The LCF system was the first mechanical theorem prover to be user-programmable via a metalanguage, ML, from which the functional programming language Standard ML has been developed. Paulson has demonstrated how a modular rewriting engine can be implemented in LCF. This provides both clarity and flexibility. This paper shows that the same modular approach (using higher-order functions) allows transparent optimisation of the rewriting engine; performance can be improved while few, if any, changes are required to code written using these functions. The techniques described have been implemented in the HOL system, a descendant of LCF, and some are now in daily use. Comparative results are given. Some of the techniques described, in particular ones to avoid processing parts of a data structure that do not need to be changed, may be of more general use in functional programming and beyond.

This paper discusses the principles of implementing an LCF-style proof assistant using a purely functional metalanguage. Two approaches are described; in both, signatures are treated as ordinary values, rather than as mutable components within an abstract datatype. The first approach treats the object logic as a partial algebra and represents it as a partial datatype, that is, a datatype in which the domains of the constructors are restricted by predicate functions. This results in a compact, executable specification of the logic. The second approach uses an abstract type to allow an efficient representation of the logic, whilst keeping the same interface. A case study describes how these principles were put into practice in implementing a fairly complex dependently-sorted logic.

The design of theorem provers, especially in the LCF-prover family, has strongly profited from functional programming. This paper attempts to develop a metaphor suited to visualize the LCF-style prover design, and a methodology for the implementation of graphical user interfaces for these provers and encapsulations of formal methods. In this problem domain, particular attention has to be paid to the need to construct a variety of objects, keep track of their interdependencies and provide support for their reconstruction as a consequence of changes. We present a prototypical implementation of a generic and open interface system architecture, and show how it can be instantiated to an interface for Isabelle, called IsaWin, as well as to a tailored tool for transformational program development, called TAS.

HOLCF is the definitional extension of Church's Higher-Order Logic with Scott's Logic for Computable Functions that has been implemented in the theorem prover Isabelle. This results in a flexible setup for reasoning about functional programs. HOLCF supports standard domain theory (in particular fixpoint reasoning and recursive domain equations), but also coinductive arguments about lazy datatypes. This paper describes in detail how domain theory is embedded in HOL, and presents applications from functional programming, concurrency and denotational semantics.

Proof by mathematical induction plays a crucial role in reasoning about functional programs. A generalization step often holds the key to discovering an inductive proof. We present a generalization technique which is particularly applicable when reasoning about functional programs involving accumulating parameters. We provide empirical evidence for the success of our technique and show how it is contributing to the ongoing development of a parallelizing compiler for Standard ML.