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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 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*)
+
+Require Le.
+Require Lt.
+Require Plus.
+V7only [Import nat_scope.].
+Open Local Scope nat_scope.
+
+Implicit Variables Type m,n,p:nat.
+
+(** Order and successor *)
+
+Theorem gt_Sn_O : (n:nat)(gt (S n) O).
+Proof.
+  Auto with arith.
+Qed.
+Hints Resolve gt_Sn_O : arith v62.
+
+Theorem gt_Sn_n : (n:nat)(gt (S n) n).
+Proof.
+  Auto with arith.
+Qed.
+Hints Resolve gt_Sn_n : arith v62.
+
+Theorem gt_n_S : (n,m:nat)(gt n m)->(gt (S n) (S m)).
+Proof.
+  Auto with arith.
+Qed.
+Hints Resolve gt_n_S : arith v62.
+
+Lemma gt_S_n : (n,p:nat)(gt (S p) (S n))->(gt p n).
+Proof.
+  Auto with arith.
+Qed.
+Hints Immediate gt_S_n : arith v62.
+
+Theorem gt_S : (n,m:nat)(gt (S n) m)->((gt n m)\/(m=n)).
+Proof.
+  Intros n m H; Unfold gt; Apply le_lt_or_eq; Auto with arith.
+Qed.
+
+Lemma gt_pred : (n,p:nat)(gt p (S n))->(gt (pred p) n).
+Proof.
+  Auto with arith.
+Qed.
+Hints Immediate gt_pred : arith v62.
+
+(** Irreflexivity *)
+
+Lemma gt_antirefl : (n:nat)~(gt n n).
+Proof lt_n_n.
+Hints Resolve gt_antirefl : arith v62.
+
+(** Asymmetry *)
+
+Lemma gt_not_sym : (n,m:nat)(gt n m) -> ~(gt m n).
+Proof [n,m:nat](lt_not_sym m n).
+
+Hints Resolve gt_not_sym : arith v62.
+
+(** Relating strict and large orders *)
+
+Lemma le_not_gt : (n,m:nat)(le n m) -> ~(gt n m).
+Proof le_not_lt.
+Hints Resolve le_not_gt : arith v62.
+
+Lemma gt_not_le : (n,m:nat)(gt n m) -> ~(le n m).
+Proof.
+Auto with arith.
+Qed.
+
+Hints Resolve gt_not_le : arith v62.
+
+Theorem le_S_gt : (n,m:nat)(le (S n) m)->(gt m n).
+Proof.
+  Auto with arith.
+Qed.
+Hints Immediate le_S_gt : arith v62.
+
+Lemma gt_S_le : (n,p:nat)(gt (S p) n)->(le n p).
+Proof.
+  Intros n p; Exact (lt_n_Sm_le n p).
+Qed.
+Hints Immediate gt_S_le : arith v62.
+
+Lemma gt_le_S : (n,p:nat)(gt p n)->(le (S n) p).
+Proof.
+  Auto with arith.
+Qed.
+Hints Resolve gt_le_S : arith v62.
+
+Lemma le_gt_S : (n,p:nat)(le n p)->(gt (S p) n).
+Proof.
+  Auto with arith.
+Qed.
+Hints Resolve le_gt_S : arith v62.
+
+(** Transitivity *)
+
+Theorem le_gt_trans : (n,m,p:nat)(le m n)->(gt m p)->(gt n p).
+Proof.
+  Red; Intros; Apply lt_le_trans with m; Auto with arith.
+Qed.
+
+Theorem gt_le_trans : (n,m,p:nat)(gt n m)->(le p m)->(gt n p).
+Proof.
+  Red; Intros; Apply le_lt_trans with m; Auto with arith.
+Qed.
+
+Lemma gt_trans : (n,m,p:nat)(gt n m)->(gt m p)->(gt n p).
+Proof.
+  Red; Intros n m p H1 H2.
+  Apply lt_trans with m; Auto with arith.
+Qed.
+
+Theorem gt_trans_S : (n,m,p:nat)(gt (S n) m)->(gt m p)->(gt n p).
+Proof.
+  Red; Intros; Apply lt_le_trans with m; Auto with arith.
+Qed.
+
+Hints Resolve gt_trans_S le_gt_trans gt_le_trans : arith v62.
+
+(** Comparison to 0 *)
+
+Theorem gt_O_eq : (n:nat)((gt n O)\/(O=n)).
+Proof.
+  Intro n ; Apply gt_S ; Auto with arith.
+Qed.
+
+(** Simplification and compatibility *)
+
+Lemma simpl_gt_plus_l : (n,m,p:nat)(gt (plus p n) (plus p m))->(gt n m).
+Proof.
+  Red; Intros n m p H; Apply simpl_lt_plus_l with p; Auto with arith.
+Qed.
+
+Lemma gt_reg_l : (n,m,p:nat)(gt n m)->(gt (plus p n) (plus p m)).
+Proof.
+  Auto with arith.
+Qed.
+Hints Resolve gt_reg_l : arith v62.