# Int

By Brandon Wu, May 2020

int is the SML type of integers.

## Values

The underlying representation of integers can be likened to the following:

datatype int = ... | ~2 | ~1 | 0 | 1 | 2 | ...


This signifies that the type int is inhabited by infinitely many values, all corresponding to whole numbers. In particular, every integer forms its own constant constructor for the int type, meaning that they each individually can be pattern matched upon. Note that the use of ~ above denotes negativity. Additionally, ~ is a valid function of type int -> int that negates a number.

While in practice, computers can only represent a finite number of integers, for the purposes of this class we will generally assume the integers to be unbounded. This means that we can do induction on SML integers in exactly the same way as we would do induction on the natural numbers, and that we do not have to worry about the consequences of edge case behavior. This allows us to ignore pedantic implementation details and explore mathematically interesting properties of programs.

## Production

Integers have all the familiar arithmetic operations available to them. Note that some of these functions are also overloaded to work with real types - this is further discussed in the Real page.

(op +)   : int * int -> int
(op -)   : int * int -> int
(op *)   : int * int -> int
(op div) : int * int -> int
(op mod) : int * int -> int


All of these functions are infixed, so instead of being applied as +(2, 3), we write 2 + 3. Additionally, div and mod are not defined when the second argument is 0, and will raise a Div exception.

## Combination

Integers are also eligible for comparison, including equality and inequality. (In other words, integers are an equality type).

(op =)  : int * int -> bool
(op <>) : int * int -> bool (* Inequality *)

(op <)  : int * int -> bool
(op >)  : int * int -> bool
(op <=) : int * int -> bool
(op >=) : int * int -> bool


## From the Structure

The structure Int is bound as part of the SML Basis. It includes helpful functions such as

Int.toString : int -> string
Int.compare  : int * int -> order
Int.min      : int * int -> int
Int.max      : int * int -> int


where Int.toString is the function that returns the string representation of a given integer, and Int.compare has return type order, which is inhabited only by values LESS, EQUAL, and GREATER. Int.compare (x, y) returns LESS only if x < y, EQUAL if x = y, and GREATER if x > y. Additionally, Int.min and Int.max are just the corresponding min and max functions for integers.