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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
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(*         *     GNU Lesser General Public License Version 2.1          *)
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(** * Binary Numerical Datatypes *)

Set Implicit Arguments.

(** [positive] is a datatype representing the strictly positive integers
   in a binary way. Starting from 1 (represented by [xH]), one can
   add a new least significant digit via [xO] (digit 0) or [xI] (digit 1).
   Numbers in [positive] will also be denoted using a decimal notation;
   e.g. [6%positive] will abbreviate [xO (xI xH)] *)

Inductive positive : Set :=
  | xI : positive -> positive
  | xO : positive -> positive
  | xH : positive.

Declare Scope positive_scope.
Delimit Scope positive_scope with positive.
Bind Scope positive_scope with positive.
Arguments xO _%positive.
Arguments xI _%positive.

Register positive as num.pos.type.
Register xI as num.pos.xI.
Register xO as num.pos.xO.
Register xH as num.pos.xH.

(** [N] is a datatype representing natural numbers in a binary way,
    by extending the [positive] datatype with a zero.
    Numbers in [N] will also be denoted using a decimal notation;
    e.g. [6%N] will abbreviate [Npos (xO (xI xH))] *)

Inductive N : Set :=
  | N0 : N
  | Npos : positive -> N.

Declare Scope N_scope.
Delimit Scope N_scope with N.
Bind Scope N_scope with N.
Arguments Npos _%positive.

Register N as num.N.type.
Register N0 as num.N.N0.
Register Npos as num.N.Npos.

(** [Z] is a datatype representing the integers in a binary way.
    An integer is either zero or a strictly positive number
    (coded as a [positive]) or a strictly negative number
    (whose opposite is stored as a [positive] value).
    Numbers in [Z] will also be denoted using a decimal notation;
    e.g. [(-6)%Z] will abbreviate [Zneg (xO (xI xH))] *)

Inductive Z : Set :=
  | Z0 : Z
  | Zpos : positive -> Z
  | Zneg : positive -> Z.

Declare Scope Z_scope.
Delimit Scope Z_scope with Z.
Bind Scope Z_scope with Z.
Arguments Zpos _%positive.
Arguments Zneg _%positive.

Register Z as num.Z.type.
Register Z0 as num.Z.Z0.
Register Zpos as num.Z.Zpos.
Register Zneg as num.Z.Zneg.