%--------------------------------------------------%
% vim: ts=4 sw=4 et ft=mercury
%--------------------------------------------------%
% Copyright (C) 2012-2014 YesLogic Pty. Ltd.
% Copyright (C) 2014-2015, 2017-2018 The Mercury team.
% This file is distributed under the terms specified in COPYING.LIB.
%--------------------------------------------------%
%
% File: diet.m.
% Author: wangp.
% Stability: medium.
%
% Discrete Interval Encoding Trees are a highly efficient set implementation
% for fat sets, i.e. densely populated sets over a discrete linear order.
%
% M. Erwig: Diets for Fat Sets,
% Journal of Functional Programming, Vol. 8, No. 6, pp. 627-632.
%
% O. Friedmann, M. Lange: More on Balanced Diets,
% Journal of Functional Programming, volume 21, issue 02, pp. 135-157.
%
%--------------------------------------------------%

:- module diet.
:- interface.

:- import_module bool.
:- import_module enum.
:- import_module list.

%--------------------------------------------------%

:- type diet(T). % <= diet_element(T).

:- typeclass diet_element(T) where [
    % less_than(X, Y) succeeds iff X < Y.
    pred less_than(T::in, T::in) is semidet,

    % successor(X) returns the successor of X, e.g. X + 1.
    func successor(T) = T,

    % predecessor(X) returns the predecessor of X, e.g. X - 1.
    func predecessor(T) = T
].

:- instance diet_element(int).

%--------------------------------------------------%
%
% Initial creation of sets.
%

    % Return an empty set.
    %
:- func init = diet(T).
:- pred init(diet(T)::out) is det.

    % `make_singleton_set(Elem)' returns a set containing just the single
    % element `Elem'.
    %
:- func make_singleton_set(T) = diet(T) <= diet_element(T).

    % `make_interval_set(X, Y)' returns a set containing just the elements in
    % the interval [X, Y]. Throws an exception if Y < X.
    %
:- func make_interval_set(T, T) = diet(T) <= diet_element(T).

%--------------------------------------------------%
%
% Emptiness and singleton-ness tests.
%

:- pred empty(diet(T)).
:- mode empty(in) is semidet.
:- mode empty(out) is det.
:- pragma obsolete(pred(empty/1), [init/0, init/1, is_empty/1]).

:- pred is_empty(diet(T)::in) is semidet.

:- pred is_non_empty(diet(T)::in) is semidet.

    % `is_singleton(Set, X)' is true iff `Set' is a singleton containing the
    % element `X'.
    %
:- pred is_singleton(diet(T)::in, T::out) is semidet <= diet_element(T).

%--------------------------------------------------%
%
% Membership tests.
%

    % `member(X, Set)' is true iff `X' is a member of `Set'.
    %
:- pred member(T, diet(T)) <= diet_element(T).
:- mode member(in, in) is semidet.
:- mode member(out, in) is nondet.

    % `contains(Set, X)' is true iff `X' is a member of `Set'.
    %
:- pred contains(diet(T)::in, T::in) is semidet <= diet_element(T).

%--------------------------------------------------%
%
% Insertions and deletions.
%

    % `insert(X, Set0, Set)' is true iff `Set' is the union of
    % `Set0' and the set containing only `X'.
    %
:- func insert(diet(T), T) = diet(T) <= diet_element(T).
:- pred insert(T::in, diet(T)::in, diet(T)::out) is det <= diet_element(T).

    % `insert_interval(X, Y, Set0, Set)' is true iff `Set' is the union of
    % `Set0' and the set containing only the elements of the interval [X, Y].
    % Throws an exception if Y < X.
    %
:- pred insert_interval(T::in, T::in, diet(T)::in, diet(T)::out) is det
    <= diet_element(T).

    % `insert_new(X, Set0, Set)' is true iff `Set0' does not contain
    % `X', and `Set' is the union of `Set0' and the set containing only `X'.
    %
:- pred insert_new(T::in, diet(T)::in, diet(T)::out) is semidet
    <= diet_element(T).

