10.11 List Operations—library(lists)

This library module provides operations on lists. Exported predicates:

select(?Element, ?Set, ?Residue)

is true when Set is a list, Element occurs in Set, and Residue is everything in Set except Element (things stay in the same order).

selectchk(+Element, +Set, ?Residue)

is to select/3 what memberchk/2 is to member/2. That is, it locates the first occurrence of Element in Set, and deletes it, giving Residue. It is steadfast in Residue.

append(+ListOfLists, -List)

is true when ListOfLists is a list [L1,...,Ln] of lists, List is a list, and appending L1, ..., Ln together yields List. The ListOfLists must be a proper list. Additionally, either List should be a proper list, or each of L1, ..., Ln should be a proper list. The behaviour on non-lists is undefined. ListOfLists must be proper because for any given solution, infinitely many more can be obtained by inserting nils ([]) into ListOfList.

append(?Prefix, ?Tail1, ?List1, ?Tail2, ?List2)

is true when append(Prefix, Tail1, List1) and append(Prefix, Tail2, List2) are both true. You could call append/3 twice, but that is order- dependent. This will terminate if Prefix is a proper list or if either List1 or List2 is a proper list.

correspond(?X, ?Xlist, ?Ylist, ?Y)

is true when Xlist and Ylist are lists, X is an element of Xlist, Y is an element of Ylist, and X and Y are in similar places in their lists. No relation is implied between other elements of Xlist and Ylist. For a similar predicate without the cut, see select/4.

delete(+List, +Kill, -Residue)

is true when List is a list, in which Kill may or may not occur, and Residue is a copy of List with all elements equal to Kill deleted. To extract a single copy of Kill, use select(Kill, List, Residue). If List is not proper, delete/3 will fail. Kill and the elements of List should be sufficiently instantiated for \= to be sound.

delete(+List, +Kill, +Count, -Residue)

is true when List is a list, in which Kill may or may not occur, and Count is a non-negative integer, and Residue is a copy of List with the first Count elements equal to Kill deleted. If List has fewer than Count elements equal to Count, all of them are deleted. If List is not proper, delete/4 may fail. Kill and the elements of List should be sufficiently instantiated for \= to be sound.

is_list(+List)

succeeds when List is a proper list. That is, List is nil ([]) or a cons cell ([Head|Tail]) whose Tail is a proper list. A variable, or a list whose final tail is a variable, will fail this test.

keys_and_values(?[K1-V1,...,Kn-Vn], ?[K1,...,Kn], ?[V1,...,Vn])

is true when its arguments look like the picture above. It is meant for splitting a list of Key-Value pairs (such as keysort/2 wants and produces) into separate lists of Keys and of Values. It may just as well be used for building a list of pairs from a pair of lists. In fact one usually wants just the keys or just the values, but you can supply _ as the other argument. For example, suppose you wanted to sort a list without having duplicates removed. You could do

              keys_and_values(RawPairs, RawKeys, _),
              keysort(RawPairs, OrdPairs),
              keys_and_values(OrdPairs, OrdKeys, _).
     

last(+List, -Last)

is true when List is a List and Last is its last element. This could be defined as

              last(L, X) :- append(_, [X], L).
     

nextto(?X, ?Y, +List)

is true when X and Y appear side-by-side in List. It could be written as

              nextto(X, Y, List) :- append(_, [X,Y|_], List).
     

It may be used to enumerate successive pairs from the list. List should be proper, otherwise nextto/3 will generate it.

nth0(?N, ?List, ?Elem)

is true when Elem is the Nth member of List, counting the first as element 0. That is, throw away the first N elements and unify Elem with the next. E.g. nth0(0, [H|T], H). Either N should be an integer, or List should be proper.

nth1(?N, ?List, ?Element)

is true when Elem is the Nth member of List, counting the first as element 1. That is, throw away the first N-1 elements and unify Elem with the next element (the Nth). E.g. nth1(1, [H|T], H). This is just like nth0/3 except that it counts from 1 instead of 0. Either N should be an integer, or List should be proper.

nth0(?N, ?List, ?Elem, ?Rest)

unifies Elem with the Nth element of List, counting from 0, and Rest with the other elements. It can be used to select the Nth element of List (yielding Elem and Rest), or to insert Elem before the Nth (counting from 0) element of Rest, when it yields List, e.g. nth0(2, List, c, [a,b,d,e]) unifies List with [a,b,c,d,e]. This can be seen as inserting Elem after the Nth element of Rest if you count from 1 rather than 0. Either N should be an integer, or List or Rest should be proper.

