bound: Haskell 98 Locally-Nameless Generalized de Bruijn Terms

[ bsd3, compilers-interpreters, language, library ] [ Propose Tags ]

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 worked 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 type extensions, including polymorphic kinds to ensure type safety. In general, the approach taken by Derived seems to deliver a better power to weight ratio.

Versions [faq] 0.1, 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, 1.0.5, 1.0.6, 1.0.7, 2, 2.0.1
Dependencies base (==4.*), bifunctors (>=0.1.3 && <0.2), prelude-extras (==0.2.*), transformers (>=0.2 && <0.4) [details]
License BSD-3-Clause
Copyright Copyright (C) 2012 Edward A. Kmett
Author Edward A. Kmett
Maintainer Edward A. Kmett <>
Category Language, Compilers/Interpreters
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Source repo head: git clone git://
Uploaded by EdwardKmett at Sat Jun 16 23:51:19 UTC 2012
Distributions LTSHaskell:2.0.1, NixOS:2.0.1, Stackage:2.0.1
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