GHC 7.6.3 has quite convenient type-level numeric literals. We can use numbers like 1 and 2 in types. However, using the type level numeral has been quite a bit of a challenge, illustrated, for example, by the following SO question. http://stackoverflow.com/questions/20809998/type-level-nats-with-literals-an... It seems the challenge has the reason: the type level numerals and their operations provided in GHC.TypeLits have a consistency issue. The following code demonstrates the issue: it constructs two distinct values of the type Sing 2. Singletons aren't singletons. {-# LANGUAGE KindSignatures #-} {-# LANGUAGE DataKinds #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE TypeFamilies #-} {-# LANGUAGE GADTs #-} {-# LANGUAGE PolyKinds #-} module NotSing where import GHC.TypeLits -- GHC strangely enough lets us define instances for (+) type instance 1 + 1 = 0 type instance 0 + 1 = 2 -- A singular representative of 1::Nat s1 :: Sing 1 s1 = withSing id -- A singular representative of 2::Nat s2 :: Sing 2 s2 = withSing id is2 :: IsZero 0 is2 = IsSucc s1 tran :: IsZero 0 -> Sing 2 tran (IsSucc x) = case isZero x of IsSucc y -> case isZero y of IsZero -> x -- Another singular representative of 2::Nat -- A singular representative of 2::Nat s2' :: Sing 2 s2' = tran is2 {- *NotSing> s2 2 *NotSing> s2' 1 -}