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cleananditasks
cleanplatform
Commits
0ee632f0
Commit
0ee632f0
authored
May 14, 2020
by
Peter Achten
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Data.SetBy is the higherorder version of Data.Set
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src/libraries/OSIndependent/Data/SetBy.dcl
src/libraries/OSIndependent/Data/SetBy.dcl
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src/libraries/OSIndependent/Data/SetBy.icl
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0ee632f0
definition
module
Data
.
SetBy
/**
* An efficient implementation of sets.
*
* This version is the same as Data.Set, except that the overloaded API is replaced
* with a higherorder function API.
*
* For all documentation, please consult Data.Set.
*
* The `morally equivalent` function from Data.Set is added in the comment.
*
* When using the functions in Data.SetBy, make sure to use the same higherorder
* function parameter for the same data structure to ensure internal integrity.
* This higherorder function represents the < ordering on your set elements and
* should have the usual ordering properties:
*
*  if a < b and b < c then a < c
*  if a < b then not (b < a)
*  if not (a < b) and not (b < a) then a and b are considered to 'equal'
*
*/
from
StdOverloaded
import
class
==,
class
<
(..)
from
StdClass
import
class
Ord
(..),
<=,
>
from
StdList
import
foldl
,
map
from
Data
.
Maybe
import
::
Maybe
from
StdBool
import
not
,
&&
from
Data
.
GenLexOrd
import
::
LexOrd
import
qualified
Data
.
Foldable
from
Data
.
Foldable
import
class
Foldable
::
SetBy
a
=
TipBy

BinBy
!
Int
!
a
!(
SetBy
a
)
!(
SetBy
a
)
instance
==
(
SetBy
a
)

