# 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.

Declare Scope hex_positive_scope.
Delimit Scope hex_positive_scope with xpositive.

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.

Declare Scope hex_N_scope.
Delimit Scope hex_N_scope with xN.

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.

Declare Scope hex_Z_scope.
Delimit Scope hex_Z_scope with xZ.

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