Rewriting rules for event model decomposition

The purpose of this page is to list and justify the rewriting / simplification rules applied in the event model decomposition, when building the actions of an external event of a sub-machine from those of an initial event in the non-decomposed machine.

Equivalence relation

It is possible to define an equivalence relation on the Event-B actions, and by restriction on the Event-B assignments. Two actions are considered as being equivalent if the proof obligations generated for these actions are logically equivalent.

This relation is represented with the $\defi$ symbol.

Rewriting rules on Event-B assignments

As detailed in the modelling language, the Event-B assignments are formed of two parts:

• A left-hand side, which is a list of free identifiers.
• A right-hand side.

There are various kinds of assignments:

• The $\bcmsuch$ ("becomes such that") assignment is the most general (non-deterministic) assignment, where a predicate is given on the before and after values of assigned identifiers. The after values of the assigned identifiers are denoted by a primed identifier whose prefix is the assigned identifier.
• The $\bcmeq$ ("becomes equal to") assignment is the deterministic assignment where an expression is given for each assigned identifier.
• The $\bcmin$ ("becomes member of") assignment is the set-based (non-deterministic) assignment, where a set expression is given for the assigned identifier.

Let $v$ and $w$ be variables, and $E$ and $F$ be expressions. In the following table, the left-hand assignments are equivalent ($\defi$) to the right-hand ones (see the B-book):

$v \bcmeq E$	$v \bcmsuch v' = E$	Rule 1
$v(E) \bcmeq F$	$v \bcmeq v \ovl \{E \mapsto F\}$	Rule 2
$v \bcmin E$	$v \bcmsuch v' \in E$	Rule 3

Rewriting rules on Event-B actions

Let $v$ and $w$ be variables, $E$ and $F$ be expressions, and $P$ and $Q$ be predicates. The left-hand actions are equivalent ($\defi$) to the right-hand ones:

$\begin{array}{ll}v\!\!\! &\bcmsuch P(v,v')\\ w\!\!\! &\bcmsuch Q(w,w') \end{array}$	$v,w \bcmsuch P(v,v') \land Q(w,w')$	Rule 4
$v,w \bcmeq E,F$	$\begin{array}{ll}v\!\!\! &\bcmeq E\\ w\!\!\! &\bcmeq F \end{array}$	Rule 5

• Rule 4 could be more general since P can refer to other before variables than v, e.g. w or x /= v, w. Similarly for Q.

$\begin{array}{ll}v\!\!\! &\bcmsuch P(v,w,x,v')\\ w\!\!\! &\bcmsuch Q(w,v,x,w') \end{array}$

$v,w \bcmsuch P(v,w,x,v') \land Q(w,v,x,w')$

Rule 4

• I (Son) thinks this Rule 5 is better to include the Rule 4 so that there is only one action, and it could be a general form of Rule 1.

$v,w \bcmeq E,F$

$\begin{array}{ll}v,w \bcmsuch v'\!\!\! = E \land w'\!\!\! = F \end{array}$

Rule 5

Simplification rules on Event-B predicates

Let $x_i$, $y$ and $z$ be variables, and $P$ and $Q$ be predicates.

• Rule 6: If $P(x_1,...,x_n,y)~$ is equal to $y = Q(x_1,...,x_n)~$, then the $\exists y.P(x1,...,x_n,y)$ predicate is true, and it may be deleted in conjunctive predicates ($\land$) where it appears.
• Rule 7: The $(\exists z \qdot P(x_1,...,x_n,z) \land Q(y_1,...,y_m))$ predicate, where $z \notin \{x_1,...,x_n,y_1,...,y_m\}$, may be rewritten as $(\exists z \qdot P(x_1,...,x_n,z)) \land Q(y_1,...,y_m)$.
• Rule 8: The $\exists z \qdot P(x_1,...,x_n,z)$ predicate may be deleted in conjunctive predicates where it appears if the $y \bcmsuch P(x_1,...,x_n,y')$ assignment is among the actions of the initial event. It indeed is nothing else but the feasibility (FIS) proof obligation for such an assignment, and a model to be decomposed is assumed to be proved (see the section related to the proof obligations in the event model decomposition).

Example

Let $a$, $b$ and $x$ be variables, and $P$ and $Q$ be predicates.

$a,b \bcmsuch \exists x \qdot P(a,a',x) \land Q(b,b')$

$\defi$ (Rule 7)

$a,b \bcmsuch (\exists x \qdot P(a,a',x)) \land Q(b,b')$

$\defi$ (Rule 4)

$\begin{array}{ll}a\!\!\! &\bcmsuch \exists x \qdot P(a,a',x)\\ b\!\!\! &\bcmsuch Q(b,b')\end{array}$