Safe Haskell	Safe
Language	Haskell2010

Data.Number.Peano

Description

Value-level Peano arithmetic.

Synopsis

Documentation

data Nat Source

Lazy Peano numbers. Allow calculation with infinite values.

Constructors

Z	Zero
S Nat	Successor

Instances

Bounded Nat Source	The lower bound is zero, the upper bound is infinity.
Enum Nat Source	The `pred` function is bounded at Zero.
Eq Nat Source	`==` and `/=` work as long as at least one operand is finite.
Integral Nat Source	Since negative numbers are not allowed, `quot = div` and `rem = mod`. The methods `quot`, `rem`, `div`, `mod`, `quotRem` and `divMod` will terminate as long as their first argument is finite. Infinities in their second arguments are permitted and are handled as follows: n `quot` infinity = n `div` infinity = 0 n `rem` infinity = n `mod` infinity = n
Num Nat Source	Addition, multiplication, and subtraction are lazy in both arguments, meaning that, in the case of infinite values, they can produce an infinite stream of S-constructors. As long as the callers of these functions only consume a finite amount of these, the program will not hang. `fromInteger` is not injective in case of `Nat`, since negative integers are all converted to zero (`Z`).
Ord Nat Source	All methods work as long as at least one operand is finite.
Real Nat Source	Since `toRational` returns a `Ratio Integer`, it WILL NOT terminate on infinities.
Show Nat Source
Peano Nat Source	Peano class instance

data Sign Source

Sign for whole numbers.

Constructors

Pos
Neg

Instances

Bounded Sign Source
Enum Sign Source
Eq Sign Source
Ord Sign Source
Read Sign Source
Show Sign Source

data Whole Source

Whole numbers (Z).

Constructors

Whole Nat Sign

Construct a whole number out of a magnitue and a sign.

Instances

Bounded Whole Source	The bounds are negative and positive infinity.
Enum Whole Source	`succ` and `pred` work according to the total order on the whole numbers, i.e. `succ n = n+1` and `pred n = n-1`.
Eq Whole Source	Positive and negative zero are considered equal.
Integral Whole Source	Integer conversions and division
Num Whole Source	Implements arithmetics for Whole numbers
Ord Whole Source	The ordering is the standard total order on Z. Positive and negative zero are equal.
Real Whole Source	Since `toRational` returns a `Ratio Integer`, it WILL NOT terminate on infinities.
Peano Whole Source	Peano class instance defines infinity (positive) and other functions handling the sign

class Enum a => Peano a where Source

The class of Peano-like constructions (i.e. Nat and Whole).

Methods

isZero :: a -> Bool Source

Test for zero.

infinity :: a Source

An unobservable infinity. For all finite numbers n, n < infinity must hold, but there need not be a total function that tests whether a number is infinite.

fromPeano :: a -> Integer Source

Converts the number to an Integer.

decr :: a -> a Source

Reduces the absolute value of the number by 1. If isZero n, then decr n = n and vice versa.

Instances

Peano Whole Source	Peano class instance defines infinity (positive) and other functions handling the sign
Peano Nat Source	Peano class instance

isSucc :: Peano n => n -> Bool Source

Negation of isZero.

takeNat :: (Num a, Enum a, Ord a, Peano n) => a -> n -> (a, n) Source

Removes at most S constructors from a Peano number. Outputs the number of removed constructors and the remaining number.

toNat :: Whole -> Nat Source

Extract the Nat value of a Whole

(//) :: Integral a => a -> a -> a Source

Alias to quot

natLength :: Foldable f => f a -> Nat Source

Returns the length of a foldable container as Nat. The number is generated lazily and thus, infinitely large containers are supported.