#### 34.9.7 Execution of Checking Indexicals

Consider the definition of a constraint C containing a checking
indexical `X in R`

. Let \FV(X,C,S) denote the
set of values for X that can make C false in some
ground extension of the store S. Then the indexical should
obey the following coding rules:

- all arguments of C except X should occur in R
- if R is ground in S, S(R) = \TV(X,C,S)

If the coding rules are observed, S(R) can be proven to exclude
\FV(X,C,S) for all stores in which R is anti-monotone.
Hence it is natural for the implementation to wait until R
becomes anti-monotone before admitting the checking indexical for
execution. The execution of `X in R`

thus involves
the following:

- If D(X,S) is contained in S(R), none of the possible values for
X can make C false, and so C is detected as entailed.
- Otherwise, if D(X,S) is disjoint from S(R) and R is
ground in S, all possible values for X will make
C false, and so C is detected as disentailed.
- Otherwise, D(X,S) contains some values that could make C true
and some that could make C false, and the indexical suspends.

A checking indexical is scheduled for execution as follows:

- it is evaluated initially as soon as it has become anti-monotone
- it is re-evaluated when one of the following conditions occurs:
- the domain of X has been pruned, or X has been assigned
- the domain of a variable Y that occurs as
`dom(Y)`

or `card(Y)`

in R has been pruned
- the lower bound of a variable Y that occurs as
`min(Y)`

in R has been increased
- the upper bound of a variable Y that occurs as
`max(Y)`

in R has been decreased