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By Martha Garcia-Murillo

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The author and P. , 7] in his "specification logic," a version of Hoare's logic that allows modular reasoning about programs in Algol-like languages with higher-order procedures. *Supported by research grants from the Natural Sciencesand Engineering Research Council of Canada and the Information Technology Research Centre of Ontario. 35 An essential technique is to allow "changes" of world whose only effect is to constrain state transformations to be compatible with some pre-ordering on the set of states.

5 L e m m a (D,D---~D oyalg. + distributivity principal ideals ~ D has property I3) Suppose D and D--~D are ca-algebraic cpo^. Then (1) If d e D then Sd is an ca-algebraic lattice. (2) If de D o and $d is distributive then Sd is finite. (3) If for each de D o $d is distributive then D has property I. Proof (1) Consider the cpo^ Sd with the order induced by D. Show: ($d) o = Sdc~D o. Next we show that any two compact elements have a lub in Sd. This is enough to guarantee the existence of arbitrary lubs and therefore of arbitrary glbs.

7 we k n o w --~(ySy'). Let S~Y such that y e S and y'~S. There are uncountably m a n y of such S. (e_

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