Formalizing Knuth-Bendix Orders and Knuth-Bendix Completion

We present extensions of our Isabelle Formalization of Rewriting that cover two historically related concepts: the Knuth-Bendix order and the Knuth-Bendix completion procedure. The former, besides being the first development of its kind in a proof assistant, is based on a generalized version of the Knuth-Bendix order. We compare our version to variants from the literature and show all properties required to certify termination proofs of TRSs. The latter comprises the formalization of important facts that are related to completion, like Birkhoff\u27s theorem, the critical pair theorem, and a soundness proof of completion, showing that the strict encompassment condition is superfluous for finite runs. As a result, we are able to certify completion proofs

AC Dependency Pairs Revisited

Rewriting modulo AC, i.e., associativity and/or commutativity of certain symbols, is among the most frequently used extensions of term rewriting by equational theories. In this paper we present a generalization of the dependency pair framework for termination analysis to rewriting modulo AC. It subsumes existing variants of AC dependency pairs, admits standard dependency graph analyses, and in particular enjoys the minimality property in the standard sense. As a direct benefit, important termination techniques are easily extended; we describe usable rules and the subterm criterion for AC termination, which properly generalize the non-AC versions. We also perform these extensions within IsaFoR - the Isabelle formalization of rewriting - and thereby provide the first formalization of AC dependency pairs. Consequently, our certifier CeTA now supports checking proofs of AC termination

Homotopy Type Theory in Lean

We discuss the homotopy type theory library in the Lean proof assistant. The library is especially geared toward synthetic homotopy theory. Of particular interest is the use of just a few primitive notions of higher inductive types, namely quotients and truncations, and the use of cubical methods.Comment: 17 pages, accepted for ITP 201

ProverX: rewriting and extending prover9

O propósito principal deste projecto é tornar o demonstrador automático de teoremas Prover9 programável e, por conseguinte, extensível. Este propósito foi conseguido acrescentando um interpretador de Python, uma linha de comandos e uma biblioteca de módulos, objectos e funções escritos em Python para interagir com ficheiros de Prover9 e Mace4. Foi também criada uma “interface” gráfica de utilizador (GUI) sob a forma de uma aplicação web para trazer aos utilizadores um meio mais eficiente e rápido de trabalhar com demonstrações automáticas de teoremas. A nova biblioteca de “scripting” oferece aos utilizadores novas funcionalidades tais como correr várias sessões simultâneas de Prover9 parando automaticamente quando uma demonstração (ou um contraexemplo) é encontrada, elaborar estratégias para aumentar a velocidade com que as demonstrações são encontradas ou diminuir o tamanho das mesmas. Outro módulo permite interagir com o sistema de álgebra GAP. Sobre esta biblioteca, muitas outras funcionalidades podem ser facilmente acrescentadas pois o objectivo principal é dar aos utilizadores a capacidade de acrescentar novas funcionalidades ao Prover9. Resumindo, o objectivo deste projecto é oferecer à comunidade matemática um ambiente integrado para trabalhar com demonstração automática de teoremas.The primary purpose of this project is to extend Prover9 with a scripting language. This was achieved by adding a Python interpreter, an interactive command line and a special scripting library to interact with Prover9 and Mace4 files. A user interface in the form of a web application was also created to help users achieve a more rapid and efficient way of working with automated theorem proving. The new scripting library offers utilities that allows a user to run several Prover9 sessions concurrently and to create strategies for increasing the effectiveness of the proof search or to search for shorter proofs. Another module allows to interact with the algebra system GAP. Based on the library, many more functionalities can be easily added, as the main goal is to give users the ability to extend the functionality of Prover9 the way they see fit. In conclusion, the aim of this project is to offer to the mathematical community an integrated environment for working with automated reasonin

ProofWatch: Watchlist Guidance for Large Theories in E

Watchlist (also hint list) is a mechanism that allows related proofs to guide a proof search for a new conjecture. This mechanism has been used with the Otter and Prover9 theorem provers, both for interactive formalizations and for human-assisted proving of open conjectures in small theories. In this work we explore the use of watchlists in large theories coming from first-order translations of large ITP libraries, aiming at improving hammer-style automation by smarter internal guidance of the ATP systems. In particular, we (i) design watchlist-based clause evaluation heuristics inside the E ATP system, and (ii) develop new proof guiding algorithms that load many previous proofs inside the ATP and focus the proof search using a dynamically updated notion of proof matching. The methods are evaluated on a large set of problems coming from the Mizar library, showing significant improvement of E's standard portfolio of strategies, and also of the previous best set of strategies invented for Mizar by evolutionary methods.Comment: 19 pages, 10 tables, submitted to ITP 2018 at FLO

A verified prover based on ordered resolution

International audienceThe superposition calculus, which underlies first-order theorem provers such as E, SPASS, and Vampire, combines ordered resolution and equality reasoning. As a step towards verifying modern provers, we specify, using Isabelle/HOL, a purely functional first-order ordered resolution prover and establish its soundness and refutational completeness. Methodologically, we apply stepwise refinement to obtain, from an abstract nondeterministic specification, a verified de-terministic program, written in a subset of Isabelle/HOL from which we extract purely functional Standard ML code that constitutes a semidecision procedure for first-order logic

Combined decision procedures for nonlinear arithmetics, real and complex

We describe contributions to algorithmic proof techniques for deciding the satisfiability of boolean combinations of many-variable nonlinear polynomial equations and inequalities over the real and complex numbers. In the first half, we present an abstract theory of Grobner basis construction algorithms for algebraically closed fields of characteristic zero and use it to introduce and prove the correctness of Grobner basis methods tailored to the needs of modern satisfiability modulo theories (SMT) solvers. In the process, we use the technique of proof orders to derive a generalisation of S-polynomial superfluousness in terms of transfinite induction along an ordinal parameterised by a monomial order. We use this generalisation to prove the abstract (“strategy-independent”) admissibility of a number of superfluous S-polynomial criteria important for efficient basis construction. Finally, we consider local notions of proof minimality for weak Nullstellensatz proofs and give ideal-theoretic methods for computing complex “unsatisfiable cores” which contribute to efficient SMT solving in the context of nonlinear complex arithmetic. In the second half, we consider the problem of effectively combining a heterogeneous collection of decision techniques for fragments of the existential theory of real closed fields. We propose and investigate a number of novel combined decision methods and implement them in our proof tool RAHD (Real Algebra in High Dimensions). We build a hierarchy of increasingly powerful combined decision methods, culminating in a generalisation of partial cylindrical algebraic decomposition (CAD) which we call Abstract Partial CAD. This generalisation incorporates the use of arbitrary sound but possibly incomplete proof procedures for the existential theory of real closed fields as first-class functional parameters for “short-circuiting” expensive computations during the lifting phase of CAD. Identifying these proof procedure parameters formally with RAHD proof strategies, we implement the method in RAHD for the case of full-dimensional cell decompositions and investigate its efficacy with respect to the Brown-McCallum projection operator. We end with some wishes for the future

Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL

The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic