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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: Gt.v,v 1.8.2.1 2004/07/16 19:31:00 herbelin Exp $ i*)
+
+Require Import Le.
+Require Import Lt.
+Require Import Plus.
+Open Local Scope nat_scope.
+
+Implicit Types m n p : nat.
+
+(** Order and successor *)
+
+Theorem gt_Sn_O : forall n, S n > 0.
+Proof.
+  auto with arith.
+Qed.
+Hint Resolve gt_Sn_O: arith v62.
+
+Theorem gt_Sn_n : forall n, S n > n.
+Proof.
+  auto with arith.
+Qed.
+Hint Resolve gt_Sn_n: arith v62.
+
+Theorem gt_n_S : forall n m, n > m -> S n > S m.
+Proof.
+  auto with arith.
+Qed.
+Hint Resolve gt_n_S: arith v62.
+
+Lemma gt_S_n : forall n m, S m > S n -> m > n.
+Proof.
+  auto with arith.
+Qed.
+Hint Immediate gt_S_n: arith v62.
+
+Theorem gt_S : forall n m, S n > m -> n > m \/ m = n.
+Proof.
+  intros n m H; unfold gt in |- *; apply le_lt_or_eq; auto with arith.
+Qed.
+
+Lemma gt_pred : forall n m, m > S n -> pred m > n.
+Proof.
+  auto with arith.
+Qed.
+Hint Immediate gt_pred: arith v62.
+
+(** Irreflexivity *)
+
+Lemma gt_irrefl : forall n, ~ n > n.
+Proof lt_irrefl.
+Hint Resolve gt_irrefl: arith v62.
+
+(** Asymmetry *)
+
+Lemma gt_asym : forall n m, n > m -> ~ m > n.
+Proof fun n m => lt_asym m n.
+
+Hint Resolve gt_asym: arith v62.
+
+(** Relating strict and large orders *)
+
+Lemma le_not_gt : forall n m, n <= m -> ~ n > m.
+Proof le_not_lt.
+Hint Resolve le_not_gt: arith v62.
+
+Lemma gt_not_le : forall n m, n > m -> ~ n <= m.
+Proof.
+auto with arith.
+Qed.
+
+Hint Resolve gt_not_le: arith v62.
+
+Theorem le_S_gt : forall n m, S n <= m -> m > n.
+Proof.
+  auto with arith.
+Qed.
+Hint Immediate le_S_gt: arith v62.
+
+Lemma gt_S_le : forall n m, S m > n -> n <= m.
+Proof.
+  intros n p; exact (lt_n_Sm_le n p).
+Qed.
+Hint Immediate gt_S_le: arith v62.
+
+Lemma gt_le_S : forall n m, m > n -> S n <= m.
+Proof.
+  auto with arith.
+Qed.
+Hint Resolve gt_le_S: arith v62.
+
+Lemma le_gt_S : forall n m, n <= m -> S m > n.
+Proof.
+  auto with arith.
+Qed.
+Hint Resolve le_gt_S: arith v62.
+
+(** Transitivity *)
+
+Theorem le_gt_trans : forall n m p, m <= n -> m > p -> n > p.
+Proof.
+  red in |- *; intros; apply lt_le_trans with m; auto with arith.
+Qed.
+
+Theorem gt_le_trans : forall n m p, n > m -> p <= m -> n > p.
+Proof.
+  red in |- *; intros; apply le_lt_trans with m; auto with arith.
+Qed.
+
+Lemma gt_trans : forall n m p, n > m -> m > p -> n > p.
+Proof.
+  red in |- *; intros n m p H1 H2.
+  apply lt_trans with m; auto with arith.
+Qed.
+
+Theorem gt_trans_S : forall n m p, S n > m -> m > p -> n > p.
+Proof.
+  red in |- *; intros; apply lt_le_trans with m; auto with arith.
+Qed.
+
+Hint Resolve gt_trans_S le_gt_trans gt_le_trans: arith v62.
+
+(** Comparison to 0 *)
+
+Theorem gt_O_eq : forall n, n > 0 \/ 0 = n.
+Proof.
+  intro n; apply gt_S; auto with arith.
+Qed.
+
+(** Simplification and compatibility *)
+
+Lemma plus_gt_reg_l : forall n m p, p + n > p + m -> n > m.
+Proof.
+  red in |- *; intros n m p H; apply plus_lt_reg_l with p; auto with arith.
+Qed.
+
+Lemma plus_gt_compat_l : forall n m p, n > m -> p + n > p + m.
+Proof.
+  auto with arith.
+Qed.
+Hint Resolve plus_gt_compat_l: arith v62.
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