Abstract

The HOL system is a fully-expansive theorem prover: Proofs generated in the system are composed of applications of the primitive inference rules of the underlying logic. This has two main advantages. First, the soundness of the system depends only on the implementations of the primitive rules. Second, users can be given the freedom to write their own proof procedures without the risk of making the system unsound. A full functional programming language is provided for this purpose. The disadvantage with the approach is that performance is compromised. This is partly due to the inherent cost of fully expanding a proof but, as demonstrated in this thesis, much of the observed inefficiency is due to the way the derived proof procedures are written. This thesis seeks to identify sources of non-inherent inefficiency in the HOL system and proposes some general-purpose and some specialised techniques for eliminating it. One area that seems to be particularly amenable to optimisation is equational reasoning. This is significant because equational reasoning constitutes large portions of many proofs. A number of techniques are proposed that transparently optimise equational reasoning. Existing programs in the HOL system require little or no modification to work faster. The other major contribution of this thesis is a framework in which part of the computation involved in HOL proofs can be postponed. This enables users to make better use of their time. The technique exploits a form of lazy evaluation. The critical feature is the separation of the code that generates the structure of a theorem from the code that justifies it logically. Delaying the justification allows some non-local optimisations to be performed in equational reasoning. None of the techniques sacrifice the securit...

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