nth1(?N, ?List, ?Elem, ?Rest)

unifies Elem with the Nth element of List, counting from 1, and Rest with the other elements. It can be used to select the Nth element of List (yielding Elem and Rest), or to insert Elem before the Nth (counting from 1) element of Rest, when it yields List, e.g. nth1(2, List, b, [a,c,d,e]) unifies List with [a,b,c,d,e]. Either N should be an integer, or List or Rest should be proper.

one_longer(?Longer, ?Shorter)

is true when

              length(Longer,N), length(Shorter,M), succ(M,N)
     

for some integers M, N. It was written to make {nth0,nth1}/4 able to find the index, just as same_length/2 is useful for making things invertible.

perm(+List, ?Perm)

is true when List and Perm are permutations of each other. The main use of perm/2 is to generate permutations. You should not use this predicate in new programs; use permutation/2 instead. List must be a proper list. Perm may be partly instantiated.

permutation(?List, ?Perm)

is true when List and Perm are permuations of each other. Unlike perm/2, it will work even when List is not a proper list. It even acts in a marginally sensible way when Perm isn't proper either, but it will still backtrack forever. Be careful: this is quite efficient, but the number of permutations of an N-element list is N!, and even for a 7-element list that is 5040.

perm2(?A,?B, ?C,?D)

is true when {A,B} = {C,D}. It is very useful for writing pattern matchers over commutative operators.

proper_length(+List, ?Length)

succeeds when List is a proper list, binding Length to its length.

remove_dups(+List, ?Pruned)

removes duplicated elements from List, which should be a proper list. If List has non-ground elements, Pruned may contain elements which unify; two elements will remain separate iff there is a substitution which makes them different. E.g. [X,X] -> [X] but [X,Y] -> [X,Y].

reverse(?List, ?Reversed)

is true when List and Reversed are lists with the same elements but in opposite orders. Either List or Reversed should be a proper list: given either argument the other can be found. If both are incomplete reverse/2 can backtrack forever trying ever longer lists.

rev(+List, ?Reversed)

is a version of reverse/2 which only works one way around. Its List argument must be a proper list whatever Reversed is. You should use reverse/2 in new programs, though rev/2 is faster when it is safe to use it.

same_length(?List1, ?List2)

is true when List1 and List2 are both lists and have the same number of elements. No relation between the values of their elements is implied. It may be used to generate either list given the other, or indeed to generate two lists of the same length, in which case the arguments will be bound to lists of length 0, 1, 2, ...

same_length(?List1, ?List2, ?Length)

is true when List1 and List2 are both lists, Length is a non-negative integer, and both List1 and List2 have exactly Length elements. No relation between the elements of the lists is implied. If Length is instantiated, or if either List1 or List2 is bound to a proper list, same_length is determinate and terminating.

select(?X, ?Xlist, ?Y, ?Ylist)

is true when X is the Kth member of Xlist and Y the Kth element of Ylist for some K, and apart from that Xlist and Ylist are the same. You can use it to replace X by Y or vice versa. Either Xlist or Ylist should be a proper list.

selectchk(?X, +Xlist, ?Y, +Ylist)

is to select/4 as memberhck/2 is to member/2. That is, it finds the first K such that X unifies with the Kth element of Xlist and Y with the Kth element of Ylist, and it commits to the bindings thus found. If you have Keys and Values in "parallel" lists, you can use this to find the Value associated with a particular Key (much better methods exist). Except for argument order, this is identical to correspond/4, but selectchk/4 is a member of a coherent family. Note that the arguments are like the arguments of memberchk/2, twice.

shorter_list(?Short, ?Long)

is true when Short is a list is strictly shorter than Long. Long doesn't have to be a proper list provided it is long enough. This can be used to generate lists shorter than Long, lengths 0, 1, 2... will be tried, but backtracking will terminate with a list that is one element shorter than Long. It cannot be used to generate lists longer than Short, because it doesn't look at all the elements of the longer list.

subseq(?Sequence, ?SubSequence, ?Complement)

is true when SubSequence and Complement are both subsequences of the list Sequence (the order of corresponding elements being preserved) and every element of Sequence which is not in SubSequence is in the Complement and vice versa. That is, length(Sequence) = length(SubSequence)+length(Complement), e.g. subseq([1,2,3,4], [1,3,4], [2]). This was written to generate subsets and their complements together, but can also be used to interleave two lists in all possible ways.

subseq0(+Sequence, ?SubSequence)

is true when SubSequence is a subsequence of Sequence, but may be Sequence itself. Thus subseq0([a,b], [a,b]) is true as well as subseq0([a,b], [a]). Sequence must be a proper list, since there are infinitely many lists with a given SubSequence.

