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Diffstat (limited to 'lib/tools/test/prof_bench_SUITE_data/sofs_copy.erl')
| -rw-r--r-- | lib/tools/test/prof_bench_SUITE_data/sofs_copy.erl | 2809 |
1 files changed, 2809 insertions, 0 deletions
diff --git a/lib/tools/test/prof_bench_SUITE_data/sofs_copy.erl b/lib/tools/test/prof_bench_SUITE_data/sofs_copy.erl new file mode 100644 index 0000000000..2a9b19177e --- /dev/null +++ b/lib/tools/test/prof_bench_SUITE_data/sofs_copy.erl @@ -0,0 +1,2809 @@ +%% +%% %CopyrightBegin% +%% +%% Copyright Ericsson AB 2001-2017. All Rights Reserved. +%% +%% Licensed under the Apache License, Version 2.0 (the "License"); +%% you may not use this file except in compliance with the License. +%% You may obtain a copy of the License at +%% +%% http://www.apache.org/licenses/LICENSE-2.0 +%% +%% Unless required by applicable law or agreed to in writing, software +%% distributed under the License is distributed on an "AS IS" BASIS, +%% WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +%% See the License for the specific language governing permissions and +%% limitations under the License. +%% +%% %CopyrightEnd% +%% +-module(sofs_copy). + +-export([from_term/1, from_term/2, from_external/2, empty_set/0, + is_type/1, set/1, set/2, from_sets/1, relation/1, relation/2, + a_function/1, a_function/2, family/1, family/2, + to_external/1, type/1, to_sets/1, no_elements/1, + specification/2, union/2, intersection/2, difference/2, + symdiff/2, symmetric_partition/2, product/1, product/2, + constant_function/2, is_equal/2, is_subset/2, is_sofs_set/1, + is_set/1, is_empty_set/1, is_disjoint/2]). + +-export([union/1, intersection/1, canonical_relation/1]). + +-export([relation_to_family/1, domain/1, range/1, field/1, + relative_product/1, relative_product/2, relative_product1/2, + converse/1, image/2, inverse_image/2, strict_relation/1, + weak_relation/1, extension/3, is_a_function/1]). + +-export([composite/2, inverse/1]). + +-export([restriction/2, restriction/3, drestriction/2, drestriction/3, + substitution/2, projection/2, partition/1, partition/2, + partition/3, multiple_relative_product/2, join/4]). + +-export([family_to_relation/1, family_specification/2, + union_of_family/1, intersection_of_family/1, + family_union/1, family_intersection/1, + family_domain/1, family_range/1, family_field/1, + family_union/2, family_intersection/2, family_difference/2, + partition_family/2, family_projection/2]). + +-export([family_to_digraph/1, family_to_digraph/2, + digraph_to_family/1, digraph_to_family/2]). + +%% Shorter names of some functions. +-export([fam2rel/1, rel2fam/1]). + +-import(lists, + [any/2, append/1, flatten/1, foreach/2, + keysort/2, last/1, map/2, mapfoldl/3, member/2, merge/2, + reverse/1, reverse/2, sort/1, umerge/1, umerge/2, usort/1]). + +-compile({inline, [{family_to_relation,1}, {relation_to_family,1}]}). + +-compile({inline, [{rel,2},{a_func,2},{fam,2},{term2set,2}]}). + +-compile({inline, [{external_fun,1},{element_type,1}]}). + +-compile({inline, + [{unify_types,2}, {match_types,2}, + {test_rel,3}, {symdiff,3}, + {subst,3}]}). + +-compile({inline, [{fam_binop,3}]}). + +%% Nope, no is_member, del_member or add_member. +%% +%% See also "Naive Set Theory" by Paul R. Halmos. +%% +%% By convention, erlang:error/1 is called from exported functions. + +-define(TAG, 'Set'). +-define(ORDTAG, 'OrdSet'). + +-record(?TAG, {data = [] :: list(), type = type :: term()}). +-record(?ORDTAG, {orddata = {} :: tuple() | atom(), + ordtype = type :: term()}). + +-define(LIST(S), (S)#?TAG.data). +-define(TYPE(S), (S)#?TAG.type). +-define(SET(L, T), #?TAG{data = L, type = T}). +-define(IS_SET(S), is_record(S, ?TAG)). +-define(IS_UNTYPED_SET(S), ?TYPE(S) =:= ?ANYTYPE). + +%% Ordered sets and atoms: +-define(ORDDATA(S), (S)#?ORDTAG.orddata). +-define(ORDTYPE(S), (S)#?ORDTAG.ordtype). +-define(ORDSET(L, T), #?ORDTAG{orddata = L, ordtype = T}). +-define(IS_ORDSET(S), is_record(S, ?ORDTAG)). +-define(ATOM_TYPE, atom). +-define(IS_ATOM_TYPE(T), is_atom(T)). % true for ?ANYTYPE... + +%% When IS_SET is true: +-define(ANYTYPE, '_'). +-define(BINREL(X, Y), {X, Y}). +-define(IS_RELATION(R), is_tuple(R)). +-define(REL_ARITY(R), tuple_size(R)). +-define(REL_TYPE(I, R), element(I, R)). +-define(SET_OF(X), [X]). +-define(IS_SET_OF(X), is_list(X)). +-define(FAMILY(X, Y), ?BINREL(X, ?SET_OF(Y))). + +-export_type([anyset/0, binary_relation/0, external_set/0, a_function/0, + family/0, relation/0, set_of_sets/0, set_fun/0, spec_fun/0, + type/0]). +-export_type([ordset/0, a_set/0]). + +-type(anyset() :: ordset() | a_set()). +-type(binary_relation() :: relation()). +-type(external_set() :: term()). +-type(a_function() :: relation()). +-type(family() :: a_function()). +-opaque(ordset() :: #?ORDTAG{}). +-type(relation() :: a_set()). +-opaque(a_set() :: #?TAG{}). +-type(set_of_sets() :: a_set()). +-type(set_fun() :: pos_integer() + | {external, fun((external_set()) -> external_set())} + | fun((anyset()) -> anyset())). +-type(spec_fun() :: {external, fun((external_set()) -> boolean())} + | fun((anyset()) -> boolean())). +-type(type() :: term()). + +-type(tuple_of(_T) :: tuple()). + +%% +%% Exported functions +%% + +%%% +%%% Create sets +%%% + +-spec(from_term(Term) -> AnySet when + AnySet :: anyset(), + Term :: term()). +from_term(T) -> + Type = case T of + _ when is_list(T) -> [?ANYTYPE]; + _ -> ?ANYTYPE + end, + try setify(T, Type) + catch _:_ -> erlang:error(badarg) + end. + +-spec(from_term(Term, Type) -> AnySet when + AnySet :: anyset(), + Term :: term(), + Type :: type()). +from_term(L, T) -> + case is_type(T) of + true -> + try setify(L, T) + catch _:_ -> erlang:error(badarg) + end; + false -> + erlang:error(badarg) + end. + +-spec(from_external(ExternalSet, Type) -> AnySet when + ExternalSet :: external_set(), + AnySet :: anyset(), + Type :: type()). +from_external(L, ?SET_OF(Type)) -> + ?SET(L, Type); +from_external(T, Type) -> + ?ORDSET(T, Type). + +-spec(empty_set() -> Set when + Set :: a_set()). +empty_set() -> + ?SET([], ?ANYTYPE). + +-spec(is_type(Term) -> Bool when + Bool :: boolean(), + Term :: term()). +is_type(Atom) when ?IS_ATOM_TYPE(Atom), Atom =/= ?ANYTYPE -> + true; +is_type(?SET_OF(T)) -> + is_element_type(T); +is_type(T) when tuple_size(T) > 0 -> + is_types(tuple_size(T), T); +is_type(_T) -> + false. + +-spec(set(Terms) -> Set when + Set :: a_set(), + Terms :: [term()]). +set(L) -> + try usort(L) of + SL -> ?SET(SL, ?ATOM_TYPE) + catch _:_ -> erlang:error(badarg) + end. + +-spec(set(Terms, Type) -> Set when + Set :: a_set(), + Terms :: [term()], + Type :: type()). +set(L, ?SET_OF(Type)) when ?IS_ATOM_TYPE(Type), Type =/= ?ANYTYPE -> + try usort(L) of + SL -> ?SET(SL, Type) + catch _:_ -> erlang:error(badarg) + end; +set(L, ?SET_OF(_) = T) -> + try setify(L, T) + catch _:_ -> erlang:error(badarg) + end; +set(_, _) -> + erlang:error(badarg). + +-spec(from_sets(ListOfSets) -> Set when + Set :: a_set(), + ListOfSets :: [anyset()]; + (TupleOfSets) -> Ordset when + Ordset :: ordset(), + TupleOfSets :: tuple_of(anyset())). +from_sets(Ss) when is_list(Ss) -> + case set_of_sets(Ss, [], ?ANYTYPE) of + {error, Error} -> + erlang:error(Error); + Set -> + Set + end; +from_sets(Tuple) when is_tuple(Tuple) -> + case ordset_of_sets(tuple_to_list(Tuple), [], []) of + error -> + erlang:error(badarg); + Set -> + Set + end; +from_sets(_) -> + erlang:error(badarg). + +-spec(relation(Tuples) -> Relation when + Relation :: relation(), + Tuples :: [tuple()]). +relation([]) -> + ?SET([], ?BINREL(?ATOM_TYPE, ?ATOM_TYPE)); +relation(Ts = [T | _]) when is_tuple(T) -> + try rel(Ts, tuple_size(T)) + catch _:_ -> erlang:error(badarg) + end; +relation(_) -> + erlang:error(badarg). + +-spec(relation(Tuples, Type) -> Relation when + N :: integer(), + Type :: N | type(), + Relation :: relation(), + Tuples :: [tuple()]). +relation(Ts, TS) -> + try rel(Ts, TS) + catch _:_ -> erlang:error(badarg) + end. + +-spec(a_function(Tuples) -> Function when + Function :: a_function(), + Tuples :: [tuple()]). +a_function(Ts) -> + try func(Ts, ?BINREL(?ATOM_TYPE, ?ATOM_TYPE)) of + Bad when is_atom(Bad) -> + erlang:error(Bad); + Set -> + Set + catch _:_ -> erlang:error(badarg) + end. + +-spec(a_function(Tuples, Type) -> Function when + Function :: a_function(), + Tuples :: [tuple()], + Type :: type()). +a_function(Ts, T) -> + try a_func(Ts, T) of + Bad when is_atom(Bad) -> + erlang:error(Bad); + Set -> + Set + catch _:_ -> erlang:error(badarg) + end. + +-spec(family(Tuples) -> Family when + Family :: family(), + Tuples :: [tuple()]). +family(Ts) -> + try fam2(Ts, ?FAMILY(?ATOM_TYPE, ?ATOM_TYPE)) of + Bad when is_atom(Bad) -> + erlang:error(Bad); + Set -> + Set + catch _:_ -> erlang:error(badarg) + end. + +-spec(family(Tuples, Type) -> Family when + Family :: family(), + Tuples :: [tuple()], + Type :: type()). +family(Ts, T) -> + try fam(Ts, T) of + Bad when is_atom(Bad) -> + erlang:error(Bad); + Set -> + Set + catch _:_ -> erlang:error(badarg) + end. + +%%% +%%% Functions on sets. +%%% + +-spec(to_external(AnySet) -> ExternalSet when + ExternalSet :: external_set(), + AnySet :: anyset()). +to_external(S) when ?IS_SET(S) -> + ?LIST(S); +to_external(S) when ?IS_ORDSET(S) -> + ?ORDDATA(S). + +-spec(type(AnySet) -> Type when + AnySet :: anyset(), + Type :: type()). +type(S) when ?IS_SET(S) -> + ?SET_OF(?TYPE(S)); +type(S) when ?IS_ORDSET(S) -> + ?ORDTYPE(S). + +-spec(to_sets(ASet) -> Sets when + ASet :: a_set() | ordset(), + Sets :: tuple_of(AnySet) | [AnySet], + AnySet :: anyset()). +to_sets(S) when ?IS_SET(S) -> + case ?TYPE(S) of + ?SET_OF(Type) -> list_of_sets(?LIST(S), Type, []); + Type -> list_of_ordsets(?LIST(S), Type, []) + end; +to_sets(S) when ?IS_ORDSET(S), is_tuple(?ORDTYPE(S)) -> + tuple_of_sets(tuple_to_list(?ORDDATA(S)), tuple_to_list(?ORDTYPE(S)), []); +to_sets(S) when ?IS_ORDSET(S) -> + erlang:error(badarg). + +-spec(no_elements(ASet) -> NoElements when + ASet :: a_set() | ordset(), + NoElements :: non_neg_integer()). +no_elements(S) when ?IS_SET(S) -> + length(?LIST(S)); +no_elements(S) when ?IS_ORDSET(S), is_tuple(?ORDTYPE(S)) -> + tuple_size(?ORDDATA(S)); +no_elements(S) when ?IS_ORDSET(S) -> + erlang:error(badarg). + +-spec(specification(Fun, Set1) -> Set2 when + Fun :: spec_fun(), + Set1 :: a_set(), + Set2 :: a_set()). +specification(Fun, S) when ?IS_SET(S) -> + Type = ?TYPE(S), + R = case external_fun(Fun) of + false -> + spec(?LIST(S), Fun, element_type(Type), []); + XFun -> + specification(?LIST(S), XFun, []) + end, + case R of + SL when is_list(SL) -> + ?SET(SL, Type); + Bad -> + erlang:error(Bad) + end. + +-spec(union(Set1, Set2) -> Set3 when + Set1 :: a_set(), + Set2 :: a_set(), + Set3 :: a_set()). +union(S1, S2) when ?IS_SET(S1), ?IS_SET(S2) -> + case unify_types(?TYPE(S1), ?TYPE(S2)) of + [] -> erlang:error(type_mismatch); + Type -> ?SET(umerge(?LIST(S1), ?LIST(S2)), Type) + end. + +-spec(intersection(Set1, Set2) -> Set3 when + Set1 :: a_set(), + Set2 :: a_set(), + Set3 :: a_set()). +intersection(S1, S2) when ?IS_SET(S1), ?IS_SET(S2) -> + case unify_types(?TYPE(S1), ?TYPE(S2)) of + [] -> erlang:error(type_mismatch); + Type -> ?SET(intersection(?LIST(S1), ?LIST(S2), []), Type) + end. + +-spec(difference(Set1, Set2) -> Set3 when + Set1 :: a_set(), + Set2 :: a_set(), + Set3 :: a_set()). +difference(S1, S2) when ?IS_SET(S1), ?IS_SET(S2) -> + case unify_types(?TYPE(S1), ?TYPE(S2)) of + [] -> erlang:error(type_mismatch); + Type -> ?SET(difference(?LIST(S1), ?LIST(S2), []), Type) + end. + +-spec(symdiff(Set1, Set2) -> Set3 when + Set1 :: a_set(), + Set2 :: a_set(), + Set3 :: a_set()). +symdiff(S1, S2) when ?IS_SET(S1), ?IS_SET(S2) -> + case unify_types(?TYPE(S1), ?TYPE(S2)) of + [] -> erlang:error(type_mismatch); + Type -> ?SET(symdiff(?LIST(S1), ?LIST(S2), []), Type) + end. + +-spec(symmetric_partition(Set1, Set2) -> {Set3, Set4, Set5} when + Set1 :: a_set(), + Set2 :: a_set(), + Set3 :: a_set(), + Set4 :: a_set(), + Set5 :: a_set()). +symmetric_partition(S1, S2) when ?IS_SET(S1), ?IS_SET(S2) -> + case unify_types(?TYPE(S1), ?TYPE(S2)) of + [] -> erlang:error(type_mismatch); + Type -> sympart(?LIST(S1), ?LIST(S2), [], [], [], Type) + end. + +-spec(product(Set1, Set2) -> BinRel when + BinRel :: binary_relation(), + Set1 :: a_set(), + Set2 :: a_set()). +product(S1, S2) when ?IS_SET(S1), ?IS_SET(S2) -> + if + ?TYPE(S1) =:= ?ANYTYPE -> S1; + ?TYPE(S2) =:= ?ANYTYPE -> S2; + true -> + F = fun(E) -> {0, E} end, + T = ?BINREL(?TYPE(S1), ?TYPE(S2)), + ?SET(relprod(map(F, ?LIST(S1)), map(F, ?LIST(S2))), T) + end. + +-spec(product(TupleOfSets) -> Relation when + Relation :: relation(), + TupleOfSets :: tuple_of(a_set())). +product({S1, S2}) -> + product(S1, S2); +product(T) when is_tuple(T) -> + Ss = tuple_to_list(T), + try sets_to_list(Ss) of + [] -> + erlang:error(badarg); + L -> + Type = types(Ss, []), + case member([], L) of + true -> + empty_set(); + false -> + ?SET(reverse(prod(L, [], [])), Type) + end + catch _:_ -> erlang:error(badarg) + end. + +-spec(constant_function(Set, AnySet) -> Function when + AnySet :: anyset(), + Function :: a_function(), + Set :: a_set()). +constant_function(S, E) when ?IS_SET(S) -> + case {?TYPE(S), is_sofs_set(E)} of + {?ANYTYPE, true} -> S; + {Type, true} -> + NType = ?BINREL(Type, type(E)), + ?SET(constant_function(?LIST(S), to_external(E), []), NType); + _ -> erlang:error(badarg) + end; +constant_function(S, _) when ?IS_ORDSET(S) -> + erlang:error(badarg). + +-spec(is_equal(AnySet1, AnySet2) -> Bool when + AnySet1 :: anyset(), + AnySet2 :: anyset(), + Bool :: boolean()). +is_equal(S1, S2) when ?IS_SET(S1), ?IS_SET(S2) -> + case match_types(?TYPE(S1), ?TYPE(S2)) of + true -> ?LIST(S1) == ?LIST(S2); + false -> erlang:error(type_mismatch) + end; +is_equal(S1, S2) when ?IS_ORDSET(S1), ?IS_ORDSET(S2) -> + case match_types(?ORDTYPE(S1), ?ORDTYPE(S2)) of + true -> ?ORDDATA(S1) == ?ORDDATA(S2); + false -> erlang:error(type_mismatch) + end; +is_equal(S1, S2) when ?IS_SET(S1), ?IS_ORDSET(S2) -> + erlang:error(type_mismatch); +is_equal(S1, S2) when ?IS_ORDSET(S1), ?IS_SET(S2) -> + erlang:error(type_mismatch). + +-spec(is_subset(Set1, Set2) -> Bool when + Bool :: boolean(), + Set1 :: a_set(), + Set2 :: a_set()). +is_subset(S1, S2) when ?IS_SET(S1), ?IS_SET(S2) -> + case match_types(?TYPE(S1), ?TYPE(S2)) of + true -> subset(?LIST(S1), ?LIST(S2)); + false -> erlang:error(type_mismatch) + end. + +-spec(is_sofs_set(Term) -> Bool when + Bool :: boolean(), + Term :: term()). +is_sofs_set(S) when ?IS_SET(S) -> + true; +is_sofs_set(S) when ?IS_ORDSET(S) -> + true; +is_sofs_set(_S) -> + false. + +-spec(is_set(AnySet) -> Bool when + AnySet :: anyset(), + Bool :: boolean()). +is_set(S) when ?IS_SET(S) -> + true; +is_set(S) when ?IS_ORDSET(S) -> + false. + +-spec(is_empty_set(AnySet) -> Bool when + AnySet :: anyset(), + Bool :: boolean()). +is_empty_set(S) when ?IS_SET(S) -> + ?LIST(S) =:= []; +is_empty_set(S) when ?IS_ORDSET(S) -> + false. + +-spec(is_disjoint(Set1, Set2) -> Bool when + Bool :: boolean(), + Set1 :: a_set(), + Set2 :: a_set()). +is_disjoint(S1, S2) when ?IS_SET(S1), ?IS_SET(S2) -> + case match_types(?TYPE(S1), ?TYPE(S2)) of + true -> + case ?LIST(S1) of + [] -> true; + [A | As] -> disjoint(?LIST(S2), A, As) + end; + false -> erlang:error(type_mismatch) + end. + +%%% +%%% Functions on set-of-sets. +%%% + +-spec(union(SetOfSets) -> Set when + Set :: a_set(), + SetOfSets :: set_of_sets()). +union(Sets) when ?IS_SET(Sets) -> + case ?TYPE(Sets) of + ?SET_OF(Type) -> ?SET(lunion(?LIST(Sets)), Type); + ?ANYTYPE -> Sets; + _ -> erlang:error(badarg) + end. + +-spec(intersection(SetOfSets) -> Set when + Set :: a_set(), + SetOfSets :: set_of_sets()). +intersection(Sets) when ?IS_SET(Sets) -> + case ?LIST(Sets) of + [] -> erlang:error(badarg); + [L | Ls] -> + case ?TYPE(Sets) of + ?SET_OF(Type) -> + ?SET(lintersection(Ls, L), Type); + _ -> erlang:error(badarg) + end + end. + +-spec(canonical_relation(SetOfSets) -> BinRel when + BinRel :: binary_relation(), + SetOfSets :: set_of_sets()). +canonical_relation(Sets) when ?IS_SET(Sets) -> + ST = ?TYPE(Sets), + case ST of + ?SET_OF(?ANYTYPE) -> empty_set(); + ?SET_OF(Type) -> + ?SET(can_rel(?LIST(Sets), []), ?BINREL(Type, ST)); + ?ANYTYPE -> Sets; + _ -> erlang:error(badarg) + end. + +%%% +%%% Functions on binary relations only. +%%% + +-spec(rel2fam(BinRel) -> Family when + Family :: family(), + BinRel :: binary_relation()). +rel2fam(R) -> + relation_to_family(R). + +-spec(relation_to_family(BinRel) -> Family when + Family :: family(), + BinRel :: binary_relation()). +%% Inlined. +relation_to_family(R) when ?IS_SET(R) -> + case ?TYPE(R) of + ?BINREL(DT, RT) -> + ?SET(rel2family(?LIST(R)), ?FAMILY(DT, RT)); + ?ANYTYPE -> R; + _Else -> erlang:error(badarg) + end. + +-spec(domain(BinRel) -> Set when + BinRel :: binary_relation(), + Set :: a_set()). +domain(R) when ?IS_SET(R) -> + case ?TYPE(R) of + ?BINREL(DT, _) -> ?SET(dom(?LIST(R)), DT); + ?ANYTYPE -> R; + _Else -> erlang:error(badarg) + end. + +-spec(range(BinRel) -> Set when + BinRel :: binary_relation(), + Set :: a_set()). +range(R) when ?IS_SET(R) -> + case ?TYPE(R) of + ?BINREL(_, RT) -> ?SET(ran(?LIST(R), []), RT); + ?ANYTYPE -> R; + _ -> erlang:error(badarg) + end. + +-spec(field(BinRel) -> Set when + BinRel :: binary_relation(), + Set :: a_set()). +%% In "Introduction to LOGIC", Suppes defines the field of a binary +%% relation to be the union of the domain and the range (or +%% counterdomain). +field(R) -> + union(domain(R), range(R)). + +-spec(relative_product(ListOfBinRels) -> BinRel2 when + ListOfBinRels :: [BinRel, ...], + BinRel :: binary_relation(), + BinRel2 :: binary_relation()). +%% The following clause is kept for backward compatibility. +%% The list is due to Dialyzer's specs. +relative_product(RT) when is_tuple(RT) -> + relative_product(tuple_to_list(RT)); +relative_product(RL) when is_list(RL) -> + case relprod_n(RL, foo, false, false) of + {error, Reason} -> + erlang:error(Reason); + Reply -> + Reply + end. + +-spec(relative_product(ListOfBinRels, BinRel1) -> BinRel2 when + ListOfBinRels :: [BinRel, ...], + BinRel :: binary_relation(), + BinRel1 :: binary_relation(), + BinRel2 :: binary_relation(); + (BinRel1, BinRel2) -> BinRel3 when + BinRel1 :: binary_relation(), + BinRel2 :: binary_relation(), + BinRel3 :: binary_relation()). +relative_product(R1, R2) when ?IS_SET(R1), ?IS_SET(R2) -> + relative_product1(converse(R1), R2); +%% The following clause is kept for backward compatibility. +%% The list is due to Dialyzer's specs. +relative_product(RT, R) when is_tuple(RT), ?IS_SET(R) -> + relative_product(tuple_to_list(RT), R); +relative_product(RL, R) when is_list(RL), ?IS_SET(R) -> + EmptyR = case ?TYPE(R) of + ?BINREL(_, _) -> ?LIST(R) =:= []; + ?ANYTYPE -> true; + _ -> erlang:error(badarg) + end, + case relprod_n(RL, R, EmptyR, true) of + {error, Reason} -> + erlang:error(Reason); + Reply -> + Reply + end. + +-spec(relative_product1(BinRel1, BinRel2) -> BinRel3 when + BinRel1 :: binary_relation(), + BinRel2 :: binary_relation(), + BinRel3 :: binary_relation()). +relative_product1(R1, R2) when ?IS_SET(R1), ?IS_SET(R2) -> + {DTR1, RTR1} = case ?TYPE(R1) of + ?BINREL(_, _) = R1T -> R1T; + ?ANYTYPE -> {?ANYTYPE, ?ANYTYPE}; + _ -> erlang:error(badarg) + end, + {DTR2, RTR2} = case ?TYPE(R2) of + ?BINREL(_, _) = R2T -> R2T; + ?ANYTYPE -> {?ANYTYPE, ?ANYTYPE}; + _ -> erlang:error(badarg) + end, + case match_types(DTR1, DTR2) of + true when DTR1 =:= ?ANYTYPE -> R1; + true when DTR2 =:= ?ANYTYPE -> R2; + true -> ?SET(relprod(?LIST(R1), ?LIST(R2)), ?BINREL(RTR1, RTR2)); + false -> erlang:error(type_mismatch) + end. + +-spec(converse(BinRel1) -> BinRel2 when + BinRel1 :: binary_relation(), + BinRel2 :: binary_relation()). +converse(R) when ?IS_SET(R) -> + case ?TYPE(R) of + ?BINREL(DT, RT) -> ?SET(converse(?LIST(R), []), ?BINREL(RT, DT)); + ?ANYTYPE -> R; + _ -> erlang:error(badarg) + end. + +-spec(image(BinRel, Set1) -> Set2 when + BinRel :: binary_relation(), + Set1 :: a_set(), + Set2 :: a_set()). +image(R, S) when ?IS_SET(R), ?IS_SET(S) -> + case ?TYPE(R) of + ?BINREL(DT, RT) -> + case match_types(DT, ?TYPE(S)) of + true -> + ?SET(usort(restrict(?LIST(S), ?LIST(R))), RT); + false -> + erlang:error(type_mismatch) + end; + ?ANYTYPE -> R; + _ -> erlang:error(badarg) + end. + +-spec(inverse_image(BinRel, Set1) -> Set2 when + BinRel :: binary_relation(), + Set1 :: a_set(), + Set2 :: a_set()). +inverse_image(R, S) when ?IS_SET(R), ?IS_SET(S) -> + case ?TYPE(R) of + ?BINREL(DT, RT) -> + case match_types(RT, ?TYPE(S)) of + true -> + NL = restrict(?LIST(S), converse(?LIST(R), [])), + ?SET(usort(NL), DT); + false -> + erlang:error(type_mismatch) + end; + ?ANYTYPE -> R; + _ -> erlang:error(badarg) + end. + +-spec(strict_relation(BinRel1) -> BinRel2 when + BinRel1 :: binary_relation(), + BinRel2 :: binary_relation()). +strict_relation(R) when ?IS_SET(R) -> + case ?TYPE(R) of + Type = ?BINREL(_, _) -> + ?SET(strict(?LIST(R), []), Type); + ?ANYTYPE -> R; + _ -> erlang:error(badarg) + end. + +-spec(weak_relation(BinRel1) -> BinRel2 when + BinRel1 :: binary_relation(), + BinRel2 :: binary_relation()). +weak_relation(R) when ?IS_SET(R) -> + case ?TYPE(R) of + ?BINREL(DT, RT) -> + case unify_types(DT, RT) of + [] -> + erlang:error(badarg); + Type -> + ?SET(weak(?LIST(R)), ?BINREL(Type, Type)) + end; + ?ANYTYPE -> R; + _ -> erlang:error(badarg) + end. + +-spec(extension(BinRel1, Set, AnySet) -> BinRel2 when + AnySet :: anyset(), + BinRel1 :: binary_relation(), + BinRel2 :: binary_relation(), + Set :: a_set()). +extension(R, S, E) when ?IS_SET(R), ?IS_SET(S) -> + case {?TYPE(R), ?TYPE(S), is_sofs_set(E)} of + {T=?BINREL(DT, RT), ST, true} -> + case match_types(DT, ST) and match_types(RT, type(E)) of + false -> + erlang:error(type_mismatch); + true -> + RL = ?LIST(R), + case extc([], ?LIST(S), to_external(E), RL) of + [] -> + R; + L -> + ?SET(merge(RL, reverse(L)), T) + end + end; + {?ANYTYPE, ?ANYTYPE, true} -> + R; + {?ANYTYPE, ST, true} -> + case type(E) of + ?SET_OF(?ANYTYPE) -> + R; + ET -> + ?SET([], ?BINREL(ST, ET)) + end; + {_, _, true} -> + erlang:error(badarg) + end. + +-spec(is_a_function(BinRel) -> Bool when + Bool :: boolean(), + BinRel :: binary_relation()). +is_a_function(R) when ?IS_SET(R) -> + case ?TYPE(R) of + ?BINREL(_, _) -> + case ?LIST(R) of + [] -> true; + [{V,_} | Es] -> is_a_func(Es, V) + end; + ?ANYTYPE -> true; + _ -> erlang:error(badarg) + end. + +-spec(restriction(BinRel1, Set) -> BinRel2 when + BinRel1 :: binary_relation(), + BinRel2 :: binary_relation(), + Set :: a_set()). +restriction(Relation, Set) -> + restriction(1, Relation, Set). + +-spec(drestriction(BinRel1, Set) -> BinRel2 when + BinRel1 :: binary_relation(), + BinRel2 :: binary_relation(), + Set :: a_set()). +drestriction(Relation, Set) -> + drestriction(1, Relation, Set). + +%%% +%%% Functions on functions only. +%%% + +-spec(composite(Function1, Function2) -> Function3 when + Function1 :: a_function(), + Function2 :: a_function(), + Function3 :: a_function()). +composite(Fn1, Fn2) when ?IS_SET(Fn1), ?IS_SET(Fn2) -> + ?BINREL(DTF1, RTF1) = case ?TYPE(Fn1)of + ?BINREL(_, _) = F1T -> F1T; + ?ANYTYPE -> {?ANYTYPE, ?ANYTYPE}; + _ -> erlang:error(badarg) + end, + ?BINREL(DTF2, RTF2) = case ?TYPE(Fn2) of + ?BINREL(_, _) = F2T -> F2T; + ?ANYTYPE -> {?ANYTYPE, ?ANYTYPE}; + _ -> erlang:error(badarg) + end, + case match_types(RTF1, DTF2) of + true when DTF1 =:= ?ANYTYPE -> Fn1; + true when DTF2 =:= ?ANYTYPE -> Fn2; + true -> + case comp(?LIST(Fn1), ?LIST(Fn2)) of + SL when is_list(SL) -> + ?SET(sort(SL), ?BINREL(DTF1, RTF2)); + Bad -> + erlang:error(Bad) + end; + false -> erlang:error(type_mismatch) + end. + +-spec(inverse(Function1) -> Function2 when + Function1 :: a_function(), + Function2 :: a_function()). +inverse(Fn) when ?IS_SET(Fn) -> + case ?TYPE(Fn) of + ?BINREL(DT, RT) -> + case inverse1(?LIST(Fn)) of + SL when is_list(SL) -> + ?SET(SL, ?BINREL(RT, DT)); + Bad -> + erlang:error(Bad) + end; + ?ANYTYPE -> Fn; + _ -> erlang:error(badarg) + end. + +%%% +%%% Functions on relations (binary or other). +%%% + +-spec(restriction(SetFun, Set1, Set2) -> Set3 when + SetFun :: set_fun(), + Set1 :: a_set(), + Set2 :: a_set(), + Set3 :: a_set()). +%% Equivalent to range(restriction(inverse(substitution(Fun, S1)), S2)). +restriction(I, R, S) when is_integer(I), ?IS_SET(R), ?IS_SET(S) -> + RT = ?TYPE(R), + ST = ?TYPE(S), + case check_for_sort(RT, I) of + empty -> + R; + error -> + erlang:error(badarg); + Sort -> + RL = ?LIST(R), + case {match_types(?REL_TYPE(I, RT), ST), ?LIST(S)} of + {true, _SL} when RL =:= [] -> + R; + {true, []} -> + ?SET([], RT); + {true, [E | Es]} when Sort =:= false -> % I =:= 1 + ?SET(reverse(restrict_n(I, RL, E, Es, [])), RT); + {true, [E | Es]} -> + ?SET(sort(restrict_n(I, keysort(I, RL), E, Es, [])), RT); + {false, _SL} -> + erlang:error(type_mismatch) + end + end; +restriction(SetFun, S1, S2) when ?IS_SET(S1), ?IS_SET(S2) -> + Type1 = ?TYPE(S1), + Type2 = ?TYPE(S2), + SL1 = ?LIST(S1), + case external_fun(SetFun) of + false when Type2 =:= ?ANYTYPE -> + S2; + false -> + case subst(SL1, SetFun, element_type(Type1)) of + {NSL, NewType} -> % NewType can be ?ANYTYPE + case match_types(NewType, Type2) of + true -> + NL = sort(restrict(?LIST(S2), converse(NSL, []))), + ?SET(NL, Type1); + false -> + erlang:error(type_mismatch) + end; + Bad -> + erlang:error(Bad) + end; + _ when Type1 =:= ?ANYTYPE -> + S1; + _XFun when ?IS_SET_OF(Type1) -> + erlang:error(badarg); + XFun -> + FunT = XFun(Type1), + try check_fun(Type1, XFun, FunT) of + Sort -> + case match_types(FunT, Type2) of + true -> + R1 = inverse_substitution(SL1, XFun, Sort), + ?SET(sort(Sort, restrict(?LIST(S2), R1)), Type1); + false -> + erlang:error(type_mismatch) + end + catch _:_ -> erlang:error(badarg) + end + end. + +-spec(drestriction(SetFun, Set1, Set2) -> Set3 when + SetFun :: set_fun(), + Set1 :: a_set(), + Set2 :: a_set(), + Set3 :: a_set()). +drestriction(I, R, S) when is_integer(I), ?IS_SET(R), ?IS_SET(S) -> + RT = ?TYPE(R), + ST = ?TYPE(S), + case check_for_sort(RT, I) of + empty -> + R; + error -> + erlang:error(badarg); + Sort -> + RL = ?LIST(R), + case {match_types(?REL_TYPE(I, RT), ST), ?LIST(S)} of + {true, []} -> + R; + {true, _SL} when RL =:= [] -> + R; + {true, [E | Es]} when Sort =:= false -> % I =:= 1 + ?SET(diff_restrict_n(I, RL, E, Es, []), RT); + {true, [E | Es]} -> + ?SET(diff_restrict_n(I, keysort(I, RL), E, Es, []), RT); + {false, _SL} -> + erlang:error(type_mismatch) + end + end; +drestriction(SetFun, S1, S2) when ?IS_SET(S1), ?IS_SET(S2) -> + Type1 = ?TYPE(S1), + Type2 = ?TYPE(S2), + SL1 = ?LIST(S1), + case external_fun(SetFun) of + false when Type2 =:= ?ANYTYPE -> + S1; + false -> + case subst(SL1, SetFun, element_type(Type1)) of + {NSL, NewType} -> % NewType can be ?ANYTYPE + case match_types(NewType, Type2) of + true -> + SL2 = ?LIST(S2), + NL = sort(diff_restrict(SL2, converse(NSL, []))), + ?SET(NL, Type1); + false -> + erlang:error(type_mismatch) + end; + Bad -> + erlang:error(Bad) + end; + _ when Type1 =:= ?ANYTYPE -> + S1; + _XFun when ?IS_SET_OF(Type1) -> + erlang:error(badarg); + XFun -> + FunT = XFun(Type1), + try check_fun(Type1, XFun, FunT) of + Sort -> + case match_types(FunT, Type2) of + true -> + R1 = inverse_substitution(SL1, XFun, Sort), + SL2 = ?LIST(S2), + ?SET(sort(Sort, diff_restrict(SL2, R1)), Type1); + false -> + erlang:error(type_mismatch) + end + catch _:_ -> erlang:error(badarg) + end + end. + +-spec(projection(SetFun, Set1) -> Set2 when + SetFun :: set_fun(), + Set1 :: a_set(), + Set2 :: a_set()). +projection(I, Set) when is_integer(I), ?IS_SET(Set) -> + Type = ?TYPE(Set), + case check_for_sort(Type, I) of + empty -> + Set; + error -> + erlang:error(badarg); + _ when I =:= 1 -> + ?SET(projection1(?LIST(Set)), ?REL_TYPE(I, Type)); + _ -> + ?SET(projection_n(?LIST(Set), I, []), ?REL_TYPE(I, Type)) + end; +projection(Fun, Set) -> + range(substitution(Fun, Set)). + +-spec(substitution(SetFun, Set1) -> Set2 when + SetFun :: set_fun(), + Set1 :: a_set(), + Set2 :: a_set()). +substitution(I, Set) when is_integer(I), ?IS_SET(Set) -> + Type = ?TYPE(Set), + case check_for_sort(Type, I) of + empty -> + Set; + error -> + erlang:error(badarg); + _Sort -> + NType = ?REL_TYPE(I, Type), + NSL = substitute_element(?LIST(Set), I, []), + ?SET(NSL, ?BINREL(Type, NType)) + end; +substitution(SetFun, Set) when ?IS_SET(Set) -> + Type = ?TYPE(Set), + L = ?LIST(Set), + case external_fun(SetFun) of + false when L =/= [] -> + case subst(L, SetFun, element_type(Type)) of + {SL, NewType} -> + ?SET(reverse(SL), ?BINREL(Type, NewType)); + Bad -> + erlang:error(Bad) + end; + false -> + empty_set(); + _ when Type =:= ?ANYTYPE -> + empty_set(); + _XFun when ?IS_SET_OF(Type) -> + erlang:error(badarg); + XFun -> + FunT = XFun(Type), + try check_fun(Type, XFun, FunT) of + _Sort -> + SL = substitute(L, XFun, []), + ?SET(SL, ?BINREL(Type, FunT)) + catch _:_ -> erlang:error(badarg) + end + end. + +-spec(partition(SetOfSets) -> Partition when + SetOfSets :: set_of_sets(), + Partition :: a_set()). +partition(Sets) -> + F1 = relation_to_family(canonical_relation(Sets)), + F2 = relation_to_family(converse(F1)), + range(F2). + +-spec(partition(SetFun, Set) -> Partition when + SetFun :: set_fun(), + Partition :: a_set(), + Set :: a_set()). +partition(I, Set) when is_integer(I), ?IS_SET(Set) -> + Type = ?TYPE(Set), + case check_for_sort(Type, I) of + empty -> + Set; + error -> + erlang:error(badarg); + false -> % I =:= 1 + ?SET(partition_n(I, ?LIST(Set)), ?SET_OF(Type)); + true -> + ?SET(partition_n(I, keysort(I, ?LIST(Set))), ?SET_OF(Type)) + end; +partition(Fun, Set) -> + range(partition_family(Fun, Set)). + +-spec(partition(SetFun, Set1, Set2) -> {Set3, Set4} when + SetFun :: set_fun(), + Set1 :: a_set(), + Set2 :: a_set(), + Set3 :: a_set(), + Set4 :: a_set()). +partition(I, R, S) when is_integer(I), ?IS_SET(R), ?IS_SET(S) -> + RT = ?TYPE(R), + ST = ?TYPE(S), + case check_for_sort(RT, I) of + empty -> + {R, R}; + error -> + erlang:error(badarg); + Sort -> + RL = ?LIST(R), + case {match_types(?REL_TYPE(I, RT), ST), ?LIST(S)} of + {true, _SL} when RL =:= [] -> + {R, R}; + {true, []} -> + {?SET([], RT), R}; + {true, [E | Es]} when Sort =:= false -> % I =:= 1 + [L1 | L2] = partition3_n(I, RL, E, Es, [], []), + {?SET(L1, RT), ?SET(L2, RT)}; + {true, [E | Es]} -> + [L1 | L2] = partition3_n(I, keysort(I,RL), E, Es, [], []), + {?SET(L1, RT), ?SET(L2, RT)}; + {false, _SL} -> + erlang:error(type_mismatch) + end + end; +partition(SetFun, S1, S2) when ?IS_SET(S1), ?IS_SET(S2) -> + Type1 = ?TYPE(S1), + Type2 = ?TYPE(S2), + SL1 = ?LIST(S1), + case external_fun(SetFun) of + false when Type2 =:= ?ANYTYPE -> + {S2, S1}; + false -> + case subst(SL1, SetFun, element_type(Type1)) of + {NSL, NewType} -> % NewType can be ?ANYTYPE + case match_types(NewType, Type2) of + true -> + R1 = converse(NSL, []), + [L1 | L2] = partition3(?LIST(S2), R1), + {?SET(sort(L1), Type1), ?SET(sort(L2), Type1)}; + false -> + erlang:error(type_mismatch) + end; + Bad -> + erlang:error(Bad) + end; + _ when Type1 =:= ?ANYTYPE -> + {S1, S1}; + _XFun when ?IS_SET_OF(Type1) -> + erlang:error(badarg); + XFun -> + FunT = XFun(Type1), + try check_fun(Type1, XFun, FunT) of + Sort -> + case match_types(FunT, Type2) of + true -> + R1 = inverse_substitution(SL1, XFun, Sort), + [L1 | L2] = partition3(?LIST(S2), R1), + {?SET(sort(L1), Type1), ?SET(sort(L2), Type1)}; + false -> + erlang:error(type_mismatch) + end + catch _:_ -> erlang:error(badarg) + end + end. + +-spec(multiple_relative_product(TupleOfBinRels, BinRel1) -> BinRel2 when + TupleOfBinRels :: tuple_of(BinRel), + BinRel :: binary_relation(), + BinRel1 :: binary_relation(), + BinRel2 :: binary_relation()). +multiple_relative_product(T, R) when is_tuple(T), ?IS_SET(R) -> + case test_rel(R, tuple_size(T), eq) of + true when ?TYPE(R) =:= ?ANYTYPE -> + empty_set(); + true -> + MProd = mul_relprod(tuple_to_list(T), 1, R), + relative_product(MProd); + false -> + erlang:error(badarg) + end. + +-spec(join(Relation1, I, Relation2, J) -> Relation3 when + Relation1 :: relation(), + Relation2 :: relation(), + Relation3 :: relation(), + I :: pos_integer(), + J :: pos_integer()). +join(R1, I1, R2, I2) + when ?IS_SET(R1), ?IS_SET(R2), is_integer(I1), is_integer(I2) -> + case test_rel(R1, I1, lte) and test_rel(R2, I2, lte) of + false -> erlang:error(badarg); + true when ?TYPE(R1) =:= ?ANYTYPE -> R1; + true when ?TYPE(R2) =:= ?ANYTYPE -> R2; + true -> + L1 = ?LIST(raise_element(R1, I1)), + L2 = ?LIST(raise_element(R2, I2)), + T = relprod1(L1, L2), + F = case (I1 =:= 1) and (I2 =:= 1) of + true -> + fun({X,Y}) -> join_element(X, Y) end; + false -> + fun({X,Y}) -> + list_to_tuple(join_element(X, Y, I2)) + end + end, + ?SET(replace(T, F, []), F({?TYPE(R1), ?TYPE(R2)})) + end. + +%% Inlined. +test_rel(R, I, C) -> + case ?TYPE(R) of + Rel when ?IS_RELATION(Rel), C =:= eq, I =:= ?REL_ARITY(Rel) -> true; + Rel when ?IS_RELATION(Rel), C =:= lte, I>=1, I =< ?REL_ARITY(Rel) -> + true; + ?ANYTYPE -> true; + _ -> false + end. + +%%% +%%% Family functions +%%% + +-spec(fam2rel(Family) -> BinRel when + Family :: family(), + BinRel :: binary_relation()). +fam2rel(F) -> + family_to_relation(F). + +-spec(family_to_relation(Family) -> BinRel when + Family :: family(), + BinRel :: binary_relation()). +%% Inlined. +family_to_relation(F) when ?IS_SET(F) -> + case ?TYPE(F) of + ?FAMILY(DT, RT) -> + ?SET(family2rel(?LIST(F), []), ?BINREL(DT, RT)); + ?ANYTYPE -> F; + _ -> erlang:error(badarg) + end. + +-spec(family_specification(Fun, Family1) -> Family2 when + Fun :: spec_fun(), + Family1 :: family(), + Family2 :: family()). +family_specification(Fun, F) when ?IS_SET(F) -> + case ?TYPE(F) of + ?FAMILY(_DT, Type) = FType -> + R = case external_fun(Fun) of + false -> + fam_spec(?LIST(F), Fun, Type, []); + XFun -> + fam_specification(?LIST(F), XFun, []) + end, + case R of + SL when is_list(SL) -> + ?SET(SL, FType); + Bad -> + erlang:error(Bad) + end; + ?ANYTYPE -> F; + _ -> erlang:error(badarg) + end. + +-spec(union_of_family(Family) -> Set when + Family :: family(), + Set :: a_set()). +union_of_family(F) when ?IS_SET(F) -> + case ?TYPE(F) of + ?FAMILY(_DT, Type) -> + ?SET(un_of_fam(?LIST(F), []), Type); + ?ANYTYPE -> F; + _ -> erlang:error(badarg) + end. + +-spec(intersection_of_family(Family) -> Set when + Family :: family(), + Set :: a_set()). +intersection_of_family(F) when ?IS_SET(F) -> + case ?TYPE(F) of + ?FAMILY(_DT, Type) -> + case int_of_fam(?LIST(F)) of + FU when is_list(FU) -> + ?SET(FU, Type); + Bad -> + erlang:error(Bad) + end; + _ -> erlang:error(badarg) + end. + +-spec(family_union(Family1) -> Family2 when + Family1 :: family(), + Family2 :: family()). +family_union(F) when ?IS_SET(F) -> + case ?TYPE(F) of + ?FAMILY(DT, ?SET_OF(Type)) -> + ?SET(fam_un(?LIST(F), []), ?FAMILY(DT, Type)); + ?ANYTYPE -> F; + _ -> erlang:error(badarg) + end. + +-spec(family_intersection(Family1) -> Family2 when + Family1 :: family(), + Family2 :: family()). +family_intersection(F) when ?IS_SET(F) -> + case ?TYPE(F) of + ?FAMILY(DT, ?SET_OF(Type)) -> + case fam_int(?LIST(F), []) of + FU when is_list(FU) -> + ?SET(FU, ?FAMILY(DT, Type)); + Bad -> + erlang:error(Bad) + end; + ?ANYTYPE -> F; + _ -> erlang:error(badarg) + end. + +-spec(family_domain(Family1) -> Family2 when + Family1 :: family(), + Family2 :: family()). +family_domain(F) when ?IS_SET(F) -> + case ?TYPE(F) of + ?FAMILY(FDT, ?BINREL(DT, _)) -> + ?SET(fam_dom(?LIST(F), []), ?FAMILY(FDT, DT)); + ?ANYTYPE -> F; + ?FAMILY(_, ?ANYTYPE) -> F; + _ -> erlang:error(badarg) + end. + +-spec(family_range(Family1) -> Family2 when + Family1 :: family(), + Family2 :: family()). +family_range(F) when ?IS_SET(F) -> + case ?TYPE(F) of + ?FAMILY(DT, ?BINREL(_, RT)) -> + ?SET(fam_ran(?LIST(F), []), ?FAMILY(DT, RT)); + ?ANYTYPE -> F; + ?FAMILY(_, ?ANYTYPE) -> F; + _ -> erlang:error(badarg) + end. + +-spec(family_field(Family1) -> Family2 when + Family1 :: family(), + Family2 :: family()). +family_field(F) -> + family_union(family_domain(F), family_range(F)). + +-spec(family_union(Family1, Family2) -> Family3 when + Family1 :: family(), + Family2 :: family(), + Family3 :: family()). +family_union(F1, F2) -> + fam_binop(F1, F2, fun fam_union/3). + +-spec(family_intersection(Family1, Family2) -> Family3 when + Family1 :: family(), + Family2 :: family(), + Family3 :: family()). +family_intersection(F1, F2) -> + fam_binop(F1, F2, fun fam_intersect/3). + +-spec(family_difference(Family1, Family2) -> Family3 when + Family1 :: family(), + Family2 :: family(), + Family3 :: family()). +family_difference(F1, F2) -> + fam_binop(F1, F2, fun fam_difference/3). + +%% Inlined. +fam_binop(F1, F2, FF) when ?IS_SET(F1), ?IS_SET(F2) -> + case unify_types(?TYPE(F1), ?TYPE(F2)) of + [] -> + erlang:error(type_mismatch); + ?ANYTYPE -> + F1; + Type = ?FAMILY(_, _) -> + ?SET(FF(?LIST(F1), ?LIST(F2), []), Type); + _ -> erlang:error(badarg) + end. + +-spec(partition_family(SetFun, Set) -> Family when + Family :: family(), + SetFun :: set_fun(), + Set :: a_set()). +partition_family(I, Set) when is_integer(I), ?IS_SET(Set) -> + Type = ?TYPE(Set), + case check_for_sort(Type, I) of + empty -> + Set; + error -> + erlang:error(badarg); + false -> % when I =:= 1 + ?SET(fam_partition_n(I, ?LIST(Set)), + ?BINREL(?REL_TYPE(I, Type), ?SET_OF(Type))); + true -> + ?SET(fam_partition_n(I, keysort(I, ?LIST(Set))), + ?BINREL(?REL_TYPE(I, Type), ?SET_OF(Type))) + end; +partition_family(SetFun, Set) when ?IS_SET(Set) -> + Type = ?TYPE(Set), + SL = ?LIST(Set), + case external_fun(SetFun) of + false when SL =/= [] -> + case subst(SL, SetFun, element_type(Type)) of + {NSL, NewType} -> + P = fam_partition(converse(NSL, []), true), + ?SET(reverse(P), ?BINREL(NewType, ?SET_OF(Type))); + Bad -> + erlang:error(Bad) + end; + false -> + empty_set(); + _ when Type =:= ?ANYTYPE -> + empty_set(); + _XFun when ?IS_SET_OF(Type) -> + erlang:error(badarg); + XFun -> + DType = XFun(Type), + try check_fun(Type, XFun, DType) of + Sort -> + Ts = inverse_substitution(?LIST(Set), XFun, Sort), + P = fam_partition(Ts, Sort), + ?SET(reverse(P), ?BINREL(DType, ?SET_OF(Type))) + catch _:_ -> erlang:error(badarg) + end + end. + +-spec(family_projection(SetFun, Family1) -> Family2 when + SetFun :: set_fun(), + Family1 :: family(), + Family2 :: family()). +family_projection(SetFun, F) when ?IS_SET(F) -> + case ?TYPE(F) of + ?FAMILY(_, _) when [] =:= ?LIST(F) -> + empty_set(); + ?FAMILY(DT, Type) -> + case external_fun(SetFun) of + false -> + case fam_proj(?LIST(F), SetFun, Type, ?ANYTYPE, []) of + {SL, NewType} -> + ?SET(SL, ?BINREL(DT, NewType)); + Bad -> + erlang:error(Bad) + end; + _ -> + erlang:error(badarg) + end; + ?ANYTYPE -> F; + _ -> erlang:error(badarg) + end. + +%%% +%%% Digraph functions +%%% + +-spec(family_to_digraph(Family) -> Graph when + Graph :: digraph:graph(), + Family :: family()). +family_to_digraph(F) when ?IS_SET(F) -> + case ?TYPE(F) of + ?FAMILY(_, _) -> fam2digraph(F, digraph:new()); + ?ANYTYPE -> digraph:new(); + _Else -> erlang:error(badarg) + end. + +-spec(family_to_digraph(Family, GraphType) -> Graph when + Graph :: digraph:graph(), + Family :: family(), + GraphType :: [digraph:d_type()]). +family_to_digraph(F, Type) when ?IS_SET(F) -> + case ?TYPE(F) of + ?FAMILY(_, _) -> ok; + ?ANYTYPE -> ok; + _Else -> erlang:error(badarg) + end, + try digraph:new(Type) of + G -> case catch fam2digraph(F, G) of + {error, Reason} -> + true = digraph:delete(G), + erlang:error(Reason); + _ -> + G + end + catch + error:badarg -> erlang:error(badarg) + end. + +-spec(digraph_to_family(Graph) -> Family when + Graph :: digraph:graph(), + Family :: family()). +digraph_to_family(G) -> + try digraph_family(G) of + L -> ?SET(L, ?FAMILY(?ATOM_TYPE, ?ATOM_TYPE)) + catch _:_ -> erlang:error(badarg) + end. + +-spec(digraph_to_family(Graph, Type) -> Family when + Graph :: digraph:graph(), + Family :: family(), + Type :: type()). +digraph_to_family(G, T) -> + case {is_type(T), T} of + {true, ?SET_OF(?FAMILY(_,_) = Type)} -> + try digraph_family(G) of + L -> ?SET(L, Type) + catch _:_ -> erlang:error(badarg) + end; + _ -> + erlang:error(badarg) + end. + +%% +%% Local functions +%% + +%% Type = OrderedSetType +%% | SetType +%% | atom() except '_' +%% OrderedSetType = {Type, ..., Type} +%% SetType = [ElementType] % list of exactly one element +%% ElementType = '_' % any type (implies empty set) +%% | Type + +is_types(0, _T) -> + true; +is_types(I, T) -> + case is_type(?REL_TYPE(I, T)) of + true -> is_types(I-1, T); + false -> false + end. + +is_element_type(?ANYTYPE) -> + true; +is_element_type(T) -> + is_type(T). + +set_of_sets([S | Ss], L, T0) when ?IS_SET(S) -> + case unify_types([?TYPE(S)], T0) of + [] -> {error, type_mismatch}; + Type -> set_of_sets(Ss, [?LIST(S) | L], Type) + end; +set_of_sets([S | Ss], L, T0) when ?IS_ORDSET(S) -> + case unify_types(?ORDTYPE(S), T0) of + [] -> {error, type_mismatch}; + Type -> set_of_sets(Ss, [?ORDDATA(S) | L], Type) + end; +set_of_sets([], L, T) -> + ?SET(usort(L), T); +set_of_sets(_, _L, _T) -> + {error, badarg}. + +ordset_of_sets([S | Ss], L, T) when ?IS_SET(S) -> + ordset_of_sets(Ss, [?LIST(S) | L], [[?TYPE(S)] | T]); +ordset_of_sets([S | Ss], L, T) when ?IS_ORDSET(S) -> + ordset_of_sets(Ss, [?ORDDATA(S) | L], [?ORDTYPE(S) | T]); +ordset_of_sets([], L, T) -> + ?ORDSET(list_to_tuple(reverse(L)), list_to_tuple(reverse(T))); +ordset_of_sets(_, _L, _T) -> + error. + +%% Inlined. +rel(Ts, [Type]) -> + case is_type(Type) and atoms_only(Type, 1) of + true -> + rel(Ts, tuple_size(Type), Type); + false -> + rel_type(Ts, [], Type) + end; +rel(Ts, Sz) -> + rel(Ts, Sz, erlang:make_tuple(Sz, ?ATOM_TYPE)). + +atoms_only(Type, I) when ?IS_ATOM_TYPE(?REL_TYPE(I, Type)) -> + atoms_only(Type, I+1); +atoms_only(Type, I) when I > tuple_size(Type), ?IS_RELATION(Type) -> + true; +atoms_only(_Type, _I) -> + false. + +rel(Ts, Sz, Type) when Sz >= 1 -> + SL = usort(Ts), + rel(SL, SL, Sz, Type). + +rel([T | Ts], L, Sz, Type) when tuple_size(T) =:= Sz -> + rel(Ts, L, Sz, Type); +rel([], L, _Sz, Type) -> + ?SET(L, Type). + +rel_type([E | Ts], L, Type) -> + {NType, NE} = make_element(E, Type, Type), + rel_type(Ts, [NE | L], NType); +rel_type([], [], ?ANYTYPE) -> + empty_set(); +rel_type([], SL, Type) when ?IS_RELATION(Type) -> + ?SET(usort(SL), Type). + +%% Inlined. +a_func(Ts, T) -> + case {T, is_type(T)} of + {[?BINREL(DT, RT) = Type], true} when ?IS_ATOM_TYPE(DT), + ?IS_ATOM_TYPE(RT) -> + func(Ts, Type); + {[Type], true} -> + func_type(Ts, [], Type, fun(?BINREL(_,_)) -> true end) + end. + +func(L0, Type) -> + L = usort(L0), + func(L, L, L, Type). + +func([{X,_} | Ts], X0, L, Type) when X /= X0 -> + func(Ts, X, L, Type); +func([{X,_} | _Ts], X0, _L, _Type) when X == X0 -> + bad_function; +func([], _X0, L, Type) -> + ?SET(L, Type). + +%% Inlined. +fam(Ts, T) -> + case {T, is_type(T)} of + {[?FAMILY(DT, RT) = Type], true} when ?IS_ATOM_TYPE(DT), + ?IS_ATOM_TYPE(RT) -> + fam2(Ts, Type); + {[Type], true} -> + func_type(Ts, [], Type, fun(?FAMILY(_,_)) -> true end) + end. + +fam2([], Type) -> + ?SET([], Type); +fam2(Ts, Type) -> + fam2(sort(Ts), Ts, [], Type). + +fam2([{I,L} | T], I0, SL, Type) when I /= I0 -> + fam2(T, I, [{I,usort(L)} | SL], Type); +fam2([{I,L} | T], I0, SL, Type) when I == I0 -> + case {usort(L), SL} of + {NL, [{_I,NL1} | _]} when NL == NL1 -> + fam2(T, I0, SL, Type); + _ -> + bad_function + end; +fam2([], _I0, SL, Type) -> + ?SET(reverse(SL), Type). + +func_type([E | T], SL, Type, F) -> + {NType, NE} = make_element(E, Type, Type), + func_type(T, [NE | SL], NType, F); +func_type([], [], ?ANYTYPE, _F) -> + empty_set(); +func_type([], SL, Type, F) -> + true = F(Type), + NL = usort(SL), + check_function(NL, ?SET(NL, Type)). + +setify(L, ?SET_OF(Atom)) when ?IS_ATOM_TYPE(Atom), Atom =/= ?ANYTYPE -> + ?SET(usort(L), Atom); +setify(L, ?SET_OF(Type0)) -> + try is_no_lists(Type0) of + N when is_integer(N) -> + rel(L, N, Type0); + Sizes -> + make_oset(L, Sizes, L, Type0) + catch + _:_ -> + {?SET_OF(Type), Set} = create(L, Type0, Type0, []), + ?SET(Set, Type) + end; +setify(E, Type0) -> + {Type, OrdSet} = make_element(E, Type0, Type0), + ?ORDSET(OrdSet, Type). + +is_no_lists(T) when is_tuple(T) -> + Sz = tuple_size(T), + is_no_lists(T, Sz, Sz, []). + +is_no_lists(_T, 0, Sz, []) -> + Sz; +is_no_lists(_T, 0, Sz, L) -> + {Sz, L}; +is_no_lists(T, I, Sz, L) when ?IS_ATOM_TYPE(?REL_TYPE(I, T)) -> + is_no_lists(T, I-1, Sz, L); +is_no_lists(T, I, Sz, L) -> + is_no_lists(T, I-1, Sz, [{I,is_no_lists(?REL_TYPE(I, T))} | L]). + +create([E | Es], T, T0, L) -> + {NT, S} = make_element(E, T, T0), + create(Es, NT, T0, [S | L]); +create([], T, _T0, L) -> + {?SET_OF(T), usort(L)}. + +make_element(C, ?ANYTYPE, _T0) -> + make_element(C); +make_element(C, Atom, ?ANYTYPE) when ?IS_ATOM_TYPE(Atom), + not is_list(C), not is_tuple(C) -> + {Atom, C}; +make_element(C, Atom, Atom) when ?IS_ATOM_TYPE(Atom) -> + {Atom, C}; +make_element(T, TT, ?ANYTYPE) when tuple_size(T) =:= tuple_size(TT) -> + make_tuple(tuple_to_list(T), tuple_to_list(TT), [], [], ?ANYTYPE); +make_element(T, TT, T0) when tuple_size(T) =:= tuple_size(TT) -> + make_tuple(tuple_to_list(T), tuple_to_list(TT), [], [], tuple_to_list(T0)); +make_element(L, [LT], ?ANYTYPE) when is_list(L) -> + create(L, LT, ?ANYTYPE, []); +make_element(L, [LT], [T0]) when is_list(L) -> + create(L, LT, T0, []). + +make_tuple([E | Es], [T | Ts], NT, L, T0) when T0 =:= ?ANYTYPE -> + {ET, ES} = make_element(E, T, T0), + make_tuple(Es, Ts, [ET | NT], [ES | L], T0); +make_tuple([E | Es], [T | Ts], NT, L, [T0 | T0s]) -> + {ET, ES} = make_element(E, T, T0), + make_tuple(Es, Ts, [ET | NT], [ES | L], T0s); +make_tuple([], [], NT, L, _T0s) when NT =/= [] -> + {list_to_tuple(reverse(NT)), list_to_tuple(reverse(L))}. + +%% Derive type. +make_element(C) when not is_list(C), not is_tuple(C) -> + {?ATOM_TYPE, C}; +make_element(T) when is_tuple(T) -> + make_tuple(tuple_to_list(T), [], []); +make_element(L) when is_list(L) -> + create(L, ?ANYTYPE, ?ANYTYPE, []). + +make_tuple([E | Es], T, L) -> + {ET, ES} = make_element(E), + make_tuple(Es, [ET | T], [ES | L]); +make_tuple([], T, L) when T =/= [] -> + {list_to_tuple(reverse(T)), list_to_tuple(reverse(L))}. + +make_oset([T | Ts], Szs, L, Type) -> + true = test_oset(Szs, T, T), + make_oset(Ts, Szs, L, Type); +make_oset([], _Szs, L, Type) -> + ?SET(usort(L), Type). + +%% Optimization. Avoid re-building (nested) tuples. +test_oset({Sz,Args}, T, T0) when tuple_size(T) =:= Sz -> + test_oset_args(Args, T, T0); +test_oset(Sz, T, _T0) when tuple_size(T) =:= Sz -> + true. + +test_oset_args([{Arg,Szs} | Ss], T, T0) -> + true = test_oset(Szs, ?REL_TYPE(Arg, T), T0), + test_oset_args(Ss, T, T0); +test_oset_args([], _T, _T0) -> + true. + +list_of_sets([S | Ss], Type, L) -> + list_of_sets(Ss, Type, [?SET(S, Type) | L]); +list_of_sets([], _Type, L) -> + reverse(L). + +list_of_ordsets([S | Ss], Type, L) -> + list_of_ordsets(Ss, Type, [?ORDSET(S, Type) | L]); +list_of_ordsets([], _Type, L) -> + reverse(L). + +tuple_of_sets([S | Ss], [?SET_OF(Type) | Types], L) -> + tuple_of_sets(Ss, Types, [?SET(S, Type) | L]); +tuple_of_sets([S | Ss], [Type | Types], L) -> + tuple_of_sets(Ss, Types, [?ORDSET(S, Type) | L]); +tuple_of_sets([], [], L) -> + list_to_tuple(reverse(L)). + +spec([E | Es], Fun, Type, L) -> + case Fun(term2set(E, Type)) of + true -> + spec(Es, Fun, Type, [E | L]); + false -> + spec(Es, Fun, Type, L); + _ -> + badarg + end; +spec([], _Fun, _Type, L) -> + reverse(L). + +specification([E | Es], Fun, L) -> + case Fun(E) of + true -> + specification(Es, Fun, [E | L]); + false -> + specification(Es, Fun, L); + _ -> + badarg + end; +specification([], _Fun, L) -> + reverse(L). + +%% Elements from the first list are kept. +intersection([H1 | T1], [H2 | T2], L) when H1 < H2 -> + intersection1(T1, T2, L, H2); +intersection([H1 | T1], [H2 | T2], L) when H1 == H2 -> + intersection(T1, T2, [H1 | L]); +intersection([H1 | T1], [_H2 | T2], L) -> + intersection2(T1, T2, L, H1); +intersection(_, _, L) -> + reverse(L). + +intersection1([H1 | T1], T2, L, H2) when H1 < H2 -> + intersection1(T1, T2, L, H2); +intersection1([H1 | T1], T2, L, H2) when H1 == H2 -> + intersection(T1, T2, [H1 | L]); +intersection1([H1 | T1], T2, L, _H2) -> + intersection2(T1, T2, L, H1); +intersection1(_, _, L, _) -> + reverse(L). + +intersection2(T1, [H2 | T2], L, H1) when H1 > H2 -> + intersection2(T1, T2, L, H1); +intersection2(T1, [H2 | T2], L, H1) when H1 == H2 -> + intersection(T1, T2, [H1 | L]); +intersection2(T1, [H2 | T2], L, _H1) -> + intersection1(T1, T2, L, H2); +intersection2(_, _, L, _) -> + reverse(L). + +difference([H1 | T1], [H2 | T2], L) when H1 < H2 -> + diff(T1, T2, [H1 | L], H2); +difference([H1 | T1], [H2 | T2], L) when H1 == H2 -> + difference(T1, T2, L); +difference([H1 | T1], [_H2 | T2], L) -> + diff2(T1, T2, L, H1); +difference(L1, _, L) -> + reverse(L, L1). + +diff([H1 | T1], T2, L, H2) when H1 < H2 -> + diff(T1, T2, [H1 | L], H2); +diff([H1 | T1], T2, L, H2) when H1 == H2 -> + difference(T1, T2, L); +diff([H1 | T1], T2, L, _H2) -> + diff2(T1, T2, L, H1); +diff(_, _, L, _) -> + reverse(L). + +diff2(T1, [H2 | T2], L, H1) when H1 > H2 -> + diff2(T1, T2, L, H1); +diff2(T1, [H2 | T2], L, H1) when H1 == H2 -> + difference(T1, T2, L); +diff2(T1, [H2 | T2], L, H1) -> + diff(T1, T2, [H1 | L], H2); +diff2(T1, _, L, H1) -> + reverse(L, [H1 | T1]). + +symdiff([H1 | T1], T2, L) -> + symdiff2(T1, T2, L, H1); +symdiff(_, T2, L) -> + reverse(L, T2). + +symdiff1([H1 | T1], T2, L, H2) when H1 < H2 -> + symdiff1(T1, T2, [H1 | L], H2); +symdiff1([H1 | T1], T2, L, H2) when H1 == H2 -> + symdiff(T1, T2, L); +symdiff1([H1 | T1], T2, L, H2) -> + symdiff2(T1, T2, [H2 | L], H1); +symdiff1(_, T2, L, H2) -> + reverse(L, [H2 | T2]). + +symdiff2(T1, [H2 | T2], L, H1) when H1 > H2 -> + symdiff2(T1, T2, [H2 | L], H1); +symdiff2(T1, [H2 | T2], L, H1) when H1 == H2 -> + symdiff(T1, T2, L); +symdiff2(T1, [H2 | T2], L, H1) -> + symdiff1(T1, T2, [H1 | L], H2); +symdiff2(T1, _, L, H1) -> + reverse(L, [H1 | T1]). + +sympart([H1 | T1], [H2 | T2], L1, L12, L2, T) when H1 < H2 -> + sympart1(T1, T2, [H1 | L1], L12, L2, T, H2); +sympart([H1 | T1], [H2 | T2], L1, L12, L2, T) when H1 == H2 -> + sympart(T1, T2, L1, [H1 | L12], L2, T); +sympart([H1 | T1], [H2 | T2], L1, L12, L2, T) -> + sympart2(T1, T2, L1, L12, [H2 | L2], T, H1); +sympart(S1, [], L1, L12, L2, T) -> + {?SET(reverse(L1, S1), T), + ?SET(reverse(L12), T), + ?SET(reverse(L2), T)}; +sympart(_, S2, L1, L12, L2, T) -> + {?SET(reverse(L1), T), + ?SET(reverse(L12), T), + ?SET(reverse(L2, S2), T)}. + +sympart1([H1 | T1], T2, L1, L12, L2, T, H2) when H1 < H2 -> + sympart1(T1, T2, [H1 | L1], L12, L2, T, H2); +sympart1([H1 | T1], T2, L1, L12, L2, T, H2) when H1 == H2 -> + sympart(T1, T2, L1, [H1 | L12], L2, T); +sympart1([H1 | T1], T2, L1, L12, L2, T, H2) -> + sympart2(T1, T2, L1, L12, [H2 | L2], T, H1); +sympart1(_, T2, L1, L12, L2, T, H2) -> + {?SET(reverse(L1), T), + ?SET(reverse(L12), T), + ?SET(reverse(L2, [H2 | T2]), T)}. + +sympart2(T1, [H2 | T2], L1, L12, L2, T, H1) when H1 > H2 -> + sympart2(T1, T2, L1, L12, [H2 | L2], T, H1); +sympart2(T1, [H2 | T2], L1, L12, L2, T, H1) when H1 == H2 -> + sympart(T1, T2, L1, [H1 | L12], L2, T); +sympart2(T1, [H2 | T2], L1, L12, L2, T, H1) -> + sympart1(T1, T2, [H1 | L1], L12, L2, T, H2); +sympart2(T1, _, L1, L12, L2, T, H1) -> + {?SET(reverse(L1, [H1 | T1]), T), + ?SET(reverse(L12), T), + ?SET(reverse(L2), T)}. + +prod([[E | Es] | Xs], T, L) -> + prod(Es, Xs, T, prod(Xs, [E | T], L)); +prod([], T, L) -> + [list_to_tuple(reverse(T)) | L]. + +prod([E | Es], Xs, T, L) -> + prod(Es, Xs, T, prod(Xs, [E | T], L)); +prod([], _Xs, _E, L) -> + L. + +constant_function([E | Es], X, L) -> + constant_function(Es, X, [{E,X} | L]); +constant_function([], _X, L) -> + reverse(L). + +subset([H1 | T1], [H2 | T2]) when H1 > H2 -> + subset(T1, T2, H1); +subset([H1 | T1], [H2 | T2]) when H1 == H2 -> + subset(T1, T2); +subset(L1, _) -> + L1 =:= []. + +subset(T1, [H2 | T2], H1) when H1 > H2 -> + subset(T1, T2, H1); +subset(T1, [H2 | T2], H1) when H1 == H2 -> + subset(T1, T2); +subset(_, _, _) -> + false. + +disjoint([B | Bs], A, As) when A < B -> + disjoint(As, B, Bs); +disjoint([B | _Bs], A, _As) when A == B -> + false; +disjoint([_B | Bs], A, As) -> + disjoint(Bs, A, As); +disjoint(_Bs, _A, _As) -> + true. + +%% Append sets that come in order, then "merge". +lunion([[_] = S]) -> % optimization + S; +lunion([[] | Ls]) -> + lunion(Ls); +lunion([S | Ss]) -> + umerge(lunion(Ss, last(S), [S], [])); +lunion([]) -> + []. + +lunion([[E] = S | Ss], Last, SL, Ls) when E > Last -> % optimization + lunion(Ss, E, [S | SL], Ls); +lunion([S | Ss], Last, SL, Ls) when hd(S) > Last -> + lunion(Ss, last(S), [S | SL], Ls); +lunion([S | Ss], _Last, SL, Ls) -> + lunion(Ss, last(S), [S], [append(reverse(SL)) | Ls]); +lunion([], _Last, SL, Ls) -> + [append(reverse(SL)) | Ls]. + +%% The empty list is always the first list, if present. +lintersection(_, []) -> + []; +lintersection([S | Ss], S0) -> + lintersection(Ss, intersection(S, S0, [])); +lintersection([], S) -> + S. + +can_rel([S | Ss], L) -> + can_rel(Ss, L, S, S); +can_rel([], L) -> + sort(L). + +can_rel(Ss, L, [E | Es], S) -> + can_rel(Ss, [{E, S} | L], Es, S); +can_rel(Ss, L, _, _S) -> + can_rel(Ss, L). + +rel2family([{X,Y} | S]) -> + rel2fam(S, X, [Y], []); +rel2family([]) -> + []. + +rel2fam([{X,Y} | S], X0, YL, L) when X0 == X -> + rel2fam(S, X0, [Y | YL], L); +rel2fam([{X,Y} | S], X0, [A,B | YL], L) -> % optimization + rel2fam(S, X, [Y], [{X0,reverse(YL,[B,A])} | L]); +rel2fam([{X,Y} | S], X0, YL, L) -> + rel2fam(S, X, [Y], [{X0,YL} | L]); +rel2fam([], X, YL, L) -> + reverse([{X,reverse(YL)} | L]). + +dom([{X,_} | Es]) -> + dom([], X, Es); +dom([] = L) -> + L. + +dom(L, X, [{X1,_} | Es]) when X == X1 -> + dom(L, X, Es); +dom(L, X, [{Y,_} | Es]) -> + dom([X | L], Y, Es); +dom(L, X, []) -> + reverse(L, [X]). + +ran([{_,Y} | Es], L) -> + ran(Es, [Y | L]); +ran([], L) -> + usort(L). + +relprod(A, B) -> + usort(relprod1(A, B)). + +relprod1([{Ay,Ax} | A], B) -> + relprod1(B, Ay, Ax, A, []); +relprod1(_A, _B) -> + []. + +relprod1([{Bx,_By} | B], Ay, Ax, A, L) when Ay > Bx -> + relprod1(B, Ay, Ax, A, L); +relprod1([{Bx,By} | B], Ay, Ax, A, L) when Ay == Bx -> + relprod(B, Bx, By, A, [{Ax,By} | L], Ax, B, Ay); +relprod1([{Bx,By} | B], _Ay, _Ax, A, L) -> + relprod2(B, Bx, By, A, L); +relprod1(_B, _Ay, _Ax, _A, L) -> + L. + +relprod2(B, Bx, By, [{Ay, _Ax} | A], L) when Ay < Bx -> + relprod2(B, Bx, By, A, L); +relprod2(B, Bx, By, [{Ay, Ax} | A], L) when Ay == Bx -> + relprod(B, Bx, By, A, [{Ax,By} | L], Ax, B, Ay); +relprod2(B, _Bx, _By, [{Ay, Ax} | A], L) -> + relprod1(B, Ay, Ax, A, L); +relprod2(_, _, _, _, L) -> + L. + +relprod(B0, Bx0, By0, A0, L, Ax, [{Bx,By} | B], Ay) when Ay == Bx -> + relprod(B0, Bx0, By0, A0, [{Ax,By} | L], Ax, B, Ay); +relprod(B0, Bx0, By0, A0, L, _Ax, _B, _Ay) -> + relprod2(B0, Bx0, By0, A0, L). + +relprod_n([], _R, _EmptyG, _IsR) -> + {error, badarg}; +relprod_n(RL, R, EmptyR, IsR) -> + case domain_type(RL, ?ANYTYPE) of + Error = {error, _Reason} -> + Error; + DType -> + Empty = any(fun is_empty_set/1, RL) or EmptyR, + RType = range_type(RL, []), + Type = ?BINREL(DType, RType), + Prod = + case Empty of + true when DType =:= ?ANYTYPE; RType =:= ?ANYTYPE -> + empty_set(); + true -> + ?SET([], Type); + false -> + TL = ?LIST((relprod_n(RL))), + Sz = length(RL), + Fun = fun({X,A}) -> {X, flat(Sz, A, [])} end, + ?SET(map(Fun, TL), Type) + end, + case IsR of + true -> relative_product(Prod, R); + false -> Prod + end + end. + +relprod_n([R | Rs]) -> + relprod_n(Rs, R). + +relprod_n([], R) -> + R; +relprod_n([R | Rs], R0) -> + T = raise_element(R0, 1), + R1 = relative_product1(T, R), + NR = projection({external, fun({{X,A},AS}) -> {X,{A,AS}} end}, R1), + relprod_n(Rs, NR). + +flat(1, A, L) -> + list_to_tuple([A | L]); +flat(N, {T,A}, L) -> + flat(N-1, T, [A | L]). + +domain_type([T | Ts], T0) when ?IS_SET(T) -> + case ?TYPE(T) of + ?BINREL(DT, _RT) -> + case unify_types(DT, T0) of + [] -> {error, type_mismatch}; + T1 -> domain_type(Ts, T1) + end; + ?ANYTYPE -> + domain_type(Ts, T0); + _ -> {error, badarg} + end; +domain_type([], T0) -> + T0. + +range_type([T | Ts], L) -> + case ?TYPE(T) of + ?BINREL(_DT, RT) -> + range_type(Ts, [RT | L]); + ?ANYTYPE -> + ?ANYTYPE + end; +range_type([], L) -> + list_to_tuple(reverse(L)). + +converse([{A,B} | X], L) -> + converse(X, [{B,A} | L]); +converse([], L) -> + sort(L). + +strict([{E1,E2} | Es], L) when E1 == E2 -> + strict(Es, L); +strict([E | Es], L) -> + strict(Es, [E | L]); +strict([], L) -> + reverse(L). + +weak(Es) -> + %% Not very efficient... + weak(Es, ran(Es, []), []). + +weak(Es=[{X,_} | _], [Y | Ys], L) when X > Y -> + weak(Es, Ys, [{Y,Y} | L]); +weak(Es=[{X,_} | _], [Y | Ys], L) when X == Y -> + weak(Es, Ys, L); +weak([E={X,Y} | Es], Ys, L) when X > Y -> + weak1(Es, Ys, [E | L], X); +weak([E={X,Y} | Es], Ys, L) when X == Y -> + weak2(Es, Ys, [E | L], X); +weak([E={X,_Y} | Es], Ys, L) -> % when X < _Y + weak2(Es, Ys, [E, {X,X} | L], X); +weak([], [Y | Ys], L) -> + weak([], Ys, [{Y,Y} | L]); +weak([], [], L) -> + reverse(L). + +weak1([E={X,Y} | Es], Ys, L, X0) when X > Y, X == X0 -> + weak1(Es, Ys, [E | L], X); +weak1([E={X,Y} | Es], Ys, L, X0) when X == Y, X == X0 -> + weak2(Es, Ys, [E | L], X); +weak1([E={X,_Y} | Es], Ys, L, X0) when X == X0 -> % when X < Y + weak2(Es, Ys, [E, {X,X} | L], X); +weak1(Es, Ys, L, X) -> + weak(Es, Ys, [{X,X} | L]). + +weak2([E={X,_Y} | Es], Ys, L, X0) when X == X0 -> % when X < _Y + weak2(Es, Ys, [E | L], X); +weak2(Es, Ys, L, _X) -> + weak(Es, Ys, L). + +extc(L, [D | Ds], C, Ts) -> + extc(L, Ds, C, Ts, D); +extc(L, [], _C, _Ts) -> + L. + +extc(L, Ds, C, [{X,_Y} | Ts], D) when X < D -> + extc(L, Ds, C, Ts, D); +extc(L, Ds, C, [{X,_Y} | Ts], D) when X == D -> + extc(L, Ds, C, Ts); +extc(L, Ds, C, [{X,_Y} | Ts], D) -> + extc2([{D,C} | L], Ds, C, Ts, X); +extc(L, Ds, C, [], D) -> + extc_tail([{D,C} | L], Ds, C). + +extc2(L, [D | Ds], C, Ts, X) when X > D -> + extc2([{D,C} | L], Ds, C, Ts, X); +extc2(L, [D | Ds], C, Ts, X) when X == D -> + extc(L, Ds, C, Ts); +extc2(L, [D | Ds], C, Ts, _X) -> + extc(L, Ds, C, Ts, D); +extc2(L, [], _C, _Ts, _X) -> + L. + +extc_tail(L, [D | Ds], C) -> + extc_tail([{D,C} | L], Ds, C); +extc_tail(L, [], _C) -> + L. + +is_a_func([{E,_} | Es], E0) when E /= E0 -> + is_a_func(Es, E); +is_a_func(L, _E) -> + L =:= []. + +restrict_n(I, [T | Ts], Key, Keys, L) -> + case element(I, T) of + K when K < Key -> + restrict_n(I, Ts, Key, Keys, L); + K when K == Key -> + restrict_n(I, Ts, Key, Keys, [T | L]); + K -> + restrict_n(I, K, Ts, Keys, L, T) + end; +restrict_n(_I, _Ts, _Key, _Keys, L) -> + L. + +restrict_n(I, K, Ts, [Key | Keys], L, E) when K > Key -> + restrict_n(I, K, Ts, Keys, L, E); +restrict_n(I, K, Ts, [Key | Keys], L, E) when K == Key -> + restrict_n(I, Ts, Key, Keys, [E | L]); +restrict_n(I, _K, Ts, [Key | Keys], L, _E) -> + restrict_n(I, Ts, Key, Keys, L); +restrict_n(_I, _K, _Ts, _Keys, L, _E) -> + L. + +restrict([Key | Keys], Tuples) -> + restrict(Tuples, Key, Keys, []); +restrict(_Keys, _Tuples) -> + []. + +restrict([{K,_E} | Ts], Key, Keys, L) when K < Key -> + restrict(Ts, Key, Keys, L); +restrict([{K,E} | Ts], Key, Keys, L) when K == Key -> + restrict(Ts, Key, Keys, [E | L]); +restrict([{K,E} | Ts], _Key, Keys, L) -> + restrict(Ts, K, Keys, L, E); +restrict(_Ts, _Key, _Keys, L) -> + L. + +restrict(Ts, K, [Key | Keys], L, E) when K > Key -> + restrict(Ts, K, Keys, L, E); +restrict(Ts, K, [Key | Keys], L, E) when K == Key -> + restrict(Ts, Key, Keys, [E | L]); +restrict(Ts, _K, [Key | Keys], L, _E) -> + restrict(Ts, Key, Keys, L); +restrict(_Ts, _K, _Keys, L, _E) -> + L. + +diff_restrict_n(I, [T | Ts], Key, Keys, L) -> + case element(I, T) of + K when K < Key -> + diff_restrict_n(I, Ts, Key, Keys, [T | L]); + K when K == Key -> + diff_restrict_n(I, Ts, Key, Keys, L); + K -> + diff_restrict_n(I, K, Ts, Keys, L, T) + end; +diff_restrict_n(I, _Ts, _Key, _Keys, L) when I =:= 1 -> + reverse(L); +diff_restrict_n(_I, _Ts, _Key, _Keys, L) -> + sort(L). + +diff_restrict_n(I, K, Ts, [Key | Keys], L, T) when K > Key -> + diff_restrict_n(I, K, Ts, Keys, L, T); +diff_restrict_n(I, K, Ts, [Key | Keys], L, _T) when K == Key -> + diff_restrict_n(I, Ts, Key, Keys, L); +diff_restrict_n(I, _K, Ts, [Key | Keys], L, T) -> + diff_restrict_n(I, Ts, Key, Keys, [T | L]); +diff_restrict_n(I, _K, Ts, _Keys, L, T) when I =:= 1 -> + reverse(L, [T | Ts]); +diff_restrict_n(_I, _K, Ts, _Keys, L, T) -> + sort([T | Ts ++ L]). + +diff_restrict([Key | Keys], Tuples) -> + diff_restrict(Tuples, Key, Keys, []); +diff_restrict(_Keys, Tuples) -> + diff_restrict_tail(Tuples, []). + +diff_restrict([{K,E} | Ts], Key, Keys, L) when K < Key -> + diff_restrict(Ts, Key, Keys, [E | L]); +diff_restrict([{K,_E} | Ts], Key, Keys, L) when K == Key -> + diff_restrict(Ts, Key, Keys, L); +diff_restrict([{K,E} | Ts], _Key, Keys, L) -> + diff_restrict(Ts, K, Keys, L, E); +diff_restrict(_Ts, _Key, _Keys, L) -> + L. + +diff_restrict(Ts, K, [Key | Keys], L, E) when K > Key -> + diff_restrict(Ts, K, Keys, L, E); +diff_restrict(Ts, K, [Key | Keys], L, _E) when K == Key -> + diff_restrict(Ts, Key, Keys, L); +diff_restrict(Ts, _K, [Key | Keys], L, E) -> + diff_restrict(Ts, Key, Keys, [E | L]); +diff_restrict(Ts, _K, _Keys, L, E) -> + diff_restrict_tail(Ts, [E | L]). + +diff_restrict_tail([{_K,E} | Ts], L) -> + diff_restrict_tail(Ts, [E | L]); +diff_restrict_tail(_Ts, L) -> + L. + +comp([], B) -> + check_function(B, []); +comp(_A, []) -> + bad_function; +comp(A0, [{Bx,By} | B]) -> + A = converse(A0, []), + check_function(A0, comp1(A, B, [], Bx, By)). + +comp1([{Ay,Ax} | A], B, L, Bx, By) when Ay == Bx -> + comp1(A, B, [{Ax,By} | L], Bx, By); +comp1([{Ay,Ax} | A], B, L, Bx, _By) when Ay > Bx -> + comp2(A, B, L, Bx, Ay, Ax); +comp1([{Ay,_Ax} | _A], _B, _L, Bx, _By) when Ay < Bx -> + bad_function; +comp1([], B, L, Bx, _By) -> + check_function(Bx, B, L). + +comp2(A, [{Bx,_By} | B], L, Bx0, Ay, Ax) when Ay > Bx, Bx /= Bx0 -> + comp2(A, B, L, Bx, Ay, Ax); +comp2(A, [{Bx,By} | B], L, _Bx0, Ay, Ax) when Ay == Bx -> + comp1(A, B, [{Ax,By} | L], Bx, By); +comp2(_A, _B, _L, _Bx0, _Ay, _Ax) -> + bad_function. + +inverse1([{A,B} | X]) -> + inverse(X, A, [{B,A}]); +inverse1([]) -> + []. + +inverse([{A,B} | X], A0, L) when A0 /= A -> + inverse(X, A, [{B,A} | L]); +inverse([{A,_B} | _X], A0, _L) when A0 == A -> + bad_function; +inverse([], _A0, L) -> + SL = [{V,_} | Es] = sort(L), + case is_a_func(Es, V) of + true -> SL; + false -> bad_function + end. + +%% Inlined. +external_fun({external, Function}) when is_atom(Function) -> + false; +external_fun({external, Fun}) -> + Fun; +external_fun(_) -> + false. + +%% Inlined. +element_type(?SET_OF(Type)) -> Type; +element_type(Type) -> Type. + +subst(Ts, Fun, Type) -> + subst(Ts, Fun, Type, ?ANYTYPE, []). + +subst([T | Ts], Fun, Type, NType, L) -> + case setfun(T, Fun, Type, NType) of + {SD, ST} -> subst(Ts, Fun, Type, ST, [{T, SD} | L]); + Bad -> Bad + end; +subst([], _Fun, _Type, NType, L) -> + {L, NType}. + +projection1([E | Es]) -> + projection1([], element(1, E), Es); +projection1([] = L) -> + L. + +projection1(L, X, [E | Es]) -> + case element(1, E) of + X1 when X == X1 -> projection1(L, X, Es); + X1 -> projection1([X | L], X1, Es) + end; +projection1(L, X, []) -> + reverse(L, [X]). + +projection_n([E | Es], I, L) -> + projection_n(Es, I, [element(I, E) | L]); +projection_n([], _I, L) -> + usort(L). + +substitute_element([T | Ts], I, L) -> + substitute_element(Ts, I, [{T, element(I, T)} | L]); +substitute_element(_, _I, L) -> + reverse(L). + +substitute([T | Ts], Fun, L) -> + substitute(Ts, Fun, [{T, Fun(T)} | L]); +substitute(_, _Fun, L) -> + reverse(L). + +partition_n(I, [E | Ts]) -> + partition_n(I, Ts, element(I, E), [E], []); +partition_n(_I, []) -> + []. + +partition_n(I, [E | Ts], K, Es, P) -> + case {element(I, E), Es} of + {K1, _} when K == K1 -> + partition_n(I, Ts, K, [E | Es], P); + {K1, [_]} -> % optimization + partition_n(I, Ts, K1, [E], [Es | P]); + {K1, _} -> + partition_n(I, Ts, K1, [E], [reverse(Es) | P]) + end; +partition_n(I, [], _K, Es, P) when I > 1 -> + sort([reverse(Es) | P]); +partition_n(_I, [], _K, [_] = Es, P) -> % optimization + reverse(P, [Es]); +partition_n(_I, [], _K, Es, P) -> + reverse(P, [reverse(Es)]). + +partition3_n(I, [T | Ts], Key, Keys, L1, L2) -> + case element(I, T) of + K when K < Key -> + partition3_n(I, Ts, Key, Keys, L1, [T | L2]); + K when K == Key -> + partition3_n(I, Ts, Key, Keys, [T | L1], L2); + K -> + partition3_n(I, K, Ts, Keys, L1, L2, T) + end; +partition3_n(I, _Ts, _Key, _Keys, L1, L2) when I =:= 1 -> + [reverse(L1) | reverse(L2)]; +partition3_n(_I, _Ts, _Key, _Keys, L1, L2) -> + [sort(L1) | sort(L2)]. + +partition3_n(I, K, Ts, [Key | Keys], L1, L2, T) when K > Key -> + partition3_n(I, K, Ts, Keys, L1, L2, T); +partition3_n(I, K, Ts, [Key | Keys], L1, L2, T) when K == Key -> + partition3_n(I, Ts, Key, Keys, [T | L1], L2); +partition3_n(I, _K, Ts, [Key | Keys], L1, L2, T) -> + partition3_n(I, Ts, Key, Keys, L1, [T | L2]); +partition3_n(I, _K, Ts, _Keys, L1, L2, T) when I =:= 1 -> + [reverse(L1) | reverse(L2, [T | Ts])]; +partition3_n(_I, _K, Ts, _Keys, L1, L2, T) -> + [sort(L1) | sort([T | Ts ++ L2])]. + +partition3([Key | Keys], Tuples) -> + partition3(Tuples, Key, Keys, [], []); +partition3(_Keys, Tuples) -> + partition3_tail(Tuples, [], []). + +partition3([{K,E} | Ts], Key, Keys, L1, L2) when K < Key -> + partition3(Ts, Key, Keys, L1, [E | L2]); +partition3([{K,E} | Ts], Key, Keys, L1, L2) when