Faithful Logic Embeddings in HOL -- A recipe to have it all: deep and shallow, automated and interactive, heavy and light, proofs and counterexamples, meta and object level

Deep and shallow embeddings of non-classical logics in classical higher-order logic have been explored, implemented, and used in various automated reasoning tools in recent years. This paper presents a recipe for the simultaneous deployment of different forms of deep and shallow embeddings in classical higher-order logic, enabling not only flexible interactive and automated theorem proving and counterexample finding at meta and object level, but also automated faithfulness proofs between the logic embeddings. The approach, which is fruitful for logic education, research and application, is deliberately illustrated here using simple propositional modal logic. However, the work presented is conceptual in nature and not limited to such a simple logic context. Keywords: Logic embeddings Faithfulness Automated Reasoning 1 Motivation and Introduction Deep embeddings of logics, or more generally of domain-specific languages, in a suitable metalogic, such as the classical higher-order logic (HOL) [23,4], are typically based on explicitly introduced abstract data types that essentially axioma-tize the inductively defined character of the new language.