LAoP is a library for algebraic (inductive) construction and manipulation of matrices in Haskell. See my Msc Thesis for the motivation behind the library, the underlying theory, and implementation details.

This module provides the Natural data type. The semantic associated with this data type is that it's meant to be a restricted Int value.

Utility module that provides the Natural data type. The semantic associated with this data type is that it's meant to be a restricted Int value. For example the type Natural 1 6 can only be instanciated with nat n where 1 <= n <= 6. Why, You might ask, because with normal Ints it is not possible to have a decent Enum (Int, Int) instance. See the following probabilistic programming model as and example:

We want to calculate the probability of the sum of two dice throws. To do this we start by defining the sample space:

type SampleSpace = Int -- We think Int are enough

die :: Dist Int 6
die = unifrom [1..6] 

-- Promote Int addition to a matrix
addM = fromF (uncurry (+)) -- Impossible

The last line is impossible because (Int, Int) does not have a good Enum instance: [(0, 1), (0, 2), .. (0, maxBound), (1, 0), ..]. And we'd like the addition matrix to be of 36 columns by 12 rows but limited to integers up to 6!

One way to solve this issue is by defining and auxilary data type to represent the sample space:

data SampleSpace = S1 | S2 | S3 | S4 | S5 | S6
  deriving (Show, Eq, Enum, Bounded) -- Enum and Bounded are
  important

And write the sample space addition function:

ssAdd :: SampleSpace -> SampleSpace -> Int
ssAdd a b = (fromEnum a + 1) + (fromEnum b + 1)

And then promote that function to matrix and everything is alright:

ssAddM = fromF' (uncurry ssAdd)

dieSumProb = ssAddM comp (khatri die die)

This is a nice solution for small sample spaces. But for larger ones it is not feasible to write a data type with hundreds of constructors and then write manipulation functions that need to deal with them. To mitigate this limitation the Natural type comes a long way and allows one to model the sample in an easier way. See for instance:

ssAdd :: Natural 1 6 -> Natural 1 6 -> Natural 1 12
ssAdd = coerceNat (+) 

ssAddM = fromF' (uncurry sumSS)

die :: Dist (Natural 1 6) 6
die = uniform [nat 1 6 1 .. nat 6]

dieSumProb = ssAddM comp (khatri die die)

• Natural data type