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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        *)
+(************************************************************************)
+
+(*i $Id: Le.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*)
+
+(** Order on natural numbers *)
+V7only [Import nat_scope.].
+Open Local Scope nat_scope.
+
+Implicit Variables Type m,n,p:nat.
+
+(** Reflexivity *)
+
+Theorem le_refl : (n:nat)(le n n).
+Proof.
+Exact le_n.
+Qed.
+
+(** Transitivity *)
+
+Theorem le_trans : (n,m,p:nat)(le n m)->(le m p)->(le n p).
+Proof.
+  NewInduction 2; Auto.
+Qed.
+Hints Resolve le_trans : arith v62.
+
+(** Order, successor and predecessor *)
+
+Theorem le_n_S : (n,m:nat)(le n m)->(le (S n) (S m)).
+Proof.
+  NewInduction 1; Auto.
+Qed.
+
+Theorem le_n_Sn : (n:nat)(le n (S n)).
+Proof.
+  Auto.
+Qed.
+
+Theorem le_O_n : (n:nat)(le O n).
+Proof.
+  NewInduction n ; Auto.
+Qed.
+
+Hints Resolve le_n_S le_n_Sn le_O_n le_n_S : arith v62.
+
+Theorem le_pred_n : (n:nat)(le (pred n) n).
+Proof.
+NewInduction n ; Auto with arith.
+Qed.
+Hints Resolve le_pred_n : arith v62.
+
+Theorem le_trans_S : (n,m:nat)(le (S n) m)->(le n m).
+Proof.
+Intros n m H ; Apply le_trans with (S n); Auto with arith.
+Qed.
+Hints Immediate le_trans_S : arith v62.
+
+Theorem le_S_n : (n,m:nat)(le (S n) (S m))->(le n m).
+Proof.
+Intros n m H ; Change (le (pred (S n)) (pred (S m))).
+Elim H ; Simpl ; Auto with arith.
+Qed.
+Hints Immediate le_S_n : arith v62.
+
+Theorem le_pred : (n,m:nat)(le n m)->(le (pred n) (pred m)).
+Proof.
+NewInduction n as [|n IHn]. Simpl. Auto with arith.
+NewDestruct m as [|m]. Simpl. Intro H. Inversion H.
+Simpl. Auto with arith.
+Qed.
+
+(** Comparison to 0 *)
+
+Theorem le_Sn_O : (n:nat)~(le (S n) O).
+Proof.
+Red ; Intros n H.
+Change (IsSucc O) ; Elim H ; Simpl ; Auto with arith.
+Qed.
+Hints Resolve le_Sn_O : arith v62.
+
+Theorem le_n_O_eq : (n:nat)(le n O)->(O=n).
+Proof.
+NewInduction n; Auto with arith.
+Intro; Contradiction le_Sn_O with n.
+Qed.
+Hints Immediate le_n_O_eq : arith v62.
+
+(** Negative properties *)
+
+Theorem le_Sn_n : (n:nat)~(le (S n) n).
+Proof.
+NewInduction n; Auto with arith.
+Qed.
+Hints Resolve le_Sn_n : arith v62.
+
+(** Antisymmetry *)
+
+Theorem le_antisym : (n,m:nat)(le n m)->(le m n)->(n=m).
+Proof.
+Intros n m h ; NewDestruct h as [|m0]; Auto with arith.
+Intros H1.
+Absurd (le (S m0) m0) ; Auto with arith.
+Apply le_trans with n ; Auto with arith.
+Qed.
+Hints Immediate le_antisym : arith v62.
+
+(** A different elimination principle for the order on natural numbers *)
+
+Lemma le_elim_rel : (P:nat->nat->Prop)
+     ((p:nat)(P O p))->
+     ((p,q:nat)(le p q)->(P p q)->(P (S p) (S q)))->
+     (n,m:nat)(le n m)->(P n m).
+Proof.
+NewInduction n; Auto with arith.
+Intros m Le.
+Elim Le; Auto with arith.
+Qed.