System Description: an Interface Between CLAM and HOL

. The CLaM proof planner has been interfaced to the HOL interactive theorem prover to provide the power of proof planning to people using HOL for formal verification, etc. The interface sends HOL goals to CLaM for planning and translates plans back into HOL tactics that solve the initial goals. The project homepage can be found at http://www.cl.cam.ac.uk/Research/HVG/Clam.HOL/intro.html. 1 Introduction CLaM [2] is a proof planning system for Oyster, a tactic-based implementation of the constructive type theory of Martin-Lof. CLaM works by using formalized pre- and post-conditions of Oyster tactics as the basis of plan search. These specifications of tactics are called methods. When a plan for a goal is found, the expectation is that the resulting tactic will solve the goal. Experience shows that the search space for plans is often tractable: CLaM has been able to automatically plan many proofs. A particular emphasis of research with CLaM has been the automation of inductive proo..

System Description: an Interface Between CLAM and HOL

The CLAM proof planner has been interfaced to the HOL interactive theorem prover to provide the power of proof planning to people using HOL for formal verification, etc. The interface sends HOL goals to CLAM for planning and translates plans back into HOL tactics that solve the initial goals. The project homepage can be found at http://www.cl.cam.ac.uk/Research/HVG/Clam.HOL/intro.html

Learning-Assisted Automated Reasoning with Flyspeck

The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the proofs, producing an AI system capable of answering a wide range of mathematical queries automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39% of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided

Hammering towards QED

This paper surveys the emerging methods to automate reasoning over large libraries developed with formal proof assistants. We call these methods hammers. They give the authors of formal proofs a strong “one-stroke” tool for discharging difficult lemmas without the need for careful and detailed manual programming of proof search. The main ingredients underlying this approach are efficient automatic theorem provers that can cope with hundreds of axioms, suitable translations of the proof assistant’s logic to the logic of the automatic provers, heuristic and learning methods that select relevant facts from large libraries, and methods that reconstruct the automatically found proofs inside the proof assistants. We outline the history of these methods, explain the main issues and techniques, and show their strength on several large benchmarks. We also discuss the relation of this technology to the QED Manifesto and consider its implications for QED-like efforts.Blanchette’s Sledgehammer research was supported by the Deutsche Forschungs- gemeinschaft projects Quis Custodiet (grants NI 491/11-1 and NI 491/11-2) and Hardening the Hammer (grant NI 491/14-1). Kaliszyk is supported by the Austrian Science Fund (FWF) grant P26201. Sledgehammer was originally supported by the UK’s Engineering and Physical Sciences Research Council (grant GR/S57198/01). Urban’s work was supported by the Marie-Curie Outgoing International Fellowship project AUTOKNOMATH (grant MOIF-CT-2005-21875) and by the Netherlands Organisation for Scientific Research (NWO) project Knowledge-based Automated Reasoning (grant 612.001.208).This is the final published version. It first appeared at http://jfr.unibo.it/article/view/4593/5730?acceptCookies=1

Supporting formal reasoning about functional programs

It is often claimed that functional programming languages, and in particular pure functional languages are suitable for formal reasoning. This claim is supported by the fact that many people in the functional programming community do reason about languages and programs in a formal or semi-formal way. Different reasoning principles such as equational reasoning, induction and co-induction, are used, depending on the nature of the problem. Using a computer program to check the application of rules and to mechanise the tedious bookkeeping involved can simplify proofs and provide more confidence in their correctness. When reasoning about programs, this can also allow experiments with new rules and reasoning styles, where a user may not be confident about structuring a proof on paper. Checking the applicability of a rule can eliminate the risk of mistakes caused by misunderstanding the theory being used. Just as there are different ways in which formal or informal reasoning can be applied in functional programming, there are different ways in which tools can be provided to support this reasoning. This thesis describes an investigation of how to develop a mechanised reasoning system to allow reasoning about algorithms as a functional programmer would write them, not an encoding of the algorithm into a significantly different form. In addition, this work aims to develop a system to support a user who is not a theorem proving expert or an expert in the theoretical foundations of functional programming. The work is aimed towards a system that could be used by a functional programmer developing real programs and wishing to prove some or all of the programs correct or to prove that two programs are equivalent

A Proof Planning Framework For Isabelle

Centre for Intelligent Systems and their ApplicationsProof planning is a paradigm for the automation of proof that focuses on encoding intelligence to guide the proof process. The idea is to capture common patterns of reasoning which can be used to derive abstract descriptions of proofs known as proof plans. These can then be executed to provide fully formal proofs. This thesis concerns the development and analysis of a novel approach to proof planning that focuses on an explicit representation of choices during search. We embody our approach as a proof planner for the generic proof assistant Isabelle and use the Isar language, which is human-readable and machine-checkable, to represent proof plans. Within this framework we develop an inductive theorem prover as a case study of our approach to proof planning. Our prover uses the difference reduction heuristic known as rippling to automate the step cases of the inductive proofs. The development of a flexible approach to rippling that supports its various modifications and extensions is the second major focus of this thesis. Here, our inductive theorem prover provides a context in which to evaluate rippling experimentally. This work results in an efficient and powerful inductive theorem prover for Isabelle as well as proposals for further improving the efficiency of rippling. We also draw observations in order to direct further work on proof planning. Overall, we aim to make it easier for mathematical techniques, and those specific to mechanical theorem proving, to be encoded and applied to problems

Supporting dependently typed functional programming with proof automation and testing

Dependent types can be used to capture useful properties about programs at compile time. However, developing dependently typed programs can be difficult in current systems. Capturing interesting program properties usually requires the user to write proofs, where constructing the latter can be both a difficult and tedious process. Additionally, finding and fixing errors in program scripts can be challenging. This thesis concerns ways in which functional programming with dependent types can be made easier. In particular, we focus on providing help for developing programs that incorporate user-defined types and user-defined functions. For the purpose of supporting dependently typed programming, we have designed a framework that provides improved proof automation and error feedback. Proof automation is provided with the use of heuristic based tactics that automate common patterns of proofs that arise when programming with dependent types. In particular, we use heuristics for generalising goals and employ the rippling heuristic for guiding inductive and non-inductive proofs. The automation we describe includes features for caching and reusing lemmas proven during proof search and, whenever proof search fails, the user can assist the prover by providing high-level hints. We concentrate on providing improved feedback for the errors that occur when there is a mismatch between the specification of a program, described with the use of dependent types, and the behaviour of the program. We employ a QuickCheck-like testing tool for automatically identifying these forms of errors, where the counter examples generated are used as error messages. To demonstrate the effectiveness of our framework for supporting dependently typed programming, we have developed a prototype based around the Coq theorem prover. We demonstrate that the framework as a whole makes program development easier by conducting a series of case studies. In these case studies, which involved verifying properties of tail recursive functions, sorting functions and a binary adder, a significant number of the proofs required were automated