==
a
/**
* True iff first set is `smaller` than second set, according to
* first argument (assuming the two sets are ordered with the
* same first function argument).
*
* Morally equivalent function: instance < (Set a)  < a
*/
isOrderedBy
::
!(
a
a
>
Bool
)
!(
SetBy
a
)
!(
SetBy
a
)
>
Bool
/**
* EQ iff the two sets have the same number of elements, occurring in the
* same order.
* LT iff the first set is the common prefix of the second set or the common
* prefix is followed in the first set with an element that is considered
* than the corresponding element in the second set.
* GT iff the second set is the common prefix of the first set or the common
* prefix is followed in the second set with an element that is considered
* greater than the corresponding element in the first set.
* The comparison of elements is done with the first function argument.
*
* Morally equivalent function: derive gLexOrd Set
*/
lexOrdBy
::
!(
a
a
>
Bool
)
!(
SetBy
a
)
!(
SetBy
a
)
>
LexOrd
instance
Foldable
SetBy
/**
* True iff this is the empty set.
* @type (SetBy a) > Bool
* @property equivalence with size 0: A.s :: SetBy a:
* size s == 0 <==> null s
* @property equivalence with newSet: A.s :: SetBy a:
* s == newSetBy <==> null s
*/
null
s
:==
case
s
of
TipBy
>
True
(
BinBy
sz
_
_
_)
>
False
/**
* The number of elements in the set.
* @type (SetBy a) > Int
* @property correctness: A.s :: SetBy a:
* size s =.= length (toList s)
*/
size
s
:==
case
s
of
TipBy
>
0
(
BinBy
sz
_
_
_)
>
sz
/**
* Is the element in the set?
*
* Morally equivalent function: Data.Set.member x s = Data.SetBy.memberBy (<) x s
*/
memberBy
::
!(
a
a
>
Bool
)
!
a
!(
SetBy
a
)
>
Bool
/**
* Checks if an element is not in the set.
*/
notMemberBy
comp
x
t
:==
not
(
memberBy
comp
x
t
)
/**
* Is t1 a subset of t2?
*
* Morally equivalent function: Data.Set.isSubsetOf s1 s2 = Data.SetBy.isSubsetOfBy (<) s1 s2
*/
isSubsetOfBy
comp
t1
t2
:==
(
size
t1
<=
size
t2
)
&&
(
isSubsetOfXBy
comp
t1
t2
)
isSubsetOfXBy
::
!(
a
a
>
Bool
)
!(
SetBy
a
)
!(
SetBy
a
)
>
Bool
/**
* Is t1 a proper subset of t2?
*
* Morally equivalent function: Data.Set.isProperSubsetOf s1 s2 = Data.SetBy.isProperSubsetOfBy (<) s1 s2
*/
isProperSubsetOfBy
comp
s1
s2
:==
(
size
s1
<
size
s2
)
&&
(
isSubsetOfBy
comp
s1
s2
)
/**
* The empty set.
* @complexity O(1)
* @property is null:
* null newSetBy
*/
newSetBy
::
SetBy
a
/**
* Create a singleton set.
* @complexity O(1)
*/
singletonBy
::
!
u
:
a
>
w
:(
SetBy
u
:
a
),
[
w
<=
u
]
/**
* Insert an element in a set. If the set already contains an element equal to
* the given value, it is replaced with the new value.
*
* Morally equivalent function: Data.Set.insert x s = Data.SetBy.insertBy (<) x s
*/
insertBy
::
!(
a
a
>
Bool
)
!
a
!.(
SetBy
a
)
>
SetBy
a
/**
* Delete an element from a set.
*
* Morally equivalent function: Data.Set.delete x s = Data.SetBy (<) x s
*/
deleteBy
::
!(
a
a
>
Bool
)
!
a
!.(
SetBy
a
)
>
SetBy
a
/**
* The minimal element of a set.
*
* Morally equivalent function: Data.Set.findMin
*/
findMin
::
!(
SetBy
a
)
>
a
/**
* The maximal element of a set.
*
* Morally equivalent function: Data.Set.findMax
*/
findMax
::
!(
SetBy
a
)
>
a
/**
* Delete the minimal element.
*
* Morally equivalent function: Data.Set.deleteMin
*/
deleteMin
::
!.(
SetBy
a
)
>
SetBy
a
/**
* Delete the maximal element.
*
* Morally equivalent function: Data.Set.deleteMax
*/
deleteMax
::
!.(
SetBy
a
)
>
SetBy
a
/**
* deleteFindMin set = (findMin set, deleteMin set)
*/
deleteFindMin
::
!.(
SetBy
a
)
>
(!
a
,
!
SetBy
a
)
/**
* deleteFindMax set = (findMax set, deleteMax set)
*/
deleteFindMax
::
!.(
SetBy
a
)
>
(!
a
,
!
SetBy
a
)
/**
* Retrieves the minimal key of the set, and the set stripped of that element,
* or 'Nothing' if passed an empty set.
*/
minView
::
!.(
SetBy
a
)
>
.(
Maybe
(!
a
,
!
SetBy
a
))
/**
* Retrieves the maximal key of the set, and the set stripped of that element,
* or 'Nothing' if passed an empty set.
*/
maxView
::
!.(
SetBy
a
)
>
.(
Maybe
(!
a
,
!
SetBy
a
))
/**
* The union of two sets, preferring the first set when equal elements are
* encountered.
*
* Morally equivalent function: Data.Set.union s1 s2 = Data.SetBy.unionBy (<) s1 s2
*/
unionBy
::
!(
a
a
>
Bool
)
!
u
:(
SetBy
a
)
!
u
:(
SetBy
a
)
>
SetBy
a
/**
* The union of a list of sets.
*
* Morally equivalent function: Data.Set.unions ts = Data.SetBy.unionsBy (<) ts
*/
unionsBy
ts
:==
foldl
unionBy
newSetBy
ts
/**
* Difference of two sets.
*
* Morally equivalent function: Data.Set.difference s1 s2 = Data.SetBy.differenceBy (<) s1 s2
*/
differenceBy
::
!(
a
a
>
Bool
)
!(
SetBy
a
)
!(
SetBy
a
)
>
SetBy
a
/**
* The intersection of two sets.
* Elements of the result come from the first set.
*
* Morally equivalent function: Data.Set.intersection s1 s2 = Data.SetBy.intersectionBy (<) s1 s2
*/
intersectionBy
::
!(
a
a
>
Bool
)
!(
SetBy
a
)
!(
SetBy
a
)
>
SetBy
a
/**
* The intersection of a list of sets.
* Elements of the result come from the first set
*
* Morally equivalent function: Data.Set.intersections ts = Data.SetBy.intersectionsBy (<) ts
*/
intersectionsBy
::
!(
a
a
>
Bool
)
![
SetBy
a
]
>
SetBy
a
/**
* Filter all elements that satisfy the predicate.
*
* Morally equivalent function: Data.Set.filter
*/
filter
::
!(
a
>
Bool
)
!(
SetBy
a
)
>
SetBy
a
/**
* Partition the set into two sets, one with all elements that satisfy the
* predicate and one with all elements that don't satisfy the predicate.
*
* Morally equivalent function: Data.Set.partition
*/
partition
::
!(
a
>
Bool
)
!(
SetBy
a
)
>
(!
SetBy
a
,
!
SetBy
a
)
/**
* Split a set in elements less and elements greater than a certain pivot.
*
* Morally equivalent function: Data.Set.split x s = Data.SetBy.splitBy (<) x s
*/
splitBy
::
!(
a
a
>
Bool
)
!
a
!(
SetBy
a
)
>
(!
SetBy
a
,
!
SetBy
a
)
/**
* Performs a 'split' but also returns whether the pivot element was found in
* the original set.
*
* Morally equivalent function: Data.Set.splitMember x s = Data.SetBy.splitMemberBy (<) x s
*/
splitMemberBy
::
!(
a
a
>
Bool
)
!
a
!(
SetBy
a
)
>
(!
SetBy
a
,
!
Bool
,
!
SetBy
a
)
/**
* Convert the set to an ascending list of elements.
*/
toList
s
:==
toAscList
s
/**
* Same as toList.
*/
toAscList
t
:==
'
Data
.
Foldable
'.
foldr`
(\
a
as
>
[
a
:
as
])
[]
t
/**
* Create a set from a list of elements.
*
* Morally equivalent function: Data.Set.fromList xs = Data.SetBy.fromListBy (<) xs
*/
fromListBy
::
!(
a
a
>
Bool
)
![
a
]
>
SetBy
a
/**
* Map a function to all elements in a set.
*
* Morally equivalent function: Data.Set.mapSet f s = Data.SetBy.mapSetBy (<) f s
*/
mapSetBy
comp_b
f
s
:==
fromListBy
comp_b
(
map
f
(
toList
s
))
/**
* Map a set without converting it to and from a list.
*
* Morally equivalent function: Data.Set.mapSetMonotonic
*/
mapSetByMonotonic
::
!(
a
>
b
)
!(
SetBy
a
)
>
SetBy
b
src/libraries/OSIndependent/Data/SetBy.icl
0 → 100644
View file @
0ee632f0
implementation
module
Data
.
SetBy
import
StdClass
,
StdMisc
,
StdBool
,
StdFunc
,
StdInt
import
Data
.
Maybe
from
Data
.
GenLexOrd
import
::
LexOrd
(..)
import
Data
.
Monoid
from
Data
.
Foldable
import
class
Foldable
(..)
import
qualified
StdList
from
StdList
import
instance
==
[
a
]
// PA: this is a dangerous function... shouldn't be part of the API
mapSetByMonotonic
::
!(
a
>
b
)
!(
SetBy
a
)
>
SetBy
b
mapSetByMonotonic
_
TipBy
=
TipBy
mapSetByMonotonic
f
(
BinBy
n
x
l
r
)
=
BinBy
n
(
f
x
)
(
mapSetByMonotonic
f
l
)
(
mapSetByMonotonic
f
r
)
/*
* Sets are size balanced trees.
* A set of values @a@.
*/
::
SetBy
a
=
TipBy

