Plainly partible
Contents
What is partible?
Partible types are specific forms of pseudodata (a generalisation of oracle streams) whose values satisfy the following properties:
 they are all unique: no two values will ever be the same;
 they are monousal: if it is used, each value can only be used once;
 their splitting is disjoint: the resulting new values are independent.
Why splittable isn't always enough
Further developments
Since its advent, pseudodata (or aspects thereof) have appeared, or can be recognised in other contexts:
 In Simon Peyton Jones's The implementation of functional programming languages (section 9.6 on page 188 of 458), Peter Hancock provides a simpler interface for generating new type variables (of type tvname) for a type checker, using the type name_supply:
 next_name :: name_supply > tvname
 deplete :: name_supply > name_supply
 split :: name_supply > (name_supply, name_supply)
 The crucial point here is the absence of trees  they have been reduced to an implementation detail, oblivious to the users of name_supply values.
 In Efficient and Portable Combined Random Number Generators, Pierre L'Ecuyer suggests the disjoint splitting of random numbers into independent subsequences as needed.
 As previously specified, if pseudodata is used then it remains constant  reusing it doesn't change its value. Lennart Augustsson, Mikael Rittri and Dan Synek take this to an extreme in their functional pearl On generating unique names with their concept implementation for a singleuse variant of Hancock's uniquename supply  each one can only be used once, if at all:
 abstype uniquesupply
 with
 new_uniquesupply :: uniquesupply
 split_uniquesupply :: uniquesupply > (uniquesupply, uniquesupply)
 get_unique :: uniquesupply > unique
 uniquesupply ::= US
 new_uniquesupply = US
 split_uniquesupply US = (US, US)
 get_unique s = gensym(s)
 unique == int
  Not a regular definition!
 gensym :: * > unique
 In contrast to the example by John Launchbury and Simon Peyton Jones in State in Haskell (see pages 3940 of 51), this monousal strategy completely obviates the need for trees (or other intermediary structured values such as streams).
 Nobuo Yamashita uses a type reminiscent of pseudodata in his
IO
alternative oi package: see the Data.OI.Internal module for the details.
A matter of nomenclature
As mentioned earlier, L'Ecuyer suggests the splitting of random numbers be disjoint. But for entities like uniquename supplies, disjoint splitting is an absolute necessity! To avoid having to repeatedly specify it, an alternate terminology is needed  one which clearly indicates that for some *pseudodata* types, the disjointedness of splitting is mandatory, instead of just being very convenient.
The Partible
class
Depending on how it's corresponding partible type is defined, the disjoint splitting of an unused value can either be a pair or list of new values:
 part_uniquesupply :: uniquesupply > (uniquesupply, uniquesupply)
  or 
 parts_uniquesupply :: uniquesupply > [uniquesupply]
As each method can be defined using the other:
 part_uniquesupply u = (u1, u2) where u1:u2:_ = parts_uniquesupply u
  or 
 parts_uniquesupply u = u1 : parts_uniquesupply u2 where (u1, u2) = part_uniquesupply u
they can both be overloaded in Haskell using default definitions:
class Partible a where
part :: a > (a, a)
parts :: a > [a]
 Minimal complete definition: part or parts
part u = case parts u of u1:u2:_ > (u1, u2)
parts u = case part u of (u1, u2) > u1 : parts u2
(Of course if it's more efficient to do so, part
and parts
can both be defined.)
Ideally, each partible type in Haskell should also satisfy the partible laws.
Examples
 Using State in Haskell as a guide, an encapsulated singleuse type can be defined as follows:
{# LANGUAGE BangPatterns, RankNTypes, UnboxedTuples, MagicHash #} module UseOnce(UO, Monomo, part, runUO, useUO, asUO) where import Prelude(String, Eq(..)) import Prelude((.), ($), (++), error, all) import Data.Char(isSpace) import Partible import Monomo import GHC.Base(State#, MutVar#) import GHC.Base(runRW#, newMutVar#, noDuplicate#) import GHC.Exts(atomicModifyMutVar#) import GHC.ST(ST(..), STRep) data UO s = UO (UO# s) instance Partible (UO s) where part = partUO partUO :: UO s > (UO s, UO s) partUO (UO h) = let !(# h1, h2 #) = partUO# h in (UO h1, UO h2) runUO :: (forall s . UO s > a) > a runUO g = let !(# _, r #) = runRW# (useUO# (g . UO)) in r useUO :: (UO s > a) > ST s a useUO g = ST (\s > useUO# (g . UO) s) asUO :: Monomo a => String > ST s a > UO s > a asUO name (ST act) (UO h) = asUO# name act h  local definitions   type UO# s = String > State# s partUO# :: UO# s > (# UO# s, UO# s #) partUO# h = let !s = h "partUO" !(# s', h1 #) = dispense# s !(# _, h2 #) = dispense# s' in (# h1, h2 #) useUO# :: (UO# s > a) > STRep s a useUO# g s = let !(# s', h #) = dispense# s !r = g h in (# s', r #) dispense# :: STRep s (UO# s) dispense# s = let !(# s', r #) = newMutVar# () s in (# s', expire# s' r #) expire# :: State# s > MutVar# s () > String > State# s expire# s r name = let !(# s', () #) = atomicModifyMutVar# r use s in s' where use x = (error nowUsed, x) nowUsed = name' ++ ": already expired" name' = if all isSpace name then "(unknown)" else name asUO# :: Monomo a => String > STRep s a > UO# s > a asUO# name act h = let !(# _, t #) = act (noDuplicate# (h name)) in t
 Some notes:
 the elementary reuseerror reporting is optional;
 the use of
Monomo
inasUO
leverage Haskell's type system to provide an extra measure of safety, by restricting any typepolymorphism in the result: for more information, look into the history of Standard ML.
