Prelude.SafeEnum

A redefinition of the Prelude's Enum class in order to render it safe. That is, the Haskell Language Report defines pred, succ, fromEnum and toEnum to be partial functions when the type is Bounded[1], but this is unacceptable. So these classes are offered as a replacement, correcting the types of those functions. We intentionally clash with the names of the Prelude's class; if you wish to use both in the same file, then import this module (or the Prelude) qualified.

While we're at it, we also generalize the notion of enumeration. Rather than requiring that the type is linearly enumerable, we distinguish between forward enumeration (which allows for multiple predecessors) and backward enumeration (which allows for multiple successors). Moreover, we do not require that the enumeration order coincides with the Ord ordering (if one exists), though it's advisable that they do (for your sanity). However, we also ensure that the notion of enumeration (in either direction) is well-defined, which rules out instances for Float and Double, and renders instances for Ratio problematic. Ratio instances can be provided so long as the base type is enumerable (and Integral, naturally); but they must be done in an obscure order[2] that does not coincide with Ord, which is not what people expect.

The MagicHash extension is only actually required if on GHC. This extension is used only so that the implementation of the instances for Char match those of the Prelude's Enum. I have not benchmarked to determine whether this low-level hackery is actually still necessary.

1

http://www.haskell.org/onlinereport/haskell2010/haskellch6.html#x13-1310006.3.4

2

Jeremy Gibbons, David Lester, and Richard Bird (2006). Enumerating the Rationals. JFP 16(3):281--291. DOI:10.1017/S0956796806005880 http://www.cs.ox.ac.uk/jeremy.gibbons/publications/rationals.pdf

Documentation

class UpwardEnum a where

A class for upward enumerable types. That is, we can enumerate larger and larger values, eventually getting all of them. We require that succeeds forms a strict partial order. That is, it must obey the following laws (N.B., if the first two laws hold, then the third one follows for free):

 if x `succeeds` y && y `succeeds` z then x `succeeds` z
 if x `succeeds` y then not (y `succeeds` x)
 not (x `succeeds` x)

Moreover, we require that succeeds agrees with succ, and that succ is exhaustive for succeeds (assuming Eq a, by magic if need be):

 if succ x == Just y then y `succeeds` x
 if x `succeeds` y   then x `elem` enumFrom y

Minimal complete definition: succ, succeeds.

Methods

succ :: a -> Maybe a

The successor of a value, or Nothing is there isn't one. For the numeric types in the Prelude, succ adds 1.

succeeds :: a -> a -> Bool

A variant of (>) with regards to the enumeration order.

enumFrom :: a -> [a]

Return x followed by all it's successors, in order. The resulting list is always non-empty, since it includes x. If the resulting list is always finite, then the succeeds ordering is converse well-founded. In GHC, the default implementation is a "good producer" for list fusion.

enumFromTo :: a -> a -> [a]

Return the elements of enumFrom x, filtering out everything that succeeds z. If x succeeds z, then the resulting list is empty; otherwise, it is non-empty, since it includes x. In GHC, the default implementation is a "good producer" for list fusion.

Instances

UpwardEnum Bool

UpwardEnum Char

UpwardEnum Int

UpwardEnum Integer

UpwardEnum Ordering

UpwardEnum ()

Integral a => UpwardEnum (CalkinWilf a)

class DownwardEnum a where

A class for downward enumerable types. That is, we can enumerate smaller and smaller values, eventually getting all of them. We require that precedes forms a strict partial order. That is, it must obey the following laws (N.B., if the first two laws hold, then the third one follows for free):

 if x `precedes` y && y `precedes` z then x `precedes` z
 if x `precedes` y then not (y `precedes` x)
 not (x `precedes` x)

Moreover, we require that precedes agrees with pred, and that pred is exhaustive for precedes (assuming Eq a, by magic if need be):

 if pred x == Just y then y `precedes` x
 if x `precedes` y   then x `elem` enumDownFrom y

Minimal complete definition: pred, precedes.

