. We present a denition of untyped -terms using a heterogeneous datatype, i.e. an inductively dened operator. This operator can be extended to a Kleisli triple, which is a concise way to verify the substitution laws for -calculus. We also observe that repetitions in the denition of the monad as well as in the proofs can be avoided by using well-founded recursion and induction instead of structural induction. We extend the construction to the simply typed -calculus using dependent types, and show that this is an instance of a generalization of Kleisli triples. The proofs for the untyped case have been checked using the LEGO system. Keywords. Type Theory, inductive types, -calculus, category theory. 1 Introduction The metatheory of substitution for -calculi is interesting maybe because it seems intuitively obvious but becomes quite intricate if we take a closer look. [Hue92] states seven formal properties of substitution which are then used to prove a general substitution theor...

Deforestation optimises a functional program by transforming it into another one that does not create certain intermediate data structures. In [Chi99] we presented a type-inference based deforestation algorithm which performs extensive inlining. However, across module boundaries only limited inlining is practically feasible. Furthermore, inlining is a non-trivial transformation which is therefore best implemented as a separate optimisation pass. To perform short cut deforestation (nearly) without inlining, Gill suggested to split definitions into workers and wrappers and inline only the small wrappers, which transfer the information needed for deforestation. We show that Gill’s use of a function build limits deforestation and note that his reasons for using build do not apply to our approach. Hence we develop a more general worker/wrapper scheme without build. We give a type-inference based algorithm which splits definitions into workers and wrappers. Finally, we show that we can deforest more expressions with the worker/wrapper scheme than the algorithm with inlining. 1 Type-Inference-Based Short Cut Deforestation In lazy functional programs two functions are often glued together by an intermediate data structure that is produced by one function and consumed by the other. For example, the function any, which tests whether any element of a list xs satisfies a given predicate p, may be defined as follows in Haskell [PH + 99]: any p xs = or (map p xs) The function map applies p to all elements of xs yielding a list of boolean values. The function or combines these boolean values with the logical or operation (||). Although lazy evaluation makes this modular programming style practicable [Hug89], it does not come for free. Each list cell has to be allocated, filled, taken apart and finally garbage collected. The following monolithic definition of any is more efficient. any p [] = False any p (x:xs) = p x | | any p xs It is the aim of deforestation algorithms to automatically transform a functional program into another one that does not create such intermediate data structures. We say that the producer and the consumer of the data structure are fused.

We show how to use intersection types for building models of a #-calculus enriched with recursive terms, whose intended meaning is of minimal fixed points. As a by-product we prove an interesting consistency result.