interval-algebra-1.1.2: src/IntervalAlgebra/Core.hs
{-|
Module : Interval Algebra
Description : Implementation of Allen's interval algebra
Copyright : (c) NoviSci, Inc 2020
License : BSD3
Maintainer : bsaul@novisci.com
The @IntervalAlgebra@ module provides data types and related classes for the
interval-based temporal logic described in [Allen (1983)](https://doi.org/10.1145/182.358434)
and axiomatized in [Allen and Hayes (1987)](https://doi.org/10.1111/j.1467-8640.1989.tb00329.x).
A good primer on Allen's algebra can be [found here](https://thomasalspaugh.org/pub/fnd/allen.html).
= Design
The module is built around three typeclasses designed to separate concerns of
constructing, relating, and combining types that contain @'Interval'@s:
1. @'Intervallic'@ provides an interface to the data structures which contain an
@'Interval'@.
2. @'IntervalCombinable'@ provides an interface to methods of combining two
@'Interval's@.
3. @'IntervalSizeable'@ provides methods for measuring and modifying the size of
an interval.
-}
{-# LANGUAGE Safe #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies #-}
{-# LANGUAGE NoImplicitPrelude #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE AllowAmbiguousTypes #-}
module IntervalAlgebra.Core(
-- * Intervals
Interval
, Intervallic(..)
, ParseErrorInterval(..)
, begin
, end
-- ** Create new intervals
, parseInterval
, beginerval
, enderval
-- ** Modify intervals
, expand
, expandl
, expandr
-- * Interval Algebra
-- ** Interval Relations and Predicates
, IntervalRelation(..)
{- |
=== Meets, Metby
> x `meets` y
> y `metBy` x
@
x: |-----|
y: |-----|
@
-}
, meets , metBy
{- |
=== Before, After
> x `before` y
> y `after` x
@
x: |-----|
y: |-----|
@
-}
, before , after
{- |
=== Overlaps, OverlappedBy
> x `overlaps` y
> y `overlappedBy` x
@
x: |-----|
y: |-----|
@
-}
, overlaps , overlappedBy
{- |
=== Finishes, FinishedBy
> x `finishes` y
> y `finishedBy` x
@
x: |---|
y: |-----|
@
-}
, finishedBy , finishes
{- |
=== During, Contains
> x `during` y
> y `contains` x
@
x: |-|
y: |-----|
@
-}
, contains , during
{- |
=== Starts, StartedBy
> x `starts` y
> y `startedBy` x
@
x: |---|
y: |-----|
@
-}
, starts , startedBy
{- |
=== Equal
> x `equal` y
> y `equal` x
@
x: |-----|
y: |-----|
@
-}
, equals
-- ** Additional predicates and utilities
, precedes, precededBy
, disjoint , notDisjoint, concur
, within, enclose, enclosedBy
, (<|>)
, predicate, unionPredicates
, disjointRelations, withinRelations
, strictWithinRelations
, ComparativePredicateOf1
, ComparativePredicateOf2
, beginervalFromEnd
, endervalFromBegin
, beginervalMoment
, endervalMoment
, diffFromBegin
, diffFromEnd
, momentize
-- ** Algebraic operations
, intervalRelations
, relate
, compose
, complement
, union
, intersection
, converse
-- * Combine two intervals
, IntervalCombinable(..)
, extenterval
-- * Measure an interval
, IntervalSizeable(..)
) where
import Prelude ( Eq, Show, Enum(..), Bounded(..)
, Maybe(..), Either(..), String, Bool(..)
