The Theory Behind TheoryMine

Abstract. We describe the technology behind the TheoryMine novelty gift company, which sells the rights to name novel mathematical theorems. A tower of four computer systems is used to generate recursive theories, then to speculate conjectures in those theories and then to prove these conjectures. All stages of the process are entirely automatic. The process guarantees large numbers of sound, novel theorems of some intrinsic merit.

Automating Inductive Proofs using Theory Exploration

HipSpec is a system for automatically deriving and proving properties about functional programs. It uses a novel approach, combining theory exploration, counterexample testing and inductive theorem proving. HipSpec automatically generates a set of equational theorems about the available recursive functions of a program. These equational properties make up an algebraic specification for the program and can in addition be used as a background theory for proving additional user-stated properties. Experimental results are encouraging: HipSpec compares favourably to other inductive theorem provers and theory exploration systems

Best-First Rippling

Rippling is a form of rewriting that guides search by only performing steps that reduce the syntactic differences between formulae. Termination is normally ensured by a measure that is decreases with each rewrite step. Because of this restriction, rippling will fail to prove theorems about, for example, mutual recursion as steps that temporarily increase the differences are necessary. Best-first rippling is an extension to rippling where the restrictions have been recast as heuristic scores for use in best-first search. If nothing better is available, previously illegal steps can be considered, making best-first rippling more flexible than ordinary rippling. We have implemented best-first rippling in the IsaPlanner system together with a mechanism for caching proof-states that helps remove symmetries in the search space, and machinery to ensure termination based on term embeddings. Our experiments show that the implementation of best-first rippling is faster on average than IsaPlanner’s version of traditional depth-first rippling, and solves a range of problems where ordinary rippling fails

Case-Analysis for Rippling and Inductive Proof

Rippling is a heuristic used to guide rewriting and is typically used for inductive theorem proving. We introduce a method to support case-analysis within rippling. Like earlier work, this allows goals containing if-statements to be proved automatically. The new contribution is that our method also supports case-analysis on datatypes. By locating the case-analysis as a step within rippling we also maintain the termination. The work has been implemented in IsaPlanner and used to extend the existing inductive proof method. We evaluate this extended prover on a large set of examples from Isabelle’s theory library and from the inductive theorem proving literature. We find that this leads to a significant improvement in the coverage of inductive theorem proving. The main limitations of the extended prover are identified, highlight the need for advances in the treatment of assumptions during rippling and when conjecturing lemmas

Automated discovery of inductive lemmas

The discovery of unknown lemmas, case-splits and other so called eureka steps are challenging problems for automated theorem proving and have generally been assumed to require user intervention. This thesis is mainly concerned with the automated discovery of inductive lemmas. We have explored two approaches based on failure recovery and theory formation, with the aim of improving automation of firstand higher-order inductive proofs in the IsaPlanner system. We have implemented a lemma speculation critic which attempts to find a missing lemma using information from a failed proof-attempt. However, we found few proofs for which this critic was applicable and successful. We have also developed a program for inductive theory formation, which we call IsaCoSy. IsaCoSy was evaluated on different inductive theories about natural numbers, lists and binary trees, and found to successfully produce many relevant theorems and lemmas. Using a background theory produced by IsaCoSy, it was possible for IsaPlanner to automatically prove more new theorems than with lemma speculation. In addition to the lemma discovery techniques, we also implemented an automated technique for case-analysis. This allows IsaPlanner to deal with proofs involving conditionals, expressed as if- or case-statements. ii

Dynamic Rippling, Middle-Out Reasoning and Lemma Discovery

We present a succinct account of dynamic rippling, a technique used to guide the automation of inductive proofs. This simplifies termination proofs for rippling and hence facilitates extending the technique in ways that preserve termination. We illustrate this by extending rippling with a terminating version of middle-out reasoning for lemma speculation. This supports automatic speculation of schematic lemmas which are incrementally instantiated by unification as the rippling proof progresses. Middle-out reasoning and lemma speculation have been implemented in higher-order logic and evaluated on typical libraries of formalised mathematics. This reveals that, when applied, the technique often finds the needed lemmas to complete the proof, but it is not as frequently applicable as initially expected. In comparison, we show that theory formation methods, combined with simpler proof methods, offer an effective alternative

A proof-centric approach to mathematical assistants

We present an approach to mathematical assistants which uses readable, executable proof scripts as the central language for interaction. We examine an implementation that combines the Isar language, the Isabelle theorem prover and the IsaPlanner proof planner. We argue that this synergy provides a flexible environment for the exploration, certification, and presentation of mathematical proof

Conditional Lemma Discovery and Recursion Induction in Hipster

Hipster is a theory exploration tool for the proof assistant Isabelle/HOL. It automatically discovers lemmas about given recursive functions and datatypes and proves them by induction. Previously, only equational properties could be discovered. Conditional lemmas, for example required when reasoning about sorting, has been beyond the scope of theory exploration. In this paper we describe an extension to Hipster to also support discovery and proof of conditional lemmas. We also present a new automated tactic, which uses recursion induction. Recursion induction follows the recursive structure of a function definition through its termina- tion order, as opposed to structural induction, which follows that of the datatype. We find that the addition of recursion induction increases the number of proofs completed automatically, both for conditional and equational statements.