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Structured derivations
Structured derivations (SD)〔Ralph-Johan Back. Structured derivations: a unified proof style for teaching mathematics. Formal aspects of computing, vol. 22, n. 5, 2010, pp. 629–661.〕 is a logic-based format for presenting mathematical solutions and proofs created by Prof. Ralph-Johan Back and Joakim von Wright at Åbo Akademi University, Turku, Finland. The format was originally introduced as a way for presenting proofs in programming logic, but was later adapted to provide a practical approach to presenting proofs and derivations in mathematics education including exact formalisms. A structured derivation has a precise mathematical interpretation, and the syntax and the layout are precisely defined. The standardized syntax renders the format suitable for presenting and manipulating mathematics digitally.
SD is a further development of the calculational proof format introduced by Edsger W. Dijkstra and others in the early 1990s. In essence, three main extensions have been made. First, a mechanism for decomposing proofs through the use of subderivations has been added. The calculational approach is limited to writing proof fragments, and longer derivations are commonly decomposed into several separate subproofs. Using SD with subderivations, on the other hand, the presentation of a complete proof or solution is kept together, as subproofs can be presented exactly where they are needed. In addition, SD makes it possible to handle assumptions and observations in proofs. As such, the format can be seen as combining the benefits of the calculational style with the decomposition facilities of natural deduction.
==Examples==
The following three examples will be used to illustrate the most central features of structured derivations.
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