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When a change of data representation, as from bool to Z in class Flag, is involved, we have to resort to different notions of refinement and equivalence and their associated laws. The equivalence notion of class declarations used so far, cdsa =cds,c cdsc, is just an abbreviation for the program equivalence cdsa cds • c = cdsc cds • c, which simply corresponds to program refinement in both directions: cdsa cds • c cdsc cds • c and cdsc cds • c cdsa cds • c Similarly, cdsa =cds,c cdsc corresponds to cdsa cds,c cdsc and cdsc which are abbreviations for the program refinements just presented.

A. If a is declared in B , then the cast is not necessary for compilation, but once it is there, it cannot simply be eliminated, because a cast also has a run time effect. At run time, if the value of a cast expression does not have the required type, its evaluation results in error, and the command in which it appears aborts. In the previous example, if the attribute a is in class B , although the cast could be eliminated regarding its static effect, it still has a dynamic effect when the object value of x happens to be of type B , but not of type C .

The next two laws allow us to change the type of a parameter; they are similar to Law 21. The first law handles value parameters. Law 25 (change value parameter type). class C extends D ads meth m = val x : T ; pds ops end • b =cds,c class C extends D ads meth m = val x : T ; pds • b ops end provided (↔) T ≤ T and every non-assignable occurrence of x in expressions of b are cast with T or any subtype of T ; (←) (1) every actual parameter associated with x in ops, cds and c is of type T or any subtype of it; (2) every expression assigned to x in b is of type T or any subtype of T ; (3) every use of x as result argument in b is for a corresponding formal parameter of type T or any subtype of T .

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