Expansion Postponement via Cut Elimination in Sequent Calculi for Pure Type Systems

Gutiérrez, Francisco; Ruiz, Blas C.

doi:10.1007/3-540-45061-0_74

Cited by 3 publications

(2 citation statements)

References 13 publications

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“…Unfortunately, the properties of this relation prove difficult to establish. (See Gutiérrez & Ruiz (2003) for the latest progress on expansion postponement. )…”

Section: Completing the Theorymentioning

confidence: 99%

Pure type systems with judgemental equality

Adams¹

2005

J. Funct. Prog.

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In a typing system, there are two approaches that may be taken to the notion of equality. One can use some external relation of convertibility defined on the terms of the grammar, such as β-convertibility or βη-convertibility; or one can introduce a judgement form for equality into the rules of the typing system itself. For quite some time, it has been an open problem whether the two systems produced by these two choices are equivalent. This problem is essentially the problem of proving that the Subject Reduction property holds in the system with judgemental equality. In this paper, we shall prove that the equivalence holds for all functional Pure Type Systems (PTSs). The proof essentially consists of proving the Church-Rosser Theorem for a typed version of parallel one-step reduction. This method should generalise easily to many typing systems which satisfy the Uniqueness of Types property.

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“…Unfortunately, the properties of this relation prove difficult to establish. (See Gutiérrez & Ruiz (2003) for the latest progress on expansion postponement. )…”

Section: Completing the Theorymentioning

confidence: 99%

Pure type systems with judgemental equality

Adams¹

2005

J. Funct. Prog.

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“…Our choice here is motivated by a tighter connection with natural deduction and by the tactics currently used in proof assistants such as Coq and Lego. While [Pym95] does identify permutations of inference rules which would allow the recovery of a goal-directed strategy, [GR03c] focuses instead on the elimination of a cut-rule which then sheds a surprising light on the open problem of Expansion Postponement [GR03b].…”

Section: Introductionmentioning

confidence: 99%

A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems

Lengrand¹,

Dyckhoff²,

McKinna³

2011

Logical Methods in Computer Science

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Abstract. Basic proof-search tactics in logic and type theory can be seen as the root-first applications of rules in an appropriate sequent calculus, preferably without the redundancies generated by permutation of rules. This paper addresses the issues of defining such sequent calculi for Pure Type Systems (PTS, which were originally presented in natural deduction style) and then organizing their rules for effective proof-search. We introduce the idea of Pure Type Sequent Calculus with meta-variables (PTSCα), by enriching the syntax of a permutation-free sequent calculus for propositional logic due to Herbelin, which is strongly related to natural deduction and already well adapted to proof-search. The operational semantics is adapted from Herbelin's and is defined by a system of local rewrite rules as in cut-elimination, using explicit substitutions. We prove confluence for this system. Restricting our attention to PTSC, a type system for the ground terms of this system, we obtain the Subject Reduction property and show that each PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising iff the latter is. We show how to make the logical rules of PTSC into a syntax-directed system PS for proof-search, by incorporating the conversion rules as in syntax-directed presentations of the PTS rules for type-checking. Finally, we consider how to use the explicitly scoped meta-variables of PTSCα to represent partial proof-terms, and use them to analyse interactive proof construction. This sets up a framework PE in which we are able to study proof-search strategies, type inhabitant enumeration and (higher-order) unification.1998 ACM Subject Classification: F.4.1.

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Cut Elimination in a Class of Sequent Calculi for Pure Type Systems

Gutiérrez

Ruiz

2003

Electronic Notes in Theoretical Computer Science

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Expansion Postponement via Cut Elimination in Sequent Calculi for Pure Type Systems

Cited by 3 publications

References 13 publications

Pure type systems with judgemental equality

Pure type systems with judgemental equality

A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems

Cut Elimination in a Class of Sequent Calculi for Pure Type Systems

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