Proof-relevant resolution - the foundations of constructive proof automation

Abstract

Dependent type theory is an expressive programming language. This language allows to write programs that carry proofs of their properties. This in turn gives high confidence in such programs, making the software trustworthy. Yet, the trustworthiness comes for a price: type inference involves an increasing number of proof obligations.

Automation of this process becomes necessary for any system with dependent types that aims to be usable in practice. At the same time, implementation of automation in a verified manner is prohibitively complex. Sometimes, external solvers are used to aid the automation. These solvers may be based on classical logic and may not be themselves verified, thus compromising the guarantees provided by constructive nature of type theory. In this thesis, we explore the idea of proof relevant resolution that allows automation of type inference in type theory in a verifiable and constructive manner, hence to restore the confidence in programs and the trustworthiness of software.

Technical content of this thesis is threefold. First, we propose a novel framework for proof-relevant resolution. We take two constructive logics, Horn-clause and hereditary Harrop formulae logics as a starting point. We formulate the standard big-step operational semantics of these logics. We expose their Curry-Howard nature by treating formulae of these logics as types and proofs as terms thus developing a theory of proof-relevant resolution. We develop small-step operational semantics of proof-relevant resolution and prove it sound with respect to the big-step operational semantics.

Secondly, we demonstrate our approach on an example of type inference in Logical Framework (LF). We translate a type-inference problem in LF into resolution in proof-relevant Horn-clause logic. Such resolution provides, besides an answer substitution to logic variables, a proof term that captures the resolution tree. We interpret the proof term as a derivation of well-formedness judgement of the object in the original problem. This allows for a straightforward implementation of type checking of the resolved solution since type checking is reduced to verifying the derivation captured by the proof term. The theoretical development is substantiated by an implementation.

Finally, we demonstrate that our approach allows to reason about semantic properties of code. Type class resolution has been well-known to be a proof-relevant fragment of Horn-clause logic, and recently its coinductive extensions were introduced. In this thesis, we show that all of these extensions amalgamate with the theoretical framework we introduce. Our novel result here is exposing that the coinductive extensions are actually based on hereditary Harrop logic, rather than Horn-clause logic. We establish a number of soundness and completeness results for them. We also discuss soundness of program transformation that are allowed by proof-relevant presentation of type class resolution.