    % `insert_list(Xs, Set0, Set)' is true iff `Set' is the union of
    % `Set0' and the set containing only the members of `Xs'.
    %
:- func insert_list(diet(T), list(T)) = diet(T) <= diet_element(T).
:- pred insert_list(list(T)::in, diet(T)::in, diet(T)::out) is det
    <= diet_element(T).

    % `delete(X, Set0, Set)' is true iff `Set' is the relative
    % complement of `Set0' and the set containing only `X', i.e.
    % if `Set' is the set which contains all the elements of `Set0'
    % except `X'.
    %
:- func delete(diet(T), T) = diet(T) <= diet_element(T).
:- pred delete(T::in, diet(T)::in, diet(T)::out) is det <= diet_element(T).

    % `delete_list(Set, X)' returns the difference of `Set' and the set
    % containing only the members of `X'. Same as
    % `difference(Set, list_to_set(X))', but may be more efficient.
    %
:- func delete_list(diet(T), list(T)) = diet(T) <= diet_element(T).
:- pred delete_list(list(T)::in, diet(T)::in, diet(T)::out) is det
    <= diet_element(T).

    % `remove(X, Set0, Set)' is true iff `Set0' contains `X',
    % and `Set' is the relative complement of `Set0' and the set
    % containing only `X', i.e.  if `Set' is the set which contains
    % all the elements of `Set0' except `X'.
    %
:- pred remove(T::in, diet(T)::in, diet(T)::out) is semidet <= diet_element(T).

    % `remove_list(X, Set0, Set)' returns in `Set' the difference of `Set0'
    % and the set containing all the elements of `X', failing if any element
    % of `X' is not in `Set0'. Same as `subset(list_to_set(X), Set0),
    % difference(Set0, list_to_set(X), Set)', but may be more efficient.
    %
:- pred remove_list(list(T)::in, diet(T)::in, diet(T)::out) is semidet
    <= diet_element(T).

    % `remove_least(X, Set0, Set)' is true iff `X' is the least element in
    % `Set0', and `Set' is the set which contains all the elements of `Set0'
    % except `X'.
    %
:- pred remove_least(T::out, diet(T)::in, diet(T)::out) is semidet
    <= diet_element(T).

%--------------------------------------------------%
%
% Comparisons between sets.
%

    % `equal(SetA, SetB)' is true iff `SetA' and `SetB' contain the same
    % elements.
    %
:- pred equal(diet(T)::in, diet(T)::in) is semidet <= diet_element(T).

    % `subset(Subset, Set)' is true iff `Subset' is a subset of `Set'.
    %
:- pred subset(diet(T)::in, diet(T)::in) is semidet <= diet_element(T).

    % `superset(Superset, Set)' is true iff `Superset' is a superset of `Set'.
    %
:- pred superset(diet(T)::in, diet(T)::in) is semidet <= diet_element(T).

%--------------------------------------------------%
%
% Operations on two or more sets.
%

    % `union(SetA, SetB, Set)' is true iff `Set' is the union of
    % `SetA' and `SetB'.
    %
:- func union(diet(T), diet(T)) = diet(T) <= diet_element(T).
:- pred union(diet(T)::in, diet(T)::in, diet(T)::out) is det
    <= diet_element(T).

    % `union_list(Sets, Set)' returns the union of all the sets in Sets.
    %
:- func union_list(list(diet(T))) = diet(T) <= diet_element(T).
:- pred union_list(list(diet(T))::in, diet(T)::out) is det <= diet_element(T).

    % `intersect(SetA, SetB, Set)' is true iff `Set' is the
    % intersection of `SetA' and `SetB'.
    %
:- func intersect(diet(T), diet(T)) = diet(T) <= diet_element(T).
:- pred intersect(diet(T)::in, diet(T)::in, diet(T)::out) is det
    <= diet_element(T).

