First-Order Automated Reasoning with Theories: When Deduction Modulo Theory Meets Practice

Abstract

We discuss the practical results obtained by the first generation of automated theorem provers based on Deduction modulo theory. In particular, we demonstrate the concrete improvements such a framework can bring to firstorder theorem provers with the introduction of a rewrite feature. Deduction modulo theory is an extension of predicate calculus with rewriting both on terms and propositions. It is well suited for proof search in theories because it turns many axioms into rewrite rules. We introduce two automated reasoning

Goéland: A Concurrent Tableau-Based Theorem Prover (System Description)

“…However, we have implemented an extension that implements deduction modulo theory [9], i.e., transforms axioms into rewrite rules over propositions and terms. Deduction modulo theory has proved very useful to improve proof search when integrated into usual automated proof techniques [5], and also produces excellent results with manually-defined rewrite rules [6,7]. In Goéland, deduction modulo theory selects some axioms on the basis of a simple syntactic criterion and replaces them by rewrite rules.…”

“…After both children have returned a substitution (1), the parent arbitrarily chooses one of them, starting with X → b, and sends it to the children (2). Since this substitution prevents closure in the right branch (3), the parent later backtracks and sends the other substitution X → a (4), allowing both children (5) and then the parent to close successfully.…”

Safe, fast, concurrent proof checking for the lambda-pi calculus modulo rewriting

“…ATP datasets consist of theory files that can be checked independently, whereas ITP datasets consist of theory files that depend on each other. Among the ATP datasets, I evaluate proofs of TPTP problems generated by iProver Modulo and proofs of theorems from B method set theory generated by Zenon Modulo [9]. For the ITP datasets, I evaluate parts of the standard libraries from HOL Light (up to finite Cartesian products) and Isabelle/HOL (up to HOL.List), as well as Fermat's little theorem proved in Matita [31].…”

“…There exist several proof verifiers for Metamath, one of the smallest being written in 308 LOC of Python. 9 Furthermore, Metamath allows to import OpenTheory proofs and thus to verify proofs from HOL Light, HOL4, and Isabelle [10].…”

Safe, Fast, Concurrent Proof Checking for the lambda-Pi Calculus Modulo Rewriting

“…ATP datasets consist of theory files that can be checked independently, whereas ITP datasets consist of theory files that depend on each other. Among the ATP datasets, I evaluate proofs of TPTP problems generated by iProver Modulo and proofs of theorems from B method set theory generated by Zenon Modulo [9]. For the ITP datasets, I evaluate parts of the standard libraries from HOL Light (up to finite Cartesian products) and Isabelle/HOL (up to HOL.List), as well as Fermat's little theorem proved in Matita [34].…”