BinBy
!
Int
!
a
!(
SetBy
a
)
!(
SetBy
a
)
instance
==
(
SetBy
a
)

==
a
where
(==)
t1
t2
=
size
t1
==
size
t2
&&
toAscList
t1
==
toAscList
t2
isOrderedBy
::
!(
a
a
>
Bool
)
!(
SetBy
a
)
!(
SetBy
a
)
>
Bool
isOrderedBy
comp
s1
s2
=
compare
comp
(
toAscList
s1
)
(
toAscList
s2
)
where
compare
::
!(
a
a
>
Bool
)
![
a
]
![
a
]
>
Bool
compare
_
[]
[]
=
False
compare
_
[]
_
=
True
compare
_
[_:_]
[]
=
False
compare
comp
[
a
:
as
]
[
b
:
bs
]

comp
a
b
=
True

comp
b
a
=
False

otherwise
=
compare
comp
as
bs
lexOrdBy
::
!(
a
a
>
Bool
)
!(
SetBy
a
)
!(
SetBy
a
)
>
LexOrd
lexOrdBy
comp
s1
s2
=
ordby
comp
(
toAscList
s1
)
(
toAscList
s2
)
where
ordby
::
!(
a
a
>
Bool
)
![
a
]
![
a
]
>
LexOrd
ordby
_
[]
[]
=
EQ
ordby
_
[]
_
=
LT
ordby
_
[_:_]
[]
=
GT
ordby
comp
[
a
:
as
]
[
b
:
bs
]