    % `intersect_list(Sets, Set)' returns the intersection of all the sets
    % in Sets.
    %
:- func intersect_list(list(diet(T))) = diet(T) <= diet_element(T).
:- pred intersect_list(list(diet(T))::in, diet(T)::out) is det
    <= diet_element(T).

    % `difference(SetA, SetB)' returns the set containing all the elements
    % of `SetA' except those that occur in `SetB'.
    %
:- func difference(diet(T), diet(T)) = diet(T) <= diet_element(T).
:- pred difference(diet(T)::in, diet(T)::in, diet(T)::out) is det
    <= diet_element(T).

%--------------------------------------------------%
%
% Operations that divide a set into two parts.
%

    % `split(X, Set, Lesser, IsPresent, Greater)' is true iff
    % `Lesser' is the set of elements in `Set' which are less than `X' and
    % `Greater' is the set of elements in `Set' which are greater than `X'.
    % `IsPresent' is `yes' if `Set' contains `X', and `no' otherwise.
    %
:- pred split(T::in, diet(T)::in, diet(T)::out, bool::out, diet(T)::out) is det
    <= diet_element(T).

    % divide(Pred, Set, InPart, OutPart):
    % InPart consists of those elements of Set for which Pred succeeds;
    % OutPart consists of those elements of Set for which Pred fails.
    %
:- pred divide(pred(T)::in(pred(in) is semidet), diet(T)::in,
    diet(T)::out, diet(T)::out) is det <= diet_element(T).

    % divide_by_set(DivideBySet, Set, InPart, OutPart):
    % InPart consists of those elements of Set which are also in DivideBySet;
    % OutPart consists of those elements of Set which are not in DivideBySet.
    %
:- pred divide_by_set(diet(T)::in, diet(T)::in, diet(T)::out, diet(T)::out)
    is det <= diet_element(T).

%--------------------------------------------------%
%
% Converting lists to sets.
%

    % `list_to_set(List)' returns a set containing only the members of `List'.
    %
:- func list_to_set(list(T)) = diet(T) <= diet_element(T).
:- pred list_to_set(list(T)::in, diet(T)::out) is det <= diet_element(T).

:- func from_list(list(T)) = diet(T) <= diet_element(T).
:- pred from_list(list(T)::in, diet(T)::out) is det <= diet_element(T).

    % `from_interval_list(Intervals, Set)' returns a Set containing the
    % elements of all intervals [X, Y] in Intervals, where each interval is
    % represented by a tuple. Throws an exception if any interval has Y < X.
    % The intervals may overlap.
    %
:- pred from_interval_list(list({T, T})::in, diet(T)::out) is det
    <= diet_element(T).

    % `sorted_list_to_set(List)' returns a set containing only the members
    % of `List'. `List' must be sorted.
    %
:- func sorted_list_to_set(list(T)) = diet(T) <= diet_element(T).
:- pred sorted_list_to_set(list(T)::in, diet(T)::out) is det
    <= diet_element(T).

%--------------------------------------------------%
%
% Converting sets to lists.
%

    % `to_sorted_list(Set)' returns a list containing all the members of `Set',
    % in sorted order.
    %
:- func to_sorted_list(diet(T)) = list(T) <= diet_element(T).
:- pred to_sorted_list(diet(T)::in, list(T)::out) is det <= diet_element(T).

    % `to_sorted_interval_list(Set)' returns a list of intervals in `Set'
    % in sorted order, where each interval is represented by a tuple.
    % The intervals do not overlap.
    %
:- pred to_sorted_interval_list(diet(T)::in, list({T, T})::out) is det
    <= diet_element(T).

%--------------------------------------------------%
%
% Counting.
%

    % `count(Set)' returns the number of elements in Set.
    %
:- func count(diet(T)) = int <= enum(T).

