{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies #-}
{-# LANGUAGE NoImplicitPrelude #-}
{-# LANGUAGE AllowAmbiguousTypes #-}

{-|
Module      : Interval Algebra
Description : Implementation of Allen's interval algebra
Copyright   : (c) NoviSci, Inc 2020
License     : BSD3
Maintainer  : bsaul@novisci.com
Stability   : experimental

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 four typeclasses designed to separate concerns of 
constructing, relating, and combining @'Interval'@s: 

1. @'Intervallic'@ provides an interface to the data structure of an @'Interval'@, 
   defining how an @'Interval' a@ is constructed.
2. @'IntervalAlgebraic'@ provides an interface to the @'IntervalRelation's@, 
   the workhorse of Allen's temporal logic.
3. @'IntervalCombinable'@ provides an interface to methods of combining two
   @'Interval's@.
4. @'IntervalSizeable'@ and the related @'Moment'@ provides methods for 
   measuring and modifying the size of an interval.
   collections of intervals.

An advantage of nested typeclass design is that developers can define an 
@'Interval'@ of type @a@ with just the amount of structure that they need.

== Total Ordering of @Interval@s 

The modules makes the (opinionated) choice of a total ordering for @'Intervallic'@ 
@'Interval'@s. Namely, the ordering is based on first ordering the 'begin's 
then the 'end's.

= Development

This module is under development and the API may change in the future.
-}

module IntervalAlgebra(

    -- * Classes
      Intervallic(..)
    , IntervalAlgebraic(..)
    , IntervalCombinable(..)
    , Moment(..)
    , IntervalSizeable(..)

    -- * Types
    , Interval
    , IntervalRelation(..)
    , ComparativePredicateOf
) where

import Prelude (Eq, Ord, Show, Read, Enum(..), Bounded(..), Ordering (LT)
               , Maybe(..), Either(..), String, Integer, Int, Bool(..), Num
               , Foldable (maximum, minimum, foldMap, foldr)
               , map, otherwise, flip, show, fst, snd, min, max, any, negate, not, replicate
               , (++), (==), (&&), (<), (>), (<=), ($), (+), (-), (.), (!!))
import Data.Time as DT ( Day, addDays, diffDays, addGregorianYearsClip, calendarYear )
import Data.Semigroup ( Semigroup((<>)) )
import Data.Set(Set, fromList, difference, intersection, union, map, toList)
import Data.Ord( Ord(..), Ordering(..))
import GHC.Base (Applicative(pure))

{- | An @'Interval' a@ is a pair of @a@s \( (x, y) \text{ where } x < y\). The
@'Intervallic'@ class provides a safe @'parseInterval'@ function that returns a 
@'Left'@ error if \(y < x\) and 'unsafeInterval' as constructor for creating an
interval that may not be valid. 
-}
newtype Interval a = Interval (a, a) deriving (Interval a -> Interval a -> Bool
(Interval a -> Interval a -> Bool)
-> (Interval a -> Interval a -> Bool) -> Eq (Interval a)
forall a. Eq a => Interval a -> Interval a -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
/= :: Interval a -> Interval a -> Bool
$c/= :: forall a. Eq a => Interval a -> Interval a -> Bool
== :: Interval a -> Interval a -> Bool
$c== :: forall a. Eq a => Interval a -> Interval a -> Bool
Eq)

{- | 

The 'IntervalRelation' type enumerates the thirteen possible ways that two 
@'Interval' a@ objects can relate according to the interval algebra.

=== Meets, Metby

> x `meets` y
> y `metBy` x

@ 
x: |-----|
y:       |-----| 
@

=== Before, After

> x `before` y
> y `after` x

@ 
x: |-----|  
y:          |-----|
@


=== Overlaps, OverlappedBy

> x `overlaps` y
> y `overlappedBy` x

@ 
x: |-----|
y:     |-----|
@

=== Starts, StartedBy

> x `starts` y
> y `startedBy` x

@ 
x: |---| 
y: |-----|
@

=== Finishes, FinishedBy

> x `finishes` y
> y `finishedBy` x

@ 
x:   |---| 
y: |-----|
@

=== During, Contains

> x `during` y
> y `contains` x

@ 
x:   |-| 
y: |-----|
@

=== Equal

> x `equal` y
> y `equal` x

@ 
x: |-----| 
y: |-----|
@

-}
data IntervalRelation a =
      Meets
    | MetBy
    | Before
    | After
    | Overlaps
    | OverlappedBy
    | Starts
    | StartedBy
    | Finishes
    | FinishedBy
    | During
    | Contains
    | Equals
    deriving (IntervalRelation a -> IntervalRelation a -> Bool
(IntervalRelation a -> IntervalRelation a -> Bool)
-> (IntervalRelation a -> IntervalRelation a -> Bool)
-> Eq (IntervalRelation a)
forall a. IntervalRelation a -> IntervalRelation a -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
/= :: IntervalRelation a -> IntervalRelation a -> Bool
$c/= :: forall a. IntervalRelation a -> IntervalRelation a -> Bool
== :: IntervalRelation a -> IntervalRelation a -> Bool
$c== :: forall a. IntervalRelation a -> IntervalRelation a -> Bool
Eq, Int -> IntervalRelation a -> ShowS
[IntervalRelation a] -> ShowS
IntervalRelation a -> String
(Int -> IntervalRelation a -> ShowS)
-> (IntervalRelation a -> String)
-> ([IntervalRelation a] -> ShowS)
-> Show (IntervalRelation a)
forall a. Int -> IntervalRelation a -> ShowS
forall a. [IntervalRelation a] -> ShowS
forall a. IntervalRelation a -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
showList :: [IntervalRelation a] -> ShowS
$cshowList :: forall a. [IntervalRelation a] -> ShowS
show :: IntervalRelation a -> String
$cshow :: forall a. IntervalRelation a -> String
showsPrec :: Int -> IntervalRelation a -> ShowS
$cshowsPrec :: forall a. Int -> IntervalRelation a -> ShowS
Show, ReadPrec [IntervalRelation a]
ReadPrec (IntervalRelation a)
Int -> ReadS (IntervalRelation a)
ReadS [IntervalRelation a]
(Int -> ReadS (IntervalRelation a))
-> ReadS [IntervalRelation a]
-> ReadPrec (IntervalRelation a)
-> ReadPrec [IntervalRelation a]
-> Read (IntervalRelation a)
forall a. ReadPrec [IntervalRelation a]
forall a. ReadPrec (IntervalRelation a)
forall a. Int -> ReadS (IntervalRelation a)
forall a. ReadS [IntervalRelation a]
forall a.
(Int -> ReadS a)
-> ReadS [a] -> ReadPrec a -> ReadPrec [a] -> Read a
readListPrec :: ReadPrec [IntervalRelation a]
$creadListPrec :: forall a. ReadPrec [IntervalRelation a]
readPrec :: ReadPrec (IntervalRelation a)
$creadPrec :: forall a. ReadPrec (IntervalRelation a)
readList :: ReadS [IntervalRelation a]
$creadList :: forall a. ReadS [IntervalRelation a]
readsPrec :: Int -> ReadS (IntervalRelation a)
$creadsPrec :: forall a. Int -> ReadS (IntervalRelation a)
Read)

