diff --git a/test/Spec.hs b/test/Spec.hs
index bc318eb..11412d0 100644
--- a/test/Spec.hs
+++ b/test/Spec.hs
@@ -1,4 +1,6 @@
 {-# OPTIONS_GHC -Wall                   #-}
+-- Unfortunately, using Fin types breaks the warnings for incomplete patterns at this time
+{-# OPTIONS_GHC -fno-warn-incomplete-patterns #-}
 {-# LANGUAGE UnicodeSyntax              #-}
 {-# LANGUAGE OverloadedStrings          #-}
 
@@ -21,28 +23,62 @@ suite ∷ Test ()
 suite = tests [ scope "DFA.empty"     . expect $ Config.language DFA.empty          == ([]   ∷ [[Bool]])
               , scope "DFA.epsilon"   . expect $ Config.language DFA.epsilon        == ([[]] ∷ [[Bool]])
               , scope "DFA.literal"   . expect $ Config.language (DFA.literal True) == [[True]]
-              -- https://math.stackexchange.com/questions/871662/finding-right-quotient-of-languages
-              , scope "DFA.rquotient" . expect $ and [ Config.accepts e3L1 [C, A, R, R, O, T]
-                                                     , Config.accepts e3L2 [T]
-                                                     , Config.accepts e3L2 [O, T]
-                                                     , Config.accepts e3L1L2 [C, A, R, R, O]
-                                                     , Config.accepts e3L1L2 [C, A, R, R]
-                                                     , Prelude.take 2 (Config.language e3L1L2) == [[C, A, R, R], [C, A, R, R, O]]
-                                                     ]
-              , scope "DFA.quotient"  . expect $ minimal `DFA.equal` (quotient minimal) && size' (useful (quotient minimal)) < size' (useful (minimal))
+              , scope "DFA.quotient"  . expect $ minimal `DFA.equal` quotient minimal && size' (useful (quotient minimal)) < size' (useful (minimal))
               , scope "DFA.toRE"      . expect $ toRE by5 `RE.equivalent` by5'
+              , testDFArquotient
+              , testDFAinvhomimage
               ]
-    where e3L1 ∷ DFA Fin₈ Alpha
-          e3L1   = DFA { delta = δ
-                        , q0 = 0
-                        , fs = singleton 6
-                        } where δ (0, C) = 1
-                                δ (1, A) = 2
-                                δ (2, R) = 3
-                                δ (3, R) = 4
-                                δ (4, O) = 5
-                                δ (5, T) = 6
-                                δ _      = 7
-          e3L2   = DFA.union (right e3L1 4) (DFA.literal T)
-          -- e3L2 = DFA.union (right e3L1 4) (right e3L1 5)
-          e3L1L2 = DFA.rquotient e3L1 e3L2
+ 
+-- https://math.stackexchange.com/questions/871662/finding-right-quotient-of-languages
+testDFArquotient ∷ Test ()
+testDFArquotient = scope "DFA.rquotient" . expect $ and [ Config.accepts e3L1 [C, A, R, R, O, T]
+                                                        , Config.accepts e3L2 [T]
+                                                        , Config.accepts e3L2 [O, T]
+                                                        , Config.accepts e3L1L2 [C, A, R, R, O]
+                                                        , Config.accepts e3L1L2 [C, A, R, R]
+                                                        , Prelude.take 2 (Config.language e3L1L2) == [[C, A, R, R], [C, A, R, R, O]]
+                                                        ]
+             where e3L1 ∷ DFA Fin₈ Alpha
+                   e3L1   = DFA { delta = δ
+                                , q0 = 0
+                                , fs = singleton 6
+                                } where δ (0, C) = 1
+                                        δ (1, A) = 2
+                                        δ (2, R) = 3
+                                        δ (3, R) = 4
+                                        δ (4, O) = 5
+                                        δ (5, T) = 6
+                                        δ _      = 7
+                   e3L2   = DFA.union (right e3L1 4) (DFA.literal T)
+                   -- e3L2 = DFA.union (right e3L1 4) (right e3L1 5)
+                   e3L1L2 = DFA.rquotient e3L1 e3L2
+
+testDFAinvhomimage ∷ Test ()
+testDFAinvhomimage = scope "DFA.invhomimage" . expect $ DFA.invhomimage h slide22 `DFA.equal` expected
+               where -- slide 22 http://infolab.stanford.edu/~ullman/ialc/spr10/slides/rs2.pdf
+                     slide22 ∷ DFA Fin₃ Fin₂
+                     slide22 = DFA { delta = δ
+                                   , q0    = 0
+                                   , fs    = singleton 2
+                                   } where δ (0, 0) = 1
+                                           δ (0, 1) = 2
+                                           δ (1, 0) = 0
+                                           δ (1, 1) = 2
+                                           δ (2, 0) = 0
+                                           δ (2, 1) = 1        
+                     h ∷ Bool → [Fin₂]
+                     h False = [0,1]
+                     h True  = []
+                     expected ∷ DFA Fin₃ Bool 
+                     expected = DFA { delta = δ
+                                    , q0    = 0
+                                    , fs    = singleton 2
+                                    } where δ (0, False) = 2
+                                            δ (0, True)  = 0
+                                            δ (1, False) = 2
+                                            δ (1, True)  = 1
+                                            δ (2, False) = 2
+                                            δ (2, True)  = 2
+
+
+