Object-Level Reasoning with Logics Encoded in HOL Light

TL;DR

This paper introduces a flexible framework within HOL Light for object-level reasoning across various logics, automating structural reasoning and enabling proof-based type checking and computational process construction.

Contribution

It provides a generic, automated approach to encode and reason about different logics within HOL Light, supporting proof construction and type-theoretic correspondences.

Findings

Successfully implemented for propositional logic and lambda-calculus

Automates structural reasoning and term matching

Supports reasoning with linear logic and session types

Abstract

We present a generic framework that facilitates object level reasoning with logics that are encoded within the Higher Order Logic theorem proving environment of HOL Light. This involves proving statements in any logic using intuitive forward and backward chaining in a sequent calculus style. It is made possible by automated machinery that take care of the necessary structural reasoning and term matching automatically. Our framework can also handle type theoretic correspondences of proofs, effectively allowing the type checking and construction of computational processes via proof. We demonstrate our implementation using a simple propositional logic and its Curry-Howard correspondence to the lambda-calculus, and argue its use with linear logic and its various correspondences to session types.