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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
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(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(** Nota : this file is OBSOLETE, and left only for compatibility. Please consider using [Nat.div2] directly, and results about it (see file PeanoNat). *)
Local Open Scope nat_scope.
Implicit Type n : nat.
(** Here we define [n/2] and prove some of its properties *)
Notation div2 := Nat.div2 (only parsing).
(** Since [div2] is recursively defined on [0], [1] and [(S (S n))], it is useful to prove the corresponding induction principle *)
(** [0 <n  =>  n/2 < n] *)
Hint Resolve even_div2 div2_even odd_div2 div2_odd: arith.
(** Properties related to the double ([2n]) *)
Notation double := Nat.double (only parsing).
Hint Unfold double Nat.double: arith.
Hint Resolve double_S: arith.
(** Specializations *)
Proof proj1 (proj1 (even_odd_double n)).
Proof proj2 (proj1 (even_odd_double n)).
Proof proj1 (proj2 (even_odd_double n)).

Lemma double_odd n : n = S (double (div2 n)) -> odd n.