Jeremy Avigad

A machine-checked proof of the odd order theorem

This paper reports on a six-year collaborative effort that culminated in a complete formalization of a proof of the Feit-Thompson Odd Order Theorem in the Coq proof assistant. The formalized proof is constructive, and relies on nothing but the axioms and rules of the foundational framework implemented by Coq. To support the formalization, we developed a comprehensive set of reusable libraries of formalized mathematics, including results in finite group theory, linear algebra, Galois theory, and the theories of the real and complex algebraic numbers.

A FORMAL SYSTEM FOR EUCLID'S ELEMENTS

We present a formal system, E, which provides a faithful model of the proofs in 3 Euclids Elements, including the use of diagrammatic reasoning. 4

A formally verified proof of the prime number theorem

The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdös in 1948. We describe a formally verified version of Selbergs proof, obtained using the Isabelle proof assistant.

Delta-Decidability over the Reals

Given any collection F of computable functions over the reals, we show that there exists an algorithm that, given any sentence A containing only bounded quantifiers and functions in F, and any positive rational number delta, decides either “A is true”, or “a delta-strengthening of A is false”. Moreover, if F can be computed in complexity class C, then under mild assumptions, this “delta-decision problem” for bounded Sigma k-sentences resides in Sigma k(C). The results stand in sharp contrast to the well-known undecidability of the general first-order theories with these functions, and serve as a theoretical basis for the use of numerical methods in decision procedures for formulas over the reals.

Building a push-button RESOLVE verifier: Progress and challenges

A central objective of the verifying compiler grand challenge is to develop a push-button verifier that generates proofs of correctness in a syntax-driven fashion similar to the way an ordinary compiler generates machine code. The software developer’s role is then to provide suitable specifications and annotated code, but otherwise to have no direct involvement in the verification step. However, the general mathematical developments and results upon which software correctness is based may be established through a separate formal proof process in which proofs might be mechanically checked, but not necessarily automatically generated. While many ideas that could conceivably form the basis for software verification have been known “in principle” for decades, and several tools to support an aspect of verification have been devised, practical fully automated verification of full software behavior remains a grand challenge. This paper explains how RESOLVE takes a step towards addressing this challenge by integrating foundational and practical elements of software engineering, programming languages, and mathematical logic into a coherent framework. Current versions of the RESOLVE verifier generate verification conditions (VCs) for the correctness of component-based software in a modular fashion—one component at a time. The VCs are currently verified using automated capabilities of the Isabelle proof assistant, the SMT solver Z3, a minimalist rewrite prover, and some specialized decision procedures. Initial experiments with the tools and further analytic considerations show both the progress that has been made and the challenges that remain.

The Lean Theorem Prover (System Description)

Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. Lean is an ongoing and long-term effort, but it already provides many useful components, integrated development environments, and a rich API which can be used to embed it into other systems. It is currently being used to formalize category theory, homotopy type theory, and abstract algebra. We describe the project goals, system architecture, and main features, and we discuss applications and continuing work.

Interpreting Classical Theories in Constructive Ones

A number of classical theories are interpreted in analogous theories that are based on intuitionistic logic. The classical theories considered include subsystems of firstand second-order arithmetic, bounded arithmetic, and admissible set theory. ?

Formalizing forcing arguments in subsystems of second-order arithmetic

Abstract We show that certain model-theoretic forcing arguments involving subsystems of second-order arithmetic can be formalized in the base theory, thereby converting them to effective proof-theoretic arguments. We use this method to sharpen the conservation theorems of Harrington and Brown-Simpson, giving an effective proof that WKL+0 is conservative over RCA0 with no significant increase in the lengths of proofs.

Mathematical Method and Proof

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.

Formally verified mathematics

With the help of computational proof assistants, formal verification could become the new standard for rigor in mathematics.