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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: Compare_dec.v,v 1.13.2.1 2004/07/16 19:31:00 herbelin Exp $ i*)
+
+Require Import Le.
+Require Import Lt.
+Require Import Gt.
+Require Import Decidable.
+
+Open Local Scope nat_scope.
+
+Implicit Types m n x y : nat.
+
+Definition zerop : forall n, {n = 0} + {0 < n}.
+destruct n; auto with arith.
+Defined.
+
+Definition lt_eq_lt_dec : forall n m, {n < m} + {n = m} + {m < n}.
+Proof.
+induction n; simple destruct m; auto with arith.
+intros m0; elim (IHn m0); auto with arith.
+induction 1; auto with arith.
+Defined.
+
+Lemma gt_eq_gt_dec : forall n m, {m > n} + {n = m} + {n > m}.
+Proof lt_eq_lt_dec.
+
+Lemma le_lt_dec : forall n m, {n <= m} + {m < n}.
+Proof.
+induction n.
+auto with arith.
+induction m.
+auto with arith.
+elim (IHn m); auto with arith.
+Defined.
+
+Definition le_le_S_dec : forall n m, {n <= m} + {S m <= n}.
+Proof.
+exact le_lt_dec.
+Defined.
+
+Definition le_ge_dec : forall n m, {n <= m} + {n >= m}.
+Proof.
+intros; elim (le_lt_dec n m); auto with arith.
+Defined.
+
+Definition le_gt_dec : forall n m, {n <= m} + {n > m}.
+Proof.
+exact le_lt_dec.
+Defined.
+
+Definition le_lt_eq_dec : forall n m, n <= m -> {n < m} + {n = m}.
+Proof.
+intros; elim (lt_eq_lt_dec n m); auto with arith.
+intros; absurd (m < n); auto with arith.
+Defined.
+
+(** Proofs of decidability *)
+
+Theorem dec_le : forall n m, decidable (n <= m).
+intros x y; unfold decidable in |- *; elim (le_gt_dec x y);
+ [ auto with arith | intro; right; apply gt_not_le; assumption ].
+Qed.
+
+Theorem dec_lt : forall n m, decidable (n < m).
+intros x y; unfold lt in |- *; apply dec_le.
+Qed.
+
+Theorem dec_gt : forall n m, decidable (n > m).
+intros x y; unfold gt in |- *; apply dec_lt.
+Qed.
+
+Theorem dec_ge : forall n m, decidable (n >= m).
+intros x y; unfold ge in |- *; apply dec_le.
+Qed.
+
+Theorem not_eq : forall n m, n <> m -> n < m \/ m < n.
+intros x y H; elim (lt_eq_lt_dec x y);
+ [ intros H1; elim H1;
+    [ auto with arith | intros H2; absurd (x = y); assumption ]
+ | auto with arith ].
+Qed.
+
+
+Theorem not_le : forall n m, ~ n <= m -> n > m.
+intros x y H; elim (le_gt_dec x y);
+ [ intros H1; absurd (x <= y); assumption | trivial with arith ].
+Qed.
+
+Theorem not_gt : forall n m, ~ n > m -> n <= m.
+intros x y H; elim (le_gt_dec x y);
+ [ trivial with arith | intros H1; absurd (x > y); assumption ].
+Qed.
+
+Theorem not_ge : forall n m, ~ n >= m -> n < m.
+intros x y H; exact (not_le y x H).
+Qed.
+
+Theorem not_lt : forall n m, ~ n < m -> n >= m.
+intros x y H; exact (not_gt y x H). 
+Qed.