Methods

pred :: a -> Maybe a

The predecessor of a value, or Nothing is there isn't one. For the numeric types in the Prelude, pred subtracts 1.

precedes :: a -> a -> Bool

A variant of (<) with regards to the enumeration order.

enumDownFrom :: a -> [a]

Return x followed by all it's predecessors, in (reverse) order. The resulting list is always non-empty, since it includes x. If the resulting list is always finite, then the precedes ordering is well-founded. In GHC, the default implementation is a "good producer" for list fusion.

enumDownFromTo :: a -> a -> [a]

Return the elements of enumDownFrom x, filtering out everything that precedes z. If x precedes z, then the resulting list is empty; otherwise, it is non-empty, since it includes x. In GHC, the default implementation is a "good producer" for list fusion.

Instances

DownwardEnum Bool

DownwardEnum Char

DownwardEnum Int

DownwardEnum Integer

DownwardEnum Ordering

DownwardEnum ()

Integral a => DownwardEnum (CalkinWilf a)

class (UpwardEnum a, DownwardEnum a) => Enum a where

A class for types with a linear enumeration order. We require that the partial orders of the superclasses agree:

 x `succeeds` y  ==  y `precedes` x

That the enumeration order is preserved/reflected:

 i `succeeds` j  ==  toEnum   i `succeeds` toEnum   j
 x `succeeds` y  ==  fromEnum x `succeeds` fromEnum y

And that toEnum and fromEnum form a weak isomorphism; i.e., for some p and q, the following must hold:

 fromEnum <=< toEnum    ==  (\i -> if p i then Just i else Nothing)
 toEnum   <=< fromEnum  ==  (\x -> if q x then Just x else Nothing)

In other words, the following type-restricted functions form an isomorphism of linear orderings.

 toEnum'   :: {i :: Int | toEnum   i == Just _} -> a
 fromEnum' :: {x :: a   | fromEnum x == Just _} -> Int

Minimal complete definition: toEnum, fromEnum. N.B., the default definitions for enumFromThen and enumFromThenTo only make sense when the type a is "smaller" than Int (i.e., fromEnum always succeeds); if fromEnum ever fails, then you must override the defaults in order to correctly infer the stride for values which cannot be converted to Int.

Methods

toEnum :: Int -> Maybe a

Convert from an Int.

fromEnum :: a -> Maybe Int

Convert to an Int.

enumFromThen :: a -> a -> [a]

Enumerate values with an inferred stride. The resulting list is always non-empty, since it includes x. Naturally, this should agree with enumFrom and enumDownFrom (assuming Eq a, by magic if need be):

 if succ x == Just y then enumFromThen x y == enumFrom x
 if pred x == Just y then enumFromThen x y == enumDownFrom x

In the default implementation: if fromEnum fails on either argument, then the result is exactly [x]; and if toEnum fails on any of the enumerated integers, then the first failure terminates the enumeration. If either of these properties is inappropriate, then you should override the default. In GHC, the default implementation is a "good producer" for list fusion.

enumFromThenTo :: a -> a -> a -> [a]

Enumerate values with an inferred stride and a given limit. If x precedes y (and therefore we're enumerating forward) but x succeeds z (and therefore is past the limit), then the result is empty. Similarly, if x succeeds y (and therefore we're enumerating backward) but x precedes z (and therefore is past the limit), then the result is empty. Otherwise the result is non-empty since it contains x. Naturally, this should agree with enumFromTo and enumDownFromTo (assuming Eq a, by magic if need be):

 if succ x == Just y then enumFromThenTo x y z == enumFromTo x z
 if pred x == Just y then enumFromThenTo x y z == enumDownFromTo x z

In the default implementation: if fromEnum fails on any argument, then the result is either [] or [x] (as appropriate); and if toEnum fails on any of the enumerated integers, then the first failure terminates the enumeration. If either of these properties is inappropriate, then you should override the default. In GHC, the default implementation is a "good producer" for list fusion.

Instances

Enum Bool

Enum Char

Enum Int

Enum Integer

Enum Ordering

Enum ()

Integral a => Enum (CalkinWilf a)