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)
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