. 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...

We present a new approach to introducing an extensional propositional equality in Intensional Type Theory. Our construction is based on the observation that there is a sound, intensional setoid model in Intensional Type theory with a proof-irrelevant universe of propositions and -rules for - and -types. The Type Theory corresponding to this model is decidable, has no irreducible constants and permits large eliminations, which are essential for universes. Keywords. Type Theory, categorical models. 1. Introduction and Summary In Intensional Type Theory (see e.g. [11]) we differentiate between a decidable definitional equality (which we denote by =) and a propositional equality type (Id ( ; ) for any given type ) which requires proof. Typing only depends on definitional equality and hence is decidable. In Intensional Type Theory the type corresponding to the principle of extensionality Ext x2:(x) f;g2(x2:(x)) ( x2 Id (x) (f(x); g(x))) ! Id x2:(x) (f; g) is not...