Copyright	(c) Ross Paterson 2021
License	BSD-style (see the file LICENSE)
Maintainer	R.Paterson@city.ac.uk
Stability	provisional
Portability	portable
Safe Haskell	None
Language	Haskell2010

Data.YAP.Peano

Description

An example instance of the algebraic classes: lazy Peano numerals.

Synopsis

data Peano
- = Zero
- | Succ Peano
peano :: a -> (a -> a) -> Peano -> a
maybeMinus :: Peano -> Peano -> Maybe Peano

Documentation

data Peano Source #

Lazy Peano numerals: a unary representation of the natural numbers, which is very inefficient, but does offer lazy semantics.

Constructors

Zero
Succ Peano

Instances

Instances details

Read Peano Source #
Instance details Defined in Data.YAP.Peano Methods readsPrec :: Int -> ReadS Peano # readList :: ReadS [Peano] # readPrec :: ReadPrec Peano # readListPrec :: ReadPrec [Peano] #
Show Peano Source #
Instance details Defined in Data.YAP.Peano Methods showsPrec :: Int -> Peano -> ShowS # show :: Peano -> String # showList :: [Peano] -> ShowS #
Eq Peano Source #
Instance details Defined in Data.YAP.Peano Methods (==) :: Peano -> Peano -> Bool # (/=) :: Peano -> Peano -> Bool #
Ord Peano Source #
Instance details Defined in Data.YAP.Peano Methods compare :: Peano -> Peano -> Ordering # (<) :: Peano -> Peano -> Bool # (<=) :: Peano -> Peano -> Bool # (>) :: Peano -> Peano -> Bool # (>=) :: Peano -> Peano -> Bool # max :: Peano -> Peano -> Peano # min :: Peano -> Peano -> Peano #
AdditiveMonoid Peano Source #	`(+)` is lazy in the second argument
Instance details Defined in Data.YAP.Peano Methods (+) :: Peano -> Peano -> Peano # zero :: Peano # atimes :: ToInteger b => b -> Peano -> Peano #
Euclidean Peano Source #
Instance details Defined in Data.YAP.Peano Methods div :: Peano -> Peano -> Peano # mod :: Peano -> Peano -> Peano # divMod :: Peano -> Peano -> (Peano, Peano) # euclideanNorm :: Peano -> Natural #
Semiring Peano Source #
Instance details Defined in Data.YAP.Peano Methods (*) :: Peano -> Peano -> Peano # one :: Peano # fromNatural :: Natural -> Peano # rescale :: Peano -> Peano -> (Peano, Peano, Peano -> Peano) #
StandardAssociate Peano Source #	The only unit is `one`. `stdAssociate` is the identity.
Instance details Defined in Data.YAP.Peano Methods stdAssociate :: Peano -> Peano # stdUnit :: Peano -> Peano # stdRecip :: Peano -> Peano #
ToInteger Peano Source #
Instance details Defined in Data.YAP.Peano Methods toInteger :: Peano -> Integer #
ToRational Peano Source #
Instance details Defined in Data.YAP.Peano Methods toRational :: Peano -> Rational #

peano :: a -> (a -> a) -> Peano -> a Source #

Eliminator for Peano numerals: peano z s replaces Zero with z and Succ with s.

maybeMinus :: Peano -> Peano -> Maybe Peano Source #

Partial subtraction