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-rw-r--r--lib/stdlib/src/sofs.erl545
1 files changed, 441 insertions, 104 deletions
diff --git a/lib/stdlib/src/sofs.erl b/lib/stdlib/src/sofs.erl
index a83f803330..d38b8ab37a 100644
--- a/lib/stdlib/src/sofs.erl
+++ b/lib/stdlib/src/sofs.erl
@@ -1,19 +1,19 @@
%%
%% %CopyrightBegin%
-%%
-%% Copyright Ericsson AB 2001-2009. All Rights Reserved.
-%%
+%%
+%% Copyright Ericsson AB 2001-2011. All Rights Reserved.
+%%
%% The contents of this file are subject to the Erlang Public License,
%% Version 1.1, (the "License"); you may not use this file except in
%% compliance with the License. You should have received a copy of the
%% Erlang Public License along with this software. If not, it can be
%% retrieved online at http://www.erlang.org/.
-%%
+%%
%% Software distributed under the License is distributed on an "AS IS"
%% basis, WITHOUT WARRANTY OF ANY KIND, either express or implied. See
%% the License for the specific language governing rights and limitations
%% under the License.
-%%
+%%
%% %CopyrightEnd%
%%
-module(sofs).
@@ -40,11 +40,11 @@
substitution/2, projection/2, partition/1, partition/2,
partition/3, multiple_relative_product/2, join/4]).
--export([family_to_relation/1, family_specification/2,
+-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,
+ family_union/2, family_intersection/2, family_difference/2,
partition_family/2, family_projection/2]).
-export([family_to_digraph/1, family_to_digraph/2,
@@ -64,9 +64,9 @@
-compile({inline, [{external_fun,1},{element_type,1}]}).
--compile({inline,
+-compile({inline,
[{unify_types,2}, {match_types,2},
- {test_rel,3}, {symdiff,3},
+ {test_rel,3}, {symdiff,3},
{subst,3}]}).
-compile({inline, [{fam_binop,3}]}).
@@ -80,13 +80,13 @@
-define(TAG, 'Set').
-define(ORDTAG, 'OrdSet').
--record(?TAG, {data = [], type = type}).
--record(?ORDTAG, {orddata = {}, ordtype = type}).
+-record(?TAG, {data = [] :: list(), type = type :: term()}).
+-record(?ORDTAG, {orddata = {} :: tuple(), ordtype = type :: term()}).
-define(LIST(S), (S)#?TAG.data).
-define(TYPE(S), (S)#?TAG.type).
-%%-define(SET(L, T),
-%% case is_type(T) of
+%%-define(SET(L, T),
+%% case is_type(T) of
%% true -> #?TAG{data = L, type = T};
%% false -> erlang:error(badtype, [T])
%% end
@@ -113,14 +113,40 @@
-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];
@@ -133,6 +159,10 @@ from_term(T) ->
Set
end.
+-spec(from_term(Term, Type) -> AnySet when
+ AnySet :: anyset(),
+ Term :: term(),
+ Type :: type()).
from_term(L, T) ->
case is_type(T) of
true ->
@@ -146,14 +176,23 @@ from_term(L, T) ->
erlang:error(badarg, [L, T])
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)) ->
@@ -163,19 +202,26 @@ is_type(T) when tuple_size(T) > 0 ->
is_type(_T) ->
false.
+-spec(set(Terms) -> Set when
+ Set :: a_set(),
+ Terms :: [term()]).
set(L) ->
case catch usort(L) of
{'EXIT', _} ->
erlang:error(badarg, [L]);
- SL ->
+ SL ->
?SET(SL, ?ATOM_TYPE)
end.
+-spec(set(Terms, Type) -> Set when
+ Set :: a_set(),
+ Terms :: [term()],
+ Type :: type()).
set(L, ?SET_OF(Type) = T) when ?IS_ATOM_TYPE(Type), Type =/= ?ANYTYPE ->
case catch usort(L) of
{'EXIT', _} ->
erlang:error(badarg, [L, T]);
- SL ->
+ SL ->
?SET(SL, Type)
end;
set(L, ?SET_OF(_) = T) ->
@@ -188,6 +234,12 @@ set(L, ?SET_OF(_) = T) ->
set(L, T) ->
erlang:error(badarg, [L, T]).
+-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} ->
@@ -205,6 +257,9 @@ from_sets(Tuple) when is_tuple(Tuple) ->
from_sets(T) ->
erlang:error(badarg, [T]).
+-spec(relation(Tuples) -> Relation when
+ Relation :: relation(),
+ Tuples :: [tuple()]).
relation([]) ->
?SET([], ?BINREL(?ATOM_TYPE, ?ATOM_TYPE));
relation(Ts = [T | _]) when is_tuple(T) ->
@@ -217,6 +272,11 @@ relation(Ts = [T | _]) when is_tuple(T) ->
relation(E) ->
erlang:error(badarg, [E]).
+-spec(relation(Tuples, Type) -> Relation when
+ N :: integer(),
+ Type :: N | type(),
+ Relation :: relation(),
+ Tuples :: [tuple()]).
relation(Ts, TS) ->
case catch rel(Ts, TS) of
{'EXIT', _} ->
@@ -225,6 +285,9 @@ relation(Ts, TS) ->
Set
end.
+-spec(a_function(Tuples) -> Function when
+ Function :: a_function(),
+ Tuples :: [tuple()]).
a_function(Ts) ->
case catch func(Ts, ?BINREL(?ATOM_TYPE, ?ATOM_TYPE)) of
{'EXIT', _} ->
@@ -235,6 +298,10 @@ a_function(Ts) ->
Set
end.
+-spec(a_function(Tuples, Type) -> Function when
+ Function :: a_function(),
+ Tuples :: [tuple()],
+ Type :: type()).
a_function(Ts, T) ->
case catch a_func(Ts, T) of
{'EXIT', _} ->
@@ -245,6 +312,9 @@ a_function(Ts, T) ->
Set
end.
+-spec(family(Tuples) -> Family when
+ Family :: family(),
+ Tuples :: [tuple()]).
family(Ts) ->
case catch fam2(Ts, ?FAMILY(?ATOM_TYPE, ?ATOM_TYPE)) of
{'EXIT', _} ->
@@ -255,6 +325,10 @@ family(Ts) ->
Set
end.
+-spec(family(Tuples, Type) -> Family when
+ Family :: family(),
+ Tuples :: [tuple()],
+ Type :: type()).
family(Ts, T) ->
case catch fam(Ts, T) of
{'EXIT', _} ->
@@ -265,20 +339,30 @@ family(Ts, T) ->
Set
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, []);
@@ -289,6 +373,9 @@ to_sets(S) when ?IS_ORDSET(S), is_tuple(?ORDTYPE(S)) ->
to_sets(S) when ?IS_ORDSET(S) ->
erlang:error(badarg, [S]).
+-spec(no_elements(ASet) -> NoElements when
+ ASet :: a_set() | ordset(),
+ NoElements :: pos_integer()).
no_elements(S) when ?IS_SET(S) ->
length(?LIST(S));
no_elements(S) when ?IS_ORDSET(S), is_tuple(?ORDTYPE(S)) ->
@@ -296,6 +383,10 @@ no_elements(S) when ?IS_ORDSET(S), is_tuple(?ORDTYPE(S)) ->
no_elements(S) when ?IS_ORDSET(S) ->
erlang:error(badarg, [S]).
+-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
@@ -311,36 +402,62 @@ specification(Fun, S) when ?IS_SET(S) ->
erlang:error(Bad, [Fun, S])
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, [S1, S2]);
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, [S1, S2]);
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, [S1, S2]);
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, [S1, S2]);
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, [S1, S2]);
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;
@@ -351,6 +468,9 @@ product(S1, S2) when ?IS_SET(S1), ?IS_SET(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) ->
@@ -365,11 +485,15 @@ product(T) when is_tuple(T) ->
case member([], L) of
true ->
empty_set();
- false ->
+ false ->
?SET(reverse(prod(L, [], [])), Type)
end
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;
@@ -381,6 +505,10 @@ constant_function(S, E) when ?IS_SET(S) ->
constant_function(S, E) when ?IS_ORDSET(S) ->
erlang:error(badarg, [S, E]).
