The Meta-Theory of Q_0 in the Calculus of Inductive Constructions, Master\u27s Thesis, May 2006

The notion of a proof is central to all of mathematics. In the language of formal logic, a proof is a finite sequence of inferences from a set of axioms, and any statement one yields from such a finitistic procedure is called a theorem. For better or for worse, this is far from the form a traditional mathematical proof takes. Mathematicians write proofs that omit routine logical steps, and details deemed tangential to the central result are often elided. These proofs are fuzzy and human-centric, and a great amount of context is assumed on the part of the reader. While traditional proofs are not overly symbolic or syntactic, and hence are easily understood, such informal proofs are susceptible to logical errors -- Fermat\u27s Last Theorem and the Four Color Theorem being prime examples. In light of this, there has been significant interest in producing formal proofs of mathematical theorems: proofs in which every intermediate logical step is supplied. Drawing on ideas from Computational Logic, Type Theory and the theory of Automated Deduction, we are able to guaranteed the correctness of these proofs. The formalization of mathematics is an endeavor that has enjoyed very encouraging progress in recent years. Major achievements include the complete formalization of the Four Color Theorem, the Prime Number Theorem, Goedel\u27s Incompleteness Theorem, the Jordan Curve Theorem (all within the past five years!). This thesis presents our work in formalizing the meta-theory of Peter Andrews\u27 classical higher-order logic in a higher-order typed lambda calculus. Our development is a completely formal one -- in addition to formalizing logical meta-theory, we have also developed and formalized the syntactic meta-theory. Our syntactic meta-theory allows for the reasoning of notions such as variable occurrences, scope and variable binding, linear replacement, etc. Our formalization is carried out in the interactive proof assistant Coq, developed as part of the LogiCal Project in INRIA. Coq is built upon the Calculus of Inductive Constructions, an extension of Coquand and Huet\u27s seminal Calculus of Construction with support for inductive data types. As far as we know, this thesis presents the first effort to formalize Andrews\u27 logical system

Combination of convex theories: Modularity, deduction completeness, and explanation

AbstractDecision procedures are key components of theorem provers and constraint satisfaction systems. Their modular combination is of prime interest for building efficient systems, but their effective use is often limited by poor interface capabilities, when such procedures only provide a simple “sat/unsat” answer. In this paper, we develop a framework to design cooperation schemas between such procedures while maintaining modularity of their interfaces. First, we use the framework to specify and prove the correctness of classic combination schemas by Nelson–Oppen and Shostak. Second, we introduce the concept of deduction complete satisfiability procedures, we show how to build them for large classes of theories, then we provide a schema to modularly combine them. Third, we consider the problem of modularly constructing explanations for combinations by re-using available proof-producing procedures for the component theories

Knuth-Bendix Completion with Modern Termination Checking, Master\u27s Thesis, August 2006

Knuth-Bendix completion is a technique for equational automated theorem proving based on term rewriting. This classic procedure is parametrized by an equational theory and a (well-founded) reduction order used at runtime to ensure termination of intermediate rewriting systems. Any reduction order can be used in principle, but modern completion tools typically implement only a few classes of such orders (e.g., recursive path orders and polynomial orders). Consequently, the theories for which completion can possibly succeed are limited to those compatible with an instance of an implemented class of orders. Finding and specifying a compatible order, even among a small number of classes, is challenging in practice and crucial to the success of the method. In this thesis, a new variant on the Knuth-Bendix completion procedure is developed in which no order is provided by the user. Modern termination-checking methods are instead used to verify termination of rewriting systems. We prove the new method correct and also present an implementation called Slothrop which obtains solutions for theories that do not admit typical orders and that have not previously been solved by a fully automatic tool

Programming Up to Congruence (Extended version)

This paper presents the design of ZOMBIE, a dependently-typed programming language that uses an adaptation of a congruence closure algorithm for proof and type inference. This algorithm allows the type checker to automatically use equality assumptions from the context when reasoning about equality. Most dependently typed languages automatically use equalities that follow from -reduction during type checking; however, such reasoning is incompatible with congruence closure. In contrast, ZOMBIE does not use automatic -reduction because types may contain potentially diverging terms. Therefore ZOMBIE provides a unique opportunity to explore an alternative definition of equivalence in dependently typed language design. Our work includes the specification of the language via a bidirectional type system, which works “up-to-congruence,” and an algorithm for elaborating expressions in this language to an explicitly typed core language. We prove that our elaboration algorithm is complete with respect to the source type system, and always produces well typed terms in the core language. This algorithm has been implemented in the ZOMBIE language, which includes general recursion, irrelevant arguments, heterogeneous equality and data types

A Dependently Typed Language with Nontermination

We propose a full-spectrum dependently typed programming language, Zombie, which supports general recursion natively. The Zombie implementation is an elaborating typechecker. We prove type saftey for a large subset of the Zombie core language, including features such as computational irrelevance, CBV-reduction, and propositional equality with a heterogeneous, completely erased elimination form. Zombie does not automatically beta-reduce expressions, but instead uses congruence closure for proof and type inference. We give a specification of a subset of the surface language via a bidirectional type system, which works up-to-congruence, and an algorithm for elaborating expressions in this language to an explicitly typed core language. We prove that our elaboration algorithm is complete with respect to the source type system. Zombie also features an optional termination-checker, allowing nonterminating programs returning proofs as well as external proofs about programs

The Algebra of Equality Proofs

Proofs of equalities may be built from assumptions using proof rules for reflexivity, symmetry, and transitivity. Reflexivity is an axiom proving x=x for any x; symmetry is a 1-premise rule taking a proof of x=y and returning a proof of y=x; and transitivity is a 2-premise rule taking proofs of x=y and y=z, and returning a proof of x=z. Define an equivalence relation to hold between proofs iff they prove a theorem in common. The main theoretical result of the paper is that if all assumptions are independent, this equivalence relation is axiomatized by the standard axioms of group theory: reflexivity is the unit of the group, symmetry is the inverse, and transitivity is the multiplication. Using a standard completion of the group axioms, we obtain a rewrite system which puts equality proofs into canonical form. Proofs in this canonical form use the fewest possible assumptions, and a proof can be canonized in linear time using a simple strategy. This result is applied to obtain a simple extension of the union-find algorithm for ground equational reasoning which produces minimal proofs. The time complexity of the original union-find operations is preserved, and minimal proofs are produced in worst-case time O(n log 2 3), where n is the number of expressions being equated. As a second application, the approach is used to achieve significant performance improvements for the CVC cooperating decision procedure