, Integer, Int, Num, Rational
, map, otherwise, show
, any, negate, not
, replicate
, fromRational
, toRational
, fromInteger
, toInteger
, (++), (==), (&&), (+), (-), (!!), realToFrac)
import Data.Function ( ($), id, (.), flip )
import Data.Functor ( Functor(fmap) )
import Data.Ord ( Ord(..), Ordering(..), min, max )
import Data.Semigroup ( Semigroup((<>)) )
import qualified Data.Set ( Set
, fromList
, difference
, intersection
, union
, map
, toList )
import Data.Tuple ( fst, snd )
import Data.Fixed ( Pico )
import Data.Time as DT ( Day
, UTCTime
, NominalDiffTime
, DiffTime
, addUTCTime
, diffUTCTime
, secondsToNominalDiffTime
, nominalDiffTimeToSeconds
, addDays
, diffDays )
import Control.Applicative ( Applicative(pure) )
{- | An @'Interval' a@ is a pair \( (x, y) \text{ such that } x < y\). To create
intervals use the @'parseInterval'@, @'beginerval'@, or @'enderval'@ functions.
-}
newtype Interval a = Interval (a, a) deriving (Eq)
-- | A type identifying interval parsing errors.
newtype ParseErrorInterval = ParseErrorInterval String
deriving (Eq, Show)
-- | Safely parse a pair of @a@s to create an @'Interval' a@.
--
-- >>> parseInterval 0 1
-- Right (0, 1)
--
-- >>> parseInterval 1 0
-- Left "0<1"
--
parseInterval :: (Show a, Ord a) => a -> a -> Either ParseErrorInterval (Interval a)
parseInterval x y
| x < y = Right $ Interval (x, y)
| otherwise = Left $ ParseErrorInterval $ show y ++ "<=" ++ show x
intervalBegin :: (Ord a) => Interval a -> a
intervalBegin (Interval x) = fst x
intervalEnd :: (Ord a) => Interval a -> a
intervalEnd (Interval x) = snd x
instance Functor Interval where
fmap f (Interval (x, y)) = Interval (f x, f y)
instance (Show a, Ord a) => Show (Interval a) where
show x = "(" ++ show (begin x) ++ ", " ++ show (end x) ++ ")"
{- |
The @'Intervallic'@ typeclass defines how to get and set the 'Interval' content
of a data structure. It also includes functions for getting the endpoints of the
'Interval' via @'begin'@ and @'end'@.
>>> getInterval (Interval (0, 10))
(0, 10)
>>> begin (Interval (0, 10))
0
>>> end (Interval (0, 10))
10
-}
class (Ord a) => Intervallic i a where
-- | Get the interval from an @i a@.
getInterval :: i a -> Interval a
-- | Set the interval in an @i a@.
setInterval :: i a -> Interval a -> i a
-- | Access the endpoints of an @i a@ .
begin, end :: Intervallic i a => i a -> a
begin = intervalBegin . getInterval
end = intervalEnd . getInterval
{- |
The 'IntervalRelation' type and the associated predicate functions enumerate
the thirteen possible ways that two @'Interval'@ objects may 'relate' according
to Allen's interval algebra. Constructors are shown with their corresponding
predicate function.
-}
data IntervalRelation =
Before -- ^ `before`
| Meets -- ^ `meets`
| Overlaps -- ^ `overlaps`
| FinishedBy -- ^ `finishedBy`
| Contains -- ^ `contains`
| Starts -- ^ `starts`
| Equals -- ^ `equals`
| StartedBy -- ^ `startedBy`
| During -- ^ `during`
| Finishes -- ^ `finishes`
| OverlappedBy -- ^ `overlappedBy`
| MetBy -- ^ `metBy`
| After -- ^ `after`
deriving (Eq, Show, Enum)
instance Bounded IntervalRelation where
minBound = Before
maxBound = After
instance Ord IntervalRelation where
compare x y = compare (fromEnum x) (fromEnum y)
-- | Does x `meets` y? Is x metBy y?
meets, metBy :: (Intervallic i0 a, Intervallic i1 a)=>
ComparativePredicateOf2 (i0 a) (i1 a)
meets x y = end x == begin y
metBy = flip meets
-- | Is x before y? Is x after y?
before, after, precedes, precededBy :: (Intervallic i0 a, Intervallic i1 a)=>
ComparativePredicateOf2 (i0 a) (i1 a)
before x y = end x < begin y
after = flip before
precedes = before
precededBy = after
-- | Does x overlap y? Is x overlapped by y?
overlaps, overlappedBy :: (Intervallic i0 a, Intervallic i1 a)=>
ComparativePredicateOf2 (i0 a) (i1 a)
overlaps x y = begin x < begin y && end x < end y && end x > begin y
overlappedBy = flip overlaps
-- | Does x start y? Is x started by y?
starts, startedBy :: (Intervallic i0 a, Intervallic i1 a)=>
ComparativePredicateOf2 (i0 a) (i1 a)
starts x y = begin x == begin y && end x < end y
startedBy = flip starts
-- | Does x finish y? Is x finished by y?
finishes, finishedBy :: (Intervallic i0 a, Intervallic i1 a)=>
ComparativePredicateOf2 (i0 a) (i1 a)
finishes x y = begin x > begin y && end x == end y
finishedBy = flip finishes
-- | Is x during y? Does x contain y?
during, contains :: (Intervallic i0 a, Intervallic i1 a)=>
ComparativePredicateOf2 (i0 a) (i1 a)
during x y = begin x > begin y && end x < end y
contains = flip during
-- | Does x equal y?
equals :: (Intervallic i0 a, Intervallic i1 a)=>
ComparativePredicateOf2 (i0 a) (i1 a)
equals x y = begin x == begin y && end x == end y
-- | Operator for composing the union of two predicates
(<|>) :: (Intervallic i0 a, Intervallic i1 a)=>
ComparativePredicateOf2 (i0 a) (i1 a)
-> ComparativePredicateOf2 (i0 a) (i1 a)
-> ComparativePredicateOf2 (i0 a) (i1 a)
(<|>) f g = unionPredicates [f, g]
-- | The set of @IntervalRelation@ meaning two intervals are disjoint.
disjointRelations :: Data.Set.Set IntervalRelation
disjointRelations = toSet [Before, After, Meets, MetBy]
-- | The set of @IntervalRelation@ meaning one interval is within the other.
withinRelations :: Data.Set.Set IntervalRelation
withinRelations = toSet [Starts, During, Finishes, Equals]
-- | The set of @IntervalRelation@ meaning one interval is *strictly* within the other.
strictWithinRelations :: Data.Set.Set IntervalRelation
strictWithinRelations = Data.Set.difference withinRelations (toSet [Equals])
-- | Are x and y disjoint ('before', 'after', 'meets', or 'metBy')?
disjoint :: (Intervallic i0 a, Intervallic i1 a)=>
ComparativePredicateOf2 (i0 a) (i1 a)
disjoint = predicate disjointRelations
-- | Are x and y not disjoint (concur); i.e. do they share any support? This is
-- the 'complement' of 'disjoint'.
notDisjoint, concur :: (Intervallic i0 a, Intervallic i1 a)=>
ComparativePredicateOf2 (i0 a) (i1 a)
notDisjoint = predicate (complement disjointRelations)
concur = notDisjoint
-- | Is x entirely *within* (enclosed by) the endpoints of y? That is, 'during',
-- 'starts', 'finishes', or 'equals'?
within, enclosedBy:: (Intervallic i0 a, Intervallic i1 a)=>
ComparativePredicateOf2 (i0 a) (i1 a)
within = predicate withinRelations
enclosedBy = within
-- | Does x enclose y? That is, is y 'within' x?
enclose :: (Intervallic i0 a, Intervallic i1 a)=>
ComparativePredicateOf2 (i0 a) (i1 a)
enclose = flip enclosedBy
-- | The 'Data.Set.Set' of all 'IntervalRelation's.
intervalRelations :: Data.Set.Set IntervalRelation
intervalRelations = Data.Set.fromList (Prelude.map toEnum [0..12] ::[IntervalRelation])
-- | Find the converse of a single 'IntervalRelation'
converseRelation :: IntervalRelation -> IntervalRelation
converseRelation x = toEnum (12 - fromEnum x)
-- | Shortcut to creating a 'Set IntervalRelation' from a list.
toSet :: [IntervalRelation ] -> Data.Set.Set IntervalRelation
toSet = Data.Set.fromList
-- | Compose a list of interval relations with _or_ to create a new
-- @'ComparativePredicateOf1' i a@. For example,
-- @unionPredicates [before, meets]@ creates a predicate function determining
-- if one interval is either before or meets another interval.
unionPredicates :: [ComparativePredicateOf2 a b] -> ComparativePredicateOf2 a b
unionPredicates fs x y = any (\ f -> f x y) fs
-- | Maps an 'IntervalRelation' to its corresponding predicate function.
toPredicate :: (Intervallic i0 a, Intervallic i1 a) =>
IntervalRelation
-> ComparativePredicateOf2 (i0 a) (i1 a)
toPredicate r =
case r of
Before -> before
Meets -> meets
Overlaps -> overlaps
FinishedBy -> finishedBy
Contains -> contains
Starts -> starts
Equals -> equals
StartedBy -> startedBy
During -> during
Finishes -> finishes
OverlappedBy -> overlappedBy
MetBy -> metBy
After -> after
-- | Given a set of 'IntervalRelation's return a list of 'predicate' functions
-- corresponding to each relation.
predicates :: (Intervallic i0 a, Intervallic i1 a)=>
Data.Set.Set IntervalRelation
-> [ComparativePredicateOf2 (i0 a) (i1 a)]
predicates x = Prelude.map toPredicate (Data.Set.toList x)
-- | Forms a predicate function from the union of a set of 'IntervalRelation's.
predicate :: (Intervallic i0 a, Intervallic i1 a)=>
Data.Set.Set IntervalRelation
-> ComparativePredicateOf2 (i0 a) (i1 a)
predicate = unionPredicates.predicates
-- | The lookup table for the compositions of interval relations.
composeRelationLookup :: [[[IntervalRelation]]]
composeRelationLookup =
[ [p , p , p , p , p , p , p , p , pmosd, pmosd, pmosd, pmosd, full ]
, [p , p , p , p , p , m , m , m , osd , osd , osd , fef , dsomp]
, [p , p , pmo , pmo , pmofd, o , o , ofd , osd , osd , cncr , dso , dsomp]
, [p , m , o , f' , d' , o , f', d' , osd , fef , dso , dso , dsomp]
, [pmofd, ofd , ofd , d' , d' , ofd , d', d' , cncr , dso , dso , dso , dsomp]
, [p , p , pmo , pmo , pmofd, s , s , ses , d , d , dfo , m' , p' ]
, [p , m , o , f' , d' , s , e , s' , d , f , o' , m' , p' ]
, [pmofd, ofd , ofd , d' , d' , ses , s', s' , dfo , o' , o' , m' , p' ]
, [p , p , pmosd, pmosd, full , d , d , dfomp, d , d , dfomp, p' , p' ]
, [p , m , osd , fef , dsomp, d , f , omp , d , f , omp , p' , p' ]
, [pmofd, ofd , cncr , dso , dsomp, dfo , o', omp , dfo , o' , omp , p' , p' ]
, [pmofd, ses , dfo , m' , p' , dfo , m', p' , dfo , m' , p' , p' , p' ]
, [full , dfomp, dfomp, p' , p' , dfomp, p', p' , dfomp, p' , p' , p' , p' ]
]
where p = [Before]
m = [Meets]
o = [Overlaps]
f' = [FinishedBy]
d' = [Contains]
s = [Starts]
e = [Equals]
s' = [StartedBy]
d = [During]
f = [Finishes]
o' = [OverlappedBy]
m' = [MetBy]
p' = [After]
ses = s ++ e ++ s'
fef = f' ++ e ++ f
pmo = p ++ m ++ o
pmofd = pmo ++ f' ++ d'
osd = o ++ s ++ d
ofd = o ++ f' ++ d'
omp = o' ++ m' ++ p'
dfo = d ++ f ++ o'
dfomp = dfo ++ m' ++ p'
dso = d' ++ s' ++ o'
dsomp = dso ++ m' ++ p'
pmosd = p ++ m ++ osd
cncr = o ++ f' ++ d' ++ s ++ e ++ s' ++ d ++ f ++ o'
full = p ++ m ++ cncr ++ m' ++ p'
-- | Compare two @i a@ to determine their 'IntervalRelation'.
