diff --git a/lib/Logic/Statement/Laws.hs b/lib/Logic/Statement/Laws.hs
index cd5ccf6..407b661 100644
--- a/lib/Logic/Statement/Laws.hs
+++ b/lib/Logic/Statement/Laws.hs
@@ -6,6 +6,9 @@ import Logic.Statement.Parse (stmt)
 import Logic.Statement.Serialize (serialize, SerializeFormat(Plain))
 import Logic.Graph (bfs, verifyPath, VerifyPathError)
 
+import Data.Map (Map)
+import qualified Data.Map as Map (empty, insert, lookup)
+
 import Data.Either (fromRight, rights)
 import Data.Maybe (fromJust, listToMaybe)
 
@@ -72,15 +75,15 @@ match
   -- ^ pattern
   -> Statement
   -- ^ statement to search within
-  -> Maybe [(String, Statement)]
+  -> Maybe (Map String Statement)
   -- ^ mapping from pattern-statement atoms to search-statement parts
-match = aux []
+match = aux Map.empty
   where
     aux
-      :: [(String, Statement)]
+      :: Map String Statement
       -> Statement
       -> Statement
-      -> Maybe [(String, Statement)]
+      -> Maybe (Map String Statement)
     aux mapping (Atom key) s = add mapping (key, s)
     aux mapping (Not p) (Not s) = aux mapping p s
     aux mapping (And     p1 p2) (And     s1 s2) = binary mapping (p1, s1) (p2, s2)
@@ -90,21 +93,21 @@ match = aux []
     aux mapping pattern s = Nothing
 
     add
-      :: [(String, Statement)]
+      :: Map String Statement
       -> (String, Statement)
-      -> Maybe [(String, Statement)]
+      -> Maybe (Map String Statement)
     add mapping entry@(key, s') =
-      case lookup key mapping of
-        Nothing -> Just (entry:mapping)
+      case Map.lookup key mapping of
+        Nothing -> Just $ Map.insert key s' mapping
         Just existing
           | existing == s' -> Just mapping
           | otherwise -> Nothing
 
     binary
-      :: [(String, Statement)]
+      :: Map String Statement
       -> (Statement, Statement)
       -> (Statement, Statement)
-      -> Maybe [(String, Statement)]
+      -> Maybe (Map String Statement)
     binary mapping (p1, s1) (p2, s2) = do
       mapping' <- aux mapping p1 s1
       aux mapping' p2 s2
@@ -142,8 +145,8 @@ swap p1 p2 s = do
   mapping <- maybe (Left NonMatchingPattern) Right $ match p1 s
   maybe (Left IndeterminateSwap) Right $ aux mapping p2
   where
-    aux :: [(String, Statement)] -> Statement -> Maybe Statement
-    aux mapping (Atom key) = lookup key mapping
+    aux :: Map String Statement -> Statement -> Maybe Statement
+    aux mapping (Atom key) = Map.lookup key mapping
     aux mapping (Not p) = Not <$> aux mapping p
     aux mapping (And     p1 p2) = And     <$> aux mapping p1 <*> aux mapping p2
     aux mapping (Or      p1 p2) = Or      <$> aux mapping p1 <*> aux mapping p2
@@ -190,6 +193,7 @@ data ReplaceError
   = IndeterminateReplace
   deriving Show
 
+-- Replace pattern p1 with pattern p2 at all possible depths
 replace :: Statement -> Statement -> Statement -> Either ReplaceError [Statement]
 replace p1 p2 = firstLeft . aux
   where