K == Key -> + partition3(Ts, Key, Keys, [E | L1], L2); +partition3([{K,E} | Ts], _Key, Keys, L1, L2) -> + partition3(Ts, K, Keys, L1, L2, E); +partition3(_Ts, _Key, _Keys, L1, L2) -> + [L1 | L2]. + +partition3(Ts, K, [Key | Keys], L1, L2, E) when K > Key -> + partition3(Ts, K, Keys, L1, L2, E); +partition3(Ts, K, [Key | Keys], L1, L2, E) when K == Key -> + partition3(Ts, Key, Keys, [E | L1], L2); +partition3(Ts, _K, [Key | Keys], L1, L2, E) -> + partition3(Ts, Key, Keys, L1, [E | L2]); +partition3(Ts, _K, _Keys, L1, L2, E) -> + partition3_tail(Ts, L1, [E | L2]). + +partition3_tail([{_K,E} | Ts], L1, L2) -> + partition3_tail(Ts, L1, [E | L2]); +partition3_tail(_Ts, L1, L2) -> + [L1 | L2]. + +replace([E | Es], F, L) -> + replace(Es, F, [F(E) | L]); +replace(_, _F, L) -> + sort(L). + +mul_relprod([T | Ts], I, R) when ?IS_SET(T) -> + P = raise_element(R, I), + F = relative_product1(P, T), + [F | mul_relprod(Ts, I+1, R)]; +mul_relprod([], _I, _R) -> + []. + +raise_element(R, I) -> + L = sort(I =/= 1, rearr(?LIST(R), I, [])), + Type = ?TYPE(R), + ?SET(L, ?BINREL(?REL_TYPE(I, Type), Type)). + +rearr([E | Es], I, L) -> + rearr(Es, I, [{element(I, E), E} | L]); +rearr([], _I, L) -> + L. + +join_element(E1, E2) -> + [_ | L2] = tuple_to_list(E2), + list_to_tuple(tuple_to_list(E1) ++ L2). + +join_element(E1, E2, I2) -> + tuple_to_list(E1) ++ join_element2(tuple_to_list(E2), 1, I2). + +join_element2([B | Bs], C, I2) when C =/= I2 -> + [B | join_element2(Bs, C+1, I2)]; +join_element2([_ | Bs], _C, _I2) -> + Bs. + +family2rel([{X,S} | F], L) -> + fam2rel(F, L, X, S); +family2rel([], L) -> + reverse(L). + +fam2rel(F, L, X, [Y | Ys]) -> + fam2rel(F, [{X,Y} | L], X, Ys); +fam2rel(F, L, _X, _) -> + family2rel(F, L). + +fam_spec([{_,S}=E | F], Fun, Type, L) -> + case Fun(?SET(S, Type)) of + true -> + fam_spec(F, Fun, Type, [E | L]); + false -> + fam_spec(F, Fun, Type, L); + _ -> + badarg + end; +fam_spec([], _Fun, _Type, L) -> + reverse(L). + +fam_specification([{_,S}=E | F], Fun, L) -> + case Fun(S) of + true -> + fam_specification(F, Fun, [E | L]); + false -> + fam_specification(F, Fun, L); + _ -> + badarg + end; +fam_specification([], _Fun, L) -> + reverse(L). + +un_of_fam([{_X,S} | F], L) -> + un_of_fam(F, [S | L]); +un_of_fam([], L) -> + lunion(sort(L)). + +int_of_fam([{_,S} | F]) -> + int_of_fam(F, [S]); +int_of_fam([]) -> + badarg. + +int_of_fam([{_,S} | F], L) -> + int_of_fam(F, [S | L]); +int_of_fam([], [L | Ls]) -> + lintersection(Ls, L). + +fam_un([{X,S} | F], L) -> + fam_un(F, [{X, lunion(S)} | L]); +fam_un([], L) -> + reverse(L). + +fam_int([{X, [S | Ss]} | F], L) -> + fam_int(F, [{X, lintersection(Ss, S)} | L]); +fam_int([{_X,[]} | _F], _L) -> + badarg; +fam_int([], L) -> + reverse(L). + +fam_dom([{X,S} | F], L) -> + fam_dom(F, [{X, dom(S)} | L]); +fam_dom([], L) -> + reverse(L). + +fam_ran([{X,S} | F], L) -> + fam_ran(F, [{X, ran(S, [])} | L]); +fam_ran([], L) -> + reverse(L). + +fam_union(F1 = [{A,_AS} | _AL], [B1={B,_BS} | BL], L) when A > B -> + fam_union(F1, BL, [B1 | L]); +fam_union([{A,AS} | AL], [{B,BS} | BL], L) when A == B -> + fam_union(AL, BL, [{A, umerge(AS, BS)} | L]); +fam_union([A1 | AL], F2, L) -> + fam_union(AL, F2, [A1 | L]); +fam_union(_, F2, L) -> + reverse(L, F2). + +fam_intersect(F1 = [{A,_AS} | _AL], [{B,_BS} | BL], L) when A > B -> + fam_intersect(F1, BL, L); +fam_intersect([{A,AS} | AL], [{B,BS} | BL], L) when A == B -> + fam_intersect(AL, BL, [{A, intersection(AS, BS, [])} | L]); +fam_intersect([_A1 | AL], F2, L) -> + fam_intersect(AL, F2, L); +fam_intersect(_, _, L) -> + reverse(L). + +fam_difference(F1 = [{A,_AS} | _AL], [{B,_BS} | BL], L) when A > B -> + fam_difference(F1, BL, L); +fam_difference([{A,AS} | AL], [{B,BS} | BL], L) when A == B -> + fam_difference(AL, BL, [{A, difference(AS, BS, [])} | L]); +fam_difference([A1 | AL], F2, L) -> + fam_difference(AL, F2, [A1 | L]); +fam_difference(F1, _, L) -> + reverse(L, F1). + +check_function([{X,_} | XL], R) -> + check_function(X, XL, R); +check_function([], R) -> + R. + +check_function(X0, [{X,_} | XL], R) when X0 /= X -> + check_function(X, XL, R); +check_function(X0, [{X,_} | _XL], _R) when X0 == X -> + bad_function; +check_function(_X0, [], R) -> + R. + +fam_partition_n(I, [E | Ts]) -> + fam_partition_n(I, Ts, element(I, E), [E], []); +fam_partition_n(_I, []) -> + []. + +fam_partition_n(I, [E | Ts], K, Es, P) -> + case {element(I, E), Es} of + {K1, _} when K == K1 -> + fam_partition_n(I, Ts, K, [E | Es], P); + {K1, [_]} -> % optimization + fam_partition_n(I, Ts, K1, [E], [{K,Es} | P]); + {K1, _} -> + fam_partition_n(I, Ts, K1, [E], [{K,reverse(Es)} | P]) + end; +fam_partition_n(_I, [], K, [_] = Es, P) -> % optimization + reverse(P, [{K,Es}]); +fam_partition_n(_I, [], K, Es, P) -> + reverse(P, [{K,reverse(Es)}]). + +fam_partition([{K,Vs} | Ts], Sort) -> + fam_partition(Ts, K, [Vs], [], Sort); +fam_partition([], _Sort) -> + []. + +fam_partition([{K1,V} | Ts], K, Vs, P, S) when K1 == K -> + fam_partition(Ts, K, [V | Vs], P, S); +fam_partition([{K1,V} | Ts], K, [_] = Vs, P, S) -> % optimization + fam_partition(Ts, K1, [V], [{K, Vs} | P], S); +fam_partition([{K1,V} | Ts], K, Vs, P, S) -> + fam_partition(Ts, K1, [V], [{K, sort(S, Vs)} | P], S); +fam_partition([], K, [_] = Vs, P, _S) -> % optimization + [{K, Vs} | P]; +fam_partition([], K, Vs, P, S) -> + [{K, sort(S, Vs)} | P]. + +fam_proj([{X,S} | F], Fun, Type, NType, L) -> + case setfun(S, Fun, Type, NType) of + {SD, ST} -> fam_proj(F, Fun, Type, ST, [{X, SD} | L]); + Bad -> Bad + end; +fam_proj([], _Fun, _Type, NType, L) -> + {reverse(L), NType}. + +setfun(T, Fun, Type, NType) -> + case Fun(term2set(T, Type)) of + NS when ?IS_SET(NS) -> + case unify_types(NType, ?SET_OF(?TYPE(NS))) of + [] -> type_mismatch; + NT -> {?LIST(NS), NT} + end; + NS when ?IS_ORDSET(NS) -> + case unify_types(NType, NT = ?ORDTYPE(NS)) of + [] -> type_mismatch; + NT -> {?ORDDATA(NS), NT} + end; + _ -> + badarg + end. + +%% Inlined. +term2set(L, Type) when is_list(L) -> + ?SET(L, Type); +term2set(T, Type) -> + ?ORDSET(T, Type). + +fam2digraph(F, G) -> + Fun = fun({From, ToL}) -> + digraph:add_vertex(G, From), + Fun2 = fun(To) -> + digraph:add_vertex(G, To), + case digraph:add_edge(G, From, To) of + {error, {bad_edge, _}} -> + throw({error, cyclic}); + _ -> + true + end + end, + foreach(Fun2, ToL) + end, + foreach(Fun, to_external(F)), + G. + +digraph_family(G) -> + Vs = sort(digraph:vertices(G)), + digraph_fam(Vs, Vs, G, []). + +digraph_fam([V | Vs], V0, G, L) when V /= V0 -> + Ns = sort(digraph:out_neighbours(G, V)), + digraph_fam(Vs, V, G, [{V,Ns} | L]); +digraph_fam([], _V0, _G, L) -> + reverse(L). + +%% -> boolean() +check_fun(T, F, FunT) -> + true = is_type(FunT), + {NT, _MaxI} = number_tuples(T, 1), + L = flatten(tuple2list(F(NT))), + has_hole(L, 1). + +number_tuples(T, N) when is_tuple(T) -> + {L, NN} = mapfoldl(fun number_tuples/2, N, tuple_to_list(T)), + {list_to_tuple(L), NN}; +number_tuples(_, N) -> + {N, N+1}. + +tuple2list(T) when is_tuple(T) -> + map(fun tuple2list/1, tuple_to_list(T)); +tuple2list(C) -> + [C]. + +has_hole([I | Is], I0) when I =< I0 -> has_hole(Is, erlang:max(I+1, I0)); +has_hole(Is, _I) -> Is =/= []. + +%% Optimization. Same as check_fun/3, but for integers. +check_for_sort(T, _I) when T =:= ?ANYTYPE -> + empty; +check_for_sort(T, I) when ?IS_RELATION(T), I =< ?REL_ARITY(T), I >= 1 -> + I > 1; +check_for_sort(_T, _I) -> + error. + +inverse_substitution(L, Fun, Sort) -> + %% One easily sees that the inverse of the tuples created by + %% applying Fun need to be sorted iff the tuples created by Fun + %% need to be sorted. + sort(Sort, fun_rearr(L, Fun, [])). + +fun_rearr([E | Es], Fun, L) -> + fun_rearr(Es, Fun, [{Fun(E), E} | L]); +fun_rearr([], _Fun, L) -> + L. + +sets_to_list(Ss) -> + map(fun(S) when ?IS_SET(S) -> ?LIST(S) end, Ss). + +types([], L) -> + list_to_tuple(reverse(L)); +types([S | _Ss], _L) when ?TYPE(S) =:= ?ANYTYPE -> + ?ANYTYPE; +types([S | Ss], L) -> + types(Ss, [?TYPE(S) | L]). + +%% Inlined. +unify_types(T, T) -> T; +unify_types(Type1, Type2) -> + catch unify_types1(Type1, Type2). + +unify_types1(Atom, Atom) when ?IS_ATOM_TYPE(Atom) -> + Atom; +unify_types1(?ANYTYPE, Type) -> + Type; +unify_types1(Type, ?ANYTYPE) -> + Type; +unify_types1(?SET_OF(Type1), ?SET_OF(Type2)) -> + [unify_types1(Type1, Type2)]; +unify_types1(T1, T2) when tuple_size(T1) =:= tuple_size(T2) -> + unify_typesl(tuple_size(T1), T1, T2, []); +unify_types1(_T1, _T2) -> + throw([]). + +unify_typesl(0, _T1, _T2, L) -> + list_to_tuple(L); +unify_typesl(N, T1, T2, L) -> + T = unify_types1(?REL_TYPE(N, T1), ?REL_TYPE(N, T2)), + unify_typesl(N-1, T1, T2, [T | L]). + +%% inlined. +match_types(T, T) -> true; +match_types(Type1, Type2) -> match_types1(Type1, Type2). + +match_types1(Atom, Atom) when ?IS_ATOM_TYPE(Atom) -> + true; +match_types1(?ANYTYPE, _) -> + true; +match_types1(_, ?ANYTYPE) -> + true; +match_types1(?SET_OF(Type1), ?SET_OF(Type2)) -> + match_types1(Type1, Type2); +match_types1(T1, T2) when tuple_size(T1) =:= tuple_size(T2) -> + match_typesl(tuple_size(T1), T1, T2); +match_types1(_T1, _T2) -> + false. + +match_typesl(0, _T1, _T2) -> + true; +match_typesl(N, T1, T2) -> + case match_types1(?REL_TYPE(N, T1), ?REL_TYPE(N, T2)) of + true -> match_typesl(N-1, T1, T2); + false -> false + end. + +sort(true, L) -> + sort(L); +sort(false, L) -> + reverse(L). |