comp
a
b
=
LT

comp
b
a
=
GT

otherwise
=
ordby
comp
as
bs
instance
Foldable
SetBy
where
foldr
f
z
(
BinBy
_
x
l
r
)
=
foldr
f
(
f
x
(
foldr
f
z
r
))
l
foldr
_
z
_
=
z
foldr`
f
z
(
BinBy
_
x
l
r
)
=
foldr`
f
(
f
x
(
foldr`
f
z
r
))
l
foldr`
_
z
_
=
z
foldl
f
z
(
BinBy
_
x
l
r
)
=
foldl
f
(
f
(
foldl
f
z
l
)
x
)
r
foldl
_
z
_
=
z
foldl`
f
z
(
BinBy
_
x
l
r
)
=
foldl`
f
(
f
(
foldl`
f
z
l
)
x
)
r
foldl`
_
z
_
=
z
/*
* Query
**/
memberBy
::
!(
a
a
>
Bool
)
!
a
!(
SetBy
a
)
>
Bool
memberBy
comp
x
(
BinBy
_
y
l
r
)

comp
x
y
=
memberBy
comp
x
l

comp
y
x
=
memberBy
comp
x
r

otherwise
=
True
memberBy
_
_
_
=
False
/*
* Construction
**/
newSetBy
::
SetBy
a
newSetBy
=
TipBy
singletonBy
::
!
u
:
a
>
w
:(
SetBy
u
:
a
),
[
w
<=
u
]
singletonBy
x
=
BinBy
1
x
TipBy
TipBy
/*
* Insertion, Deletion
**/
insertBy
::
!(
a
a
>
Bool
)
!
a
!.(
SetBy
a
)
>
SetBy
a
insertBy
comp
x
t
=:(
BinBy
_
y
l
r
)

comp
x
y
=
balanceL
y
(
insertBy
comp
x
l
)
r

comp
y
x
=
balanceR
y
l
(
insertBy
comp
x
r
)

otherwise
=
t
insertBy
_
x
_
=
singletonBy
x
deleteBy
::
!(
a
a
>
Bool
)
!
a
!.(
SetBy
a
)
>
SetBy
a
deleteBy
comp
x
(
BinBy
_
y
l
r
)

comp
x
y
=
balanceR
y
(
deleteBy
comp
x
l
)
r

comp
y
x
=
balanceL
y
l
(
deleteBy
comp
x
r
)

otherwise
=
glue
l
r
deleteBy
_
_
tip
=
tip
/*
* Subset
**/
isSubsetOfXBy
::
!(
a
a
>
Bool
)
!(
SetBy
a
)
!(
SetBy
a
)
>
Bool
isSubsetOfXBy
comp
(
BinBy
_
x
l
r
)
t

t
=:
TipBy
=
False
#!
(
lt
,
found
,
gt
)
=
splitMemberBy
comp
x
t
=
found
&&
isSubsetOfXBy
comp
l
lt
&&
isSubsetOfXBy
comp
r
gt
isSubsetOfXBy
_
_
_
=
True
/*
* Minimal, Maximal
**/
findMin
::
!(
SetBy
a
)
>
a
findMin
(
BinBy
_
x
TipBy
_)
=
x
findMin
(
BinBy
_
_
l
_)
=
findMin
l
findMin
TipBy
=
abort
"SetBy.findMin: empty set has no minimal element"
findMax
::
!(
SetBy
a
)
>
a
findMax
(
BinBy
_
x
_
TipBy
)
=
x
findMax
(
BinBy
_
_
_
r
)
=
findMax
r
findMax
TipBy
=
abort
"SetBy.findMax: empty set has no maximal element"
deleteMin
::
!.(
SetBy
a
)
>
SetBy
a
deleteMin
(
BinBy
_
_
TipBy
r
)
=
r
deleteMin
(
BinBy
_
x
l
r
)
=
balanceR
x
(
deleteMin
l
)
r
deleteMin
TipBy
=
TipBy
deleteMax
::
!.(
SetBy
a
)
>
SetBy
a
deleteMax
(
BinBy
_
_
l
TipBy
)
=
l
deleteMax
(
BinBy
_
x
l
r
)
=
balanceL
x
l
(
deleteMax
r
)
deleteMax
TipBy
=
TipBy
/*
* Union.
**/
unionBy
::
!(
a
a
>
Bool
)
!
u
:(
SetBy
a
)
!
u
:(
SetBy
a
)
>
SetBy
a
unionBy
_
t1
TipBy
=
t1
unionBy
comp
t1
(
BinBy
_
x
TipBy
TipBy
)
=
insertBy
comp
x
t1
unionBy
comp
(
BinBy
_
x
TipBy
TipBy
)
t2
=
insertBy
comp
x
t2
unionBy
_
TipBy
t2
=
t2
unionBy
comp
t1
=:(
BinBy
_
x
l1
r1
)
t2
=
link
x
l1l2
r1r2
where
(
l2
,
r2
)
=
splitS
comp
x
t2
l1l2
=
unionBy
comp
l1
l2
r1r2
=
unionBy
comp
r1
r2
splitS
::
!(
a
a
>
Bool
)
!
a
!(
SetBy
a
)
>
(!
SetBy
a
,
!
SetBy
a
)
splitS
_
_
TipBy
=
(
TipBy
,
TipBy
)
splitS
comp
x
(
BinBy
_
y
l
r
)

