On a Dependently Typed Encoding of Matching Logic

Abstract

Matching logic is a general formal framework for reasoning about a wide range of theories, with particular emphasis on programming language semantics. Notably, the intermediate language of the K semantics framework is an extension of matching $\mu$-logic, a sorted, polyadic variant of the logic. Metatheoretic reasoning requires the logic to be expressed within a foundational theory; opting for a dependently typed one enables well-sortedness in the object theory to correspond directly to well-typedness in the host theory. In this paper, we present the first dependently typed definition of matching $\mu$-logic, ensuring well-sortedness via sorted contexts encoded in type indices. As a result, ill-sorted syntax elements are unrepresentable, and the semantics of well-sorted elements are guaranteed to lie within the domain of their associated sort.