+-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);
@@ -396,12 +524,19 @@ is_equal(S1, S2) when ?IS_SET(S1), ?IS_ORDSET(S2) ->
is_equal(S1, S2) when ?IS_ORDSET(S1), ?IS_SET(S2) ->
erlang:error(type_mismatch, [S1, S2]).
+-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, [S1, S2])
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) ->
@@ -409,16 +544,26 @@ is_sofs_set(S) when ?IS_ORDSET(S) ->
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.
-is_empty_set(S) when ?IS_SET(S) ->
+-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 ->
@@ -433,6 +578,9 @@ is_disjoint(S1, S2) when ?IS_SET(S1), ?IS_SET(S2) ->
%%% 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);
@@ -440,6 +588,9 @@ union(Sets) when ?IS_SET(Sets) ->
_ -> erlang:error(badarg, [Sets])
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, [Sets]);
@@ -451,32 +602,41 @@ intersection(Sets) when ?IS_SET(Sets) ->
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_OF(Type) ->
?SET(can_rel(?LIST(Sets), []), ?BINREL(Type, ST));
?ANYTYPE -> Sets;
_ -> erlang:error(badarg, [Sets])
end.
-%%%
+%%%
%%% Functions on binary relations only.
-%%%
+%%%
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) ->
+ ?BINREL(DT, RT) ->
?SET(rel2family(?LIST(R)), ?FAMILY(DT, RT));
?ANYTYPE -> R;
_Else -> erlang:error(badarg, [R])
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);
@@ -484,6 +644,9 @@ domain(R) when ?IS_SET(R) ->
_Else -> erlang:error(badarg, [R])
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);
@@ -491,35 +654,63 @@ range(R) when ?IS_SET(R) ->
_ -> erlang:error(badarg, [R])
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) ->
- case relprod_n(RT, foo, false, false) of
- {error, Reason} ->
- erlang:error(Reason, [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, [RL]);
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, [RT, R])
+ _ -> erlang:error(badarg, [RL, R])
end,
- case relprod_n(RT, R, EmptyR, true) of
- {error, Reason} ->
- erlang:error(Reason, [RT, R]);
+ case relprod_n(RL, R, EmptyR, true) of
+ {error, Reason} ->
+ erlang:error(Reason, [RL, R]);
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;
@@ -538,16 +729,23 @@ relative_product1(R1, R2) when ?IS_SET(R1), ?IS_SET(R2) ->
false -> erlang:error(type_mismatch, [R1, R2])
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, [R])
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) ->
+ ?BINREL(DT, RT) ->
case match_types(DT, ?TYPE(S)) of
true ->
?SET(usort(restrict(?LIST(S), ?LIST(R))), RT);
@@ -558,9 +756,13 @@ image(R, S) when ?IS_SET(R), ?IS_SET(S) ->
_ -> erlang:error(badarg, [R, S])
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) ->
+ ?BINREL(DT, RT) ->
case match_types(RT, ?TYPE(S)) of
true ->
NL = restrict(?LIST(S), converse(?LIST(R), [])),
@@ -572,17 +774,23 @@ inverse_image(R, S) when ?IS_SET(R), ?IS_SET(S) ->
_ -> erlang:error(badarg, [R, S])
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(_, _) ->
+ Type = ?BINREL(_, _) ->
?SET(strict(?LIST(R), []), Type);
?ANYTYPE -> R;
_ -> erlang:error(badarg, [R])
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) ->
+ ?BINREL(DT, RT) ->
case unify_types(DT, RT) of
[] ->
erlang:error(badarg, [R]);
@@ -592,7 +800,12 @@ weak_relation(R) when ?IS_SET(R) ->
?ANYTYPE -> R;
_ -> erlang:error(badarg, [R])
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} ->
@@ -621,9 +834,12 @@ extension(R, S, E) when ?IS_SET(R), ?IS_SET(S) ->
erlang:error(badarg, [R, S, E])
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(_, _) ->
+ ?BINREL(_, _) ->
case ?LIST(R) of
[] -> true;
[{V,_} | Es] -> is_a_func(Es, V)
@@ -632,16 +848,28 @@ is_a_function(R) when ?IS_SET(R) ->
_ -> erlang:error(badarg, [R])
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;
@@ -656,7 +884,7 @@ composite(Fn1, Fn2) when ?IS_SET(Fn1), ?IS_SET(Fn2) ->
case match_types(RTF1, DTF2) of
true when DTF1 =:= ?ANYTYPE -> Fn1;
true when DTF2 =:= ?ANYTYPE -> Fn2;
- true ->
+ true ->
case comp(?LIST(Fn1), ?LIST(Fn2)) of
SL when is_list(SL) ->
?SET(sort(SL), ?BINREL(DTF1, RTF2));
@@ -666,9 +894,12 @@ composite(Fn1, Fn2) when ?IS_SET(Fn1), ?IS_SET(Fn2) ->
false -> erlang:error(type_mismatch, [Fn1, Fn2])
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) ->
+ ?BINREL(DT, RT) ->
case inverse1(?LIST(Fn)) of
SL when is_list(SL) ->
?SET(SL, ?BINREL(RT, DT));
@@ -678,11 +909,16 @@ inverse(Fn) when ?IS_SET(Fn) ->
?ANYTYPE -> Fn;
_ -> erlang:error(badarg, [Fn])
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),
@@ -747,6 +983,11 @@ restriction(SetFun, S1, S2) when ?IS_SET(S1), ?IS_SET(S2) ->
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),
@@ -812,6 +1053,10 @@ drestriction(SetFun, S1, S2) when ?IS_SET(S1), ?IS_SET(S2) ->
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
@@ -827,6 +1072,10 @@ projection(I, Set) when is_integer(I), ?IS_SET(Set) ->
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
@@ -867,11 +1116,18 @@ substitution(SetFun, Set) when ?IS_SET(Set) ->
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
@@ -887,6 +1143,12 @@ partition(I, Set) when is_integer(I), ?IS_SET(Set) ->
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),
@@ -954,21 +1216,32 @@ partition(SetFun, S1, S2) when ?IS_SET(S1), ?IS_SET(S2) ->
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 ->
+ true ->
MProd = mul_relprod(tuple_to_list(T), 1, R),
- relative_product(list_to_tuple(MProd));
- false ->
+ relative_product(MProd);
+ false ->
erlang:error(badarg, [T, R])
end.
-join(R1, I1, R2, I2)
+-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 ->
+ false ->
erlang:error(badarg, [R1, I1, R2, I2]);
true when ?TYPE(R1) =:= ?ANYTYPE -> R1;
true when ?TYPE(R2) =:= ?ANYTYPE -> R2;
@@ -980,8 +1253,8 @@ join(R1, I1, R2, I2)
true ->
fun({X,Y}) -> join_element(X, Y) end;
false ->
- fun({X,Y}) ->
- list_to_tuple(join_element(X, Y, I2))
+ fun({X,Y}) ->
+ list_to_tuple(join_element(X, Y, I2))
end
end,
?SET(replace(T, F, []), F({?TYPE(R1), ?TYPE(R2)}))
@@ -1001,9 +1274,15 @@ test_rel(R, I, C) ->
%%% 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
@@ -1013,6 +1292,10 @@ family_to_relation(F) when ?IS_SET(F) ->
_ -> erlang:error(badarg, [F])
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 ->
@@ -1032,6 +1315,9 @@ family_specification(Fun, F) when ?IS_SET(F) ->
_ -> erlang:error(badarg, [Fun, F])
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) ->
@@ -1040,6 +1326,9 @@ union_of_family(F) when ?IS_SET(F) ->
_ -> erlang:error(badarg, [F])
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) ->
@@ -1052,6 +1341,9 @@ intersection_of_family(F) when ?IS_SET(F) ->
_ -> erlang:error(badarg, [F])
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)) ->
@@ -1060,6 +1352,9 @@ family_union(F) when ?IS_SET(F) ->
_ -> erlang:error(badarg, [F])
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)) ->
@@ -1073,6 +1368,9 @@ family_intersection(F) when ?IS_SET(F) ->
_ -> erlang:error(badarg, [F])
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, _)) ->
@@ -1082,6 +1380,9 @@ family_domain(F) when ?IS_SET(F) ->
_ -> erlang:error(badarg, [F])
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)) ->
@@ -1091,15 +1392,30 @@ family_range(F) when ?IS_SET(F) ->
_ -> erlang:error(badarg, [F])
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).