--
-- >>> relate (Interval (0::Int, 1)) (Interval (1, 2))
-- Meets
--
-- >>> relate (Interval (1::Int, 2)) (Interval (0, 1))
-- MetBy
--
relate :: (Intervallic i0 a, Intervallic i1 a) => i0 a -> i1 a -> IntervalRelation
relate x y
| x `before` y = Before
| x `after` y = After
| x `meets` y = Meets
| x `metBy` y = MetBy
| x `overlaps` y = Overlaps
| x `overlappedBy` y = OverlappedBy
| x `starts` y = Starts
| x `startedBy` y = StartedBy
| x `finishes` y = Finishes
| x `finishedBy` y = FinishedBy
| x `during` y = During
| x `contains` y = Contains
| otherwise = Equals
-- | Compose two interval relations according to the rules of the algebra.
-- The rules are enumerated according to <https://thomasalspaugh.org/pub/fnd/allen.html#BasicCompositionsTable this table>.
compose :: IntervalRelation
-> IntervalRelation
-> Data.Set.Set IntervalRelation
compose x y = toSet (composeRelationLookup !! fromEnum x !! fromEnum y)
-- | Finds the complement of a @'Data.Set.Set' 'IntervalRelation'@.
complement :: Data.Set.Set IntervalRelation -> Data.Set.Set IntervalRelation
complement = Data.Set.difference intervalRelations
-- | Find the intersection of two 'Data.Set.Set's of 'IntervalRelation's.
intersection :: Data.Set.Set IntervalRelation
-> Data.Set.Set IntervalRelation
-> Data.Set.Set IntervalRelation
intersection = Data.Set.intersection
-- | Find the union of two 'Data.Set.Set's of 'IntervalRelation's.
union :: Data.Set.Set IntervalRelation
-> Data.Set.Set IntervalRelation
-> Data.Set.Set IntervalRelation
union = Data.Set.union
-- | Find the converse of a @'Data.Set.Set' 'IntervalRelation'@.
converse :: Data.Set.Set IntervalRelation
-> Data.Set.Set IntervalRelation
converse = Data.Set.map converseRelation
{- |
The 'IntervalSizeable' typeclass provides functions to determine the size of an
'Intervallic' type and to resize an 'Interval a'.
-}
class (Ord a, Num b, Ord b) => IntervalSizeable a b | a -> b where
-- | The smallest duration for an 'Interval a'.
moment :: b
moment = 1
-- | Gives back a 'moment' based on the input's type.
moment' :: Intervallic i a => i a -> b
moment' x = moment @a
-- | Determine the duration of an @'i a'@.
duration :: Intervallic i a => i a -> b
duration x = diff (end x) (begin x)
-- | Shifts an @a@. Most often, the @b@ will be the same type as @a@.
-- But for example, if @a@ is 'Day' then @b@ could be 'Int'.
add :: b -> a -> a
-- | Takes the difference between two @a@ to return a @b@.
diff :: a -> a -> b
-- | Resize an @i a@ to by expanding to "left" by @l@ and to the
-- "right" by @r@. In the case that @l@ or @r@ are less than a 'moment'
-- the respective endpoints are unchanged.
--
-- >>> expand 0 0 (Interval (0::Int, 2::Int))
-- (0, 2)
--
-- >>> expand 1 1 (Interval (0::Int, 2::Int))
-- (-1, 3)
--
expand :: (IntervalSizeable a b, Intervallic i a) =>
b -- ^ duration to subtract from the 'begin'
-> b -- ^ duration to add to the 'end'
-> i a
-> i a
expand l r p = setInterval p i
where s = if l < moment' p then 0 else negate l
e = if r < moment' p then 0 else r
i = Interval (add s $ begin p, add e $ end p)
-- | Expands an @i a@ to "left".