          ?- setof(X, subseq0([a,b,c],X), Xs).
          Xs = [[],[a],[a,b],[a,b,c],[a,c],[b],[b,c],[c]]
          ?- bagof(X, subseq0([a,b,c,d],X), Xs).
          Xs = [[a,b,c,d],[b,c,d],[c,d],[d],[],[c],[b,d],[b],[b,c],[a,c,d],
                [a,d],[a],[a,c],[a,b,d],[a,b],[a,b,c]]
     

subseq1(+Sequence, ?SubSequence)

is true when SubSequence is a proper subsequence of Sequence, that is it contains at least one element less. Sequence must be a proper list, as SubSequence does not determine Sequence.

sumlist(+Numbers, ?Total)

is true when Numbers is a list of integers, and Total is their sum.

transpose(?X, ?Y)

is true when X is a list of the form [[X11,...,X1m],...,[Xn1,...,Xnm]] and Y is its transpose, that is, Y = [[X11,...,Xn1],...,[X1m,...,Xnm]] We insist that both lists should have this rectangular form, so that the predicate can be invertible. For the same reason, we reject empty arrays with m = 0 or n = 0.

append_length(?Prefix, ?Suffix, ?List, ?Length)

is true when

              append(Prefix, Suffix, List), length(Prefix, Length).
     

The normal use of this is to split a List into a Prefix of a given Length and the corresponding Suffix, but it can be used any way around provided that Length is instantiated, or Prefix is a proper list, or List is a proper list.

append_length(?Suffix, ?List, ?Length)

is true when there exists a list Prefix such that append_length(Prefix, Suffix, List, Length) is true. When you don't want to know the Prefix, you should call this predicate, because it doesn't construct the Prefix argument, which append_length/4 would do.

prefix_length(?List, ?Prefix, ?Length)

is true when

              prefix(List, Prefix) &
              length(Prefix, Length).
     

The normal use of this is to find the first Length elements of a given List, but it can be used any way around provided that Length is instantiated, or Prefix is a proper list, or List is a proper list. It is identical in effect to append_length(Prefix, _, List, Length).

proper_prefix_length(?List, ?Prefix, ?Length)

is true when

              proper_prefix(List, Prefix) &
              length(Prefix, Length).
     

The normal use of this is to find the first Length elements of a given List, but it can be used any way around provided that Length is instantiated, or Prefix is a proper list, or List is a proper list. It is logically equivalent to prefix(Prefix, List, Length), Length > 0.

suffix_length(+List, ?Suffix, ?Length)

is true when

              suffix(List, Suffix) &
              length(Suffix, Length).
     

The normal use of this is to return the last Length elements of a given List. For this to be sure of termination, List must be a proper list. If Length is instantiated or Suffix is a proper list, this predicate is determinate.

proper_suffix_length(+List, ?Suffix, ?Length)

is true when

              proper_suffix(List, Suffix) &
              length(Suffix, Length).
     

The normal use of this is to return the last Length elements of a given List. For this to be sure of termination, List must be a proper list. If Length is instantiated or Suffix is a proper list, this predicate is determinate.

rotate_list(+Amount, ?List, ?Rotated)

is true when List and Rotated are lists of the same length, and

              append(Prefix, Suffix, List) &
              append(Suffix, Prefix, Rotated) &
              (   Amount >= 0 & length(Prefix, Amount)
              |   Amount =< 0 & length(Suffix, Amount)
              ).
     

That is to say, List rotated LEFT by Amount is Rotated. Amount must already be instantiated. As it is a strict input, it must come first.

rotate_list(?List, ?Rotated)

is true when rotate_list(1, List, Rotated), but is a bit less heavy-handed. rotate_list(X, Y) rotates X left one place yielding Y. rotate_list(Y, X) rotates X right one place yielding Y. Either List or Rotated should be a proper list.

sublist(+Whole, ?Part, ?Before, ?Length, ?After)

is true when

• Whole is a list – it must be proper already
• Part is a list
• Whole = Alpha || Part || Omega
• length(Alpha, Before)
• length(Part, Length)
• length(Omega, After)

cons(?Head, ?Tail, ?List)

is true when Head is the head of List and Tail is its tail. i.e. append([Head], Tail, List). No restrictions.

