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Monadic Presentations of Lambda Terms Using Generalized Inductive Types
 In Computer Science Logic
, 1999
"... . 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 wel ..."
Abstract

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. 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 wellfounded 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...