Completeness in Equational Hybrid Propositional Type Theory

Abstract

Equational hybrid propositional type theory ($\mathsf {EHPTT}$) is a combination of propositional type theory, equational logic and hybrid modal logic. The structures used to interpret the language contain a hierarchy of propositional types, an algebra (a nonempty set with functions) and a Kripke frame. The main result in this paper is the proof of completeness of a calculus specifically defined for this logic. The completeness proof is based on the three proofs Henkin published last century: (i) Completeness in type theory, (ii) The completeness of the first-order functional calculus and (iii) Completeness in propositional type theory. More precisely, from (i) and (ii) we take the idea of building the model described by the maximal consistent set; in our case the maximal consistent set has to be named, $\Diamond$-saturated and extensionally algebraic-saturated due to the hybrid and equational nature of $\mathsf {EHPTT}$. From (iii), we use the result that any element in the hierarchy has a name. The challenge was to deal with all the heterogeneous components in an integrated system.