A Coinductive Approach to Proof Search through Typed Lambda-Calculi

43 pages; comparison with v3: precise discussion of how to understand coinductive syntax mathematically, presentation of the typing system as a logic of coinductive proofs, and rephrasing of results of the paper in terms of soundness and completenessInternational audienceIn reductive proof search, proofs are naturally generalized by solutions, comprising all (possibly infinite) structures generated by locally correct, bottom-up application of inference rules. We propose an extension of the Curry-Howard paradigm of representation, from proofs to solutions: to represent solutions by (possibly infinite) terms of the coinductive variant of the typed lambda-calculus that represents proofs. On this we build a new, comprehensive approach to proof search; our case study is proof search in the sequent calculus LJT for intuitionistic implication logic. A second, finitary representation is proposed, where the lambda-calculus that represents proofs is extended with a formal greatest fixed point. In the latter system, fixed-point variables enjoy a relaxed form of binding that allows the detection of cycles through the type system. Formal sums are used in both representations to express alternatives in the search process, so that not only individual solutions but actually solution spaces are expressed. Moreover, formal sums are used in the coinductive syntax to define "decontraction" (contraction bottom-up) - an operation whose theory we initiate in this paper, and that is used to assign a coinductive lambda-term to each finitary term. The main result is the existence of an equivalent finitary representation for any given solution space expressed coinductively. This result underlies an original approach to proof search, where the search builds the finitary representation of the solution space, and the a posteriori analysis typically consisting in applying a syntax-directed procedure or function. The paper illustrates the potential of the methodology to the study of inhabitation problems in the simply-typed lambda-calculus, reviewing results detailed elsewhere, and including new results that obtain extensive generalizations of the so-called monatomic theorem