Facilitating Meta-Theory Reasoning (Invited Paper)

Abstract

Structural proof theory is praised for being a symbolic approach to reasoning and proofs, in which one can define schemas for reasoning steps and manipulate proofs as a mathematical structure. For this to be possible, proof systems must be designed as a set of rules such that proofs using those rules are correct by construction. Therefore, one must consider all ways these rules can interact and prove that they satisfy certain properties which makes them "well-behaved". This is called the meta-theory of a proof system.Meta-theory proofs typically involve many cases on structures with lots of symbols. The majority of cases are usually quite similar, and when a proof fails, it might be because of a sub-case on a very specific configuration of rules. Developing these proofs by hand is tedious and error-prone, and their combinatorial nature suggests they could be automated.There are various approaches on how to automate, either partially or completely, meta-theory proofs. In this paper, I will present some techniques that I have been involved in for facilitating meta-theory reasoning.

Using Isabelle in Two Courses on Logic and Automated Reasoning

“…Harrison [19], Blanchette [5] and Reis [34] discuss aspects of formalizing the metatheory of proof systems and provers. In contrast to our work they do not consider the use of such formalizations as central components and tools in logic and automated reasoning courses.…”

Interactive Theorem Proving for Logic and Information

“…Giselle Reis [54] sees formalizing logics in proof assistants as one of several ways to facilitate meta-theory. Concretely, she looks at three methods for facilitating meta-theory: Firstly, she considers using linear logic and subexponential linear logic as a framework for meta-theoretical reasoning.…”