@@ -1108,13 +1424,17 @@ fam_binop(F1, F2, FF) when ?IS_SET(F1), ?IS_SET(F2) ->
case unify_types(?TYPE(F1), ?TYPE(F2)) of
[] ->
erlang:error(type_mismatch, [F1, F2]);
- ?ANYTYPE ->
+ ?ANYTYPE ->
F1;
- Type = ?FAMILY(_, _) ->
+ Type = ?FAMILY(_, _) ->
?SET(FF(?LIST(F1), ?LIST(F2), []), Type);
_ -> erlang:error(badarg, [F1, F2])
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
@@ -1159,8 +1479,12 @@ partition_family(SetFun, Set) when ?IS_SET(Set) ->
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
+ case ?TYPE(F) of
?FAMILY(_, _) when [] =:= ?LIST(F) ->
empty_set();
?FAMILY(DT, Type) ->
@@ -1172,7 +1496,7 @@ family_projection(SetFun, F) when ?IS_SET(F) ->
Bad ->
erlang:error(Bad, [SetFun, F])
end;
- _ ->
+ _ ->
erlang:error(badarg, [SetFun, F])
end;
?ANYTYPE -> F;
@@ -1183,6 +1507,9 @@ family_projection(SetFun, F) when ?IS_SET(F) ->
%%% Digraph functions
%%%
+-spec(family_to_digraph(Family) -> Graph when
+ Graph :: digraph(),
+ Family :: family()).
family_to_digraph(F) when ?IS_SET(F) ->
case ?TYPE(F) of
?FAMILY(_, _) -> fam2digraph(F, digraph:new());
@@ -1190,6 +1517,10 @@ family_to_digraph(F) when ?IS_SET(F) ->
_Else -> erlang:error(badarg, [F])
end.
+-spec(family_to_digraph(Family, GraphType) -> Graph when
+ Graph :: digraph(),
+ Family :: family(),
+ GraphType :: [digraph:d_type()]).
family_to_digraph(F, Type) when ?IS_SET(F) ->
case ?TYPE(F) of
?FAMILY(_, _) -> ok;
@@ -1208,12 +1539,19 @@ family_to_digraph(F, Type) when ?IS_SET(F) ->
error:badarg -> erlang:error(badarg, [F, Type])
end.
+-spec(digraph_to_family(Graph) -> Family when
+ Graph :: digraph(),
+ Family :: family()).
digraph_to_family(G) ->
case catch digraph_family(G) of
{'EXIT', _} -> erlang:error(badarg, [G]);
L -> ?SET(L, ?FAMILY(?ATOM_TYPE, ?ATOM_TYPE))
end.
+-spec(digraph_to_family(Graph, Type) -> Family when
+ Graph :: digraph(),
+ Family :: family(),
+ Type :: type()).
digraph_to_family(G, T) ->
case {is_type(T), T} of
{true, ?SET_OF(?FAMILY(_,_) = Type)} ->
@@ -1284,7 +1622,7 @@ rel(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) ->
@@ -1312,7 +1650,7 @@ rel_type([], SL, Type) when ?IS_RELATION(Type) ->
%% Inlined.
a_func(Ts, T) ->
case {T, is_type(T)} of
- {[?BINREL(DT, RT) = Type], true} when ?IS_ATOM_TYPE(DT),
+ {[?BINREL(DT, RT) = Type], true} when ?IS_ATOM_TYPE(DT),
?IS_ATOM_TYPE(RT) ->
func(Ts, Type);
{[Type], true} ->
@@ -1333,16 +1671,16 @@ func([], _X0, L, Type) ->
%% Inlined.