%--------------------------------------------------%
%
% Standard higher order functions on collections.
%

    % all_true(Pred, Set) succeeds iff Pred(Element) succeeds
    % for all the elements of Set.
    %
:- pred all_true(pred(T)::in(pred(in) is semidet), diet(T)::in) is semidet
    <= diet_element(T).

    % `filter(Pred, Set) = TrueSet' returns the elements of Set for which
    % Pred succeeds.
    %
:- func filter(pred(T), diet(T)) = diet(T) <= diet_element(T).
:- mode filter(pred(in) is semidet, in) = out is det.

    % `filter(Pred, Set, TrueSet, FalseSet)' returns the elements of Set
    % for which Pred succeeds, and those for which it fails.
    %
:- pred filter(pred(T), diet(T), diet(T), diet(T)) <= diet_element(T).
:- mode filter(pred(in) is semidet, in, out, out) is det.

    % `foldl_intervals(Pred, Set, Start)' calls Pred with each interval of
    % `Set' (in sorted order) and an accumulator (with the initial value of
    % `Start'), and returns the final value.
    %
:- pred foldl_intervals(pred(T, T, A, A), diet(T), A, A) <= diet_element(T).
:- mode foldl_intervals(pred(in, in, in, out) is det, in, in, out) is det.
:- mode foldl_intervals(pred(in, in, di, uo) is det, in, di, uo) is det.
:- mode foldl_intervals(pred(in, in, in, out) is semidet, in, in, out)
    is semidet.

    % `foldr_intervals(Pred, Set, Start)' calls Pred with each interval of
    % `Set' (in reverse sorted order) and an accumulator (with the initial
    % value of `Start'), and returns the final value.
    %
:- pred foldr_intervals(pred(T, T, A, A), diet(T), A, A) <= diet_element(T).
:- mode foldr_intervals(pred(in, in, in, out) is det, in, in, out) is det.
:- mode foldr_intervals(pred(in, in, di, uo) is det, in, di, uo) is det.
:- mode foldr_intervals(pred(in, in, in, out) is semidet, in, in, out)
    is semidet.

    % `foldl(Func, Set, Start)' calls Func with each element of `Set'
    % (in sorted order) and an accumulator (with the initial value of `Start'),
    % and returns the final value.
    %
:- func foldl(func(T, A) = A, diet(T), A) = A <= diet_element(T).

:- pred foldl(pred(T, A, A), diet(T), A, A) <= diet_element(T).
:- mode foldl(pred(in, in, out) is det, in, in, out) is det.
:- mode foldl(pred(in, mdi, muo) is det, in, mdi, muo) is det.
:- mode foldl(pred(in, di, uo) is det, in, di, uo) is det.
:- mode foldl(pred(in, in, out) is semidet, in, in, out) is semidet.
:- mode foldl(pred(in, mdi, muo) is semidet, in, mdi, muo) is semidet.
:- mode foldl(pred(in, di, uo) is semidet, in, di, uo) is semidet.

:- pred foldl2(pred(T, A, A, B, B), diet(T), A, A, B, B) <= diet_element(T).
:- mode foldl2(pred(in, in, out, in, out) is det, in,
    in, out, in, out) is det.
:- mode foldl2(pred(in, in, out, mdi, muo) is det, in,
    in, out, mdi, muo) is det.
:- mode foldl2(pred(in, in, out, di, uo) is det, in,
    in, out, di, uo) is det.
:- mode foldl2(pred(in, in, out, in, out) is semidet, in,
    in, out, in, out) is semidet.
:- mode foldl2(pred(in, in, out, mdi, muo) is semidet, in,
    in, out, mdi, muo) is semidet.
:- mode foldl2(pred(in, in, out, di, uo) is semidet, in,
    in, out, di, uo) is semidet.