instance Bounded (IntervalRelation a) where
    minBound :: IntervalRelation a
minBound = IntervalRelation a
forall a. IntervalRelation a
Before
    maxBound :: IntervalRelation a
maxBound = IntervalRelation a
forall a. IntervalRelation a
After

instance Enum (IntervalRelation a) where
    fromEnum :: IntervalRelation a -> Int
fromEnum IntervalRelation a
r = case IntervalRelation a
r of
                    IntervalRelation a
Before       -> Int
0
                    IntervalRelation a
Meets        -> Int
1
                    IntervalRelation a
Overlaps     -> Int
2
                    IntervalRelation a
FinishedBy   -> Int
3
                    IntervalRelation a
Contains     -> Int
4
                    IntervalRelation a
Starts       -> Int
5
                    IntervalRelation a
Equals       -> Int
6
                    IntervalRelation a
StartedBy    -> Int
7
                    IntervalRelation a
During       -> Int
8
                    IntervalRelation a
Finishes     -> Int
9
                    IntervalRelation a
OverlappedBy -> Int
10
                    IntervalRelation a
MetBy        -> Int
11
                    IntervalRelation a
After        -> Int
12

    toEnum :: Int -> IntervalRelation a
toEnum Int
i = case Int
i of
               Int
0  -> IntervalRelation a
forall a. IntervalRelation a
Before
               Int
1  -> IntervalRelation a
forall a. IntervalRelation a
Meets
               Int
2  -> IntervalRelation a
forall a. IntervalRelation a
Overlaps
               Int
3  -> IntervalRelation a
forall a. IntervalRelation a
FinishedBy
               Int
4  -> IntervalRelation a
forall a. IntervalRelation a
Contains
               Int
5  -> IntervalRelation a
forall a. IntervalRelation a
Starts
               Int
6  -> IntervalRelation a
forall a. IntervalRelation a
Equals
               Int
7  -> IntervalRelation a
forall a. IntervalRelation a
StartedBy
               Int
8  -> IntervalRelation a
forall a. IntervalRelation a
During
               Int
9 -> IntervalRelation a
forall a. IntervalRelation a
Finishes
               Int
10 -> IntervalRelation a
forall a. IntervalRelation a
OverlappedBy
               Int
11 -> IntervalRelation a
forall a. IntervalRelation a
MetBy
               Int
12 -> IntervalRelation a
forall a. IntervalRelation a
After

instance Ord (IntervalRelation a) where
    compare :: IntervalRelation a -> IntervalRelation a -> Ordering
compare IntervalRelation a
x IntervalRelation a
y = Int -> Int -> Ordering
forall a. Ord a => a -> a -> Ordering
compare (IntervalRelation a -> Int
forall a. Enum a => a -> Int
fromEnum IntervalRelation a
x) (IntervalRelation a -> Int
forall a. Enum a => a -> Int
fromEnum IntervalRelation a
y)

-- | The 'Set' of all 'IntervalRelation's.
intervalRelations :: Set (IntervalRelation a)
intervalRelations :: Set (IntervalRelation a)
intervalRelations = [IntervalRelation a] -> Set (IntervalRelation a)
forall a. Ord a => [a] -> Set a
fromList ((Int -> IntervalRelation a) -> [Int] -> [IntervalRelation a]
forall a b. (a -> b) -> [a] -> [b]
Prelude.map Int -> IntervalRelation a
forall a. Enum a => Int -> a
toEnum [Int
0..Int
12] ::[IntervalRelation a])

-- | Find the converse of a single 'IntervalRelation'
converseRelation :: IntervalRelation a -> IntervalRelation a
converseRelation :: IntervalRelation a -> IntervalRelation a
converseRelation IntervalRelation a
x = Int -> IntervalRelation a
forall a. Enum a => Int -> a
toEnum (Int
12 Int -> Int -> Int
forall a. Num a => a -> a -> a
- IntervalRelation a -> Int
forall a. Enum a => a -> Int
fromEnum IntervalRelation a
x)