last(?Fore, ?Last, ?List)

is true when Last is the last element of List and Fore is the list of preceding elements, e.g. append(Fore, [Last], List). Fore or Last should be proper. It is expected that List will be proper and Fore unbound, but it will work in reverse too.

head(?List, ?Head)

is true when List is a non-empty list and Head is its head. A list has only one head. No restrictions.

tail(?List, ?Tail)

is true when List is a non-empty list and Tail is its tail. A list has only one tail. No restrictions.

prefix(?List, ?Prefix)

is true when List and Prefix are lists and Prefix is a prefix of List. It terminates if either argument is proper, and has at most N+1 solutions. Prefixes are enumerated in ascending order of length.

proper_prefix(?List, ?Prefix)

is true when List and Prefix are lists and Prefix is a proper prefix of List. That is, Prefix is a prefix of List but is not List itself. It terminates if either argument is proper, and has at most N solutions. Prefixes are enumerated in ascending order of length.

suffix(?List, ?Suffix)

is true when List and Suffix are lists and Suffix is a suffix of List. It terminates only if List is proper, and has at most N+1 solutions. Suffixes are enumerated in descending order of length.

proper_suffix(?List, ?Suffix)

is true when List and Suffix are lists and Suffix is a proper suffix of List. That is, Suffix is a suffix of List but is not List itself. It terminates only if List is proper, and has at most N solutions. Suffixes are enumerated in descending order of length.

segment(?List, ?Segment)

is true when List and Segment are lists and Segment is a segment of List. That is, List = _ <> Segment <> _ . Terminates only if List is proper. If Segment is proper it enumerates all solutions. If neither argument is proper, it would have to diagonalise to find all solutions, but it doesn't, so it is then incomplete. If Segment is proper, it has at most N+1 solutions. Otherwise, it has at most (1/2)(N+1)(N+2) solutions.

proper_segment(?List, ?Segment)

is true when List and Segment are lists and Segment is a proper segment of List. It terminates only if List is proper. The only solution of segment/2 which is not a solution of proper_segment/2 is segment(List,List). So proper_segment/2 has one solution fewer.

cumlist(:Pred, +[X1,...,Xn], ?V0, ?[V1,...,Vn])

maps a ternary predicate Pred down the list [X1,...,Xn] just as scanlist/4 does, and returns a list of the results. It terminates when either list runs out. If Pred is bidirectional, it may be used to derive [X1...Xn] from V0 and [V1...Vn], e.g. cumlist(plus, [1,2,3,4], 0, /* -> */ [1,3,6,10]) and cumlist(plus, [1,1,1,1], /* <- */ 0, [1,2,3,4]).

maplist(:Pred, +List)

succeeds when Pred(X) succeeds for each element X of List.

maplist(:Pred, +OldList, ?NewList)

succeeds when Pred(Old,New) succeeds for each corresponding Old in OldList, New in NewList. Either OldList or NewList should be a proper list.

maplist(:Pred, +Xs, ?Ys, ?Zs)

is true when Xs, Ys, and Zs are lists of equal length, and Pred(X, Y, Z) is true for corresponding elements X of Xs, Y of Ys, and Z of Zs. At least one of Xs, Ys, and Zs should be a proper list.

map_product(Pred, Xs, Ys, PredOfProduct)

Just as maplist(P, Xs, L) is the analogue of Miranda's

              let L = [ P x | x <- Xs ]
     

so map_product(P, Xs, Ys, L) is the analogue of Miranda's

              let L = [ P x y | x <- Xs; y <- Ys ]
     

That is, if Xs = [X1,...,Xm], Ys = [Y1,...,Yn], and P(Xi,Yj,Zij), L = [Z11,...,Z1n,Z21,...,Z2n,...,Zm1,...,Zmn]. It is as if we formed the cartesian product of Xs and Ys and applied P to the (Xi,Yj) pairs.

scanlist(:Pred, +List, ?V1, ?V)

maps a ternary relation Pred down a list. If the list is [X1,X2,...,Xn], the computation is Pred(X1,V1,V2), Pred(X2,V2,V3), ..., Pred(Xn,Vn,V) So if Pred is plus/3, scanlist(plus, List, 0, V) puts the sum of the list elements in V. List should be a proper list. Note that the order of the arguments passed to Pred is the same as the order of the arguments following Pred. This also holds for scanlist/5 and scanlist/6, e.g. scanlist(Pred, Xs, Ys, Zs, V1, V) calls Pred(X3,Y3,Z3,V3,V4).

some(:Pred, +List)

succeeds when Pred(Elem) succeeds for some Elem in List. It will try all ways of proving Pred for each Elem, and will try each Elem in the List. somechk/2 is to some/2 as memberchk/2 is to member/2.

              member(X,L)     <-> some(=(X), L).
              memberchk(X, L) <-> somechk(=(X), L).
              some(Pred,L)    <-> member(X, L), call(Pred,X).
     

