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Embedding developments into simply typed λcalculus
"... By using an infinity of extra variables every λterm with indexed redexes is interpreted into a term in the simply typed lambda calculus à la Curry. A development becomes a usual βreduction in the simply typed lambda calculus and the corresponding properties of developments come out from the corres ..."
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By using an infinity of extra variables every λterm with indexed redexes is interpreted into a term in the simply typed lambda calculus à la Curry. A development becomes a usual βreduction in the simply typed lambda calculus and the corresponding properties of developments come out from the corresponding properties (strong normalization and ChurchRosser) 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, ChurchRosser property, simple types 1
Embedding Finiteness of developments
, 2008
"... Conclusion George Koletsos, Yiorgos Stavrinos: Embedding developments into simply typed λcalculus 2/14Developments Simply typed λcalculus Embedding Finiteness of developments Conclusion • V = {x, y, z,...} infinite set of variables Λ: set of λterms M:: = x  λx.M  MM βreduction: (λx.M)N −→β M[N ..."
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Conclusion George Koletsos, Yiorgos Stavrinos: Embedding developments into simply typed λcalculus 2/14Developments Simply typed λcalculus Embedding Finiteness of developments Conclusion • V = {x, y, z,...} infinite set of variables Λ: set of λterms M:: = x  λx.M  MM βreduction: (λx.M)N −→β M[N/x] • Development F a set of redex occurrences in M
Simplified Reducibility Proofs of ChurchRosser for β and βηreduction
"... Reducibility has been used to prove a number of properties in the λcalculus 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 ..."
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Reducibility has been used to prove a number of properties in the λcalculus 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 ChurchRosser property. In this paper, we concentrate on using the methods of reducibility and of parallel reductions for proving ChurchRosser for both β and βηreduction. Our contributions are two fold: We give a simple proof of CR for βreduction 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 βηreduction which is a generalisation of our simple proof for βreduction. Keywords:
Reducibility proofs in λcalculi with intersection types
, 2008
"... Reducibility has been used to prove a number of properties in the λcalculus 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 differen ..."
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Reducibility has been used to prove a number of properties in the λcalculus 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 λcalculi with intersection types. We show that one such result (which aims at giving reducibility proofs of ChurchRosser, standardisation and weak normalisation for the untyped λcalculus) 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 ChurchRosser of βdevelopments for untyped terms (without needing to use strong normalisation), from which ChurchRosser of βreduction easily follows. We extend the second result to encompass both βI and βηreduction rather than simply βreduction. 1
Printed in Greece c ○ Greek Mathametical Society PROPERTIES OF DEVELOPMENTS VIA SIMPLE TYPES
"... By using an infinity of extra constants every λterm with indexed redexes is interpreted into a term in the simply typed λcalculus à la Curry. A development becomes a usual βreduction in the simply typed lambda calculus and the corresponding properties of developments come out from the correspondi ..."
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By using an infinity of extra constants every λterm with indexed redexes is interpreted into a term in the simply typed λcalculus à la Curry. A development becomes a usual βreduction in the simply typed lambda calculus and the corresponding properties of developments come out from the corresponding properties (strong normalization and ChurchRosser) 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, ChurchRosser property, simple types 1.
• Generalized development P[Q/x]
, 2009
"... Conclusion Yiorgos Stavrinos: Generalized Developments in λcalculus 2/17Outline Generalized developments λcalculi with types Embedding Finiteness of gen. developments Conclusion • V = {x, y, z,...} infinite set of variables Λ: set of λterms M:: = x  λx.P  PQ βreduction: (λx.P)Q −→β redex ..."
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Conclusion Yiorgos Stavrinos: Generalized Developments in λcalculus 2/17Outline Generalized developments λcalculi with types Embedding Finiteness of gen. developments Conclusion • V = {x, y, z,...} infinite set of variables Λ: set of λterms M:: = x  λx.P  PQ βreduction: (λx.P)Q −→β redex