The Logic of Hereditary Harrop Formulas as a Specification Logic for Hybrid

Hybrid is a logical framework that supports the use of higher-order abstract syntax (HOAS) in representing formal systems or "object logics" (OLs). It is implemented in Coq and follows a two-level approach, where a specification logic (SL) is implemented as an inductive type and used to concisely and elegantly encode the inference rules of the formal systems of interest. In this paper, we develop a new higher-order specification logic for Hybrid. By increasing the expressive power of the SL beyond what was considered previously, we increase the flexibility of encoding OLs and thus extend the class of formal systems for which we can reason about efficiently. We focus on formalizing the meta-theory of the SL. We develop an abstract way in which to present an important class of meta-theorems. This class includes properties such as weakening, contraction, exchange, and the admissibility of the cut rule. The cut admissibility theorem establishes consistency and also provides justification for substituting a formula for an assumption in a context of assumptions. It can greatly simplify reasoning about OLs in systems that provide HOAS. We present the abstraction and show how it is used to prove all of these theorems.