-- | The lookup table for the compositions of interval relations.
composeRelationLookup :: [[[IntervalRelation a]]]
composeRelationLookup :: [[[IntervalRelation a]]]
composeRelationLookup =
      [ [[IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
pmosd, [IntervalRelation a]
forall a. [IntervalRelation a]
pmosd, [IntervalRelation a]
forall a. [IntervalRelation a]
pmosd, [IntervalRelation a]
forall a. [IntervalRelation a]
pmosd, [IntervalRelation a]
forall a. [IntervalRelation a]
full ]
      , [[IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
m    , [IntervalRelation a]
forall a. [IntervalRelation a]
m , [IntervalRelation a]
forall a. [IntervalRelation a]
m    , [IntervalRelation a]
forall a. [IntervalRelation a]
osd  , [IntervalRelation a]
forall a. [IntervalRelation a]
osd  , [IntervalRelation a]
forall a. [IntervalRelation a]
osd  , [IntervalRelation a]
forall a. [IntervalRelation a]
fef  , [IntervalRelation a]
forall a. [IntervalRelation a]
dsomp]
      , [[IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
pmo  , [IntervalRelation a]
forall a. [IntervalRelation a]
pmo  , [IntervalRelation a]
forall a. [IntervalRelation a]
pmofd, [IntervalRelation a]
forall a. [IntervalRelation a]
o    , [IntervalRelation a]
forall a. [IntervalRelation a]
o , [IntervalRelation a]
forall a. [IntervalRelation a]
ofd  , [IntervalRelation a]
forall a. [IntervalRelation a]
osd  , [IntervalRelation a]
forall a. [IntervalRelation a]
osd  , [IntervalRelation a]
forall a. [IntervalRelation a]
cncr , [IntervalRelation a]
forall a. [IntervalRelation a]
dso  , [IntervalRelation a]
forall a. [IntervalRelation a]
dsomp]
      , [[IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
m    , [IntervalRelation a]
forall a. [IntervalRelation a]
o    , [IntervalRelation a]
forall a. [IntervalRelation a]
f'   , [IntervalRelation a]
forall a. [IntervalRelation a]
d'   , [IntervalRelation a]
forall a. [IntervalRelation a]
o    , [IntervalRelation a]
forall a. [IntervalRelation a]
f', [IntervalRelation a]
forall a. [IntervalRelation a]
d'   , [IntervalRelation a]
forall a. [IntervalRelation a]
osd  , [IntervalRelation a]
forall a. [IntervalRelation a]
fef  , [IntervalRelation a]
forall a. [IntervalRelation a]
dso  , [IntervalRelation a]
forall a. [IntervalRelation a]
dso  , [IntervalRelation a]
forall a. [IntervalRelation a]
dsomp]
      , [[IntervalRelation a]
forall a. [IntervalRelation a]
pmofd, [IntervalRelation a]
forall a. [IntervalRelation a]
ofd  , [IntervalRelation a]
forall a. [IntervalRelation a]
ofd  , [IntervalRelation a]
forall a. [IntervalRelation a]
d'   , [IntervalRelation a]
forall a. [IntervalRelation a]
d'   , [IntervalRelation a]
forall a. [IntervalRelation a]
ofd  , [IntervalRelation a]
forall a. [IntervalRelation a]
d', [IntervalRelation a]
forall a. [IntervalRelation a]
d'   , [IntervalRelation a]
forall a. [IntervalRelation a]
cncr , [IntervalRelation a]
forall a. [IntervalRelation a]
dso  , [IntervalRelation a]
forall a. [IntervalRelation a]
dso  , [IntervalRelation a]
forall a. [IntervalRelation a]
dso  , [IntervalRelation a]
forall a. [IntervalRelation a]
dsomp]
      , [[IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
pmo  , [IntervalRelation a]
forall a. [IntervalRelation a]
pmo  , [IntervalRelation a]
forall a. [IntervalRelation a]
pmofd, [IntervalRelation a]
forall a. [IntervalRelation a]
s    , [IntervalRelation a]
forall a. [IntervalRelation a]
s , [IntervalRelation a]
forall a. [IntervalRelation a]
ses  , [IntervalRelation a]
forall a. [IntervalRelation a]
d    , [IntervalRelation a]
forall a. [IntervalRelation a]
d    , [IntervalRelation a]
forall a. [IntervalRelation a]
dfo  , [IntervalRelation a]
forall a. [IntervalRelation a]
m'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   ]
      , [[IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
m    , [IntervalRelation a]
forall a. [IntervalRelation a]
o    , [IntervalRelation a]
forall a. [IntervalRelation a]
f'   , [IntervalRelation a]
forall a. [IntervalRelation a]
d'   , [IntervalRelation a]
forall a. [IntervalRelation a]
s    , [IntervalRelation a]
forall a. [IntervalRelation a]
e , [IntervalRelation a]
forall a. [IntervalRelation a]
s'   , [IntervalRelation a]
forall a. [IntervalRelation a]
d    , [IntervalRelation a]
forall a. [IntervalRelation a]
f    , [IntervalRelation a]
forall a. [IntervalRelation a]
o'   , [IntervalRelation a]
forall a. [IntervalRelation a]
m'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   ]
      , [[IntervalRelation a]
forall a. [IntervalRelation a]
pmofd, [IntervalRelation a]
forall a. [IntervalRelation a]
ofd  , [IntervalRelation a]
forall a. [IntervalRelation a]
ofd  , [IntervalRelation a]
forall a. [IntervalRelation a]
d'   , [IntervalRelation a]
forall a. [IntervalRelation a]
d'   , [IntervalRelation a]
forall a. [IntervalRelation a]
ses  , [IntervalRelation a]
forall a. [IntervalRelation a]
s', [IntervalRelation a]
forall a. [IntervalRelation a]
s'   , [IntervalRelation a]
forall a. [IntervalRelation a]
dfo  , [IntervalRelation a]
forall a. [IntervalRelation a]
o'   , [IntervalRelation a]
forall a. [IntervalRelation a]
o'   , [IntervalRelation a]
forall a. [IntervalRelation a]
m'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   ]
      , [[IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
pmosd, [IntervalRelation a]
forall a. [IntervalRelation a]
pmosd, [IntervalRelation a]
forall a. [IntervalRelation a]
full , [IntervalRelation a]
forall a. [IntervalRelation a]
d    , [IntervalRelation a]
forall a. [IntervalRelation a]
d , [IntervalRelation a]
forall a. [IntervalRelation a]
dfomp, [IntervalRelation a]
forall a. [IntervalRelation a]
d    , [IntervalRelation a]
forall a. [IntervalRelation a]
d    , [IntervalRelation a]
forall a. [IntervalRelation a]
dfomp, [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   ]
      , [[IntervalRelation a]
forall a. [IntervalRelation a]
p    , [IntervalRelation a]
forall a. [IntervalRelation a]
m    , [IntervalRelation a]
forall a. [IntervalRelation a]
osd  , [IntervalRelation a]
forall a. [IntervalRelation a]
fef  , [IntervalRelation a]
forall a. [IntervalRelation a]
dsomp, [IntervalRelation a]
forall a. [IntervalRelation a]
d    , [IntervalRelation a]
forall a. [IntervalRelation a]
f , [IntervalRelation a]
forall a. [IntervalRelation a]
omp  , [IntervalRelation a]
forall a. [IntervalRelation a]
d    , [IntervalRelation a]
forall a. [IntervalRelation a]
f    , [IntervalRelation a]
forall a. [IntervalRelation a]
omp  , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   ]
      , [[IntervalRelation a]
forall a. [IntervalRelation a]
pmofd, [IntervalRelation a]
forall a. [IntervalRelation a]
ofd  , [IntervalRelation a]
forall a. [IntervalRelation a]
cncr , [IntervalRelation a]
forall a. [IntervalRelation a]
dso  , [IntervalRelation a]
forall a. [IntervalRelation a]
dsomp, [IntervalRelation a]
forall a. [IntervalRelation a]
dfo  , [IntervalRelation a]
forall a. [IntervalRelation a]
o', [IntervalRelation a]
forall a. [IntervalRelation a]
omp  , [IntervalRelation a]
forall a. [IntervalRelation a]
dfo  , [IntervalRelation a]
forall a. [IntervalRelation a]
o'   , [IntervalRelation a]
forall a. [IntervalRelation a]
omp  , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   ]
      , [[IntervalRelation a]
forall a. [IntervalRelation a]
pmofd, [IntervalRelation a]
forall a. [IntervalRelation a]
ses  , [IntervalRelation a]
forall a. [IntervalRelation a]
dfo  , [IntervalRelation a]
forall a. [IntervalRelation a]
m'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
dfo  , [IntervalRelation a]
forall a. [IntervalRelation a]
m', [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
dfo  , [IntervalRelation a]
forall a. [IntervalRelation a]
m'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   ]
      , [[IntervalRelation a]
forall a. [IntervalRelation a]
full , [IntervalRelation a]
forall a. [IntervalRelation a]
dfomp, [IntervalRelation a]
forall a. [IntervalRelation a]
dfomp, [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
dfomp, [IntervalRelation a]
forall a. [IntervalRelation a]
p', [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
dfomp, [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   , [IntervalRelation a]
forall a. [IntervalRelation a]
p'   ]
      ]
      where p :: [IntervalRelation a]
p  = [IntervalRelation a
forall a. IntervalRelation a
Before]
            m :: [IntervalRelation a]
m  = [IntervalRelation a
forall a. IntervalRelation a
Meets]
            o :: [IntervalRelation a]
o  = [IntervalRelation a
forall a. IntervalRelation a
Overlaps]
            f' :: [IntervalRelation a]
f' = [IntervalRelation a
forall a. IntervalRelation a
FinishedBy]
            d' :: [IntervalRelation a]
d' = [IntervalRelation a
forall a. IntervalRelation a
Contains]
            s :: [IntervalRelation a]
s  = [IntervalRelation a
forall a. IntervalRelation a
Starts]
            e :: [IntervalRelation a]
e  = [IntervalRelation a
forall a. IntervalRelation a
Equals]
            s' :: [IntervalRelation a]
s' = [IntervalRelation a
forall a. IntervalRelation a
StartedBy]
            d :: [IntervalRelation a]
d  = [IntervalRelation a
forall a. IntervalRelation a
During]
            f :: [IntervalRelation a]
f  = [IntervalRelation a
forall a. IntervalRelation a
Finishes]
            o' :: [IntervalRelation a]
o' = [IntervalRelation a
forall a. IntervalRelation a
OverlappedBy]
            m' :: [IntervalRelation a]
m' = [IntervalRelation a
forall a. IntervalRelation a
MetBy]
            p' :: [IntervalRelation a]
p' = [IntervalRelation a
forall a. IntervalRelation a
After]
            ses :: [IntervalRelation a]
ses    = [IntervalRelation a]
forall a. [IntervalRelation a]
s [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
e [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
s'
            fef :: [IntervalRelation a]
fef    = [IntervalRelation a]
forall a. [IntervalRelation a]
f' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
e [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
f
            pmo :: [IntervalRelation a]
pmo    = [IntervalRelation a]
forall a. [IntervalRelation a]
p [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
m [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
o
            pmofd :: [IntervalRelation a]
pmofd  = [IntervalRelation a]
forall a. [IntervalRelation a]
pmo [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
f' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
d'
            osd :: [IntervalRelation a]
osd    = [IntervalRelation a]
forall a. [IntervalRelation a]
o [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
s [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
d
            ofd :: [IntervalRelation a]
ofd    = [IntervalRelation a]
forall a. [IntervalRelation a]
o [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
f' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
d'
            omp :: [IntervalRelation a]
omp    = [IntervalRelation a]
forall a. [IntervalRelation a]
o' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
m' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
p'
            dfo :: [IntervalRelation a]
dfo    = [IntervalRelation a]
forall a. [IntervalRelation a]
d [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
f [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
o'
            dfomp :: [IntervalRelation a]
dfomp  = [IntervalRelation a]
forall a. [IntervalRelation a]
dfo [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
m' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
p'
            dso :: [IntervalRelation a]
dso    = [IntervalRelation a]
forall a. [IntervalRelation a]
d' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
s' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
o'
            dsomp :: [IntervalRelation a]
dsomp  = [IntervalRelation a]
forall a. [IntervalRelation a]
dso [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
m' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
p'
            pmosd :: [IntervalRelation a]
pmosd  = [IntervalRelation a]
forall a. [IntervalRelation a]
p [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
m [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
osd
            cncr :: [IntervalRelation a]
cncr = [IntervalRelation a]
forall a. [IntervalRelation a]
o [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
f' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
d' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
s [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
e [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
s' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
d [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
f [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
o'
            full :: [IntervalRelation a]
full = [IntervalRelation a]
forall a. [IntervalRelation a]
p [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
m [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
cncr [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
m' [IntervalRelation a]
-> [IntervalRelation a] -> [IntervalRelation a]
forall a. [a] -> [a] -> [a]
++ [IntervalRelation a]
forall a. [IntervalRelation a]
p'
{-
Misc
-}

