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Unnesting of Copatterns
"... Abstract. Inductive data such as finite lists and trees can elegantly be defined by constructors which allow programmers to analyze and manipulate finite data via pattern matching. Dually, coinductive data such as streams can be defined by observations such as head and tail and programmers can syn ..."
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Abstract. Inductive data such as finite lists and trees can elegantly be defined by constructors which allow programmers to analyze and manipulate finite data via pattern matching. Dually, coinductive data such as streams can be defined by observations such as head and tail and programmers can synthesize infinite data via copattern matching. This leads to a symmetric language where finite and infinite data can be nested. In this paper, we compile nested pattern and copattern matching into a core language which only supports simple nonnested (co)pattern matching. This core language may serve as an intermediate language of a compiler. We show that this translation is conservative, i.e., the multistep reduction relation in both languages coincides for terms of the original language. Furthermore, we show that the translation preserves strong normalisation: a term of the original language is strongly normalising in one language if and only if it is so in the other.
Normalisation is Insensible to λterm Identity or Difference
"... This paper analyses the computational behaviour of λterm applications. The properties we are interested in are weak normalisation (i.e. there is a terminating reduction) and strong normalisation (i.e. all reductions are terminating). One can prove that the application of a λterm M to a fixed number ..."
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This paper analyses the computational behaviour of λterm applications. The properties we are interested in are weak normalisation (i.e. there is a terminating reduction) and strong normalisation (i.e. all reductions are terminating). One can prove that the application of a λterm M to a fixed number n of copies of the same arbitrary strongly normalising λterm is strongly normalising if and only if the application of M to n different arbitrary strongly normalising λterms is strongly normalising. I.e. one has that M X
Compositional Characterisations of λterms using Intersection Types
, 2003
"... We show how to characterise compositionally a number of evaluation properties of λterms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalisation, normalisation, head normalisation, and weak head normalisation. We consider also the p ..."
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We show how to characterise compositionally a number of evaluation properties of λterms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalisation, normalisation, head normalisation, and weak head normalisation. We consider also the persistent versions of such notions. By way of example, we consider also another evaluation property, unrelated to termination, namely reducibility to a closed term. Many of these characterisation results are new, to our knowledge, or else they streamline, strengthen, or generalise earlier results in the literature. The completeness parts of the characterisations are proved uniformly for all the properties, using a settheoretical semantics of intersection types over suitable kinds of stable sets. This technique generalises Krivine's and Mitchell's methods for strong normalisation to other evaluation properties.
Operated by Universities Space Research Association
"... CÉSAR MUÑOZ∗ Abstract. We present a dependenttype system for a λcalculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak ..."
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CÉSAR MUÑOZ∗ Abstract. We present a dependenttype system for a λcalculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.
for Explicit Substitutions Abstract
"... We characterise the strongly normalising terms of a compositionfree calculus of explicit substititions (with or without garbage collection) by means of an intersection type assignment system. The main novelty is a new cutrule which allows to forget the context of the minor premise when the context ..."
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We characterise the strongly normalising terms of a compositionfree calculus of explicit substititions (with or without garbage collection) by means of an intersection type assignment system. The main novelty is a new cutrule which allows to forget the context of the minor premise when the context of the main premise does not have an assumption for the cut variable.
Dependent Types with Explicit Substitutions: A metatheoretical development
, 1997
"... We present a theory of dependent types with explicit substitutions. We follow a metatheoretical approach where open expressions expressions with metavariables are firstclass objects. The system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normal ..."
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We present a theory of dependent types with explicit substitutions. We follow a metatheoretical approach where open expressions expressions with metavariables are firstclass objects. The system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.