Certifying Proofs in the First-Order Theory of Rewriting

The first-order theory of rewriting is a decidable theory for linear variable-separated rewrite systems. The decision procedure is based on tree automata techniques and recently we completed a formalization in the Isabelle proof assistant. In this paper we present a certificate language that enables the output of software tools implementing the decision procedure to be formally verified. To show the feasibility of this approach, we present , a reincarnation of the decision tool with certifiable output, and the formally verified certifier .

Reasoning about Partial Correctness Assertions in Isabelle/HOL

Hoare Logic has a long tradition in formal verification and has been continuously developed and used to verify a broad class of programs, including sequential, object-oriented and concurrent programs. The purpose of this work is to provide a detailed and accessible exposition of the several ways the user can conduct, explore and write proofs of correctness of sequential imperative programs with Hoare logic and the ISABELLE proof assistant. With the proof language Isar, it is possible to write structured, readable proofs that are suitable for human understanding and communication.

A MACHINE-ASSISTED PROOF OF GÖDEL’S INCOMPLETENESS THEOREMS FOR THE THEORY OF HEREDITARILY FINITE SETS

A formalization of Gödel’s incompleteness theorems using the Isabelle proof assistant is described. This is apparently the first mechanical verification of the second incompleteness theorem. The work closely follows Świerczkowski (2003), who gave a detailed proof using hereditarily finite set theory. The adoption of this theory is generally beneficial, but it poses certain technical issues that do not arise for Peano arithmetic. The formalization itself should be useful to logicians, particularly concerning the second incompleteness theorem, where existing proofs are lacking in detail.