# Rewriting rules for event model decomposition

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The purpose of this page is to list and justify the transformation 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 and simplification rules

It is first necessary to introduce some equivalence and simplification rules on Event-B assignments and predicates. These rules will then help to understand the transformation rules to be applied to build the actions of the external events.

### 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.

### Equivalence 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

### Equivalence 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

Note 1: The $P$ predicate can refer to other before variables than $v$ (e.g. $w$ or $u$, where $u \notin \{v,w\}$). Similarly, $Q$ can refer to other variables than $w$.

Note 2: The following equivalence is obtained by enforcing the rules 1 and 4, and is a good replacement for the rule 5:

 $v,w \bcmeq E,F$ $\begin{array}{ll}v,w \bcmsuch v'\!\!\! = E \land w'\!\!\! = F \end{array}$ 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) $\begin{array}{ll}a\!\!\! &\bcmsuch \exists x \qdot P(a,a',x)\\ b\!\!\! &\bcmsuch Q(b,b')\end{array}$

## Transformation rules

The transformation from a given Event-B action of a sub-machine $M_i$ to another action of a sub-machine $M_j$ is subsequently be represented with the $\rightsquigarrow$ symbol. $s$ and $t$ are assumed to be variables shared between $M_i$ and $M_j$, $v$ and $w$ other variables, $E$ and $F$ expressions, and $P$ and $Q$ before-after predicates.
Let's first establish a transformation rule for generic Event-B assignments. It is then possible to deduce transformation rules for other assignments of the Event-B language.

### Generic transformation rule on Event-B assignments

The generic transformation rule on Event-B assignments is defined below:

 $v,s \bcmeq P(s,s',v,v')$ $s \bcmsuch \exists v' \qdot P(s,s',v,v')$ Rule 9

### Derived transformation rules on Event-B assignments

The transformation rules for other Event-B assignments are obtained by applying the generic rule and the equivalence / simplification rules previously introduced. More precisely:

1. The equivalence rules 1 to 5 shall be first applied as many times as possible, from left to right, to get the assignment into the generic form.
2. Then, generic transformation rule 9 shall be enforced.
3. Then, the simplification rules 6 to 8 shall be enforced.
4. Finally, the equivalence rules 1 to 3 shall be applied, from right to left. The proof obligations generated for deterministic actions are indeed more suitable than those generated for non-deterministic actions. In the same manner, for a given set $S$, proving that $\exists x \qdot x \in S$ (FIS proof obligation generated from $x \bcmsuch x' \in S$) is indeed not as "simple" as proving that $S \neq \emptyset$ (proof obligation generated from $x \bcmin S$).

#### Example

 $s,v \bcmsuch E,F$ $\defi$ (Rule 5) $s,v \bcmsuch s' = E \land v' = F$ $\rightsquigarrow$ (Rule 9) $s \bcmsuch (\exists v' \qdot s' = E \land v' = F)$ $\defi$ (Rule 7) $s \bcmsuch s' = E \land (\exists v' \qdot v' = F)$ $\defi$ (Rule 6) $s \bcmsuch s' = E$ $\defi$ (Rule 1) $s \bcmeq E$

#### Formalization

The derived transformation rules on Event-B assignments are listed below:

 $s \bcmeq E$ $s \bcmeq E$ Rule 10 $v \bcmeq E$ (empty) Rule 11 $s \bcmin E$ $s \bcmin E$ Rule 12 $v \bcmin E$ (empty) Rule 13 $s(E) \bcmeq F$ $s(E) \bcmeq F$ Rule 14 $v(E) \bcmeq F$ (empty) Rule 15 $s,v \bcmsuch E,F$ $s \bcmeq E$ Rule 16

### Derived transformation rules on Event-B actions

The transformation can be done separately for each assignment of an Event-B action, as demonstrated below:

 $\begin{array}{ll}s,v\!\!\! &\bcmsuch P(s,s',v,v')\\ t,w \!\!\! &\bcmsuch Q(t,t',w,w')\end{array}$ $\defi$ (Rule 4) $s,v,t,w \bcmsuch P(s,s',v,v') \land Q(t,t',w,w')$ $\rightsquigarrow$ (Rule 9) $s,t \bcmsuch \exists v',w' \qdot P(s,s',v,v') \land Q(t,t',w,w')$ $\defi$ (Rule 7) $s,t \bcmsuch (\exists v' \qdot P(s,s',v,v')) \land (\exists w' \qdot Q(t,t',w,w'))$ $\defi$ (Rule 4) $\begin{array}{ll}s\!\!\! &\bcmsuch \exists v' \qdot P(s,s',v,v')\\ t\!\!\! &\bcmsuch \exists w' \qdot Q(t,t',w,w')\end{array}$