A coinductive approach to proof search through typed lambda-calculi

In reductive proof search, proofs are naturally generalized by solutions, comprising all possibly infinite structures generated locally correct, bottom-up application of inference rules. We propose an extension the Curry-Howard paradigm representation, from to solutions: represent solutions terms a dedicated lambda-calculus. This new, comprehensive approach search is exemplified with sequent calculus LJT for intuitionistic implication logic. A finitary representation proposed, lambda-terms extended formal greatest fixed point, and type system that can be seen as logic coinductive proofs. system, fixed-point variables enjoy relaxed form binding allows detection cycles through system. Formal sums used express alternatives in process. Moreover, syntax define "decontraction" (contraction bottom-up) - operation whose theory we initiate this paper. As semantics, assign lambda-term each term. The main result existence equivalent any full solution space expressed coinductively. ingredient our sound complete respect semantics. These results foundation original where builds space, posteriori analysis typically consisting applying syntax-directed procedure or function. paper illustrates inhabitation problems simply-typed lambda-calculus, reviewing detailed elsewhere, including new obtain extensive generalizations so-called monatomic theorem.