Section Porder.
Load "hCoefStructure".
Load "hOrderStructure".
Load "hTerm".
Set Implicit Arguments.
Unset Strict Implicit.
Definition ltT (a b : Term A n) : Prop := ltM (T2M a) (T2M b).
Hint Unfold ltT.
Set Strict Implicit.
Unset Implicit Arguments.
unfold transitive, ltT in |- *; auto.
intros x y z H' H'0; apply (ltM_trans _ _ _ _ os) with (y := T2M y); auto.
Qed.
unfold eqT in |- *; unfold ltT in |- *; intros a b c H; rewrite H; auto.
Qed.
unfold eqT in |- *; unfold ltT in |- *; intros a b c H; rewrite H; auto.
Qed.
intros x y; unfold eqT in |- *; simpl in |- *; auto.
apply eqmon_dec.
Qed.
intros x y; exact (ltM_dec (T2M x) (T2M y)).
Qed.
unfold eqT, ltT in |- *; simpl in |- *; intros x y H; rewrite H; auto.
apply ltM_nonrefl with (1 := os).
Qed.
unfold eqT, ltT, not in |- *; simpl in |- *; intros x y H Q;
absurd (ltM (T2M x) (T2M y)); auto; rewrite Q; auto.
apply ltM_nonrefl with (1 := os).
Qed.
intros x; unfold ltT in |- *; apply ltM_nonrefl with (1 := os).
Qed.
Hint Resolve ltT_not_eqT eqT_not_ltT ltT_not_refl.
intros x y H'; red in |- *; intros H'0; absurd (ltT x x); auto.
apply ltT_trans with (y := y); auto.
Qed.
Hint Resolve ltT_not_ltT.
unfold eqT, ltT in |- *; intros a b c d R1 R2; rewrite R1; rewrite R2; auto.
Qed.
Let eqT_refl := eqT_refl A n.
intros a b c H' H'0; apply ltT_eqT with (a := a) (c := c); auto.
Qed.
intros a b c H' H'0; apply ltT_eqT with (a := c) (c := a); auto.
Qed.
intros a b c; case a; case b; case c; unfold ltT in |- *; simpl in |- *; auto.
intros a0 m a1 m0 a2 m1 H.
apply ltM_plusl with (1 := os); auto.
Qed.
intros a b c; case a; case b; case c; unfold ltT in |- *; simpl in |- *; auto.
intros; apply ltM_plusr with (1 := os); auto.
Qed.
intros a; case a; unfold ltT in |- *; simpl in |- *; auto.
intros a0 m; unfold M1 in |- *; apply (M1_min _ _ _ _ os); auto.
Qed.
intros a b c; case a; case b; case c; unfold ltT in |- *; simpl in |- *; auto.
Qed.
intros a b; case a; case b; unfold ltT in |- *; simpl in |- *; auto.
Qed.
Set Implicit Arguments.
Unset Strict Implicit.
Definition pX := cons (A:=Term A n).
Set Strict Implicit.
Unset Implicit Arguments.
Definition pO := nil (A:=Term A n).
Let consA := cons (A:=mon n).
Let nilA := nil (A:=mon n).
Let listA := list (mon n).
Hint Unfold consA nilA listA.
Fixpoint fP (a : list (Term A n)) : listA :=
match a with
| nil => nilA
| b :: p => consA (T2M b) (fP p)
end.
intros p; elim p; simpl in |- *; auto.
intros a l H' q.
rewrite (H' q); auto.
Qed.
Let DescA := Desc (mon n) ltM.
Set Implicit Arguments.
Unset Strict Implicit.
Definition olist (p : list (Term A n)) := DescA (fP p).
Hint Resolve d_nil d_one.
Hint Unfold olist DescA.
Set Strict Implicit.
Unset Implicit Arguments.
unfold olist, ltT in |- *; simpl in |- *; auto.
intros a b H'.
generalize (d_conc _ ltM (T2M b) (T2M a) nilA); simpl in |- *; auto.
Qed.
Hint Resolve olistOne.
unfold olist, ltT in |- *; simpl in |- *; auto.
red in |- *; unfold nilA in |- *; apply d_nil.
Qed.
intros x y z t p; case p; simpl in |- *; auto.
intros H'; injection H'; simpl in |- *; auto.
intros a l; case l; simpl in |- *; auto.
intros H'; discriminate H'.
intros a0 l0; case l0; simpl in |- *; auto.
intros H'; discriminate H'.
intros a1 l1 H'; discriminate H'.
Qed.
unfold ltT, olist in |- *; simpl in |- *; auto.
intros l a b.
rewrite (fP_app l (pX a (pX b pO))); simpl in |- *; auto.
intros H'.
elim (dist_Desc_concat _ ltM (fP l) (consA (T2M a) (consA (T2M b) nilA)));
[ intros H'5 H'6; try exact H'6 | idtac ]; auto.
simple inversion H'6.
discriminate H.
discriminate H.
elim (app2_inv y x (T2M a) (T2M b) l0); [ intros H'8 H'9 | idtac ]; auto.
rewrite H'9; rewrite H'8; auto.
Qed.

Theorem olist_cons :
forall (l : list _) a b, ltT b a -> olist (pX b l) -> olist (pX a (pX b l)).