discrete

A Discrete type is a set X with at least one element, along with two functions, succ :: X -> Maybe X and pred :: X -> Maybe X, such that all inhabitants of the set X can be constructed given at least a single element of the set and these two functions. The following must hold:

pred >=> succ >=> pred = pred

succ >=> pred >=> succ = succ

This means that Int is a discrete type, because given any x :: Int, one can construct any other Int with the following functions:

succ x = if x == maxBound then Nothing else Just (x + 1)

pred x = if x == minBound then Nothing else Just (x - 1)

This also means that something like Double is not a discrete type, because there are no such functions succ and pred that given any value of type Double can allow the construction of all values of type Double.

Discrete acts as a replacement for GHC.Enum.Enum. The motivation for Discrete is two-fold: firstly, totality of all typeclass instances, and secondly, that GHC.Enum.Enum is a typeclass that does too many things, and does them poorly. If succ or pred are called on maxBound or minBound, respectively, the result will be Nothing.