A Definitional Implementation of the Lax Logical Framework LLFP in Coq, for Supporting Fast and Loose Reasoning

TL;DR

This paper presents a definitional implementation of the LLFP logical framework in Coq, enabling efficient, flexible reasoning by postponing or delegating proof checks, demonstrated through case studies on fast and loose reasoning paradigms.

Contribution

It provides the first shallow, definitional Coq implementation of LLFP, illustrating principles of Lock-types and enabling extensions for flexible proof development.

Findings

Implementation illuminates Lock-types principles

Supports development of fast and loose reasoning case studies

Demonstrates potential for extending Coq with LLFP features

Abstract

The Lax Logical Framework, LLFP, was introduced, by a team including the last two authors, to provide a conceptual framework for integrating different proof development tools, thus allowing for external evidence and for postponing, delegating, or factoring-out side conditions. In particular, LLFP allows for reducing the number of times a proof-irrelevant check is performed. In this paper we give a shallow, actually definitional, implementation of LLFP in Coq, i.e. we use Coq both as host framework and oracle for LLFP. This illuminates the principles underpinning the mechanism of Lock-types and also suggests how to possibly extend Coq with the features of LLFP. The derived proof editor is then put to use for developing case-studies on an emerging paradigm, both at logical and implementation level, which we call fast and loose reasoning following Danielsson et alii [6]. This paradigm trades off efficiency for correctness and amounts to postponing, or running in parallel, tedious or computationally demanding checks, until we are really sure that the intended goal can be achieved. Typical examples are branch-prediction in CPUs and optimistic concurrency control.