(************************************************************************)
(*         *   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)           *)
(*   \VV/  **************************************************************)
(*    //   *    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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Local Open Scope Z_scope.
(** * Boolean operations from decidability of order *)
(** The decidability of equality and order relations over type [Z] gives some boolean functions with the adequate specification. *)
Definition Z_lt_ge_bool (x y:Z) := bool_of_sumbool (Z_lt_ge_dec x y).
Definition Z_ge_lt_bool (x y:Z) := bool_of_sumbool (Z_ge_lt_dec x y).
Definition Z_le_gt_bool (x y:Z) := bool_of_sumbool (Z_le_gt_dec x y).
Definition Z_gt_le_bool (x y:Z) := bool_of_sumbool (Z_gt_le_dec x y).
Definition Z_eq_bool (x y:Z) := bool_of_sumbool (Z.eq_dec x y).
Definition Z_noteq_bool (x y:Z) := bool_of_sumbool (Z_noteq_dec x y).
Definition Zeven_odd_bool (x:Z) := bool_of_sumbool (Zeven_odd_dec x).
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(** * Boolean comparisons of binary integers *)
Notation Zle_bool := Z.leb (only parsing).
Notation Zge_bool := Z.geb (only parsing).
Notation Zlt_bool := Z.ltb (only parsing).
Notation Zgt_bool := Z.gtb (only parsing).
(** We now provide a direct [Z.eqb] that doesn't refer to [Z.compare]. The old [Zeq_bool] is kept for compatibility. *)
Definition Zeq_bool (x y:Z) :=
match x ?= y with
| Eq => true
| _ => false
end.
Definition Zneq_bool (x y:Z) :=
match x ?= y with
| Eq => false
| _ => true
end.
(** Properties in term of [if ... then ... else ...] *)

Lemma Zge_cases n m : if n >=? m then n >= m else n < m.