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By J.N.P. Hume, R.C. Holt

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Example text

Also note that the rule only holds if P is stronger than QŒxnE. If this is not the case, the rule may not be invoked! P; S; Q/, then we should seek out some intermediate predicate M that we believe to be more easily attainable from the precondition state, P . This intermediate state then serves as a sort of half-way house from which to find some other code that will enable us to arrive at the required postcondition, Q. 4 Refinement Rules 39 The rule is best stated in Hoare triple notation, but relying on the extended notation previously introduced.

But a little thought will convince otherwise. xC1 > 5/ should hold. x > 5/Œxnx C 1. e. x > 4/, the weakest precondition. x > 4/, the weakest precondition, etc. x > 4/. 3 Guarded Command Language 25 So, counter-intuitive as it may initially be, the precondition given above for assignment does indeed make sense. Readers who are not fully convinced should verify this for a variety of other scenarios. This will confirm that assignment’s precondition is as stated. Here we have started off with a notion of what assignment means, and proposed and verified its weakest precondition.

7 This means that if a loop has a variant, then the loop will definitely terminate eventually. As an example, consider the following (rather trivial) loop. i > 0/ ! i WD i 1 od A variant for this loop is the expression i . This is an expression in one of the variables in the loop; it starts with value 10; and it always decreases (by 1) in every iteration. Because of the loop’s guard, it is also bounded below by 0—the value of the variant can never legitimately be less than 0 in the loop. As a result, we can confidently state that the loop will terminate.

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