fam(Ts, T) ->
case {T, is_type(T)} of
- {[?FAMILY(DT, RT) = Type], true} when ?IS_ATOM_TYPE(DT),
+ {[?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) ->
+fam2([], Type) ->
?SET([], Type);
-fam2(Ts, Type) ->
+fam2(Ts, Type) ->
fam2(sort(Ts), Ts, [], Type).
fam2([{I,L} | T], I0, SL, Type) when I /= I0 ->
@@ -1383,7 +1721,7 @@ setify(E, Type0) ->
{Type, OrdSet} = make_element(E, Type0, Type0),
?ORDSET(OrdSet, Type).
-is_no_lists(T) when is_tuple(T) ->
+is_no_lists(T) when is_tuple(T) ->
Sz = tuple_size(T),
is_no_lists(T, Sz, Sz, []).
@@ -1404,7 +1742,7 @@ create([], T, _T0, L) ->
make_element(C, ?ANYTYPE, _T0) ->
make_element(C);
-make_element(C, Atom, ?ANYTYPE) when ?IS_ATOM_TYPE(Atom),
+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) ->
@@ -1585,12 +1923,12 @@ sympart([H1 | T1], [H2 | T2], L1, L12, L2, T) when H1 == H2 ->
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(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(L1), T),
+ ?SET(reverse(L12), T),
?SET(reverse(L2, S2), T)}.
sympart1([H1 | T1], T2, L1, L12, L2, T, H2) when H1 < H2 ->
@@ -1600,8 +1938,8 @@ sympart1([H1 | T1], T2, L1, L12, L2, T, H2) when H1 == H2 ->
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(L1), T),
+ ?SET(reverse(L12), T),
?SET(reverse(L2, [H2 | T2]), T)}.
sympart2(T1, [H2 | T2], L1, L12, L2, T, H1) when H1 > H2 ->
@@ -1611,8 +1949,8 @@ sympart2(T1, [H2 | T2], L1, L12, L2, T, H1) when H1 == H2 ->
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(L1, [H1 | T1]), T),
+ ?SET(reverse(L12), T),
?SET(reverse(L2), T)}.
prod([[E | Es] | Xs], T, L) ->
@@ -1660,7 +1998,7 @@ lunion([[] | Ls]) ->
lunion(Ls);
lunion([S | Ss]) ->
umerge(lunion(Ss, last(S), [S], []));
-lunion([]) ->
+lunion([]) ->
[].
lunion([[E] = S | Ss], Last, SL, Ls) when E > Last -> % optimization
@@ -1669,7 +2007,7 @@ 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) ->
+lunion([], _Last, SL, Ls) ->
[append(reverse(SL)) | Ls].
%% The empty list is always the first list, if present.
@@ -1752,18 +2090,17 @@ relprod(B0, Bx0, By0, A0, L, Ax, [{Bx,By} | B], Ay) when Ay == Bx ->
relprod(B0, Bx0, By0, A0, L, _Ax, _B, _Ay) ->
relprod2(B0, Bx0, By0, A0, L).
