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The Refinement Calculator: Proof Support for Program Refinement
 Formal Methods Pacific ’97
, 1997
"... . We describe the Refinement Calculator, a tool which supports ..."
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. We describe the Refinement Calculator, a tool which supports
Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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Cited by 23 (0 self)
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
Structured Calculational Proof
, 1996
"... We propose a new format for writing proofs, which we call structured calculational proof. The format is similar to the calculational style of proof already familiar to many computer scientists, but extends it by allowing large proofs to be hierarchically decomposed into smaller ones. In fact, struc ..."
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Cited by 16 (9 self)
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We propose a new format for writing proofs, which we call structured calculational proof. The format is similar to the calculational style of proof already familiar to many computer scientists, but extends it by allowing large proofs to be hierarchically decomposed into smaller ones. In fact, structured calculational proof can be seen as an alternative presentation of natural deduction. Natural deduction is a well established style of reasoning which uses hierarchical decomposition to great effect, but which is traditionally expressed in a notation that is inconvenient for writing calculational proofs. The hierarchical nature of structured calculational proofs can be used for proof browsing. We comment on how browsing can increase the value of a proof, and discuss the possibilities offered by electronic publishing for the presentation and dissemination of papers containing browsable proofs. Note: This paper is also available as Australian National University Joint Computer Science Tec...
Primitive (co)recursion and courseofvalues (co)iteration, categorically
 Informatica
, 1999
"... Abstract. In the mainstream categorical approach to typed (total) functional programming, datatypes are modelled as initial algebras and codatatypes as terminal coalgebras. The basic function definition schemes of iteration and coiteration are modelled by constructions known as catamorphisms and ana ..."
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Abstract. In the mainstream categorical approach to typed (total) functional programming, datatypes are modelled as initial algebras and codatatypes as terminal coalgebras. The basic function definition schemes of iteration and coiteration are modelled by constructions known as catamorphisms and anamorphisms. Primitive recursion has been captured by a construction called paramorphisms. We draw attention to the dual construction of apomorphisms, and show on examples that primitive corecursion is a useful function definition scheme. We also put forward and study two novel constructions, viz., histomorphisms and futumorphisms, that capture the powerful schemes of courseofvalue iteration and its dual, respectively, and argue that even these are helpful.
On dynamically presenting a topology course
 Annals of Mathematics and Artificial Intelligence
, 2001
"... www.cs.mdx.ac.uk/imp Authors of traditional mathematical texts often have difficulty balancing the amount of contextual information and proof detail. We propose a simple hypermedia framework that can assist in the organisation and presentation of mathematical theorems and definitions. We describe th ..."
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Cited by 10 (5 self)
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www.cs.mdx.ac.uk/imp Authors of traditional mathematical texts often have difficulty balancing the amount of contextual information and proof detail. We propose a simple hypermedia framework that can assist in the organisation and presentation of mathematical theorems and definitions. We describe the application of the framework to convert an existing course in general topology to a webbased set of materials. A pilot study of the materials indicated a high level of user satisfaction. We discuss further lines of investigation, in particular, the presentation of larger bodies of work. 1
Functional programming with apomorphisms (corecursion
 Proceedings of the Estonian Academy of Sciences: Physics, Mathematics
, 1998
"... Abstract. In the mainstream categorical approach to typed (total) functional programming, functions with inductive source types defined by primitive recursion are called paramorphisms; the utility of primitive recursion as a scheme for defining functions in programming is well known. We draw attenti ..."
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Cited by 10 (1 self)
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Abstract. In the mainstream categorical approach to typed (total) functional programming, functions with inductive source types defined by primitive recursion are called paramorphisms; the utility of primitive recursion as a scheme for defining functions in programming is well known. We draw attention to the dual notion of apomorphisms — with coinductive target types defined by primitive corecursion and show on examples that primitive corecursion is useful in programming. Key words: typed (total) functional programming, categorical program calculation, (co)datatypes, (co)recursion forms. 1.
A theoretical analysis of hierarchical proofs
 In Asperti et al
, 2003
"... www.uclic.ucl.ac.uk/imp Abstract. Hierarchical proof presentations are ubiquitous within logic and computer science, but have made little impact on mathematics in general. The reasons for this are not currently known, and need to be understood if mathematical knowledge management systems are to gain ..."
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www.uclic.ucl.ac.uk/imp Abstract. Hierarchical proof presentations are ubiquitous within logic and computer science, but have made little impact on mathematics in general. The reasons for this are not currently known, and need to be understood if mathematical knowledge management systems are to gain acceptance in the mathematical community. We report on some initial experiments with three users of a set of webbased hierarchical proofs, which suggest that usability problems could be a factor. In order to better understand these problems we present a theoretical analysis of hierarchical proofs using Cognitive Dimensions [6]. The analysis allows us to formulate some concrete hypotheses about the usability of hierarchical proof presentations. 1
Interactive Presentations of Mathematics: A Position Paper
"... his format recursively consists of a list of key proof steps, each justied by a subproof. The result is a hierarchy of nested lists of proof steps: the higher level steps give the outline of the proof, the lower level steps the details. The obvious adaptation to hypertext is to hide or reveal the s ..."
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his format recursively consists of a list of key proof steps, each justied by a subproof. The result is a hierarchy of nested lists of proof steps: the higher level steps give the outline of the proof, the lower level steps the details. The obvious adaptation to hypertext is to hide or reveal the subproofs under the reader's control. This was rst done by Grundy [1] for a calculational proof style. However, a major drawback is that key insights may be contained within a hidden subproof  but a reader has no way of knowing this at a glance. We believe that a more sophisticated approach should involve summarising the hidden parts. In particular, references to key steps, proof methods or imported results can allow the reader to understand the proof step without seeing the details. Incorporating Context Apart from understanding the logical argument of a proof, a reader also needs to understand its signicance. This comes from both the rationale for its su