The bound package

[Tags: bsd3, library]

We represent the target language itself as an ideal monad supplied by the user, and provide a Scope monad transformer for introducing bound variables in user supplied terms. Users supply a Monad and Traversable instance, and we traverse to find free variables, and use the Monad to perform substitution that avoids bound variables.

An untyped lambda calculus:

 import Bound
 import Prelude.Extras
 infixl 9 :@
 data Exp a = V a | Exp a :@ Exp a | Lam (Scope () Exp a)
  deriving (Eq,Ord,Show,Read,Functor,Foldable,Traversable)
 instance Eq1 Exp   where (==#)      = (==)
 instance Ord1 Exp  where compare1   = compare
 instance Show1 Exp where showsPrec1 = showsPrec
 instance Read1 Exp where readsPrec1 = readsPrec
 instance Applicative Exp where pure = V; (<*>) = ap
 instance Monad Exp where
   return = V
   V a      >>= f = f a
   (x :@ y) >>= f = (x >>= f) :@ (y >>= f)
   Lam e    >>= f = Lam (e >>>= f)

 lam :: Eq a => a -> Exp a -> Exp a
 lam v b = Lam (abstract1 v b)
 whnf :: Exp a -> Exp a
 whnf (f :@ a) = case whnf f of
   Lam b -> whnf (instantiate1 a b)
   f'    -> f' :@ a
 whnf e = e

The classes from Prelude.Extras are used to facilitate the automatic deriving of Eq, Ord, Show, and Read in the presence of polymorphic recursion used inside Scope.

The goal of this package is to make it as easy as possible to deal with name binding without forcing an awkward monadic style on the user.

With generalized de Bruijn term you can lift whole trees instead of just applying succ to individual variables, weakening the all variables bound by a scope. and by giving binders more structure we can permit easy simultaneous substitution.

The approach was first elaborated upon by Richard Bird and Ross Patterson in "de Bruijn notation as a nested data type", available from

However, the combinators they used required higher rank types. Here we demonstrate that the higher rank gfold combinator they used isn't necessary to build the monad and use a monad transformer to encapsulate the novel recursion pattern in their generalized de Bruijn representation. It is named Scope to match up with the terminology and usage pattern from Conor McBride and James McKinna's "I am not a number: I am a free variable", available from, but since the set of variables is visible in the type, we can provide stronger type safety guarantees.

There are longer examples in the examples/ folder:

  1. Simple.hs provides an untyped lambda calculus with recursive let bindings and includes an evaluator for the untyped lambda calculus and a longer example taken from Lennart Augustsson's λ-calculus cooked four ways available from

2. Derived.hs shows how much of the API can be automated with DeriveTraversable and adds combinators for building binders that support pattern matching.

3. Overkill.hs provides very strongly typed pattern matching many modern language extensions, including polymorphic kinds to ensure type safety. In general, the approach taken by Derived seems to deliver a better power to weight ratio.


Versions0.1, 0.1.1, 0.1.2, 0.1.3, 0.1.4, 0.2, 0.2.1, 0.3.1, 0.3.2, 0.4, 0.5,,, 0.5.1, 0.6, 0.6.1, 0.7, 0.8, 0.8.1, 0.9,, 0.9.1,, 1.0, 1.0.1, 1.0.2, 1.0.3, 1.0.4
Dependenciesbase (==4.*), bifunctors (>=0.1.3 && <0.2), prelude-extras (==0.2.*), transformers (>=0.2 && <0.4)
CopyrightCopyright (C) 2012 Edward A. Kmett
AuthorEdward A. Kmett
MaintainerEdward A. Kmett <>
CategoryLanguage, Compilers/Interpreters
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Source repositoryhead: git clone git://
Upload dateMon Jun 18 00:42:19 UTC 2012
Uploaded byEdwardKmett
DistributionsNixOS:1.0.3, Stackage:1.0.4
Downloads3697 total (255 in last 30 days)




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