Write an implementation for the Prime machine (de?ned at the end of the assignment sheet). Provide a suitable invariant and variant for any loop you use. Provide comments in your machine stating the precondition and postcondition which the initialised loop meets.

An abstract machine, Bu?er is de?ned at the end of this assignment sheet. Provide a re?nement of this machine which stores the queue in an array:

aa : 1 . . qlen → ELEM

When elements are removed from the front of the queue, the array itself is not altered but the positions become available for overwriting. When adding elements, once the end of aa is reached, wrap-around can occur if positions at the front of aa are available. Thus forexample, if the capacity is 8, the sequence [a, b, c, d] may correspond to:

etc (where * indicates any element).

To do this you will need to introduce some further state variables to keep track of the situation in the array and to enable you to relate the state of aa to the state of bu? . Provide the linking invariant which captures this invariant and incorporate it in a re?nement machine BufferR.