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A Model for Impredicative Type Systems, Universes, Intersection Types and Subtyping
"... We introduce a new model based on coherence spaces for interpreting large impredicative type systems such as the Extended Calculus of Constructions (ECC). Moreover, we show that this model is wellsuited for interpreting intersection types and subtyping too, and we illustrate this by interpreting a ..."
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We introduce a new model based on coherence spaces for interpreting large impredicative type systems such as the Extended Calculus of Constructions (ECC). Moreover, we show that this model is wellsuited for interpreting intersection types and subtyping too, and we illustrate this by interpreting a variant of ECC with an additional intersection type binder. Furthermore, we propose a general method for interpreting the impredicative level in a nonsyntactical way, by allowing the model to be parametrized by an arbitrarily large coherence space in order to interpret inhabitants of impredicative types. As an application, we show that uncountable types such as the type of real numbers or ZermeloFrnkel sets can safely be axiomatized on the impredicative level of, say, ECC, without harm for consistency. 1
A Formalization of a Concurrent Object Calculus Up to AlphaConversion
, 1999
"... We experiment a method for representing a concurrent object calculus in the Calculus of Inductive Constructions. Terms are first defined in de Bruijn style, then names are reintroduced in binders. The terms of the calculus are formalized in the mechanized logic by suitable subsets of the de Bruijn ..."
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We experiment a method for representing a concurrent object calculus in the Calculus of Inductive Constructions. Terms are first defined in de Bruijn style, then names are reintroduced in binders. The terms of the calculus are formalized in the mechanized logic by suitable subsets of the de Bruijn terms; namely those whose de Bruijn indices are relayed beyond the scene. The ffequivalence relation is the Leibnitz equality and the substitution functions can de defined as sets of partial rewriting rules on these terms. We prove induction schemes for both the terms and some properties of the calculus which internalize the renaming of bound variables . We show that, despite that the terms which formalize the calculus are not generated by a last fixed point relation, we can prove the desire inversion lemmas. We formalize the computational part of the semantic and a simple type system of the calculus. At least, we prove a subject reduction theorem and see that the specications and proofs have the nice feature of not mixing de Bruijn technical manipulations with real proofs.
Internal Program Extraction in the Calculus of Inductive Constructions
 In 6th Argentinian Workshop in Theoretical Computer Science (WAIT'02), 31st JAIIO
, 2002
"... Based on the Calculus of Constructions extended with inductive definitions we present a Theory of Specifications with rules for simultaneously constructing programs and their correctness proofs. The theory contains types for representing specifications, whose corresponding notion of implementation i ..."
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Based on the Calculus of Constructions extended with inductive definitions we present a Theory of Specifications with rules for simultaneously constructing programs and their correctness proofs. The theory contains types for representing specifications, whose corresponding notion of implementation is that of a pair formed by a program and a correctness proof. The rules of the theory are such that in implementations the program parts appear mixed together with the proof parts. A reduction relation performs the task of separating programs from proofs. Consequently, every implementation computes to a pair composed of a program and a proof of its correctness, and so the program extraction procedure is immediate. 1
A Modal Lambda Calculus with Iteration and Case Constructs
, 1998
"... An extension of the simplytyped calculus, allowing iteration and case reasoning over terms of functional types that arise when using higher order abstract syntax, has recently been introduced by Joëlle Despeyroux, Frank Pfenning and Carsten Schürmann. This thorny mixing is achieved thanks to the h ..."
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An extension of the simplytyped calculus, allowing iteration and case reasoning over terms of functional types that arise when using higher order abstract syntax, has recently been introduced by Joëlle Despeyroux, Frank Pfenning and Carsten Schürmann. This thorny mixing is achieved thanks to the help of the operator ` ' of modal logic S4. Here we give a new presentation of their system, with reduction rules, instead of evaluation judgments, that compute the canonical forms of terms. Our presentation is based on a modal calculus that is better from the user's point of view because it requires fewer annotations in terms. Moreover we do not impose a particular strategy of reduction during the computation. Our system enjoys the decidability of typability, soundness of typed reduction with respect to typing rules, the ChurchRosser and strong normalization properties and it is a conservative extension of the simplytyped calculus.
Specification and Verification of a SteamBoiler with SignalCoq
"... Over the last decade, the increasing demand for the validation of safety critical systems has led to the development of domainspecic programming languages (e.g. synchronous languages) and automatic veri cation tools (e.g. model checkers). Conventionally, the verication of a reactive system is i ..."
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Over the last decade, the increasing demand for the validation of safety critical systems has led to the development of domainspecic programming languages (e.g. synchronous languages) and automatic veri cation tools (e.g. model checkers). Conventionally, the verication of a reactive system is implemented by specifying a discrete model of the system (i.e. a nitestate machine) and then checking this model against temporal properties (e.g. using an automatabased tool). We investigate the use of a synchronous programming language, Signal, and of a proof assistant, Coq, for the specication and the verication of coinductive properties of the wellknown steamboiler problem. By way of this largescale casestudy, the SignalCoq formal approach, i.e. the combined use of Signal and Coq, is demonstrated to be a wellsuited and practical approach for the validation of reactive systems. Indeed, the deterministic model of concurrency of Signal, for specifying systems, together with...
Formalization of a Concurrent Object Calculus Up to AlphaConversion
, 1999
"... We present a formalization of a concurrent object calculus in the Calculus of Inductive Constructions. We use de Bruijn technique in an intermediate syntax, but de Bruijn indices do not appear in the final formalization of the terms of the calculus, which are still dened up to ffconversion. We deri ..."
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We present a formalization of a concurrent object calculus in the Calculus of Inductive Constructions. We use de Bruijn technique in an intermediate syntax, but de Bruijn indices do not appear in the final formalization of the terms of the calculus, which are still dened up to ffconversion. We derive substitution rewriting rules and an inductive principle on the subset of the terms which formalize the calculus. Once a certain amount of preliminary work has been done on the intermediate syntax this induction theorem makes possible natural proofs which do not deal with de Bruijn number.