Smart matching

One of the most annoying aspects in the formalization of mathematics is the need of transforming notions to match a given, existing result. This kind of transformations, often based on a conspicuous background knowledge in the given scientific domain (mostly expressed in the form of equalities or isomorphisms), are usually implicit in the mathematical discourse, and it would be highly desirable to obtain a similar behavior in interactive provers. The paper describes the superposition-based implementation of this feature inside the Matita interactive theorem prover, focusing in particular on the so called smart application tactic, supporting smart matching between a goal and a given result.Comment: To appear in The 9th International Conference on Mathematical Knowledge Management: MKM 201

Superposition as a logical glue

The typical mathematical language systematically exploits notational and logical abuses whose resolution requires not just the knowledge of domain specific notation and conventions, but not trivial skills in the given mathematical discipline. A large part of this background knowledge is expressed in form of equalities and isomorphisms, allowing mathematicians to freely move between different incarnations of the same entity without even mentioning the transformation. Providing ITP-systems with similar capabilities seems to be a major way to improve their intelligence, and to ease the communication between the user and the machine. The present paper discusses our experience of integration of a superposition calculus within the Matita interactive prover, providing in particular a very flexible, "smart" application tactic, and a simple, innovative approach to automation.Comment: In Proceedings TYPES 2009, arXiv:1103.311

GRUNGE: A Grand Unified ATP Challenge

This paper describes a large set of related theorem proving problems obtained by translating theorems from the HOL4 standard library into multiple logical formalisms. The formalisms are in higher-order logic (with and without type variables) and first-order logic (possibly with multiple types, and possibly with type variables). The resultant problem sets allow us to run automated theorem provers that support different logical formats on corresponding problems, and compare their performances. This also results in a new "grand unified" large theory benchmark that emulates the ITP/ATP hammer setting, where systems and metasystems can use multiple ATP formalisms in complementary ways, and jointly learn from the accumulated knowledge.Comment: CADE 27 -- 27th International Conference on Automated Deductio

System Description: E 1.8

Abstract. E is a theorem prover for full first-order logic with equality. It reduces first-order problems to clause normal form and employs a saturation algorithm based on the equational superposition calculus. E is built on shared terms with cached rewriting, and employs several innovations for efficient clause indexing. Major strengths of the system are automatic problem analysis and highly flexible search heuristics. The prover can provide verifiable proof objects and answer substitutions with very little overhead. E performs well, solving more than 69 % of TPTP-5.4.0 FOF and CNF problems in automatic mode.

Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs