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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Export Compare_dec.
Require Export Peano_dec.
Require Import Sumbool.
Local Open Scope nat_scope.
Implicit Types m n x y : nat.
(** The decidability of equality and order relations over
type [nat] give some boolean functions with the adequate specification. *)
Definition notzerop n := sumbool_not _ _ (zerop n).
Definition lt_ge_dec : forall x y, {x < y} + {x >= y} :=
fun n m => sumbool_not _ _ (le_lt_dec m n).
Definition nat_lt_ge_bool x y := bool_of_sumbool (lt_ge_dec x y).
Definition nat_ge_lt_bool x y :=
bool_of_sumbool (sumbool_not _ _ (lt_ge_dec x y)).
Definition nat_le_gt_bool x y := bool_of_sumbool (le_gt_dec x y).
Definition nat_gt_le_bool x y :=
bool_of_sumbool (sumbool_not _ _ (le_gt_dec x y)).
Definition nat_eq_bool x y := bool_of_sumbool (eq_nat_dec x y).
Definition nat_noteq_bool x y :=
bool_of_sumbool (sumbool_not _ _ (eq_nat_dec x y)).
Definition zerop_bool x := bool_of_sumbool (zerop x).
Definition notzerop_bool x := bool_of_sumbool (notzerop x).
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