) plusA (n:=n) (plusTerm (A:=A) plusA (n:=n) a1 a0) a)
(plusTerm (A:=A) plusA (n:=n) a1 (plusTerm (A:=A) plusA (n:=n) a0 a)));
[ intros eqTerm0 | idtac ].
case
(zeroP_dec A A0 eqA eqA_dec n
(plusTerm (A:=A) plusA (n:=n) (plusTerm (A:=A) plusA (n:=n) a1 a0) a));
intros Z2.
rewrite <-
(pluspf_inv3a_eq (plusTerm (A:=A) plusA (n:=n) a1 a0) a (pluspf l2 l1) l0)
; auto.
rewrite <-
(pluspf_inv3a_eq a1 (plusTerm (A:=A) plusA (n:=n) a0 a) l2 (pluspf l1 l0))
; auto.
apply H' with (y := (l2, (l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
apply
zeroP_comp_eqTerm
with
(1 := cs)
(a := plusTerm (A:=A) plusA (n:=n) (plusTerm (A:=A) plusA (n:=n) a1 a0) a);
auto.
apply (eqT_trans A n) with (y := a0); auto.
apply (plusTerm_eqT2 A plusA n); auto.
rewrite <-
(pluspf_inv3b_eq (plusTerm (A:=A) plusA (n:=n) a1 a0) a (pluspf l2 l1) l0)
; auto.
rewrite <-
(pluspf_inv3b_eq a1 (plusTerm (A:=A) plusA (n:=n) a0 a) l2 (pluspf l1 l0))
; auto.
apply eqpP1; auto.
apply H' with (y := (l2, (l1, l0))); simpl in |- *; auto.
red in |- *; red in |- *; simpl in |- *; auto with arith.
apply
nzeroP_comp_eqTerm
with
(1 := cs)
(a := plusTerm (A:=A) plusA (n:=n) (plusTerm (A:=A) plusA (n:=n) a1 a0) a);
auto.
apply (eqT_trans A n) with (y := a0); auto.
apply (plusTerm_eqT2 A plusA n); auto.
apply plusTerm_assoc with (1 := cs); auto.
apply (eqT_sym A n); auto.
Qed.
intros p q r H' H'0 H'1.
apply pluspf3_assoc with (l := (p, (q, r))); auto.
Qed.
Hint Resolve pluspf_assoc.
intros p q r H'; generalize r; elim H'; clear r H'.
intros r H' H'0 H'1; rewrite <- pO_pluspf_inv1; auto.
intros ma mb p0 q0 H' H'0 H'1 r H'2 H'3 H'4.
cut (canonical A0 eqA ltM p0);
[ intros C0 | apply canonical_imp_canonical with (a := ma) ];
auto.
cut (canonical A0 eqA ltM q0);
[ intros C1 | apply canonical_imp_canonical with (a := mb) ];
auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) ma);
[ intros nZ0 | apply (canonical_nzeroP _ A0 eqA n ltM) with (p := p0) ];
auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) mb);
[ intros nZ1 | apply (canonical_nzeroP _ A0 eqA n ltM) with (p := q0) ];
auto.
generalize H'4; elim r; clear H'4 r.
intros H'4; rewrite <- pO_pluspf_inv2; rewrite <- pO_pluspf_inv2; auto.
intros a l H'4 H'5.
cut (canonical A0 eqA ltM l);
[ intros C2 | apply canonical_imp_canonical with (a := a) ];
auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros nZ2 | apply (canonical_nzeroP _ A0 eqA n ltM) with (p := l) ];
auto.
change
(eqP A eqA n (pluspf (pX ma p0) (pX a l)) (pluspf (pX mb q0) (pX a l)))
in |- *.
case (ltT_dec A n ltM ltM_dec ma a); [ intros H0; case H0; clear H0 | idtac ];
intros H0.
cut (ltT ltM mb a); [ intros Or0 | idtac ].
rewrite <- (pluspf_inv2_eq ma a p0 l); auto.
rewrite <- (pluspf_inv2_eq mb a q0 l); auto.
apply eqT_compat_ltTl with (b := ma); auto.
apply (eqTerm_imp_eqT A eqA n); auto.
cut (ltT ltM a mb); [ intros Or0 | idtac ]; auto.
rewrite <- (pluspf_inv1_eq ma a p0 l); auto.
rewrite <- (pluspf_inv1_eq mb a q0 l); auto.
apply eqT_compat_ltTr with (b := ma); auto.
apply (eqTerm_imp_eqT A eqA n); auto.
cut
(eqTerm eqA (plusTerm (A:=A) plusA (n:=n) ma a)
(plusTerm (A:=A) plusA (n:=n) mb a)); [ intros eqTerm0 | idtac ];
auto.
case (zeroP_dec A A0 eqA eqA_dec n (plusTerm (A:=A) plusA (n:=n) ma a));
intros Z0.
rewrite <- (pluspf_inv3a_eq ma a p0 l); auto.
rewrite <- (pluspf_inv3a_eq mb a q0 l); auto.
apply (eqT_trans A n) with (y := ma); auto.
apply (eqTerm_imp_eqT A eqA n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply
zeroP_comp_eqTerm with (1 := cs) (a := plusTerm (A:=A) plusA (n:=n) ma a);
auto.
rewrite <- (pluspf_inv3b_eq ma a p0 l); auto.