-- | Defines a predicate of two objects of type @a@.
type ComparativePredicateOf a = (a -> a -> Bool)

{- | 
The @'Intervallic'@ typeclass specifies how an @'Interval' a@s is constructed.
It also includes functions for getting the @'begin'@ and @'end'@ of an @'Interval' a@.
-}
class (Ord a, Show a) => Intervallic a where

    -- | Safely parse a pair of @a@s to create an @'Interval' a@.
    parseInterval :: a -> a -> Either String (Interval a)
    parseInterval a
x a
y
        -- TODO: create more general framework for error handling
        |  a
y a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
x    = String -> Either String (Interval a)
forall a b. a -> Either a b
Left  (String -> Either String (Interval a))
-> String -> Either String (Interval a)
forall a b. (a -> b) -> a -> b
$ a -> String
forall a. Show a => a -> String
show a
y String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
"<" String -> ShowS
forall a. [a] -> [a] -> [a]
++ a -> String
forall a. Show a => a -> String
show a
x
        | Bool
otherwise = Interval a -> Either String (Interval a)
forall a b. b -> Either a b
Right (Interval a -> Either String (Interval a))
-> Interval a -> Either String (Interval a)
forall a b. (a -> b) -> a -> b
$ (a, a) -> Interval a
forall a. (a, a) -> Interval a
Interval (a
x, a
y)

