✨ New type-system for integers

[LVMVRQUXL]

Based on the concept of bijection and our desire to represent real world objects with our explicit types, we would like to re-implement a new type-system for integers.

Representation

For a full coverage, this type-system should include the following types:

1. The Zero type representing the zero number.
2. The PositiveInteger type representing an integer that is greater than zero (in the [1;+∞[ range).
3. The NegativeInteger type representing an integer that is less than zero (in the ]-∞;-1] range).
4. The NonNegativeInteger type representing an integer that is greater than or equal to zero (in the [0;+∞[ range).
5. The NonPositiveInteger type representing an integer that is less than or equal to zero (in the ]-∞;0] range).
6. The NonZeroInteger type representing an integer that is other than zero (in the ]-∞;-1]∪[1;+∞[ range).
7. The Integer type representing an integer.

These types have a subset relationship with others:

• The Zero type is a subset of the NonPositiveInteger and the NonNegativeInteger types.
• The PositiveInteger type is a subset of the NonNegativeInteger and the NonZeroInteger types.
• The NegativeInteger type is a subset of the NonPositiveInteger and the NonZeroInteger types.
• The NonPositiveInteger, the NonNegativeInteger and the NonZeroInteger types are subsets of the Integer type.

Here's an example of valid representation of this type-system using Kotlin:

interface Integer
interface NonZeroInteger : Integer
interface NonPositiveInteger : Integer
interface NonNegativeInteger : Integer
interface PositiveInteger : NonNegativeInteger, NonZeroInteger
interface NegativeInteger : NonPositiveInteger, NonZeroInteger
interface Zero : NonPositiveInteger, NonNegativeInteger

Note

Since practical computers are of finite capacity, we could represent them internally using arrays.
For example, we can consider that the number 1234 is [1, 2, 3, 4].

Operations

These types should support the following basic operations:

• The unaryMinus operation (-x) for returning the additive inverse of an integer.
• The plus operation (x + y) for adding two integers.
• The minus operation (x - y) for subtracting an integer from another one.
• The times operation (x * y) multiplying an integer by another one.
• The div operation (x / y) for dividing an integer by another one.
• The rem operation (x % y) for calculating the remainder of truncating division of an integer by another one.

[LVMVRQUXL]

See the issue #661 for the PositiveInteger type.