Merge branch 'dev' of super:otp into dev

* 'dev' of super:otp: (166 commits)
  Corrected documentation error and added examples to Users Guide
  In TLS 1.1, failure to properly close a connection no longer requires that a session not be resumed.  This is a change from TLS 1.0 to conform with widespread implementation practice.  Erlang ssl will now in TLS 1.0 conform to the widespread implementation practice instead of the specification to avoid performance issues.
  Add escript to bootstrap/bin
  Remove unused variable warning in inet_res
  Remove unused variable in epmd_port
  Remove compiler warnings in inet_drv
  Add SASL test suite
  Allow same module name in multiple applications if explicitely excluded
  Fix bugs concerning the option report_missing_types
  Fix default encoding in SAX parser.
  re: remove gratuitous "it " in manpage
  Spelling in (backward *compatibility*) comment.
  Improve erl_docgen's support for Dialyzer specs and types
  dialyzer warning on mnesia_tm
  Add documentation text about majority checking
  add mnesia_majority_test suite
  where_to_wlock optimization + change_table_majority/2
  bug in mnesia_tm:needs_majority/2
  optimize sticky_lock maj. check
  check majority for sticky locks
  ...

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);