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.

Without adding any primitives to the language, we define a concurrency monad transformer in Haskell. This allows us to add a limited form of concurrency to any existing monad. The atomic actions of the new monad are lifted actions of the underlying monad. Some extra operations, such as fork, to initiate new processes, are provided. We discuss the implementation, and use some examples to illustrate the usefulness of this construction.

List homomorphisms are functions that are parallelizable using the divide-and-conquer paradigm. We study the problem of finding homomorphic representations of functions in the Bird–Meertens constructive theory of lists, by means of term rewriting and theorem proving techniques. A previous work proved that to each pair of leftward and rightward sequential representations of a function, based on cons- and snoc-lists, respectively, there is also a representation as a homomorphism. Our contribution is a mechanizable method to extract the homomorphism representation from a pair of sequential representations. The method is decomposed to a generalization problem and an inductive claim, both solvable by term rewriting techniques. To solve the former we present a sound generalization procedure which yields the required representation, and terminates under reasonable assumptions. The inductive claim is provable automatically. We illustrate the method and the procedure by the systematic parallelization of the scan-function (parallel prefix) and of the maximum segment sum problem.

Power series and stream processing were made for each other. Stream algorithms for power series are short, sweet, and compositional. Their neatness shines through in Haskell, thanks to pattern-matching, lazy lists, and operator overloading. In a short compass one can build working code from ground zero (scalar operations) up to exact calculation of generating functions and solutions of differential equations.

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.

Injective pure type systems form a large class of pure type systems for which one can compute by purely syntactic means two sorts elmt(Γ[mid ]M) and sort(Γ[mid ]M), where Γ is a pseudo-context and M is a pseudo-term, and such that for every sort s,

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By eliminating the problematic clause in the (abstraction) rule in favor of constraints over elmt(.[mid ].) and sort(.[mid ].), we provide a sound and complete type-checking algorithm for injective pure type systems. In addition, we prove expansion postponement for a variant of injective pure type systems where the problematic clause in the (abstraction) rule is replaced in favor of constraints over elmt(.[mid ].) and sort(.[mid ].).

Everybody learns about balanced binary search trees in their introductory computer science classes, but even the stouthearted tremble at the thought of actually implementing such a beast. The details surrounding rebalancing are usually just too messy. To show that this need not be the case, we present an algorithm for insertion into red-black trees (Guibas and Sedgewick, 1978) that any competent programmer should be able to implement in fifteen minutes or less.

This paper is about point-free (or ‘pointless’) calculations – calculations performed at the level of function composition instead of that of function application. We address this topic with the help of an example, namely calculating the radix-sort algorithm from a more obvious specification of sorting. The message that we hope to send is that point-free calculations are sometimes surprisingly simpler than the corresponding point-wise calculations.