By using an infinity of extra variables every λ-term with indexed redexes is interpreted into a term in the simply typed lambda calculus à la Curry. A development becomes a usual β-reduction in the simply typed lambda calculus and the corresponding properties of developments come out from the corresponding properties (strong normalization and Church-Rosser) holding in this system. In this way we obtain a complete simulation of the notion of development into the system of simply typed lambda calculus. Keywords: developments, strong normalization, Church-Rosser property, simple types 1

Conclusion George Koletsos, Yiorgos Stavrinos: Embedding developments into simply typed λ-calculus 2/14Developments Simply typed λ-calculus Embedding Finiteness of developments Conclusion • V = {x, y, z,...} infinite set of variables Λ: set of λ-terms M:: = x | λx.M | MM β-reduction: (λx.M)N −→β M[N/x] • Development F a set of redex occurrences in M

proofs in the λ-calculus

3.1 Presentation of the λ-Calculus..................... 5

Reducibility has been used to prove a number of properties in the λ-calculus and is well known to offer on one hand very general proofs which can be applied to a number of instantiations, and on the other hand, to be quite mysterious and inflexible. It has, amongst other things, been used along with the so called method of parallel reductions to prove the Church-Rosser property. In this paper, we concentrate on using the methods of reducibility and of parallel reductions for proving Church-Rosser for both β- and βη-reduction. Our contributions are two fold: We give a simple proof of CR for β-reduction which unlike the common proofs in the literature, avoids any type machinery and is solely carried out in a completely untyped setting. We give a new proof of CR for βη-reduction which is a generalisation of our simple proof for β-reduction. Keywords:

Reducibility has been used to prove a number of properties in the λ-calculus and is well known to offer on one hand very general proofs which can be applied to a number of instantiations, and on the other hand, to be quite mysterious and inflexible. In this paper, we look at two related but different results in λ-calculi with intersection types. We show that one such result (which aims at giving reducibility proofs of Church-Rosser, standardisation and weak normalisation for the untyped λ-calculus) faces serious problems which break the reducibility method and then we provide a proposal to partially repair the method. Then, we consider a second result whose purpose is to use reducibility for typed terms to show Church-Rosser of β-developments for untyped terms (without needing to use strong normalisation), from which Church-Rosser of β-reduction easily follows. We extend the second result to encompass both βI- and βη-reduction rather than simply β-reduction. 1

By using an infinity of extra constants every λ-term with indexed redexes is interpreted into a term in the simply typed λ-calculus à la Curry. A development becomes a usual β-reduction in the simply typed lambda calculus and the corresponding properties of developments come out from the corresponding properties (strong normalization and Church-Rosser) holding in this system. In this way we obtain a complete simulation of the notion of development into the system of simply typed lambda calculus. Keywords: developments, strong normalization, Church-Rosser property, simple types 1.

Conclusion Yiorgos Stavrinos: Generalized Developments in λ-calculus 2/17Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion • V = {x, y, z,...} infinite set of variables Λ: set of λ-terms M:: = x | λx.P | PQ β-reduction: (λx.P)Q −→β redex