rewrite <- (pluspf_inv3b_eq mb a q0 l); auto.
apply (eqT_trans A n) with (y := ma); auto.
apply (eqTerm_imp_eqT A eqA n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
red in |- *; intros H'6; apply Z0.
apply
zeroP_comp_eqTerm with (1 := cs) (a := plusTerm (A:=A) plusA (n:=n) mb a);
auto.
apply plusTerm_comp_l with (1 := cs); auto.
change (eqT mb a) in |- *.
apply (eqT_trans A n) with (y := ma); auto.
apply (eqTerm_imp_eqT A eqA n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply plusTerm_comp_l with (1 := cs); auto.
change (eqT mb a) in |- *.
apply (eqT_trans A n) with (y := ma); auto.
apply (eqTerm_imp_eqT A eqA n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
Qed.
intros p q r H'; generalize r; elim H'; clear r H'.
intros r H' H'0 H'1; rewrite <- pO_pluspf_inv2; auto.
intros ma mb p0 q0 H' H'0 H'1 r H'2 H'3 H'4.
cut (canonical A0 eqA ltM p0);
[ intros C0 | apply canonical_imp_canonical with (a := ma) ];
auto.
cut (canonical A0 eqA ltM q0);
[ intros C1 | apply canonical_imp_canonical with (a := mb) ];
auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) ma);
[ intros nZ0 | apply (canonical_nzeroP _ A0 eqA n ltM) with (p := p0) ];
auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) mb);
[ intros nZ1 | apply (canonical_nzeroP _ A0 eqA n ltM) with (p := q0) ];
auto.
generalize H'4; elim r; clear H'4 r.
intros H'4; rewrite <- pO_pluspf_inv1; rewrite <- pO_pluspf_inv1; auto.
intros a l H'4 H'5.
cut (canonical A0 eqA ltM l);
[ intros C2 | apply canonical_imp_canonical with (a := a) ];
auto.
cut (~ zeroP (A:=A) A0 eqA (n:=n) a);
[ intros nZ2 | apply (canonical_nzeroP _ A0 eqA n ltM) with (p := l) ];
auto.
change
(eqP A eqA n (pluspf (pX a l) (pX ma p0)) (pluspf (pX a l) (pX mb q0)))
in |- *.
case (ltT_dec A n ltM ltM_dec ma a); [ intros H0; case H0; clear H0 | idtac ];
intros H0.
cut (ltT ltM mb a); [ intros Or0 | idtac ].
rewrite <- (pluspf_inv1_eq a ma l p0); auto.
rewrite <- (pluspf_inv1_eq a mb l q0); auto.
apply eqT_compat_ltTl with (b := ma); auto.
apply (eqTerm_imp_eqT _ eqA n); auto.
cut (ltT ltM a mb); [ intros Or0 | idtac ]; auto.
rewrite <- (pluspf_inv2_eq a ma l p0); auto.
rewrite <- (pluspf_inv2_eq a mb l q0); auto.
apply eqT_compat_ltTr with (b := ma); auto.
apply (eqTerm_imp_eqT _ eqA n); auto.
cut
(eqTerm eqA (plusTerm (A:=A) plusA (n:=n) a ma)
(plusTerm (A:=A) plusA (n:=n) a mb)); [ intros eqTerm0 | idtac ];
auto.
case (zeroP_dec A A0 eqA eqA_dec n (plusTerm (A:=A) plusA (n:=n) a ma));
intros Z0.
rewrite <- (pluspf_inv3a_eq a ma l p0); auto.
rewrite <- (pluspf_inv3a_eq a mb l q0); auto.
apply (eqT_trans A n) with (y := ma); auto; apply (eqT_sym A n); auto.
apply (eqTerm_imp_eqT _ eqA n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply
zeroP_comp_eqTerm with (1 := cs) (a := plusTerm (A:=A) plusA (n:=n) a ma);
auto.
apply (eqT_sym A n); auto.
rewrite <- (pluspf_inv3b_eq a ma l p0); auto.
rewrite <- (pluspf_inv3b_eq a mb l q0); auto.
apply (eqT_trans A n) with (y := ma); auto.
apply (eqT_sym A n); auto.
apply (eqTerm_imp_eqT _ eqA n); auto.
red in |- *; intros H'6; apply Z0.
apply
zeroP_comp_eqTerm with (1 := cs) (a := plusTerm (A:=A) plusA (n:=n) a mb);
auto.
apply plusTerm_comp_r with (1 := cs); auto.
change (eqT a mb) in |- *.
apply (eqT_trans A n) with (y := ma); auto.
apply (eqT_sym A n); auto.
apply (eqTerm_imp_eqT _ eqA n); auto.
apply (eqT_sym A n); auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply (eqT_sym A n); auto.
apply plusTerm_comp_r with (1 := cs); auto.
apply (eqT_sym A n); auto.
change (eqT a mb) in |- *.
apply (eqT_trans A n) with (y := ma); auto.
apply (eqT_sym A n); auto.
apply (eqTerm_imp_eqT _ eqA n); auto.
Qed.

Theorem eqp_pluspf_com :
forall p q r t,
canonical A0 eqA ltM p ->
canonical A0 eqA ltM q ->
canonical A0 eqA ltM r ->
canonical A0 eqA ltM t ->
eqP A eqA n p r -> eqP A eqA n q t -> eqP A eqA n (pluspf p q) (pluspf r t).