--
-- >>> expandl 2 (Interval (0::Int, 2::Int))
-- (-2, 2)
--
expandl :: (IntervalSizeable a b, Intervallic i a) => b -> i a -> i a
expandl i = expand i 0
-- | Expands an @i a@ to "right".
--
-- >>> expandr 2 (Interval (0::Int, 2::Int))
-- (0, 4)
--
expandr :: (IntervalSizeable a b, Intervallic i a) => b -> i a -> i a
expandr = expand 0
-- | Safely creates an 'Interval a' using @x@ as the 'begin' and adding
-- @max 'moment' dur@ to @x@ as the 'end'.
--
-- >>> beginerval (0::Int) (0::Int)
-- (0, 1)
--
-- >>> beginerval (1::Int) (0::Int)
-- (0, 1)
--
-- >>> beginerval (2::Int) (0::Int)
-- (0, 2)
--
beginerval :: (IntervalSizeable a b) =>
b -- ^ @dur@ation to add to the 'begin'
-> a -- ^ the 'begin' point of the 'Interval'
-> Interval a
beginerval dur x = Interval (x, y)
where i = Interval (x, x)
d = max (moment' i) dur
y = add d x
{-# INLINABLE beginerval #-}
-- | Safely creates an 'Interval a' using @x@ as the 'end' and adding
-- @negate max 'moment' dur@ to @x@ as the 'begin'.
--
-- >>> enderval (0::Int) (0::Int)
-- (-1, 0)
--
-- >>> enderval (1::Int) (0::Int)
-- (-1, 0)
--
-- >>> enderval (2::Int) (0::Int)
-- (-2, 0)
--
enderval :: (IntervalSizeable a b) =>
b -- ^ @dur@ation to subtract from the 'end'
-> a -- ^ the 'end' point of the 'Interval'
-> Interval a
enderval dur x = Interval (add (negate $ max (moment' i) dur) x, x)
where i = Interval (x, x)
{-# INLINABLE enderval #-}
-- | Creates a new Interval from the 'end' of an @i a@.
beginervalFromEnd :: (IntervalSizeable a b, Intervallic i a) =>
b -- ^ @dur@ation to add to the 'end'
-> i a -- ^ the @i a@ from which to get the 'end'
-> Interval a
beginervalFromEnd d i = beginerval d (end i)
-- | Creates a new Interval from the 'begin' of an @i a@.
endervalFromBegin :: (IntervalSizeable a b, Intervallic i a) =>
b -- ^ @dur@ation to subtract from the 'begin'
-> i a -- ^ the @i a@ from which to get the 'begin'
-> Interval a
endervalFromBegin d i = enderval d (begin i)
-- | Safely creates a new @Interval@ with 'moment' length with 'begin' at @x@
--
-- >>> beginervalMoment (10 :: Int)
-- (10, 11)
--
beginervalMoment :: (IntervalSizeable a b) => a -> Interval a
beginervalMoment x = beginerval (moment' i) x
where i = Interval (x, x)
-- | Safely creates a new @Interval@ with 'moment' length with 'end' at @x@
--
-- >>> endervalMoment (10 :: Int)
-- (9, 10)
--
endervalMoment :: (IntervalSizeable a b) => a -> Interval a
endervalMoment x = enderval (moment' i) x
where i = Interval (x, x)
-- | Creates a new @Interval@ spanning the extent x and y.
--
-- >>> extenterval (Interval (0, 1)) (Interval (9, 10))
-- (0, 10)
--
extenterval :: Intervallic i a => i a -> i a -> Interval a
extenterval x y = Interval (s, e)
where s = min (begin x) (begin y)
e = max (end x) (end y)
-- | Modifies the endpoints of second argument's interval by taking the difference
-- from the first's input's 'begin'.
-- >>> diffFromBegin (Interval ((5::Int), 6)) (Interval (10, 15))
-- (5, 10)
--
-- >>> diffFromBegin (Interval ((1::Int), 2)) (Interval (3, 15))
-- (2, 14)
--
diffFromBegin :: ( IntervalSizeable a b
, Functor i1
, Intervallic i0 a ) =>
i0 a -> i1 a -> i1 b
diffFromBegin i = fmap (`diff` begin i)
-- | Modifies the endpoints of second argument's interval by taking the difference
-- from the first's input's 'end'.
