diff --git a/2 b/2
deleted file mode 100644
index b7b3e7c..0000000
--- a/2
+++ /dev/null
@@ -1,110 +0,0 @@
-% @doc a word is an ordset of ltrs
-%
-% This mathematically is a cluster of letters; so in "A + B + AB", this is one
-% of the summands.
-%
-% multiplication is implied in a word
-%
-% empty word is 1
-%
-% anything times itself equals itself, so duplication is ignored
-%
-% multiplication = inclusive union
-%
-% operations assume all inputs are valid
--module(wfc_word).
-
--export_type([
-    word/0
-]).
-
--export([
-    %% constructors
-    one/0,
-    validate/1,
-    from_list/1, to_list/1,
-    %% ops
-    mul/2, mul/1
-]).
-
--opaque word() :: {w, ordsets:ordset(wfc_ltr:ltr())}.
-
-%%----------------------------
-%% constructor
-%%----------------------------
-
--spec one() -> word().
-
-one() ->
-    {w, []}.
-
-
--spec validate(Word) -> Result
-    when Word   :: word(),
-         Result :: ok
-                 | {error, Reason :: string()}.
-% @doc
-% check each letter in the word for validity
-%
-% also check that word shape is valid
-% @end
-
-validate(W = {w, Letters}) ->
-    case from_list(Letters) of
-        {ok, Result} when Result =:= W ->
-            ok;
-        Error ->
-            Error
-    end;
-validate(X) ->
-    {error, wfc_utils:str("wfc_word:validate: malformed word: ~tp", [X])}.
-
-
-
--spec to_list(word()) -> Result
-    when Result :: {ok, [wfc_ltr:ltr()]}
-                 | {error, string()}.
-
-to_list({w, Ltrs}) -> {ok, Ltrs};
-to_list(Bad)       -> {error, wfc_utils:str("wfc_word:to_list: bad letter: ~tp", [Bad])}.
-
-
-
--spec from_list(Ltrs) -> Result
-    when Ltrs :: list(wfc_ltr:ltr()),
-         Result  :: {ok, word()}
-                  | {error, Reason :: string()}.
-
-from_list(Chars) ->
-    from_list(ordsets:from_list(Chars), []).
-
-
-%% validate each letter
-from_list([Ltr | Rest], Acc) ->
-    case wfc_ltr:validate(Ltr) of
-        ok    -> from_list(Rest, [Ltr | Acc]);
-        Error -> Error
-    end;
-% done, all good
-from_list([], Acc) ->
-    {ok, {w, lists:reverse(Acc)}}.
-
-
-%%----------------------------
-%% ops
-%%----------------------------
-
--spec mul(word(), word()) -> word().
-% @doc product of two words
-%
-% assumes the words are valid
-
-mul({w, X}, {w, Y}) ->
-    {w, ordsets:union(X, Y)}.
-
-
--spec mul([word()]) -> word().
-% @doc multiply a list of words together
-
-mul([Word | Rest]) -> mul(Word, mul(Rest));
-mul([])            -> one().
diff --git a/src/wfc.erl b/src/wfc.erl
index 8512811..31bcc00 100644
--- a/src/wfc.erl
+++ b/src/wfc.erl
@@ -7,16 +7,8 @@
 ]).
 
 -export([
-    
-    zero/0, one/0,
-    add/1, add/2,
-    mul/1, mul/2
 ]).
 
 
 -type sentence() :: wfc_sentence:sentence().
 
-
-%% constructors
--spec 
-zero() 
diff --git a/src/wfc_sentence.erl b/src/wfc_sentence.erl
index 06d019e..3587edd 100644
--- a/src/wfc_sentence.erl
+++ b/src/wfc_sentence.erl
@@ -26,7 +26,7 @@
 %% constructors
 %%--------------------------
 
--spec zero() :: sentence().
+-spec zero() -> sentence().
 
 zero() ->
     {s, []}.
@@ -60,16 +60,34 @@ to_list({s, Words}) -> {ok, Words};
 to_list(Bad)        -> {error, wfc_utils:str("wfc_sentence:to_list: bad sentence: ~tp", [Bad])}.
 
 
+
+-spec validate(sentence()) -> ok | {error, string()}.
+
+validate({s, [W | Rest]}) ->
+    case wfc_word:validate(W) of
+        ok    -> validate({s, Rest});
+        Error -> Error
+    end;
+validate({s, []}) ->
+    ok;
+validate(Bad) ->
+    {error, wfc_utils:str("wfc_sentence:validate: bad sentence: ~tp", [Bad])}.
+
+
 %%---------------------------
 %% prim ops
 %%---------------------------
 
-add({s, X}, {s, Y}) -> {s, disjoint_union(X, Y)}.
+-spec add(sentence(), sentence()) -> sentence().
 
-disjoint_union(X, Y) ->
+add({s, X}, {s, Y}) -> {s, symdiff(X, Y)}.
+
+symdiff(X, Y) ->
     ordsets:subtract(ordsets:union(X, Y),
                      ordsets:intersection(X, Y)).
 
 
+-spec add([sentence()]) -> sentence().
+
 add([S | Rest]) -> add(S, add(Rest));
 add([])         -> zero().
diff --git a/src/wfc_word.erl b/src/wfc_word.erl
index 04888ae..b7b3e7c 100644
--- a/src/wfc_word.erl
+++ b/src/wfc_word.erl
@@ -108,6 +108,3 @@ mul({w, X}, {w, Y}) ->
 
 mul([Word | Rest]) -> mul(Word, mul(Rest));
 mul([])            -> one().
-
-
-add(