Type level natural numbers and booleans.

This package is a direct port of the Haskell type-level package by Acosta, Kiselyov, Jeltsch et al. The original BSD 3-clause licence is preserved for this derivative work.

This package provides a way to represent type level natural numbers and booleans, with a useful set of operations for both.

A natural number is expressed at the type level using uninhabited types with a `D`

prefixed to the number: `D0`

, `D1`

, `D2`

etc. You can use the `:*`

type operator to combine digits: eg. `D1 :* D3 :* D3 :* D7`

makes the number 1337. Aliases are provided for two digit numbers: `D10`

, `D11`

etc. up to `D99`

. Values of each type are also provided, with the lower case prefix `d`

, so: `d0`

, `d1`

, `d2`

etc. up to `d99`

.

Two type classes are provided acting as sets: `Nat`

is the set of natural numbers (zero to infinity), and `Pos`

is the set of positive numbers (1 to infinity). `Nat`

defines the function `toInt`

, which converts type level numbers to the value level. To move integer values to the type level, use the `reifyInt`

function (which will throw a runtime exception for values which are not natural numbers).

You can express arithmetic operators as type class constraints on type level numbers. For instance, to express that a type `c`

is the sum of types `a`

and `b`

, you would use the type signature `∀ a b c. (Nat a, Nat b, Add a b c) ⇒ a → b → c`

. To express that a type `a`

must be less than another type `b`

, you would use the type signature `∀ a b. (Nat a, Nat b, Lt a b) ⇒ a → b`

.

Most of these constraints are fully relational, so that (even though a `Sub`

constraint exists) you could make `c`

the result of subtracting `b`

from `a`

by saying `∀ a b c. (Add c b a) ⇒ a → b → c`

.

Note that while most of the arithmetic operations are reasonably performant, division (`DivMod`

and friends) is not: division with a three digit divisor takes more than five minutes to type check on my machine.

Copyright (c) 2016 Bodil Stokke

Copyright (c) 2008 Alfonso Acosta, Oleg Kiselyov, Wolfgang Jeltsch and SAM Group at the School of Information and Communication Technology, (Royal Institute of Technology, Stockholm, Sweden)

All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

- Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
- Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
- Neither the name of The ForSyDe Team nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.

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