Functional Programs: Conversions between Deep and Shallow Embeddings

This paper presents a method which simplifies verification of deeply embedded functional programs. We present a technique by which proof-certified equations describing the effect of functional programs (shallow embeddings) can be automatically extracted from their operational semantics. Our method can be used in reverse, i.e. from shallow to deep embeddings, and thus for implementing certifying code synthesis: we have implemented a tool which maps HOL functions to equivalent Lisp functions, for which we have a verified Lisp runtime. A key benefit, in both directions, is that the verifier does not need to understand the operational semantics that gives meanings to the deep embeddings.