Changes to the Mathematical Language of Event-B

From Event-B
Revision as of 12:33, 10 March 2009 by imported>Nicolas (New page: This document describes the evolution of the Event-B mathematical language. The previous version of the language will still be supported. == Identity and Projections == Three operators w...)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigationJump to search

This document describes the evolution of the Event-B mathematical language. The previous version of the language will still be supported.

Identity and Projections

Three operators were still unary while they could be atomic:

  • the identity relation \id
  • the first projection \prjone
  • the second projection \prjtwo

These operators are defined as follows:


\begin{matrix}
  E\mapsto F &\in\id(S) && E\in S\;\land\; F = E\\
  (E\mapsto F)\mapsto G &\in\prjone(r)
  && E\mapsto F\in r\;\land\; G = E\\
  (E\mapsto F)\mapsto G &\in\prjtwo(r)
  && E\mapsto F\in r\;\land\; G = F    .
\end{matrix}

If we drop the parameter, we get much more straightforward definitions that capture the essence of the operator. The new definitions are


\begin{matrix}
E\mapsto F &\in\id && E = F\\
(E\mapsto F)\mapsto G &\in\prjone && E = G\\
(E\mapsto F)\mapsto G &\in\prjtwo && F = G    .
\end{matrix}

We have the following equivalence between the old and the new versions of the operators


\begin{matrix}
    \textbf{Old Version} & \textbf{New Version}\\
    \id(S)  & S\domres id\\
    \prjone(r) & r\domres\prjone\\
    \prjtwo(r) & r\domres\prjtwo  .
\end{matrix}

Moreover, in the case where the parameter is not needed, then it can be dropped altogether: no domain restriction is needed. For instance, to express that a relation r is irreflexive, one would now write r\binter\id = \emptyset.


Partition

TODO