Set Theory for Verification: II. Induction and Recursion

TL;DR

This paper formalizes recursive definitions within Isabelle's ZF set theory, enabling rigorous verification and reasoning about recursive functions, data structures, and their properties using fixedpoint theorems and well-founded recursion.

Contribution

It introduces a mechanized framework for recursive definitions in Isabelle's ZF set theory, supporting formal verification and semantics proofs with new applications of fixedpoint theorems.

Findings

Formalization of recursive definitions in Isabelle ZF

Examples include transitive closure, lists, and recursive trees

Proofs of Schröder-Bernstein Theorem and propositional logic soundness

Abstract

A theory of recursive definitions has been mechanized in Isabelle's Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning. Inductively defined sets are expressed as least fixedpoints, applying the Knaster-Tarski Theorem over a suitable set. Recursive functions are defined by well-founded recursion and its derivatives, such as transfinite recursion. Recursive data structures are expressed by applying the Knaster-Tarski Theorem to a set, such as V[omega], that is closed under Cartesian product and disjoint sum. Worked examples include the transitive closure of a relation, lists, variable-branching trees and mutually recursive trees and forests. The Schr\"oder-Bernstein Theorem and the soundness of propositional logic are proved in Isabelle sessions.