-relprod_n({}, _R, _EmptyG, _IsR) ->
+relprod_n([], _R, _EmptyG, _IsR) ->
{error, badarg};
-relprod_n(RT, R, EmptyR, IsR) ->
- RL = tuple_to_list(RT),
+relprod_n(RL, R, EmptyR, IsR) ->
case domain_type(RL, ?ANYTYPE) of
- Error = {error, _Reason} ->
+ Error = {error, _Reason} ->
Error;
DType ->
Empty = any(fun is_empty_set/1, RL) or EmptyR,
RType = range_type(RL, []),
Type = ?BINREL(DType, RType),
- Prod =
+ Prod =
case Empty of
true when DType =:= ?ANYTYPE; RType =:= ?ANYTYPE ->
empty_set();
@@ -1771,7 +2108,7 @@ relprod_n(RT, R, EmptyR, IsR) ->
?SET([], Type);
false ->
TL = ?LIST((relprod_n(RL))),
- Sz = tuple_size(RT),
+ Sz = length(RL),
Fun = fun({X,A}) -> {X, flat(Sz, A, [])} end,
?SET(map(Fun, TL), Type)
end,
@@ -1799,12 +2136,12 @@ flat(N, {T,A}, L) ->
domain_type([T | Ts], T0) when ?IS_SET(T) ->
case ?TYPE(T) of
- ?BINREL(DT, _RT) ->
+ ?BINREL(DT, _RT) ->
case unify_types(DT, T0) of
[] -> {error, type_mismatch};
T1 -> domain_type(Ts, T1)
end;
- ?ANYTYPE ->
+ ?ANYTYPE ->
domain_type(Ts, T0);
_ -> {error, badarg}
end;
@@ -1813,12 +2150,12 @@ domain_type([], T0) ->
range_type([T | Ts], L) ->
case ?TYPE(T) of
- ?BINREL(_DT, RT) ->
+ ?BINREL(_DT, RT) ->
range_type(Ts, [RT | L]);
- ?ANYTYPE ->
+ ?ANYTYPE ->
?ANYTYPE
end;
-range_type([], L) ->
+range_type([], L) ->
list_to_tuple(reverse(L)).
converse([{A,B} | X], L) ->
@@ -1861,7 +2198,7 @@ weak1([E={X,_Y} | Es], Ys, L, X0) when X == X0 -> % when X < Y
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([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).
@@ -1910,7 +2247,7 @@ restrict_n(I, [T | Ts], Key, Keys, L) ->
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 ->
@@ -1933,7 +2270,7 @@ 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 ->
@@ -1956,7 +2293,7 @@ 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 ->
@@ -1981,7 +2318,7 @@ 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 ->
@@ -2041,7 +2378,7 @@ external_fun({external, Function}) when is_atom(Function) ->
false;
external_fun({external, Fun}) ->
Fun;
-external_fun(_) ->
+external_fun(_) ->
false.
%% Inlined.
@@ -2121,7 +2458,7 @@ 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 ->
@@ -2146,7 +2483,7 @@ 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 ->
@@ -2192,7 +2529,7 @@ join_element(E1, E2, I2) ->
join_element2([B | Bs], C, I2) when C =/= I2 ->
[B | join_element2(Bs, C+1, I2)];
-join_element2([_ | Bs], _C, _I2) ->
+join_element2([_ | Bs], _C, _I2) ->
Bs.
family2rel([{X,S} | F], L) ->
@@ -2297,7 +2634,7 @@ 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 ->
@@ -2371,14 +2708,14 @@ term2set(T, Type) ->
?ORDSET(T, Type).
fam2digraph(F, G) ->
- Fun = fun({From, ToL}) ->
+ 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, _}} ->
+ {error, {bad_edge, _}} ->
throw({error, cyclic});
- _ ->
+ _ ->
true
end
end,
@@ -2397,7 +2734,7 @@ digraph_fam([V | Vs], V0, G, L) when V /= V0 ->
digraph_fam([], _V0, _G, L) ->
reverse(L).
-%% -> bool()
+%% -> boolean()
check_fun(T, F, FunT) ->
true = is_type(FunT),
{NT, _MaxI} = number_tuples(T, 1),
@@ -2424,7 +2761,7 @@ check_for_sort(T, _I) when T =:= ?ANYTYPE ->
check_for_sort(T, I) when ?IS_RELATION(T), I =< ?REL_ARITY(T), I >= 1 ->
I > 1;
check_for_sort(_T, _I) ->
- error.
+ error.
inverse_substitution(L, Fun, Sort) ->
%% One easily sees that the inverse of the tuples created by
@@ -2477,11 +2814,11 @@ match_types(Type1, Type2) -> match_types1(Type1, Type2).
match_types1(Atom, Atom) when ?IS_ATOM_TYPE(Atom) ->
true;
-match_types1(?ANYTYPE, _) ->
+match_types1(?ANYTYPE, _) ->
true;
-match_types1(_, ?ANYTYPE) ->
+match_types1(_, ?ANYTYPE) ->
true;
-match_types1(?SET_OF(Type1), ?SET_OF(Type2)) ->
+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);