    {- | Create a new @'Interval' a@. This function is __not__ safe as it does 
       not enforce that \(x < y\). Use with caution. It is meant to be helper 
       function in early prototyping of this package. This function may be 
       deprecated in future releases.
    -}
    unsafeInterval :: a -> a -> Interval a
    unsafeInterval a
x a
y = (a, a) -> Interval a
forall a. (a, a) -> Interval a
Interval (a
x, a
y)

    -- | Access the ends of an @'Interval' a@ .
    begin, end :: Interval a -> a
    begin (Interval (a, a)
x) = (a, a) -> a
forall a b. (a, b) -> a
fst (a, a)
x --  \( \text{begin}(x, y) = x \)
    end   (Interval (a, a)
x) = (a, a) -> a
forall a b. (a, b) -> b
snd (a, a)
x --  \( \text{end}(x, y) = y \)

{- |
The @'IntervalAlgebraic'@ typeclass specifies the functions and relational 
operators for interval-based temporal logic. The typeclass defines the 
relational operators for intervals, plus other useful utilities such as 
@'disjoint'@, @'within'@, and @'unionPredicates'@.
-}
class (Eq a, Intervallic a) => IntervalAlgebraic a where

    -- | Compare two intervals to determine their 'IntervalRelation'.
    relate :: Interval a -> Interval a -> IntervalRelation a
    relate Interval a
x Interval a
y
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`before` Interval a
y       = IntervalRelation a
forall a. IntervalRelation a
Before
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`after`  Interval a
y       = IntervalRelation a
forall a. IntervalRelation a
After
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`meets`  Interval a
y       = IntervalRelation a
forall a. IntervalRelation a
Meets
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`metBy`  Interval a
y       = IntervalRelation a
forall a. IntervalRelation a
MetBy
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`overlaps` Interval a
y     = IntervalRelation a
forall a. IntervalRelation a
Overlaps
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`overlappedBy` Interval a
y = IntervalRelation a
forall a. IntervalRelation a
OverlappedBy
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`starts` Interval a
y       = IntervalRelation a
forall a. IntervalRelation a
Starts
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`startedBy` Interval a
y    = IntervalRelation a
forall a. IntervalRelation a
StartedBy
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`finishes` Interval a
y     = IntervalRelation a
forall a. IntervalRelation a
Finishes
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`finishedBy` Interval a
y   = IntervalRelation a
forall a. IntervalRelation a
FinishedBy
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`during` Interval a
y       = IntervalRelation a
forall a. IntervalRelation a
During
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`contains` Interval a
y     = IntervalRelation a
forall a. IntervalRelation a
Contains
        | Bool
otherwise          = IntervalRelation a
forall a. IntervalRelation a
Equals

    -- | Maps an 'IntervalRelation' to its corresponding predicate function.
    predicate' :: IntervalRelation a -> ComparativePredicateOf (Interval a)
    predicate' IntervalRelation a
r =
        case IntervalRelation a
r of
            IntervalRelation a
Before       -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
before
            IntervalRelation a
Meets        -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
meets
            IntervalRelation a
Overlaps     -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
overlaps
            IntervalRelation a
FinishedBy   -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
finishedBy
            IntervalRelation a
Contains     -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
contains
            IntervalRelation a
Starts       -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
starts
            IntervalRelation a
Equals       -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
equals
            IntervalRelation a
StartedBy    -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
startedBy
            IntervalRelation a
During       -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
during
            IntervalRelation a
Finishes     -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
finishes
            IntervalRelation a
OverlappedBy -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
overlappedBy
            IntervalRelation a
MetBy        -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
metBy
            IntervalRelation a
After        -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
after

    -- | Given a set of 'IntervalRelation's return a list of 'predicate' functions 
    --   corresponding to each relation.
    predicates :: Set (IntervalRelation a) -> [ComparativePredicateOf (Interval a)]
    predicates Set (IntervalRelation a)
x = (IntervalRelation a -> ComparativePredicateOf (Interval a))
-> [IntervalRelation a] -> [ComparativePredicateOf (Interval a)]
forall a b. (a -> b) -> [a] -> [b]
Prelude.map IntervalRelation a -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
IntervalRelation a -> ComparativePredicateOf (Interval a)
predicate' (Set (IntervalRelation a) -> [IntervalRelation a]
forall a. Set a -> [a]
toList Set (IntervalRelation a)
x)

    -- | Forms a predicate function from the union of a set of 'IntervalRelation's.
    predicate :: Set (IntervalRelation a) -> ComparativePredicateOf (Interval a)
    predicate = [ComparativePredicateOf (Interval a)]
-> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
[ComparativePredicateOf (Interval a)]
-> ComparativePredicateOf (Interval a)
unionPredicates([ComparativePredicateOf (Interval a)]
 -> ComparativePredicateOf (Interval a))
-> (Set (IntervalRelation a)
    -> [ComparativePredicateOf (Interval a)])
-> Set (IntervalRelation a)
-> ComparativePredicateOf (Interval a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
.Set (IntervalRelation a) -> [ComparativePredicateOf (Interval a)]
forall a.
IntervalAlgebraic a =>
Set (IntervalRelation a) -> [ComparativePredicateOf (Interval a)]
predicates

    -- ** Algebraic operations on IntervalRelations

    -- | Shortcut to creating a 'Set IntervalRelation' from a list.
    toSet :: [IntervalRelation a] -> Set (IntervalRelation a)
    toSet = [IntervalRelation a] -> Set (IntervalRelation a)
forall a. Ord a => [a] -> Set a
fromList

    -- | 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 a -> IntervalRelation a -> Set (IntervalRelation a)
    compose IntervalRelation a
x IntervalRelation a
y = [IntervalRelation a] -> Set (IntervalRelation a)
forall a.
IntervalAlgebraic a =>
[IntervalRelation a] -> Set (IntervalRelation a)
toSet (([[[IntervalRelation a]]]
forall a. [[[IntervalRelation a]]]
composeRelationLookup [[[IntervalRelation a]]] -> Int -> [[IntervalRelation a]]
forall a. [a] -> Int -> a
!! IntervalRelation a -> Int
forall a. Enum a => a -> Int
fromEnum IntervalRelation a
x) [[IntervalRelation a]] -> Int -> [IntervalRelation a]
forall a. [a] -> Int -> a
!! IntervalRelation a -> Int
forall a. Enum a => a -> Int
fromEnum IntervalRelation a
y)