This acts on backtracking like member/2; List should be a proper list.

some(:Pred, +[X1,...,Xn], ?[Y1,...,Yn])

is true when Pred(Xi, Yi) is true for some i.

some(:Pred, +[X1,...,Xn], ?[Y1,...,Yn], ?[Z1,...,Zn])

is true when Pred(Xi, Yi, Zi) is true for some i.

somechk(:Pred, +[X1,...,Xn])

is true when Pred(Xi) is true for some i, and it commits to the first solution it finds (like memberchk/2).

somechk(:Pred, +[X1,...,Xn], ?[Y1,...,Yn])

is true when Pred(Xi, Yi) is true for some i, and it commits to the first solution it finds (like memberchk/2).

somechk(:Pred, +[X1,...,Xn], ?[Y1,...,Yn], ?[Z1,...,Zn])

is true when Pred(Xi, Yi, Zn) is true for some i, and it commits to the first solution it finds (like memberchk/2).

convlist(:Rewrite, +OldList, ?NewList)

is a sort of hybrid of maplist/3 and include/3. Each element of NewList is the image under Rewrite of some element of OldList, and order is preserved, but elements of OldList on which Rewrite is undefined (fails) are not represented. Thus if foo(K,X,Y) :- integer(X), Y is X+K. then convlist(foo(1), [1,a,0,joe(99),101], [2,1,102]). OldList should be a proper list.

exclude(:Pred, +List, ?SubList)

succeeds when SubList is the SubList of List containing all the elements for which Pred(Elem) is false. That is, it removes all the elements satisfying Pred. List should be a proper list.

exclude(:Pred, +Xs, +Ys, +Es)

is true when Es is the sublist of Xs containing all the elements Xi for which Pred(Xi, Yi) is *false*. Xs should be a proper list, and Ys should be at least as long as Xs (it may be longer).

exclude(:Pred, +Xs, +Ys, +Zs, +Es)

is true when Es is the sublist of Xs containing all the elements Xi for which Pred(Xi, Yi, Zi) is false. Xs should be a proper list, and Ys and Zs should be at least as long as Xs (they may be longer).

include(:Pred, +List, ?SubList)

succeeds when SubList is the SubList of List containing all the elements for which Pred(Elem) is true. That is, it retains all the elements satisfying Pred. List should be a proper list.

include(:Pred, +Xs, +Ys, +Es)

is true when Es is the sublist of Xs containing all the elements Xi for which Pred(Xi, Yi) is true. Xs should be a proper list, and Ys should be at least as long as Xs (it may be longer).

include(:Pred, +Xs, +Ys, +Zs, +Es)

is true when Es is the sublist of Xs containing all the elements Xi for which Pred(Xi, Yi, Zi) is false. Xs should be a proper list, and Ys and Zs should be at least as long as Xs (they may be longer).

partition(:Pred, +List, ?Less, ?Equal, ?Greater)

is a relative of include/3 and exclude/3 which has some pretensions to being logical. For each X in List, we call Pred(X,R), and route X to Less, Equal, or Greater according as R is <, =, or > .

group(:Pred, +List, ?Front, ?Back)

is true when append(Front, Back, List), maplist(Pred, Front), and Front is as long as possible.

group(:Pred, +Key, +List, ?Front, ?Back)

is true when append(Front, Back, List), maplist(call(Pred,Key), Front), and Front is as long as possible. Strictly speaking we don't need it; group(call(Pred,Key), List, Front, Back) would do just as well.

group(:Pred, +List, ?ListOfLists)

is true when append(ListOfLists, List), each element of ListOfLists has the form [Head|Tail] such that group(Pred, Head, Tail, Tail, []), and each element of ListOfLists is as long as possible. For example, if you have a keysorted list, and define same_key(K-_, K-_), then group(same_key, List, Buckets) will divide List up into Buckets of pairs having the same key.

