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

Abstract

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.

Sequent Calculus and Equational Programming

“…Beyond the definition of a logical foundation for a functional language in equational style, giving a proof-theoretical explanation for the way Agda is implemented requires to accomodate in the sequent calculus both dependent types and a notion of inductive definition. This is not an easy task, although there has been some work on dependent types in the sequent calculus [14] and there is a number of approaches to inductive definitions in proof theory, including focused systems [5]. For example, the system found in [14] is based on LJT but is limited to Π and does not support Σ, while [12] has both, but requires an intricate mixture of natural deduction and sequent calculus to handle Σ.…”

“…This is not an easy task, although there has been some work on dependent types in the sequent calculus [14] and there is a number of approaches to inductive definitions in proof theory, including focused systems [5]. For example, the system found in [14] is based on LJT but is limited to Π and does not support Σ, while [12] has both, but requires an intricate mixture of natural deduction and sequent calculus to handle Σ. Induction is even more complex to handle, since there are several approaches, including definitions [19] or direct least and greatest fixpoints as found in µMALL [5] and µLJ [4].…”

“…The language used in this variant can still be related to the equational approach to functional programming, but the translation between equations and terms is more involved. The generalisation of the implication into the dependent product Π(x : P).N is a straightforward operation, and the rules we use are essentially the ones found in [14] -except that it involves a data structure, corresponding to a focus on the right-hand side of a sequent. Now, the case of Σ is more complicated, as it is a priori unclear whether it should be obtained as a generalisation of the negative conjunction ∧ or of the positive product × and both solutions might even be possible.…”

Polarities & Focussing: a journey from Realisability to Automated Reasoning

“…Still in intuitionistic logic, two more recent contributions broached the topics that this dissertation approaches under the focussing angle, namely realisability and automated reasoning: In [BGL12] we develop a simple presentation of Hyland's effective topos [Hyl82] which is based on realisability concepts; in [LDM11] we developed a focussed sequent calculus that can describe proof-search in the type theory behind the proof-assistant Coq [Coq].…”

“…In [LDM11] we used a focussed sequent calculus to describe type inhabitation / proofconstruction in Pure Type Systems [Bar92] (and higher-order unification), which provides (the basis for) the type theory behind several proof assistants such as Coq [Coq] or Twelf [Twe].…”