Results 1 
5 of
5
Pure Pattern Type Systems
 In POPL’03
, 2003
"... We introduce a new framework of algebraic pure type systems in which we consider rewrite rules as lambda terms with patterns and rewrite rule application as abstraction application with builtin matching facilities. This framework, that we call “Pure Pattern Type Systems”, is particularly wellsuite ..."
Abstract

Cited by 43 (20 self)
 Add to MetaCart
We introduce a new framework of algebraic pure type systems in which we consider rewrite rules as lambda terms with patterns and rewrite rule application as abstraction application with builtin matching facilities. This framework, that we call “Pure Pattern Type Systems”, is particularly wellsuited for the foundations of programming (meta)languages and proof assistants since it provides in a fully unified setting higherorder capabilities and pattern matching ability together with powerful type systems. We prove some standard properties like confluence and subject reduction for the case of a syntactic theory and under a syntactical restriction over the shape of patterns. We also conjecture the strong normalization of typable terms. This work should be seen as a contribution to a formal connection between logics and rewriting, and a step towards new proof engines based on the CurryHoward isomorphism.
Termination Checking with Types
, 1999
"... The paradigm of typebased termination is explored for functional programming with recursive data types. The article introduces , a lambdacalculus with recursion, inductive types, subtyping and bounded quanti cation. Decorated type variables representing approximations of inductive types ..."
Abstract

Cited by 28 (6 self)
 Add to MetaCart
The paradigm of typebased termination is explored for functional programming with recursive data types. The article introduces , a lambdacalculus with recursion, inductive types, subtyping and bounded quanti cation. Decorated type variables representing approximations of inductive types are used to track the size of function arguments and return values. The system is shown to be type safe and strongly normalizing. The main novelty is a bidirectional type checking algorithm whose soundness is established formally.
Building decision procedures in the calculus of inductive constructions
 of Lecture Notes in Computer Science
, 2007
"... It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an equivalent proposition P ’ obtained from P thanks to possibly c ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an equivalent proposition P ’ obtained from P thanks to possibly complex calculations. In this paper, we investigate a new version of the calculus of inductive constructions which incorporates arbitrary decision procedures into deduction via the conversion rule of the calculus. The novelty of the problem in the context of the calculus of inductive constructions lies in the fact that the computation mechanism varies along proofchecking: goals are sent to the decision procedure together with the set of user hypotheses available from the current context. Our main result shows that this extension of the calculus of constructions does not compromise its main properties: confluence, subject reduction, strong normalization and consistency are all preserved.
Translating Combinatory Reduction Systems into the Rewriting Calculus
 in « 4th International Workshop on RuleBased Programming (RULE 2003
, 2003
"... The last few years have seen the development of the rewriting calculus (or rhocalculus, ρCal) that extends first order term rewriting and λcalculus. The integration of these two latter formalisms has been already handled either by enriching firstorder rewriting with higherorder capabilities, like ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
The last few years have seen the development of the rewriting calculus (or rhocalculus, ρCal) that extends first order term rewriting and λcalculus. The integration of these two latter formalisms has been already handled either by enriching firstorder rewriting with higherorder capabilities, like in the Combinatory Reduction Systems, or by adding to λcalculus algebraic features. The different higherorder rewriting systems and the rewriting calculus share similar concepts and have similar applications, and thus, it seems natural to compare these formalisms. We analyze in this paper the relationship between the Rewriting Calculus and the Combinatory Reduction Systems and we present a translation of CRSterms and rewrite rules into rhoterms and we show that for any CRSreduction we have a corresponding rhoreduction. 1
On the Confluence of λCalculus with Conditional Rewriting
"... The confluence of untyped #calculus with unconditional rewriting has already been studied in various directions. In this paper, we investigate the confluence of #calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
The confluence of untyped #calculus with unconditional rewriting has already been studied in various directions. In this paper, we investigate the confluence of #calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This extends results of Muller and Dougherty for unconditional rewriting. Two cases are considered, whether betareduction is allowed or not in the evaluation of conditions. Moreover, Dougherty's result is improved from the assumption of strongly normalizing #reduction to weakly normalizing #reduction. We also provide examples showing that outside these conditions, modularity of confluence is di#cult to achieve.