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.

Theorem minusTerm_ltT_l :
forall a b c, eqT a b -> ltT a c -> ltT (minusTerm minusA a b) c.