### Strong Modular Proof Assistance: Reasoning Across Theories

Formal proof technology delivers an unparalleled level of certainty and security. Nevertheless, applying proof assistants to the verification of complex theories and designs is still extremely laborious. High profile certification projects, such as seL4, CompCert, and Flyspeck require tens of person-years. We recently demonstrated that this effort can be significantly reduced by combining reasoning and learning in so called hammer systems: 40% of the Flyspeck, HOL4, Isabelle/HOL, and Mizar top-level lemmas can be proved automatically.

Today’s early generation of hammers consists of individual systems limited to very few proof assistants. The accessible knowledge repositories are isolated, and there is no reuse of hammer components. It is possible to achieve a breakthrough in proof automation by developing new AI methods that combine reasoning knowledge and techniques into a smart hammer, that works over a very large part of today’s formalized knowledge. The main goal of the project is to develop a strong and uniform learning-reasoning system available for multiple logical foundations. To achieve this, we will develop: (a) uniform learning methods, (b) reusable ATP encoding components for different foundational aspects, (c) integration of proof reconstruction, and (d) methods for knowledge extraction, reuse and content merging. The single proof advice system will be made available for multiple proof assistants and their vast heterogeneous libraries.

The ultimate outcome is an advice system able to automatically prove half of Coq, ACL2, and Isabelle/ZF top-level theorems. Additionally we will significantly improve success rates for HOL provers and Mizar. The combined smart advice method together with the vast accumulated knowledge will result in a novel kind of tool, which allows working mathematicians to automatically find proofs of many simple conjectures, paving the way for the widespread use of formal proof in mathematics and computer science.

#### Members

Cezary Kaliszyk

#### ERC Starting grant project number

714034#### Contact

cezary dot kaliszyk at uibk dot ac dot at#### Publications

Monte Carlo Tableau Proof Search

Michael Färber, Cezary Kaliszyk, Josef Urban

26th International Conference on Automated Deduction,
LNCS 10395,
pp. 563-579,
2017.

Classification of Alignments Between Concepts of Formal Mathematical Systems

Dennis Müller, Thibault Gauthier, Cezary Kaliszyk, Michael Kohlhase, Florian Rabe

10th International Conference on Intelligent Computer Mathematics,
LNCS 10383,
pp. 83-98,
2017.

Presentation and Manipulation of Mizar Properties in an Isabelle Object Logic

Cezary Kaliszyk and Karol Pąk

10th International Conference on Intelligent Computer Mathematics,
LNCS 10383,
pp. 193-207,
2017.

Deep Network Guided Proof Search

Sarah Loos, Geoffrey Irving, Christian Szegedy, Cezary Kaliszyk.

21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning,
EPiC 46,
pp. 85-105,
2017.

TacticToe: Learning to Reason with HOL4 Tactics

Thibault Gauthier, Cezary Kaliszyk, Josef Urban

21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning,
EPIC 46,
pp. 125-143,
2017.