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TickNumber :: Int -> TickValuesplot Graphics.Rendering.Plot.Figure No documentation available.
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A redefinition of the Prelude's Enum class in order to render it safe. 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, but this is unacceptable. We define a new type-class hierarchy for enumeration which is safe and also generalizes to cover types which can only be enumerated in one direction.
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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 integral and enumerable; but they must be done in an obscure order[2] that does not coincide with Ord. Since this is not what people may expect, we only provide an instance for the newtype CalkinWilf, not for Ratio itself. 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
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prelude-safeenum Prelude.SafeEnum A class for downward enumerable types. That is, we can enumerate smaller and smaller values, eventually getting every one of them; i.e., given any x, for all y such that y `precedes` x, it must be the case that y occurs within some finite prefix of enumDownFrom x. 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. class (Upward
Enum a, DownwardEnum a) => Enum aprelude-safeenum Prelude.SafeEnum 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 _} -> IntMinimal 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.-
prelude-safeenum Prelude.SafeEnum A class for upward enumerable types. That is, we can enumerate larger and larger values, eventually getting every one of them; i.e., given any x, for all y such that y `succeeds` x, it must be the case that y occurs within some finite prefix of enumFrom x. 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. enumDownFrom :: DownwardEnum a => a -> [a]prelude-safeenum Prelude.SafeEnum 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 :: DownwardEnum a => a -> a -> [a]prelude-safeenum Prelude.SafeEnum 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.
enumFrom :: UpwardEnum a => a -> [a]prelude-safeenum Prelude.SafeEnum 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.
enumFromThen :: Enum a => a -> a -> [a]prelude-safeenum Prelude.SafeEnum 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.