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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
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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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...
Assisted proof document authoring
 Mathematical Knowledge Management MKM 2005, LNAI 3863
, 2006
"... Abstract. Recently, significant advances have been made in formalised mathematical texts for large, demanding proofs. But although such large developments are possible, they still take an inordinate amount of effort and time, and there is a significant gap between the resulting formalised machinech ..."
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Abstract. Recently, significant advances have been made in formalised mathematical texts for large, demanding proofs. But although such large developments are possible, they still take an inordinate amount of effort and time, and there is a significant gap between the resulting formalised machinecheckable proof scripts and the corresponding humanreadable mathematical texts. We present an authoring system for formal proof which addresses these concerns. It is based on a central document format which, in the tradition of literate programming, allows one to extract either a formal proof script or a humanreadable document; the two may have differing structure and detail levels, but are developed together in a synchronised way. Additionally, we introduce ways to assist production of the central document, by allowing tools to contribute backflow to update and extend it. Our authoring system builds on the new PG Kit architecture for Proof General, bringing the extra advantage that it works in a uniform interface, generically across various interactive theorem provers. 1
A Browsable Format for Proof Presentation
 Mathesis Universalis
, 1996
"... The paper describes a format for presenting proofs called structured calculational proof. The format resembles calculational proof, a style of reasoning popular among computer scientists, but extended with structuring facilities. A prototype tool has been developed which allows readers to interacti ..."
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The paper describes a format for presenting proofs called structured calculational proof. The format resembles calculational proof, a style of reasoning popular among computer scientists, but extended with structuring facilities. A prototype tool has been developed which allows readers to interactively browse proofs presented in this format via the world wide web. The ability to browse a proof increases its readability, and hence its value as a proof. Computers have been used for some time to both construct and check mathematical proofs, but using them to enhance the readability of proofs is a relatively novel application. This paper was originally presented at the symposium on Logic, Mathematics and the Computer The reference is as follows: Jim Grundy. A browsable format for proof presentation. In Christoffer Gefwert, Pekka Orponen and Jouko Seppanen (editors), Logic, Mathematics and the Computer  Foundations: History, Philosophy and Applications, volume 14 of the Finnish Artifi...
Window inference in isabelle
 University of Cambridge Computer Laboratory
, 1995
"... Window inference is a transformational style of reasoning that provides an intuitive framework for managing context during the transformation of subterms under transitive relations. This report describes the design for a prototype window inference tool in Isabelle, and discusses possible directions ..."
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Cited by 6 (2 self)
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Window inference is a transformational style of reasoning that provides an intuitive framework for managing context during the transformation of subterms under transitive relations. This report describes the design for a prototype window inference tool in Isabelle, and discusses possible directions for the final tool. 1
Doing High School Mathematics Carefully
, 1997
"... We show how solutions to typical problems of High School and firstyear University mathematics can be written using structured derivations. Such a derivation extends the calculational proof format with subderivations that allow inferences to presented at different levels of detail. By using structur ..."
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We show how solutions to typical problems of High School and firstyear University mathematics can be written using structured derivations. Such a derivation extends the calculational proof format with subderivations that allow inferences to presented at different levels of detail. By using structured derivations and a minimal amount of logical syntax, we can write solution to typical problems in algebra but also in, e.g., real analysis. We argue why structured derivations give students a better grasp of problem solutions and better possibilities to reread and discuss solutions afterwards, as compared with traditional informal approaches to writing down solutions. TUCS Research Group Programming Methodology Research Group 1 Introduction We are concerned with the way in which High School mathematics is taught. In our view, a more careful use of logical derivations would make the material easier to grasp, and would enhance the manipulative skill of the students. In this paper, we fir...
An extension of the program derivation format
 Programming Concepts and Methods (PROCOMET ’98). Chapman
, 1998
"... AvG159/AB61 1 A convention is proposed for embedding program statements into Dijkstra’s calculus, with the aim of simplifying the stepwise construction of programs. ..."
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AvG159/AB61 1 A convention is proposed for embedding program statements into Dijkstra’s calculus, with the aim of simplifying the stepwise construction of programs.
TAS  A Generic Window Inference System
"... This paper presents work on technology for transformational proof and program development, as used by window inference calculi and transformation systems. The calculi are characterised by a certain class of theorems in the underlying logic. Our transformation system TAS compiles these rules to concr ..."
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This paper presents work on technology for transformational proof and program development, as used by window inference calculi and transformation systems. The calculi are characterised by a certain class of theorems in the underlying logic. Our transformation system TAS compiles these rules to concrete deduction support, complete with a graphical user interface with commandlanguagefree user interaction by gestures like drag&drop and proofbypointing, and a development management for transformational proofs. It is generic in the sense that it is completely independent of the particular window inference or transformational calculus, and can be instantiated to many different ones; three such instantiations are presented in the paper.
Unified algebra
 In preparation
, 1998
"... Abstract—Unified Algebra unifies booleans and numbers, values and types, functions and function spaces. It incorporates basic structures, such as sets and lists, and advanced structures, such as quantifications and limits. It does so with an economy of symbols and rules. The presentation is basic an ..."
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Abstract—Unified Algebra unifies booleans and numbers, values and types, functions and function spaces. It incorporates basic structures, such as sets and lists, and advanced structures, such as quantifications and limits. It does so with an economy of symbols and rules. The presentation is basic and detailed enough to serve as a foundation of that part of mathematics that serves much of computer science, with comments on what constitutes good mathematical practice. Keywords—boolean algebra, foundation of computer science, foundation of mathematics, unified algebra
Verified Calculations
, 2013
"... Calculational proofs—proofs by stepwise formula manipulation—are praised for their rigor, readability, and elegance. It seems desirable to reuse this style, often employed on paper, in the context of mechanized reasoning, and in particular, program verification. This work leverages the power of SMT ..."
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Calculational proofs—proofs by stepwise formula manipulation—are praised for their rigor, readability, and elegance. It seems desirable to reuse this style, often employed on paper, in the context of mechanized reasoning, and in particular, program verification. This work leverages the power of SMT solvers to machinecheck calculational proofs at the level of detail they are usually written by hand. It builds the support for calculations into the programming language and autoactive program verifier Dafny. The paper demonstrates that calculations integrate smoothly with other language constructs, producing concise and readable proofs in a wide range of problem domains: from mathematical theorems to correctness of imperative programs. The examples show that calculational proofs in Dafny compare favorably, in terms of readability and conciseness, with arguments written in other styles and proof languages.