    -- | Finds the complement of a 'Set IntervalRelation'.
    complement :: Set (IntervalRelation a) -> Set (IntervalRelation a)
    complement = Set (IntervalRelation a)
-> Set (IntervalRelation a) -> Set (IntervalRelation a)
forall a. Ord a => Set a -> Set a -> Set a
difference Set (IntervalRelation a)
forall a. Set (IntervalRelation a)
intervalRelations

    -- | Find the intersection of two 'Set's of 'IntervalRelation'
    intersection ::  Set (IntervalRelation a)
                  -> Set (IntervalRelation a)
                  -> Set (IntervalRelation a)
    intersection = Set (IntervalRelation a)
-> Set (IntervalRelation a) -> Set (IntervalRelation a)
forall a. Ord a => Set a -> Set a -> Set a
Data.Set.intersection

    -- | Find the union of two 'Set's of 'IntervalRelation'
    union ::  Set (IntervalRelation a)
           -> Set (IntervalRelation a)
           -> Set (IntervalRelation a)
    union = Set (IntervalRelation a)
-> Set (IntervalRelation a) -> Set (IntervalRelation a)
forall a. Ord a => Set a -> Set a -> Set a
Data.Set.union

    -- | Find the converse of a 'Set IntervalRelation'. 
    converse ::   Set (IntervalRelation a)
                  -> Set (IntervalRelation a)
    converse = (IntervalRelation a -> IntervalRelation a)
-> Set (IntervalRelation a) -> Set (IntervalRelation a)
forall b a. Ord b => (a -> b) -> Set a -> Set b
Data.Set.map IntervalRelation a -> IntervalRelation a
forall a. IntervalRelation a -> IntervalRelation a
converseRelation

    -- ** Interval algebra predicates

    -- | Does x equal y?
    equals                 :: ComparativePredicateOf (Interval a)
    equals   Interval a
x Interval a
y  = Interval a
x ComparativePredicateOf (Interval a)
forall a. Eq a => a -> a -> Bool
== Interval a
y

    -- | Does x meet y? Is y metBy x?
    meets, metBy           :: ComparativePredicateOf (Interval a)
    meets    Interval a
x Interval a
y  = Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y
    metBy         = ComparativePredicateOf (Interval a)
-> ComparativePredicateOf (Interval a)
forall a b c. (a -> b -> c) -> b -> a -> c
flip ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
meets

    -- | Is x before y? Is x after y?
    before, after          :: ComparativePredicateOf (Interval a)
    before   Interval a
x Interval a
y  = Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y
    after         = ComparativePredicateOf (Interval a)
-> ComparativePredicateOf (Interval a)
forall a b c. (a -> b -> c) -> b -> a -> c
flip ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
before

    -- | Does x overlap y? Is x overlapped by y?
    overlaps, overlappedBy :: ComparativePredicateOf (Interval a)
    overlaps Interval a
x Interval a
y  = Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y Bool -> Bool -> Bool
&& Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
y Bool -> Bool -> Bool
&& Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
> Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y
    overlappedBy  = ComparativePredicateOf (Interval a)
-> ComparativePredicateOf (Interval a)
forall a b c. (a -> b -> c) -> b -> a -> c
flip ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
overlaps

    -- | Does x start y? Is x started by y?
    starts, startedBy      :: ComparativePredicateOf (Interval a)
    starts   Interval a
x Interval a
y  = Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
x a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y Bool -> Bool -> Bool
&& (Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
y)
    startedBy     = ComparativePredicateOf (Interval a)
-> ComparativePredicateOf (Interval a)
forall a b c. (a -> b -> c) -> b -> a -> c
flip ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
starts

    -- | Synonyms for 'starts' and 'startedBy'
    precedes, precededBy      :: ComparativePredicateOf (Interval a)
    precedes      = ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
starts
    precededBy    = ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
startedBy

    -- | Does x finish y? Is x finished by y?
    finishes, finishedBy   :: ComparativePredicateOf (Interval a)
    finishes Interval a
x Interval a
y  = Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
> Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y Bool -> Bool -> Bool
&& Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
y
    finishedBy    = ComparativePredicateOf (Interval a)
-> ComparativePredicateOf (Interval a)
forall a b c. (a -> b -> c) -> b -> a -> c
flip ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
finishes

    -- | Is x during y? Does x contain y?
    during, contains       :: ComparativePredicateOf (Interval a)
    during   Interval a
x Interval a
y  = Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
> Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y Bool -> Bool -> Bool
&& Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
y
    contains      = ComparativePredicateOf (Interval a)
-> ComparativePredicateOf (Interval a)
forall a b c. (a -> b -> c) -> b -> a -> c
flip ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
during

    -- ** Interval Algebra utilities

    -- | Compose a list of interval relations with _or_ to create a new
    -- @'ComparativePredicateOf' 'Interval' a@. For example, 
    -- @unionPredicates [before, meets]@ creates a predicate function determining
    -- if one interval is either before or meets another interval.
    unionPredicates       :: [ComparativePredicateOf (Interval a)] ->
                              ComparativePredicateOf (Interval a)
    unionPredicates [ComparativePredicateOf (Interval a)]
fs Interval a
x Interval a
y = (ComparativePredicateOf (Interval a) -> Bool)
-> [ComparativePredicateOf (Interval a)] -> Bool
forall (t :: * -> *) a. Foldable t => (a -> Bool) -> t a -> Bool
any (\ ComparativePredicateOf (Interval a)
f -> ComparativePredicateOf (Interval a)
f Interval a
x Interval a
y) [ComparativePredicateOf (Interval a)]
fs

