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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.