## Structured Calculational Proof (1996)

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Citations: | 17 - 9 self |

### BibTeX

@MISC{Back96structuredcalculational,

author = {Ralph Back and Jim Grundy and Joakim von Wright and Turku Centre and Computer Science},

title = {Structured Calculational Proof},

year = {1996}

}

### Years of Citing Articles

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### Abstract

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...

### Citations

579 |
Untersuchungen über das logische Schließen
- Gentzen
- 1935
(Show Context)
Citation Context ...ion This paper presents a new format for writing proofs, which we call structured calculational proof. Two of the main inspirations we have drawn on when formulating this format are natural deduction =-=[3, 10]-=- and calculational proof [2, 4, 13]. The clarity and readability of calculational proof has made it a popular choice among computer scientists. We feel, however, that pure calculational proof provides... |

262 |
Refinement Calculus: A Systematic Introduction
- Back, Wright
- 1998
(Show Context)
Citation Context ...lational proof has been driven by the first and third author’s requirement for a proof notation that was both clear and compact while writing their book Refinement Calculus: A Systematic Introductio=-=n [1]-=-. The book applies structured calculational proof to hundreds of problems of varying complexity. We have also used structured calculational proof to solve an unbiased problem set in the form of the 19... |

127 |
Ideas and results in proof theory
- Prawitz
- 1971
(Show Context)
Citation Context ...ion This paper presents a new format for writing proofs, which we call structured calculational proof. Two of the main inspirations we have drawn on when formulating this format are natural deduction =-=[3, 10]-=- and calculational proof [2, 4, 13]. The clarity and readability of calculational proof has made it a popular choice among computer scientists. We feel, however, that pure calculational proof provides... |

114 | The collected papers of Gerhard Gentzen - Szabo - 1969 |

82 | How to write a proof
- Lamport
- 1993
(Show Context)
Citation Context ...me from not carrying out proofs in sufficient detail. He recommends that authors expand their proofs until the lowest level statements are obvious, and then continue with the proof for one more level =-=[9]-=-. This section has tried to describe what it would be like to be able to browse proofs presented in the proposed proof format. A deeper appreciation of the possibilities offered by proof browsing can ... |

47 |
Formalising the hierarchical structure of practical mathematical reasoning
- Robinson, Staples
- 1989
(Show Context)
Citation Context ...es the readability of calculational proof with the structuring facilities of natural deduction. The resulting method resembles Robinson and Staples’s window inference system of hierarchical reasonin=-=g [8, 11]-=-, but maintains a visual similarity to the more pleasing notation of calculational proof. 2 Structuring Calculational Proof In their book Predicate Calculus and Program Semantics [2], the authors Dijk... |

22 | On the Shape of Mathematical Arguments - Gasteren - 1988 |

20 |
Transformational hierarchical reasoning
- Grundy
- 1996
(Show Context)
Citation Context ...es the readability of calculational proof with the structuring facilities of natural deduction. The resulting method resembles Robinson and Staples’s window inference system of hierarchical reasonin=-=g [8, 11]-=-, but maintains a visual similarity to the more pleasing notation of calculational proof. 2 Structuring Calculational Proof In their book Predicate Calculus and Program Semantics [2], the authors Dijk... |

11 | A browsable format for proof presentation
- Grundy
- 1996
(Show Context)
Citation Context ...le to browse proofs presented in the proposed proof format. A deeper appreciation of the possibilities offered by proof browsing can be gained from the paper A Browsable Format for Proof Presentation =-=[6]-=-. This paper appears in the electronic journal Mathesis Universalis, and the proofs it contains can actually be browsed in the manner described. 10 Conclusions This paper has presented an extension to... |

2 |
A Logical Approach to Discrete Math, chapters 3–4
- Gries, Schneider
- 1993
(Show Context)
Citation Context ...rmat for writing proofs, which we call structured calculational proof. Two of the main inspirations we have drawn on when formulating this format are natural deduction [3, 10] and calculational proof =-=[2, 4, 13]-=-. The clarity and readability of calculational proof has made it a popular choice among computer scientists. We feel, however, that pure calculational proof provides too little support for the formal ... |

1 |
Predicate Calculus and Program Semantics, chapter 4
- Dijkstra, Scholten
- 1990
(Show Context)
Citation Context ...rmat for writing proofs, which we call structured calculational proof. Two of the main inspirations we have drawn on when formulating this format are natural deduction [3, 10] and calculational proof =-=[2, 4, 13]-=-. The clarity and readability of calculational proof has made it a popular choice among computer scientists. We feel, however, that pure calculational proof provides too little support for the formal ... |

1 |
Teaching math more effectively, through the design of calculational proofs
- Gries, Schneider
- 1994
(Show Context)
Citation Context ...= fdefinition of set comprehensiong (A [ B) \ (A [ C) 7sFor readers interested in a comparison, a solution to this problem using ordinary calculational proof has been presented by Gries and Schneider =-=[5]-=-. 6 Common Proof Paradigms Many common proof paradigms are not easily expressed in a purely calculational style. Such proofs are usually presented as a combination of calculation and informal explanat... |

1 |
Structured solutions to the 1995 Finnish highschool general mathematics matriculation exam
- Grundy
- 1996
(Show Context)
Citation Context ...of problems of varying complexity. We have also used structured calculational proof to solve an unbiased problem set in the form of the 1995 Finnish High School General Mathematics Matriculation Exam =-=[7]-=-. There have been three main sources of inspiration for the development of the structured calculational proof. The form of the notation is largely inspired by the original system of calculational proo... |

1 |
On The Shape of Mathematical Arguments, volume 445 of Lecture Notes in Computer Science, chapter 14, pages 90-- 120
- Gasteren
- 1990
(Show Context)
Citation Context ...rmat for writing proofs, which we call structured calculational proof. Two of the main inspirations we have drawn on when formulating this format are natural deduction [3, 10] and calculational proof =-=[2, 4, 13]-=-. The clarity and readability of calculational proof has made it a popular choice among computer scientists. We feel, however, that pure calculational proof provides too little support for the formal ... |