-- >>> diffFromEnd (Interval ((5::Int), 6)) (Interval (10, 15))
-- (4, 9)
--
-- >>> diffFromEnd (Interval ((1::Int), 2)) (Interval (3, 15))
-- (1, 13)
--
diffFromEnd :: ( IntervalSizeable a b
, Functor i1
, Intervallic i0 a ) =>
i0 a -> i1 a -> i1 b
diffFromEnd i = fmap (`diff` end i)
-- | Changes the duration of an 'Intervallic' value to a moment starting at the
-- 'begin' of the interval.
--
-- >>> momentize (Interval (6, 10))
-- (6, 7)
--
momentize :: ( IntervalSizeable a b, Intervallic i a ) =>
i a -> i a
momentize i = setInterval i (beginerval (moment' i) (begin i))
{- |
The @'IntervalCombinable'@ typeclass provides methods for (possibly) combining
two @i a@s to form a @'Maybe' i a@, or in case of @><@, a possibly different
@Intervallic@ type.
-}
class (Intervallic i a) => IntervalCombinable i a where
-- | Maybe form a new @i a@ by the union of two @i a@s that 'meets'.
(.+.) :: i a -> i a -> Maybe (i a)
(.+.) x y
| x `meets` y = Just $ setInterval y $ Interval (b, e)
| otherwise = Nothing
where b = begin x
e = end y
{-# INLINABLE (.+.) #-}
-- | If @x@ is 'before' @y@, then form a new @Just Interval a@ from the
-- interval in the "gap" between @x@ and @y@ from the 'end' of @x@ to the
-- 'begin' of @y@. Otherwise, 'Nothing'.
(><) :: i a -> i a -> Maybe (i a)
-- | If @x@ is 'before' @y@, return @f x@ appended to @f y@. Otherwise,
-- return 'extenterval' of @x@ and @y@ (wrapped in @f@). This is useful for
-- (left) folding over an *ordered* container of @Interval@s and combining
-- intervals when @x@ is *not* 'before' @y@.
(<+>):: ( Semigroup (f (i a)), Applicative f) =>
i a
-> i a
-> f (i a)
{-
Misc
-}
-- | Defines a predicate of two objects of type @a@.
type ComparativePredicateOf1 a = (a -> a -> Bool)
-- | Defines a predicate of two object of different types.
type ComparativePredicateOf2 a b = (a -> b -> Bool)
-- {-
-- Instances
-- -}
-- | Imposes a total ordering on @'Interval' a@ based on first ordering the
-- 'begin's then the 'end's.
instance (Ord a) => Ord (Interval a) where
(<=) x y
| begin x < begin y = True
| begin x == begin y = end x <= end y
| otherwise = False
(<) x y
| begin x < begin y = True
| begin x == begin y = end x < end y
| otherwise = False
instance (Ord a) => Intervallic Interval a where
getInterval = id
setInterval _ x = x
instance (Ord a) => IntervalCombinable Interval a where
(><) x y
| x `before` y = Just $ Interval (end x, begin y)
| otherwise = Nothing
{-# INLINABLE (><) #-}
(<+>) x y
| x `before` y = pure x <> pure y
| otherwise = pure ( extenterval x y )
{-# INLINABLE (<+>) #-}
instance IntervalSizeable Int Int where
moment = 1
add = (+)
diff = (-)
instance IntervalSizeable Integer Integer where
moment = 1
add = (+)
diff = (-)
instance IntervalSizeable DT.Day Integer where
moment = 1
add = addDays
diff = diffDays
-- | Note that the @moment@ of this instance is a @'Data.Fixed.Pico'@
instance IntervalSizeable DT.UTCTime NominalDiffTime where
moment = toEnum 1 :: NominalDiffTime
add = addUTCTime
diff = diffUTCTime