We have formally specified a substantial subset of the MC68020, a widely used microprocessor built by Motorola, within the mathematical logic of the automated reasoning system Nqthm, i.e., the Boyer-Moore Theorem Prover [4]. Using this MC68020 specification, we have mechanically checked the correctness of MC68020 machine code programs for Euclid's GCD, Hoare's Quick Sort, binary search, and other well-known algorithms. The machine code for these examples was generated using the Gnu C and the Verdix Ada compilers. We have developed an extensive library of proven lemmas to facilitate automated reasoning about machine code programs. We describe a two stage methodology we use to do our machine code proofs.

We briefly review a mechanical theorem-prover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the Boyer-Moore prover, contains a mechanized principle of induction and implementations of linear resolution, rewriting, and arithmetic decision procedures. We describe some applications of the prover, including a proof of the correct implementation of a higher level language on a microprocessor defined at the gate level. We also describe the ongoing project of recoding the entire prover as an applicative function within its own logic.

Abstract. Fairly deep results of Zermelo-Frænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ ⊗ κ = κ, where κ is any infinite cardinal. Proving this result required developing theories of orders, order-isomorphisms, order types, ordinal arithmetic, cardinals, etc.; this covers most of Kunen, Set Theory, Chapter I. Furthermore, we have proved the equivalence of 7 formulations of the Well-ordering Theorem and 20 formulations of AC; this covers the first two chapters of Rubin and Rubin, Equivalents of the Axiom of Choice, and involves highly technical material. The definitions used in the proofs are

The concept of locales for Isabelle enables local definition and assumption for interactive mechanical proofs. Furthermore, dependent types are constructed in Isabelle/HOL for first class representation of structure. These two concepts are introduced briefly. Although each of them has proved useful in itself, their real power lies in combination. This paper illustrates by examples from abstract algebra how this combination works and argues that it enables modular reasoning.