### A Dual-Context Sequent Calculus for S4 Modal Lambda-Term Synthesis

#### Abstract

In type-based program synthesis, the search of inhabitants in typed calculi can be seen as a process where a specification, given by a type A, is considered to be fulfilled if we can construct a λ-term M such that M : A, or more precisely if Γ ` M : A holds, that is, if under some suitable assumptions Γ the term M inhabits the type A. In this paper, we tackle this inhabitation/ synthesis problem for the case of modal types in the necessity fragment of the constructive logic S4. Our approach is human-driven in the sense of the usual reasoning procedures of modern theorem provers. To this purpose we employ a so-called dual-context sequent calculus, where the sequents have two contexts, originally proposed to capture the notions of global and local truths without resorting to any formal semantics. The use of dual-contexts allows us to define a sequent calculus which, in comparison to other related systems for the same modal logic, exhibits simpler typing inference rules for the operator. In several cases, the task of searching for a term having subterms with modal types is reduced to the quest for a term containing only subterms typed by non modal propositions.

#### Keywords

Dual-context sequent calculus, constructive necessity, type inhabitation, modal lambda calculus, program synthesis