ordered(+List)

is true when List is a list of terms [T1,T2,...,Tn] such that for all k in 2..n Tk-1 <¯ Tk, i.e. T1 <¯ T2 <¯ T3 ... The output of keysort/2 is always ordered, and so is that of sort/2. Beware: just because a list is ordered does not mean that it is the representation of an ordered set; it might contain duplicates.

ordered(+P, +[T1,T2,...,Tn])

is true when P(T1,T2) & P(T2,T3) & ... That is, if you take P as a "comparison" predicate like <¯, the list is ordered. This is good for generating prefixes of sequences, e.g. L = [1,_,_,_,_], ordered(times(2), L) yields L = [1,2,4,8,16].

max_member(?Xmax, +[X1,...,Xn])

unifies Xmax with the maximum (in the sense of <¯) of X1,...,Xn. If the list is empty, it fails quietly.

min_member(?Xmin, +[X1,...,Xn])

unifies Xmin with the minimum (in the sense of <¯) of X1,...,Xn. If the list is empty, it fails quietly.

max_member(:P, ?Xmax, +[X1,...,Xn])

unifies Xmax with the maximum element of [X1,...,Xn], as defined by the comparison predicate P, which should act like <¯ .

min_member(:P, ?Xmin, +[X1,...,Xn])

unifies Xmin with the minimum element of [X1,...,Xn], as defined by the comparison predicate P, which should act like <¯ .

select_min(?Element, +Set, ?Residue)

unifies Element with the smallest (in the sense of <¯) element of Set, and Residue with a list of all the other elements.

select_min(:Pred, ?Element, +Set, ?Residue)

find the least Element of Set, i.e. Pred(Element,X) for all X in Set.

select_max(?Element, +Set, ?Residue)

unifies Element with the (leftmost) maximum element of the Set, and Residue to the other elements in the same order.

select_max(:Pred, ?Element, +Set, ?Residue)

find the greatest Element of Set, i.e. Pred(X,Element) for all X in Set.

increasing_prefix(?Sequence, ?Prefix, ?Tail)

is true when append(Prefix, Tail, Sequence) and Prefix, together with the first element of Tail, forms a monotone non-decreasing sequence, and no longer Prefix will do. Pictorially,

          Sequence = [x1,...,xm,xm+1,...,xn]
          Prefix   = [x1,...,xm]
          Tail     = [xm+1,...,xn]
          x1 <¯ x2 <¯ ... <¯ xm <¯ xm+1
          not xm+1 <¯ xm+2
     

This is perhaps a surprising definition; you might expect that the first element of Tail would be included in Prefix. However, this way, it means that if Sequence is a strictly decreasing sequence, the Prefix will come out empty.

increasing_prefix(:Order, ?Sequence, ?Prefix, ?Tail)

is the same as increasing_prefix/3, except that it uses the binary relation Order in place of <¯.

decreasing_prefix(?Sequence, ?Prefix, ?Tail)

decreasing_prefix(:Order, ?Sequence, ?Prefix, ?Tail)

is the same, except it looks for a decreasing prefix. The order is the converse of the given order. That is, where increasing_prefix/[3,4] check X(R)Y, these routines check Y(R)X.

clumps(+Items, -Clumps)

is true when Clumps is a list of lists such that

• append(Clumps, Items)
• for each Clump in Clumps, all the elements of Clump are identical (==)

Items must be a proper list of terms for which sorting would have been sound. In fact, it usually is the result of sorting.

keyclumps(+Pairs, ?Clumps)

is true when Pairs is a list of pairs and Clumps a list of lists such that

• append(Clumps, Pairs)
• for each Clump in Clumps, all of the Key-Value pairs in Clump have identical (==) Keys.

Pairs must be a proper list of pairs for which keysorting would have been sound. In fact, it usually is the result of keysorting.

clumped(+Items, ?Counts)

is true when Counts is a list of Item-Count pairs such that if clumps(Items, Clumps), then each Item-Count pair in Counts corresponds to an element [Item/*1*/,...,Item/*Count*/] of Clumps. Items must be a proper list of terms for which sorting would have been sound. In fact, it usually is the result of sorting.

keyclumped(+Pairs, ?Groups)

is true when Pairs is a list of Key-Item pairs and Groups is a list of Key-Items pairs such that if keyclumps(Pairs, Clumps), then for each K-[I1,...,In] pair in Groups there is a [K-I1,...,K-In] clump in Clumps. Pairs must be a proper list of pairs for which keysorting would have been sound. In fact, it usually is the result of keysorting.