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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: Div.v,v 1.1.2.1 2004/07/16 19:31:23 herbelin Exp $ i*)
+
+(** Euclidean division *)
+
+V7only [Import nat_scope.].
+Open Local Scope nat_scope.
+
+Require Le.
+Require Euclid_def.
+Require Compare_dec.
+
+Implicit Variables Type n,a,b,q,r:nat.
+
+Fixpoint inf_dec [n:nat] : nat->bool :=
+      [m:nat] Cases n m of
+                  O     _     => true
+              | (S n')  O     => false
+              | (S n') (S m') => (inf_dec n' m')
+             end.
+
+Theorem div1 : (b:nat)(gt b O)->(a:nat)(diveucl a b).
+Realizer Fix div1 {div1/2: nat->nat->diveucl :=
+  [b,a]Cases a of
+         O     => (O,O)
+       | (S n) =>
+           let (q,r) = (div1 b n) in
+             if (le_gt_dec b (S r)) then ((S q),O)
+	     else (q,(S r))
+       end}.
+Program_all.
+Rewrite e.
+Replace b with (S r).
+Simpl.
+Elim plus_n_O; Auto with arith.
+Apply le_antisym; Auto with arith.
+Elim plus_n_Sm; Auto with arith.
+Qed.
+
+Theorem div2 : (b:nat)(gt b O)->(a:nat)(diveucl a b).
+Realizer Fix div1 {div1/2: nat->nat->diveucl :=
+  [b,a]Cases a of
+         O     => (O,O)
+       | (S n) =>
+           let (q,r) = (div1 b n) in
+             if (inf_dec b (S r)) :: :: { {(le b (S r))}+{(gt b (S r))} }
+             then ((S q),O)
+	     else (q,(S r))
+       end}.
+Program_all.
+Rewrite e.
+Replace b with (S r).
+Simpl.
+Elim plus_n_O; Auto with arith.
+Apply le_antisym; Auto with arith.
+Elim plus_n_Sm; Auto with arith.
+Qed.