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push = %(x,s).(x:RANGE & s:Stack | s \/ {card(s)+1|->x}) & | push = %(x,s).(x:RANGE & s:Stack | s \/ {card(s)+1|->x}) & | ||

pop: Stack <-> Stack & | pop: Stack <-> Stack & | ||

pop = %s.(s:Stack| {card(s)} <<| s) & | pop = %s.(s:Stack\ {empty} | {card(s)} <<| s) & | ||

tops: Stack <-> RANGE & | tops: Stack <-> RANGE & | ||

tops = %s.(s:Stack| s(card(s))) | tops = %s.(s:Stack\ {empty} | s(card(s))) | ||

ASSERTIONS | ASSERTIONS | ||

/* the assertions can now be checked by ProB, they used to trigger | |||

the expansion of the infinite functions above */ | |||

tops: Stack \ {empty} --> RANGE; | tops: Stack \ {empty} --> RANGE; | ||

push: (RANGE*Stack) --> Stack \ {empty}; | push: (RANGE*Stack) --> Stack \ {empty}; |

This tutorial describes how to model (and how not to model) infinite datatypes so that they can be animated with ProB. (You may also want to look at the manual page on recursive functions.) We illustrate this by modeling a Stack datatype.

Unfortunately, the following B machine does not work with ProB:

MACHINE StackAxioms1 SETS Stack CONSTANTS pop, push,empty PROPERTIES empty:Stack & pop : Stack \ {empty} --> Stack & !s,x . (s : Stack & x : NAT => pop(push(s |-> x)) = s) & push: Stack*NAT --> Stack \ {empty} END

The problem with the above model is that the `PROPERTIES` can only be satisfied if `Stack` is infinite. However, ProB does not presently support infinite deferred sets.
We need to do a more constructive definition and use the infinite set ProB knows about: `INTEGER`. This is done in the following machine, which is included in the `examples/Tutorial` directory of the ProB distribution.

MACHINE StackConstructive /* A machine that shows how to model a stack of type RANGE so that it can be animated and validated with ProB */ /* We could have used the sequence operations instead; our intention was also to show how you can model this in Event-B */ DEFINITIONS Stack == (INTEGER <-> RANGE) SETS RANGE CONSTANTS empty, push, pop, tops PROPERTIES empty : Stack & empty = {} & push : (RANGE * Stack) <-> Stack & push = %(x,s).(x:RANGE & s:Stack | s \/ {card(s)+1|->x}) & pop: Stack <-> Stack & pop = %s.(s:Stack\ {empty} | {card(s)} <<| s) & tops: Stack <-> RANGE & tops = %s.(s:Stack\ {empty} | s(card(s))) ASSERTIONS

/* the assertions can now be checked by ProB, they used to trigger

the expansion of the infinite functions above */ tops: Stack \ {empty} --> RANGE; push: (RANGE*Stack) --> Stack \ {empty}; pop: Stack \ {empty} --> Stack VARIABLES cur INVARIANT cur: Stack & cur : seq(RANGE) /* a slightly stronger invariant */ INITIALISATION cur := empty OPERATIONS Push(yy) = PRE yy:RANGE THEN cur:= push(yy,cur) END; Pop = PRE cur /= empty THEN cur := pop(cur) END; t <-- Top = PRE cur /= empty THEN t := tops(cur) END END

As the screenshot illustrates, this model can now be animated with ProB. Observe how the functions push, pop and tops are kept symbolic in the State View.