We present a new approach for proving termination of rewrite systems by innermost termination. From the resulting abstract criterion we derive concrete conditions, based on critical peak properties, under which innermost termination implies termination (and confluence). Finally, we show how to apply the main results for providing new sufficient conditions for the modularity of termination.

This paper presents an extension of the simple-typed lambda-calculus allowing iteration and case...

An extension of the simply-typed lambda-calculus allowing iteration and case reasoning over terms defined by means of 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 IS4. 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 lambda-calculus that is better from the user's point of view, is more concise and 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 Church-Rosser and strong normalization properties. Finally it is a conservative extension of the simply-typed lambda-calculus.

An extension of the simply-typed -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 Church-Rosser and strong normalization properties and it is a conservative extension of the simply-typed -calculus.

In this paper, we present an attempt to build a calculus of explicit substitution expected to be conuent on open terms, to preserve strong normalisation and to simulate one step -reduction. We show why our attempt failed and we explain how we found a counter-example to the strong normalisation or termination of the substitution calculus. As a consequence, we provide also a counter-example to the strong normalisation of another calculus, namely (the substitution calculus of ) of Ris, for which the problem was open.