 Defining a partible variant of Yamashita's singleuse type then only requires a suitable argument:
{# LANGUAGE CPP, UnboxedTuples, MagicHash #} #define FauxWorld RealWorld module OutputInput(OI, Monomo, part, runOI, invokes) where import UseOnce import GHC.Base(IO(..), FauxWorld) import GHC.ST(ST(..)) type OI = UO FauxWorld runOI :: (OI > a) > IO a runOI g = case (useUO g) of ST m > IO m invokes :: Monomo a => String > IO a > OI > a (name `invokes` IO act) u = asUO name (ST act) u
 Using the parametric
UO
type, an encapuslated uniquename supply can also be defined:
data Fresh a = forall s . Fresh (UO s > a) (UO s) instance Partible (Fresh a) where parts (Fresh g u) = [ Fresh g v  v < parts u ] afresh :: (UO s > a) > UO s > Fresh a afresh g u = Fresh g u fresh :: Fresh a > [a] fresh (Fresh g u) = [ g v  v < parts u ] instance Functor Fresh where fmap f (Fresh g u) = Fresh (f . g) u runFresh :: (forall a. Eq a => Fresh a > b) > b runFresh f = f (runUO (freshNew (\n > n))) freshNew :: (Int > a) > UO s > Fresh a freshNew conv u = let !(u1, u2) = partUO u uvar = asUO "uvar" (newSTRef 0) u1 incr n = (n + 1, n) gensym = asUO "gensym" (atomicModifySTRef uvar incr) in Fresh (conv . gensym) u2  NB: may also need to define atomicModifySTRef; check your Haskell implementation
 Another possible abstract partible type is the generic exceptionthrower:
data Throw e instance Partible (Throw e) where part = partThrow partThrow :: Throw e > (Throw e, Throw e) curb :: (Throw e > a) > (e > OI > a) > OI > a catch :: (Throw e > a) > (e > Throw e > a) > Throw e > a throw :: e > Throw e > a
 Instances for various standard Haskell types are also a simple matter:
instance (Ix a, Partible b) => Partible (Array a b) where part arr = case unzip (map part' (assocs arr)) of (al1, al2) > (new al1, new al2) where new = array (bounds arr) part' (i, u) = case part u of (u1, u2) > ((i, u1), (i, u2)) instance (Partible a, Partible b) => Partible (Either a b) where parts (Left u) = map Left (parts u) parts (Right v) = map Right (parts v) instance (Partible a, Partible b) => Partible (a, b) where parts (u, v) = zip (parts u) (parts v) instance (Partible a, Partible b, Partible c) => Partible (a, b, c) where parts (u, v, w) = zip3 (parts u) (parts v) (parts w) instance (Partible a, Partible b, Partible c, Partible d) => Partible (a, b, c, d) where parts (u, v, w, x) = zip4 (parts u) (parts v) (parts w) (parts x) instance (Partible a, Partible b, Partible c, Partible d, Partible e) => Partible (a, b, c, d, e) where parts (u, v, w, x, y) = zip5 (parts u) (parts v) (parts w) (parts x) (parts y)  etc.
No list or Maybe
instances
The unit type ()
is clearly not partible, because of its single value:
 instance Partible () where part () = ((), ()) { WRONG! }
Therefore, because of their void values:

[] :: [a]

Nothing :: Maybe a
instances for for the list or Maybe
types are at best dubious:
instance Partible a => Partible [a] where part [] = ([], [])  !? ⋮
instance Partible a => Partible (Maybe a) where part Nothing = (Nothing, Nothing)  ?! ⋮
An alternative is to repurpose their nonvoid values to form a new type e.g:
data Some a = Only a  More a (Some a) instance Partible a => Partible (Some a) where parts (Only u) = map Only (parts u) parts (More u us) = zipWith More (parts u) (parts us)