Changes to the Mathematical Language of Event-B

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This document describes the evolution of the Event-B mathematical language that happened in release 1.0.0. See Event-B_Mathematical_Language for a full description of the language.

Generic Identity and Projections

Three operators were still unary while they could be atomic and generic:

  • the identity relation <math>\id</math>
  • the first projection <math>\prjone</math>
  • the second projection <math>\prjtwo</math>

These operators are defined as follows in the old version:

<math>
\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}
</math>

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

<math>
\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}
</math>

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

<math>
\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}
</math>

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 <math>r</math> is irreflexive, one would now write <math>r\binter\id = \emptyset</math>.

Partition

A new partition predicate is introduced. It is intended to provide an easier way to enter enumerated sets, while getting rid of the <math>\frac{n(n-1)}{2}</math> axioms needed to express pairwise distinctness (or disjointness). The partition operator is defined as follows:

<math>
\begin{array}{ll}
partition(E_0, E_1, \ldots, E_n)\defi &
 E_0 = E_1\bunion \cdots\bunion E_n \\ &
 \;\land\; E_1\binter E_2=\emptyset
 \;\land\;\cdots
 \;\land\; E_{n-1}\binter E_n = \emptyset \\ &
 (\;\land\; i \ne j \limp E_i \binter E_j = \emptyset ) \\
\end{array}
</math>


where the <math>E_i</math> are expressions bearing the same type.

Then, we can enter into a context :

<math>
 \begin{array}{ll}
   \mathsf{set}  & S\\
   \mathsf{constant}  & a_1\\
   \vdots & \vdots\\
   \mathsf{constant}  & a_n\\
   \mathsf{axiom}  & partition(S, \{a_1\}, \ldots, \{a_n\})
 \end{array}
</math>

which is a particular case of a set being defined by listing all its elements.


Partition Wizard

TODO

Operator Associativity

Operators used to build sets of relations or functions, viz.

  • relation
  • total relation
  • surjective relation
  • total surjective relation
  • partial function
  • total function
  • partial injection
  • total injection
  • partial surjection
  • total surjection
  • bijection

have no more relative priorities and loose associativity. Instead, users have to make it explicit by entering parenthesis in formulas.