Structured calculational proof

Back, Ralph-Johan; Grundy, Jim; von, Joakim Wright

doi:10.1007/bf01211456

Cited by 19 publications

(20 citation statements)

References 17 publications

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“…Back and von Wright extended the calculational proof style to structured derivations, by adding nested derivations and a mechanism for handling and propagating assumptions in derivations [BaWr98,BGW98]. Similar to Dijkstra's work, structured derivations were also originally developed for proofs about program correctness, in this case in the refinement calculus [Bac80,BaWr98,Bac88].…”

Section: Introductionmentioning

confidence: 99%

Structured derivations: a unified proof style for teaching mathematics

Back

2010

Form. Asp. Comput.

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Abstract. Structured derivations were introduced by Back and von Wright as an extension of the calculational proof style originally proposed by E.W. Dijkstra and his colleagues. Structured derivations added nested subderivations and inherited assumptions to this style. This paper introduces further extensions of the structured derivation format, and gives a precise syntax and semantics for the extended proof style. The extensions provide a unification of the three main proof styles used in mathematics today: Hilbert-style forward chaining proofs, Gentzen-style backward chaining proofs and algebraic derivations and calculations (in particular, Dijkstra's calculational proof style). Each of these proof styles can now be directly presented as a structured derivation. Even more importantly, the three proof styles can be freely intermixed in a single structured derivation, allowing different proof styles to be used in different parts of the derivation, each time choosing the proof style that is most suitable for the (sub)problem at hand.We describe here (extended) structured derivations, feature by feature, and illustrate each feature with examples. We show how to model the three main proof styles as structured derivations. We give an exact syntax for structured derivations and define their semantics by showing how a structured derivation can be automatically translated into an equivalent Gentzen-style sequent calculus derivation.Structured derivations have been primarily developed for teaching mathematics at the secondary and tertiary education level. The syntax of structured derivations determines the general structure of the proof, but does not impose any restrictions on how the basic notions of the underlying mathematical domain are treated. Hence, the style can be used for any kind of proofs, calculations, derivations, and general problem solving found in mathematics education at these levels. The precise syntax makes it easy to provide computer support for structured derivations.

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Section: Introductionmentioning

confidence: 99%

Structured derivations: a unified proof style for teaching mathematics

Back

2010

Form. Asp. Comput.

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“…Our representation is an extension of the Structured Calculational Proof format [2]. The transformations on the subformulas are indented and contextual information is stored in the top row of the indented derivation.…”

Section: Extracting Context Of Subformulasmentioning

confidence: 99%

“…are not easily expressed in a purely calculational style. Although, with some effort, these proofs can be handled by the structured calculational approach [2], employing automated theorem provers greatly simplifies the proof process. We use ATP assisted tactics to automate transformation steps that may not always be amenable to the calculational style.…”

Section: Harnessing the Automated Theorem Proversmentioning

confidence: 99%

“…The tool provides a tactic based framework for carrying out program as well as formula transformations in a coherent way. (b) We have extended the Structured Calculational Proof format [2] by making the transformation relation explicit and by adding metavariable support. We have automated the mundane formula manipulation tasks and exploited the power of automated theorem proving to design powerful transformation rules (tactics) which help in shortening and simplifying the derivations without sacrificing the correctness.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Automated Theorem Prover Assisted Program Calculations

Chaudhari

Damani

2014

Lecture Notes in Computer Science

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Abstract. Calculational Style of Programming, while very appealing, has several practical difficulties when done manually. Due to the large number of proofs involved, the derivations can be cumbersome and errorprone. To address these issues, we have developed automated theorem provers assisted program and formula transformation rules, which when coupled with the ability to extract context of a subformula, help in shortening and simplifying the derivations. We have implemented this approach in a Calculational Assistant for Programming from Specifications (CAPS). With the help of simple examples, we show how the calculational assistant helps in taking the drudgery out of the derivation process while ensuring correctness.

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“…These subcalculations can be done in another place and referenced, or they can be done in-place, nicely formatted, to provide a structured calculation (structured proof). By far the best way to handle subcalculations is provided by window inference systems [21] [2], which open a new window for each subcalculation, keep track of its sense (direction), and make its context available. For example, in solving the simultaneous equations, I used the second equation to simplify the first, and then the first to simplify the second.…”

Section: Boolean Calculationmentioning

confidence: 99%

From boolean algebra to unified algebra

Hehner

2004

The Mathematical Intelligencer

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Boolean algebra is simpler than number algebra, with applications in programming, circuit design, law, specifications, mathematical proof, and reasoning in any domain. So why is number algebra taught in primary school and used routinely by scientists, engineers, economists, and the general public, while boolean algebra is not taught until university, and not routinely used by anyone? A large part of the answer may be in the terminology and symbols used, and in the explanations of boolean algebra found in textbooks. This paper points out some of the problems delaying the acceptance and use of boolean algebra, and suggests some solutions.

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Structured calculational proof

Cited by 19 publications

References 17 publications

Structured derivations: a unified proof style for teaching mathematics

Structured derivations: a unified proof style for teaching mathematics

Automated Theorem Prover Assisted Program Calculations

From boolean algebra to unified algebra

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