    -- | Operator for composing the union of two predicates
    (<|>) ::  ComparativePredicateOf (Interval a)
        -> ComparativePredicateOf (Interval a)
        -> ComparativePredicateOf (Interval a)
    (<|>) ComparativePredicateOf (Interval a)
f ComparativePredicateOf (Interval a)
g = [ComparativePredicateOf (Interval a)]
-> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
[ComparativePredicateOf (Interval a)]
-> ComparativePredicateOf (Interval a)
unionPredicates [ComparativePredicateOf (Interval a)
f, ComparativePredicateOf (Interval a)
g]

    disjointRelations :: Set (IntervalRelation a)
    disjointRelations = [IntervalRelation a] -> Set (IntervalRelation a)
forall a.
IntervalAlgebraic a =>
[IntervalRelation a] -> Set (IntervalRelation a)
toSet [IntervalRelation a
forall a. IntervalRelation a
Before, IntervalRelation a
forall a. IntervalRelation a
After, IntervalRelation a
forall a. IntervalRelation a
Meets, IntervalRelation a
forall a. IntervalRelation a
MetBy]

    withinRelations :: Set (IntervalRelation a)
    withinRelations = [IntervalRelation a] -> Set (IntervalRelation a)
forall a.
IntervalAlgebraic a =>
[IntervalRelation a] -> Set (IntervalRelation a)
toSet [IntervalRelation a
forall a. IntervalRelation a
Starts, IntervalRelation a
forall a. IntervalRelation a
During, IntervalRelation a
forall a. IntervalRelation a
Finishes, IntervalRelation a
forall a. IntervalRelation a
Equals]

    -- | Are x and y disjoint ('before', 'after', 'meets', or 'metBy')?
    disjoint               :: ComparativePredicateOf (Interval a)
    disjoint = Set (IntervalRelation a) -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
Set (IntervalRelation a) -> ComparativePredicateOf (Interval a)
predicate Set (IntervalRelation a)
forall a. IntervalAlgebraic a => Set (IntervalRelation a)
disjointRelations

    -- | Are x and y not disjoint; i.e. do they share any support?
    notDisjoint            :: ComparativePredicateOf (Interval a)
    notDisjoint = Set (IntervalRelation a) -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
Set (IntervalRelation a) -> ComparativePredicateOf (Interval a)
predicate (Set (IntervalRelation a) -> Set (IntervalRelation a)
forall a.
IntervalAlgebraic a =>
Set (IntervalRelation a) -> Set (IntervalRelation a)
complement Set (IntervalRelation a)
forall a. IntervalAlgebraic a => Set (IntervalRelation a)
disjointRelations)

    -- | A synonym for 'notDisjoint'.
    concur                 :: ComparativePredicateOf (Interval a)
    concur = ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
notDisjoint

    -- | Is x entirely *within* the endpoints of y? That is, 'during', 
    --   'starts', 'finishes', or 'equals'?
    within                 :: ComparativePredicateOf (Interval a)
    within = Set (IntervalRelation a) -> ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
Set (IntervalRelation a) -> ComparativePredicateOf (Interval a)
predicate Set (IntervalRelation a)
forall a. IntervalAlgebraic a => Set (IntervalRelation a)
withinRelations

    -- | Does x enclose y? That is, is y 'within' x?
    enclose                :: ComparativePredicateOf (Interval a)
    enclose = ComparativePredicateOf (Interval a)
-> ComparativePredicateOf (Interval a)
forall a b c. (a -> b -> c) -> b -> a -> c
flip ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
enclosedBy

    -- | Synonym for 'within'.
    enclosedBy             :: ComparativePredicateOf (Interval a)
    enclosedBy = ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
within

{- |
The 'Moment' class fixes the smallest duration of an 'Intervallic a'.
-}
class (Intervallic a, Num b, Ord b) => Moment a b| a -> b where
    moment :: b
    moment = b
1

{- |
The 'IntervalSizeable' typeclass provides functions to determine the size of
and to resize an 'Interval a'.
-}
class (Intervallic a, Moment a b, Num b, Ord b) => IntervalSizeable a b| a -> b where

    -- | Determine the duration of an 'Interval a'.
    duration :: Interval a -> b
    duration Interval a
x = a -> a -> b
forall a b. IntervalSizeable a b => a -> a -> b
diff (Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x) (Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
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 'Interval 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 :: b -> b -> Interval a -> Interval a
    expand b
l b
r Interval a
p = (a, a) -> Interval a
forall a. (a, a) -> Interval a
Interval (b -> a -> a
forall a b. IntervalSizeable a b => b -> a -> a
add b
s (a -> a) -> a -> a
forall a b. (a -> b) -> a -> b
$ Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
p, b -> a -> a
forall a b. IntervalSizeable a b => b -> a -> a
add b
e (a -> a) -> a -> a
forall a b. (a -> b) -> a -> b
$ Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
p)
      where s :: b
s = if b
l b -> b -> Bool
forall a. Ord a => a -> a -> Bool
< (forall b. Moment a b => b
forall a b. Moment a b => b
moment @a) then b
0 else b -> b
forall a. Num a => a -> a
negate b
l
            e :: b
e = if b
r b -> b -> Bool
forall a. Ord a => a -> a -> Bool
< (forall b. Moment a b => b
forall a b. Moment a b => b
moment @a) then b
0 else b
r

    -- | Expands an 'Interval a' to left by i.
    expandl :: b -> Interval a -> Interval a
    expandl b
i = b -> b -> Interval a -> Interval a
forall a b.
IntervalSizeable a b =>
b -> b -> Interval a -> Interval a
expand b
i b
0

    -- | Expands an 'Interval a' to right by i.
    expandr :: b -> Interval a -> Interval a
    expandr = b -> b -> Interval a -> Interval a
forall a b.
IntervalSizeable a b =>
b -> b -> Interval a -> Interval a
expand b
0

    -- | Safely creates an 'Interval a' using @x@ as the 'begin' and adding 
    --   @max moment dur@ to @x@ as the 'end'.
    beginerval :: b -> a -> Interval a
    beginerval b
dur a
x = (a, a) -> Interval a
forall a. (a, a) -> Interval a
Interval (a
x, b -> a -> a
forall a b. IntervalSizeable a b => b -> a -> a
add (b -> b -> b
forall a. Ord a => a -> a -> a
max (forall b. Moment a b => b
forall a b. Moment a b => b
moment @a) b
dur) a
x)

    -- | Safely creates an 'Interval a' using @x@ as the 'end' and adding
    --   @negate max moment dur@ to @x@ as the 'begin'.
    enderval :: b -> a -> Interval a
    enderval b
dur a
x = (a, a) -> Interval a
forall a. (a, a) -> Interval a
Interval (b -> a -> a
forall a b. IntervalSizeable a b => b -> a -> a
add (b -> b
forall a. Num a => a -> a
negate (b -> b) -> b -> b
forall a b. (a -> b) -> a -> b
$ b -> b -> b
forall a. Ord a => a -> a -> a
max (forall b. Moment a b => b
forall a b. Moment a b => b
moment @a) b
dur) a
x, a
x)

