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-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-
-Require Export Axioms.
-Require Export AddProps.
-Require Export NeqProps.
-
-(** This file contains basic properties of the less than relation *)
-
-
-Lemma lt_anti_sym : (x,y:N)x<y->~(y<x).
-Red; Intros x y lt1 lt2; Apply (lt_anti_refl x); EAuto with num.
-Qed.
-Hints Resolve lt_anti_refl : num.
-
-Lemma eq_not_lt : (x,y:N)(x=y)->~(x<y).
-Red; Intros x y eq1 lt1; Apply (lt_anti_refl x); EAuto with num.
-Qed.
-Hints Resolve eq_not_lt : num.
-
-Lemma lt_0_1 : (zero<one).
-EAuto with num.
-Qed.
-Hints Resolve lt_0_1 : num.
-
-
-Lemma eq_lt_x_Sy : (x,y:N)(x=y)->(x<(S y)).
-EAuto with num.
-Qed.
-Hints Resolve eq_lt_x_Sy : num.
-
-Lemma lt_lt_x_Sy : (x,y:N)(x<y)->(x<(S y)).
-EAuto with num.
-Qed.
-Hints Immediate lt_lt_x_Sy : num.
-
-Lemma lt_Sx_y_lt : (x,y:N)((S x)<y)->(x<y).
-EAuto with num.
-Qed.
-Hints Immediate lt_Sx_y_lt : num.
-
-(** Relating [<] and [=] *)
-
-Lemma lt_neq : (x,y:N)(x<y)->(x<>y).
-Red; Intros x y lt1 eq1; Apply (lt_anti_refl x); EAuto with num.
-Qed.
-Hints Immediate lt_neq : num.
-
-Lemma lt_neq_sym : (x,y:N)(y<x)->(x<>y).
-Intros x y lt1 ; Apply neq_sym; Auto with num.
-Qed.
-Hints Immediate lt_neq_sym : num.
-
-(** Application to inequalities properties *)
-
-Lemma neq_x_Sx : (x:N)x<>(S x).
-Auto with num.
-Qed.
-Hints Resolve neq_x_Sx : num.
-
-Lemma neq_0_1 : zero<>one.
-Auto with num.
-Qed.
-Hints Resolve neq_0_1 : num.
-
-(** Relating [<] and [+] *)
-
-Lemma lt_add_compat_r : (x,y,z:N)(x<y)->((z+x)<(z+y)).
-Intros x y z H; Apply lt_eq_compat with (x+z) (y+z); Auto with num.
-Qed.
-Hints Resolve lt_add_compat_r : num.
-
-Lemma lt_add_compat : (x1,x2,y1,y2:N)(x1<x2)->(y1<y2)->((x1+y1)<(x2+y2)).
-Intros; Apply lt_trans with (x1+y2); Auto with num.
-Qed.
-Hints Immediate lt_add_compat : num.
-