comp
x
y
=
let
(
lt
,
gt
)
=
splitS
comp
x
l
in
(
lt
,
link
y
gt
r
)

comp
y
x
=
let
(
lt
,
gt
)
=
splitS
comp
x
r
in
(
link
y
l
lt
,
gt
)

otherwise
=
(
l
,
r
)
/*
* Difference
**/
differenceBy
::
!(
a
a
>
Bool
)
!(
SetBy
a
)
!(
SetBy
a
)
>
SetBy
a
differenceBy
_
TipBy
_
=
TipBy
differenceBy
comp
t1
t2
=
case
t2
of
BinBy
_
x
l2
r2
>
case
splitBy
comp
x
t1
of
(
l1
,
r1
)

size
l1l2
+
size
r1r2
==
size
t1
>
t1

otherwise
>
merge
l1l2
r1r2
where
l1l2
=
differenceBy
comp
l1
l2
r1r2
=
differenceBy
comp
r1
r2
_
>
t1
/*
* Intersection
**/
intersectionsBy
::
!(
a
a
>
Bool
)
![
SetBy
a
]
>
SetBy
a
intersectionsBy
_
[
t
]
=
t
intersectionsBy
comp
[
t
:
ts
]
=
'
StdList
'.
foldl
(
intersectionBy
comp
)
t
ts
intersectionsBy
_
[]
=
abort
"SetBy.intersectionsBy called with []
\n
"
intersectionBy
::
!(
a
a
>
Bool
)
!(
SetBy
a
)
!(
SetBy
a
)
>
SetBy
a
intersectionBy
_
TipBy
_
=
TipBy
intersectionBy
_
_
TipBy
=
TipBy
intersectionBy
comp
t1
t2
=
hedgeInt
comp
NothingS
NothingS
t1
t2
hedgeInt
::
!(
a
a
>
Bool
)
!(
MaybeS
a
)
!(
MaybeS
a
)
!(
SetBy
a
)
!(
SetBy
a
)
>
SetBy
a
hedgeInt
_
_
_
_
TipBy
=
TipBy
hedgeInt
_
_
_
TipBy
_
=
TipBy
hedgeInt
comp
blo
bhi
(
BinBy
_
x
l
r
)
t2
#!
bmi
=
JustS
x
#!
l`
=
hedgeInt
comp
blo
bmi
l
(
trimBy
comp
blo
bmi
t2
)
#!
r`
=
hedgeInt
comp
bmi
bhi
r
(
trimBy
comp
bmi
bhi
t2
)
=
if
(
memberBy
comp
x
t2
)
(
link
x
l`
r`
)
(
merge
l`
r`
)
/*
* Filter and partition
**/
filter
::
!(
a
>
Bool
)
!(
SetBy
a
)
>
SetBy
a
filter
p
(
BinBy
_
x
l
r
)

p
x
=
link
x
(
filter
p
l
)
(
filter
p
r
)

otherwise
=
merge
(
filter
p
l
)
(
filter
p
r
)
filter
_
tip
=
tip
partition
::
!(
a
>
Bool
)
!(
SetBy
a
)
>
(!
SetBy
a
,
!
SetBy
a
)
partition
p
(
BinBy
_
x
l
r
)
#!
(
l1
,
l2
)
=
partition
p
l
#!
(
r1
,
r2
)
=
partition
p
r