{- |
The @'IntervalCombinable'@ typeclass provides methods for (possibly) combining
two @'Interval's@.
-}
class (IntervalAlgebraic a) => IntervalCombinable a where

    -- | Maybe form a new @'Interval'@ by the union of two @'Interval'@s that 'meets'.
    (.+.) :: Interval a -> Interval a -> Maybe (Interval a)
    (.+.) Interval a
x Interval a
y
      | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`meets` Interval a
y = Interval a -> Maybe (Interval a)
forall a. a -> Maybe a
Just (Interval a -> Maybe (Interval a))
-> Interval a -> Maybe (Interval a)
forall a b. (a -> b) -> a -> b
$ (a, a) -> Interval a
forall a. (a, a) -> Interval a
Interval (Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
x, Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
y)
      | Bool
otherwise   = Maybe (Interval a)
forall a. Maybe a
Nothing

    -- | Creates a new @Interval@ spanning the extent x and y
    extenterval :: Interval a -> Interval a -> Interval a
    extenterval Interval a
x Interval a
y = (a, a) -> Interval a
forall a. (a, a) -> Interval a
Interval (a
s, a
e)
       where s :: a
s = a -> a -> a
forall a. Ord a => a -> a -> a
min (Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
x) (Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y)
             e :: a
e = a -> a -> a
forall a. Ord a => a -> a -> a
max (Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x) (Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
y)

    -- | 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'.
    (><) ::  Interval a -> Interval a -> Maybe (Interval a)
    (><) Interval a
x Interval a
y
        | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`before` Interval a
y = Interval a -> Maybe (Interval a)
forall a. a -> Maybe a
Just (Interval a -> Maybe (Interval a))
-> Interval a -> Maybe (Interval a)
forall a b. (a -> b) -> a -> b
$ (a, a) -> Interval a
forall a. (a, a) -> Interval a
Interval ( Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x, Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y )
        | Bool
otherwise    = Maybe (Interval a)
forall a. Maybe a
Nothing

    -- | 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 (Interval a)), Applicative f) =>
            Interval a ->
            Interval a ->
            f (Interval a)
    (<+>) Interval a
x Interval a
y
      | Interval a
x ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
`before` Interval a
y = Interval a -> f (Interval a)
forall (f :: * -> *) a. Applicative f => a -> f a
pure Interval a
x f (Interval a) -> f (Interval a) -> f (Interval a)
forall a. Semigroup a => a -> a -> a
<> Interval a -> f (Interval a)
forall (f :: * -> *) a. Applicative f => a -> f a
pure Interval a
y
      | Bool
otherwise    = Interval a -> f (Interval a)
forall (f :: * -> *) a. Applicative f => a -> f a
pure ( Interval a -> Interval a -> Interval a
forall a.
IntervalCombinable a =>
Interval a -> Interval a -> Interval a
extenterval Interval a
x Interval a
y )

    -- | Forms a 'Just' new interval from the intersection of two intervals, 
    --   provided the intervals are not disjoint.
    intersect :: Interval a -> Interval a -> Maybe (Interval a)
    intersect Interval a
x Interval a
y
       | ComparativePredicateOf (Interval a)
forall a.
IntervalAlgebraic a =>
ComparativePredicateOf (Interval a)
disjoint Interval a
x Interval a
y = Maybe (Interval a)
forall a. Maybe a
Nothing
       | Bool
otherwise    = Interval a -> Maybe (Interval a)
forall a. a -> Maybe a
Just (Interval a -> Maybe (Interval a))
-> Interval a -> Maybe (Interval a)
forall a b. (a -> b) -> a -> b
$ (a, a) -> Interval a
forall a. (a, a) -> Interval a
Interval (a
b, a
e)
           where b :: a
b = a -> a -> a
forall a. Ord a => a -> a -> a
max (Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
x) (Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y)
                 e :: a
e = a -> a -> a
forall a. Ord a => a -> a -> a
min (Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x) (Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
y)



{-
Instances
-}

-- | Imposes a total ordering on @'Interval' a@ based on first ordering the 
--   'begin's then the 'end's.
instance (Intervallic a) => Ord (Interval a) where
    <= :: Interval a -> Interval a -> Bool
(<=) Interval a
x Interval a
y
      | Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
<  Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y = Bool
True
      | Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
x a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y = Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
<= Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
y
      | Bool
otherwise = Bool
False
    < :: Interval a -> Interval a -> Bool
(<)  Interval a
x Interval a
y
      | Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
<  Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y = Bool
True
      | Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
x a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
y = Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
y
      | Bool
otherwise = Bool
False

instance (Intervallic a, Show a) => Show (Interval a) where
   show :: Interval a -> String
show Interval a
x = String
"(" String -> ShowS
forall a. [a] -> [a] -> [a]
++ a -> String
forall a. Show a => a -> String
show (Interval a -> a
forall a. Intervallic a => Interval a -> a
begin Interval a
x) String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
", " String -> ShowS
forall a. [a] -> [a] -> [a]
++ a -> String
forall a. Show a => a -> String
show (Interval a -> a
forall a. Intervallic a => Interval a -> a
end Interval a
x) String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
")"

instance Intervallic Int
instance IntervalAlgebraic Int
instance IntervalCombinable Int
instance Moment Int Int
instance IntervalSizeable Int Int where
    add :: Int -> Int -> Int
add = Int -> Int -> Int
forall a. Num a => a -> a -> a
(+)
    diff :: Int -> Int -> Int
diff = (-)

instance Intervallic Integer
instance IntervalAlgebraic Integer
instance IntervalCombinable Integer
instance Moment Integer Integer
instance IntervalSizeable Integer Integer where
    add :: Integer -> Integer -> Integer
add = Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
(+)
    diff :: Integer -> Integer -> Integer
diff = (-)

instance Intervallic DT.Day
instance IntervalAlgebraic DT.Day
instance IntervalCombinable DT.Day
instance Moment DT.Day Integer
instance IntervalSizeable DT.Day Integer where
    add :: Integer -> Day -> Day
add = Integer -> Day -> Day
addDays
    diff :: Day -> Day -> Integer
diff = Day -> Day -> Integer
diffDays