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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 1.14.2.1 2004/07/16 19:31:00 herbelin Exp $ i*)
+
+(** Order on natural numbers *)
+Open Local Scope nat_scope.
+
+Implicit Types m n p : nat.
+
+(** Reflexivity *)
+
+Theorem le_refl : forall n, n <= n.
+Proof.
+exact le_n.
+Qed.
+
+(** Transitivity *)
+
+Theorem le_trans : forall n m p, n <= m -> m <= p -> n <= p.
+Proof.
+  induction 2; auto.
+Qed.
+Hint Resolve le_trans: arith v62.
+
+(** Order, successor and predecessor *)
+
+Theorem le_n_S : forall n m, n <= m -> S n <= S m.
+Proof.
+  induction 1; auto.
+Qed.
+
+Theorem le_n_Sn : forall n, n <= S n.
+Proof.
+  auto.
+Qed.
+
+Theorem le_O_n : forall n, 0 <= n.
+Proof.
+  induction n; auto.
+Qed.
+
+Hint Resolve le_n_S le_n_Sn le_O_n le_n_S: arith v62.
+
+Theorem le_pred_n : forall n, pred n <= n.
+Proof.
+induction n; auto with arith.
+Qed.
+Hint Resolve le_pred_n: arith v62.
+
+Theorem le_Sn_le : forall n m, S n <= m -> n <= m.
+Proof.
+intros n m H; apply le_trans with (S n); auto with arith.
+Qed.
+Hint Immediate le_Sn_le: arith v62.
+
+Theorem le_S_n : forall n m, S n <= S m -> n <= m.
+Proof.
+intros n m H; change (pred (S n) <= pred (S m)) in |- *.
+elim H; simpl in |- *; auto with arith.
+Qed.
+Hint Immediate le_S_n: arith v62.
+
+Theorem le_pred : forall n m, n <= m -> pred n <= pred m.
+Proof.
+induction n as [| n IHn]. simpl in |- *. auto with arith.
+destruct m as [| m]. simpl in |- *. intro H. inversion H.
+simpl in |- *. auto with arith.
+Qed.
+
+(** Comparison to 0 *)
+
+Theorem le_Sn_O : forall n, ~ S n <= 0.
+Proof.
+red in |- *; intros n H.
+change (IsSucc 0) in |- *; elim H; simpl in |- *; auto with arith.
+Qed.
+Hint Resolve le_Sn_O: arith v62.
+
+Theorem le_n_O_eq : forall n, n <= 0 -> 0 = n.
+Proof.
+induction n; auto with arith.
+intro; contradiction le_Sn_O with n.
+Qed.
+Hint Immediate le_n_O_eq: arith v62.
+
+(** Negative properties *)
+
+Theorem le_Sn_n : forall n, ~ S n <= n.
+Proof.
+induction n; auto with arith.
+Qed.
+Hint Resolve le_Sn_n: arith v62.
+
+(** Antisymmetry *)
+
+Theorem le_antisym : forall n m, n <= m -> m <= n -> n = m.
+Proof.
+intros n m h; destruct h as [| m0 H]; auto with arith.
+intros H1.
+absurd (S m0 <= m0); auto with arith.
+apply le_trans with n; auto with arith.
+Qed.
+Hint Immediate le_antisym: arith v62.
+
+(** A different elimination principle for the order on natural numbers *)
+
+Lemma le_elim_rel :
+ forall P:nat -> nat -> Prop,
+   (forall p, P 0 p) ->
+   (forall p (q:nat), p <= q -> P p q -> P (S p) (S q)) ->
+   forall n m, n <= m -> P n m.
+Proof.
+induction n; auto with arith.
+intros m Le.
+elim Le; auto with arith.
+Qed.
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