p
x
=
(
link
x
l1
r1
,
merge
l2
r2
)

otherwise
=
(
merge
l1
r1
,
link
x
l2
r2
)
partition
_
t
=
(
t
,
t
)
/*
* Lists
**/
fromListBy
::
!(
a
a
>
Bool
)
![
a
]
>
SetBy
a
fromListBy
comp
xs
=
'
StdList
'.
foldl
(
ins
comp
)
newSetBy
xs
where
ins
::
!(
a
a
>
Bool
)
!(
SetBy
a
)
!
a
>
SetBy
a
ins
comp
t
x
=
insertBy
comp
x
t
/*
Utility functions that return subranges of the original
tree. Some functions take a comparison function as argument to
allow comparisons against infinite values. A function [cmplo x]
should be read as [compare lo x].
[trimBy comp cmplo cmphi t] A tree that is either empty or where [cmplo x == LT]
and [cmphi x == GT] for the value [x] of the root.
[splitBy comp k t] Returns two trees [l] and [r] where all values
in [l] are <[k] and all keys in [r] are >[k].
[splitMemberBy comp k t] Just like [splitBy] but also returns whether [k]
was found in the tree.
*/
::
MaybeS
a
=
NothingS

JustS
!
a
/*
[trimBy comp lo hi t] trims away all subtrees that surely contain no
values between the range [lo] to [hi]. The returned tree is either
empty or the key of the root is between @lo@ and @hi@.
*/
trimBy
::
!(
a
a
>
Bool
)
!(
MaybeS
a
)
!(
MaybeS
a
)
!(
SetBy
a
)
>
SetBy
a
trimBy
_
NothingS
NothingS
t
=
t
trimBy
comp
(
JustS
lx
)
NothingS
t
=
greater
comp
lx
t
where
greater
comp
lo
(
BinBy
_
x
_
r
)

not
(
comp
lo
x
)
=
greater
comp
lo
r
greater
_
_
t`
=
t`
trimBy
comp
NothingS
(
JustS
hx
)
t
=
lesser
comp
hx
t
where
lesser
comp
hi
(
BinBy
_
x
l
_)

not
(
comp
x
hi
)
=
lesser
comp
hi
l
lesser
_
_
t`
=
t`
trimBy
comp
(
JustS
lx
)
(
JustS
hx
)
t
=
middle
comp
lx
hx
t
where
middle
comp
lo
hi
(
BinBy
_
x
_
r
)

not
(
comp
lo
x
)
=
middle
comp
lo
hi
r
middle
comp
lo
hi
(
BinBy
_
x
l
_)

not
(
comp
x
hi
)
=
middle
comp
lo
hi
l
middle
_
_
_
t`
=
t`
/*
* Split
**/
splitBy
::
!(
a
a
>
Bool
)
!
a
!(
SetBy
a
)
>
(!
SetBy
a
,
!
SetBy
a
)
splitBy
comp
x
(
BinBy
_
y
l
r
)

comp
x
y
#!
(
lt
,
gt
)
=
splitBy
comp
x
l
=
(
lt
,
link
y
gt
r
)

comp
y
x
#!
(
lt
,
gt
)
=
splitBy
comp
x
r
=
(
link
y
l
lt
,
gt
)

otherwise
=
(
l
,
r
)
splitBy
_
_
t
=
(
t
,
t
)
splitMemberBy
::
!(
a
a
>
Bool
)
!
a
!(
SetBy
a
)
>
(!
SetBy
a
,
!
Bool
,
!
SetBy
a
)
splitMemberBy
comp
x
(
BinBy
_
y
l
r
)

comp
x
y
#!
(
lt
,
found
,
gt
)
=
splitMemberBy
comp
x
l
=
(
lt
,
found
,
link
y
gt
r
)

comp
y
x
#!
(
lt
,
found
,
gt
)
=
splitMemberBy
comp
x
r
=
(
link
y
l
lt
,
found
,
gt
)

otherwise
=
(
l
,
True
,
r
)
splitMemberBy
_
_
t
=
(
t
,
False
,
t
)
/*
Utility functions that maintain the balance properties of the tree.
All constructors assume that all values in [l] < [x] and all values
in [r] > [x], and that [l] and [r] are valid trees.
In order of sophistication:
[BinBy sz x l r] The type constructor.
[bin x l r] Maintains the correct size, assumes that both [l]
and [r] are balanced with respect to each other.
[balance x l r] Restores the balance and size.