diff --git a/Number/Peano/Inf.hs b/Number/Peano/Inf.hs
--- a/Number/Peano/Inf.hs
+++ b/Number/Peano/Inf.hs
@@ -5,11 +5,35 @@
 Portability  :   portable
 
 Lazy Peano numbers including observable infinity value.
+
+Note that the following equation does /not/ hold for this number type:
+
+* @1 + a > a@,  because @1 + infinity == infinity@.
+
+The following operation is undefined:
+
+* @infinity - infinity@
+
+There are variants of @(-)@ with different behaviour regarding this, see below.
+
+The following operations are naturally undefined:
+
+* @fromEnum infinity@
+
+* @toInteger infinity@
+
+* @0 - n@, if @n > 0@
+
+* @fromInteger n@, if @n < 0@
+
+* @toEnum n@, if @n < 0@
+
+* @pred 0@
+
 -}
 module Number.Peano.Inf
-    ( Nat (Zero, Succ)
+    ( Nat
     , infinity
-    , isInfinity
     , diff
     , zeroDiff
     , infDiff
@@ -29,18 +53,12 @@
 infinity :: Nat
 infinity = Inf
 
--- | True on @(infinity)@, @(5 + 4 * infinity)@ etc. Evaluates to bottom on @(genericLength [1..])@.
 
-isInfinity :: Nat -> Bool
-isInfinity Zero = False
-isInfinity Inf  = True
-isInfinity (Succ n) = isInfinity n
-
 instance Eq Nat where
 
     Zero   == Zero   = True
     Succ n == Succ m = n == m
-    Inf    == Inf    = error "Number.Peano.Inf: infinity == infinity."
+    Inf    == Inf    = True
     Succ n == Inf    = n == Inf 
     Inf    == Succ m = Inf == m 
     _      == _      = False
@@ -51,14 +69,14 @@
     Zero   `compare` _      = LT
     _      `compare` Zero   = GT
     Succ n `compare` Succ m = n `compare` m
-    Inf    `compare` Inf    = error "Number.Peano.Inf: infinity `compare` infinity."
+    Inf    `compare` Inf    = EQ
     Inf    `compare` Succ m = Inf `compare` m
     Succ n `compare` Inf    = n `compare` Inf
 
     _      < Zero   = False
     Zero   < _      = True
     Succ n < Succ m = n < m
-    Inf    < Inf    = error "Number.Peano.Inf: infinity < infinity."
+    Inf    < Inf    = False
     Inf    < Succ m = Inf < m
     Succ n < Inf    = n < Inf
 
@@ -108,7 +126,7 @@
 Succ n `diff` Inf    = n `diff` Inf
 
 
--- | Variant of @diff@: @infinity `infDiff` infinity  ==  Left infinity@.
+-- | Variant of @diff@: @infDiff infinity infinity  ==  Left infinity@.
 infDiff
     :: Nat             -- ^ n
     -> Nat             -- ^ m
@@ -121,7 +139,7 @@
 Zero   `infDiff` m      = Right m
 
 
--- | Variant of @diff@: @infinity `zeroDiff` infinity  ==  Left Zero@.
+-- | Variant of @diff@: @zeroDiff infinity infinity  ==  Left 0@.
 zeroDiff
     :: Nat             -- ^ n
     -> Nat             -- ^ m
@@ -210,11 +228,11 @@
         Left Zero -> n: repeat n'
 
         -- decreasing sequence
-        Left d -> case m `zeroDiff` n of
+        Left d -> case m `compare` n of
 
-            Left Zero -> [n]
-            Left _    -> []
-            Right k   -> n: f (n' `zeroDiff` m) n'  where    -- n' >= m ?    n'-m = (-(m-n)) - (n-n'),  if n,m < inf
+            EQ  -> [n]
+            GT  -> []
+            LT  -> n: f (n' `zeroDiff` m) n'  where    -- n' >= m ?    n'-m = (-(m-n)) - (n-n'),  if n,m < inf
 
                 f (Right _) _                 = []
                 f (Left j)  l                 = l: f (j `zeroDiff` d) (l - d)
diff --git a/peano-inf.cabal b/peano-inf.cabal
--- a/peano-inf.cabal
+++ b/peano-inf.cabal
@@ -1,5 +1,5 @@
 name:           peano-inf
-version:        0.2
+version:        0.3
 synopsis:       Lazy Peano numbers including observable infinity value.
 description:    
     Lazy Peano numbers including observable infinity value.