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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: Max.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*)
+
+Require Arith.
+
+V7only [Import nat_scope.].
+Open Local Scope nat_scope.
+
+Implicit Variables Type m,n:nat.
+
+(** maximum of two natural numbers *)
+
+Fixpoint max [n:nat] : nat -> nat :=  
+[m:nat]Cases n m of
+          O      _     => m
+       | (S n')  O     => n
+       | (S n') (S m') => (S (max n' m'))
+       end.
+
+(** Simplifications of [max] *)
+
+Lemma max_SS : (n,m:nat)((S (max n m))=(max (S n) (S m))).
+Proof.
+Auto with arith.
+Qed.
+
+Lemma max_sym : (n,m:nat)(max n m)=(max m n).
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Qed.
+
+(** [max] and [le] *)
+
+Lemma max_l : (n,m:nat)(le m n)->(max n m)=n.
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Qed.
+
+Lemma max_r : (n,m:nat)(le n m)->(max n m)=m.
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Qed.
+
+Lemma le_max_l : (n,m:nat)(le n (max n m)).
+Proof.
+NewInduction n; Intros; Simpl; Auto with arith.
+Elim m; Intros; Simpl; Auto with arith.
+Qed.
+
+Lemma le_max_r : (n,m:nat)(le m (max n m)).
+Proof.
+NewInduction n; Simpl; Auto with arith.
+NewInduction m; Simpl; Auto with arith.
+Qed.
+Hints Resolve max_r max_l le_max_l le_max_r: arith v62.
+
+
+(** [max n m] is equal to [n] or [m] *)
+
+Lemma max_dec : (n,m:nat){(max n m)=n}+{(max n m)=m}.
+Proof.
+NewInduction n;NewInduction m;Simpl;Auto with arith.
+Elim (IHn m);Intro H;Elim H;Auto.
+Qed.
+
+Lemma max_case : (n,m:nat)(P:nat->Set)(P n)->(P m)->(P (max n m)).
+Proof.
+NewInduction n; Simpl; Auto with arith.
+NewInduction m; Intros; Simpl; Auto with arith.
+Pattern (max n m); Apply IHn ; Auto with arith.
+Qed.
+
+Lemma max_case2 : (n,m:nat)(P:nat->Prop)(P n)->(P m)->(P (max n m)).
+Proof.
+NewInduction n; Simpl; Auto with arith.
+NewInduction m; Intros; Simpl; Auto with arith.
+Pattern (max n m); Apply IHn ; Auto with arith.
+Qed.
+
+