:- pred foldl3(pred(T, A, A, B, B, C, C), diet(T),
    A, A, B, B, C, C) <= diet_element(T).
:- mode foldl3(pred(in, in, out, in, out, in, out) is det, in,
    in, out, in, out, in, out) is det.
:- mode foldl3(pred(in, in, out, in, out, mdi, muo) is det, in,
    in, out, in, out, mdi, muo) is det.
:- mode foldl3(pred(in, in, out, in, out, di, uo) is det, in,
    in, out, in, out, di, uo) is det.
:- mode foldl3(pred(in, in, out, in, out, in, out) is semidet, in,
    in, out, in, out, in, out) is semidet.
:- mode foldl3(pred(in, in, out, in, out, mdi, muo) is semidet, in,
    in, out, in, out, mdi, muo) is semidet.
:- mode foldl3(pred(in, in, out, in, out, di, uo) is semidet, in,
    in, out, in, out, di, uo) is semidet.

:- pred foldl4(pred(T, A, A, B, B, C, C, D, D), diet(T),
    A, A, B, B, C, C, D, D) <= diet_element(T).
:- mode foldl4(pred(in, in, out, in, out, in, out, in, out) is det, in,
    in, out, in, out, in, out, in, out) is det.
:- mode foldl4(pred(in, in, out, in, out, in, out, mdi, muo) is det, in,
    in, out, in, out, in, out, mdi, muo) is det.
:- mode foldl4(pred(in, in, out, in, out, in, out, di, uo) is det, in,
    in, out, in, out, in, out, di, uo) is det.
:- mode foldl4(pred(in, in, out, in, out, in, out, in, out) is semidet, in,
    in, out, in, out, in, out, in, out) is semidet.
:- mode foldl4(pred(in, in, out, in, out, in, out, mdi, muo) is semidet, in,
    in, out, in, out, in, out, mdi, muo) is semidet.
:- mode foldl4(pred(in, in, out, in, out, in, out, di, uo) is semidet, in,
    in, out, in, out, in, out, di, uo) is semidet.

:- pred foldl5(pred(T, A, A, B, B, C, C, D, D, E, E), diet(T),
    A, A, B, B, C, C, D, D, E, E) <= diet_element(T).
:- mode foldl5(
    pred(in, in, out, in, out, in, out, in, out, in, out) is det,
    in, in, out, in, out, in, out, in, out, in, out) is det.
:- mode foldl5(
    pred(in, in, out, in, out, in, out, in, out, mdi, muo) is det,
    in, in, out, in, out, in, out, in, out, mdi, muo) is det.
:- mode foldl5(
    pred(in, in, out, in, out, in, out, in, out, di, uo) is det,
    in, in, out, in, out, in, out, in, out, di, uo) is det.
:- mode foldl5(
    pred(in, in, out, in, out, in, out, in, out, in, out) is semidet,
    in, in, out, in, out, in, out, in, out, in, out) is semidet.
:- mode foldl5(
    pred(in, in, out, in, out, in, out, in, out, mdi, muo) is semidet,
    in, in, out, in, out, in, out, in, out, mdi, muo) is semidet.
:- mode foldl5(
    pred(in, in, out, in, out, in, out, in, out, di, uo) is semidet,
    in, in, out, in, out, in, out, in, out, di, uo) is semidet.

:- func foldr(func(T, A) = A, diet(T), A) = A <= diet_element(T).

:- pred foldr(pred(T, A, A), diet(T), A, A) <= diet_element(T).
:- mode foldr(pred(in, in, out) is det, in, in, out) is det.
:- mode foldr(pred(in, mdi, muo) is det, in, mdi, muo) is det.
:- mode foldr(pred(in, di, uo) is det, in, di, uo) is det.
:- mode foldr(pred(in, in, out) is semidet, in, in, out) is semidet.
:- mode foldr(pred(in, mdi, muo) is semidet, in, mdi, muo) is semidet.
:- mode foldr(pred(in, di, uo) is semidet, in, di, uo) is semidet.

%--------------------------------------------------%
%--------------------------------------------------%