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The first universal quantification states that every element in the 'small' segment is less than every element in the 'medium' segment, the second universal quantification that every element in the 'small' segment is less than every element in the 'large' segment, and the third universal quantification that every element in the 'medium' segment is less than every element in the 'large' segment. The second universal quantification is not implied by the other two in the case that the medium segment is empty. Take, for example, N to be 2 and all of 5, K and I to be 1. Let the array have elements a[0] = 20 and a[l] = 10. Then ( V i j : 0 < i < 5 A 5^ j<l A (ViJ : 0 < i < 5 A l^j<N : a[i]<a[j]) A (ViJ :s^i<l A l^j< is false (because a[0] ^ a[l] is false), whereas {ViJ : 0 ^ i < 5 A s^j<l A {ViJ : s^i<l A I ^j <N : is (vacuously) true. The first term (0 < 5 < K ^ I ^ N) implies, however, that the medium segment is indeed non-empty. So we have (ViJ :0^i<s /\ s^j <l A (V ij : s^i < I A l^j<N: a[i] <a[j]) { Q^s <K^l^N. In particular, 5 < I. So, use the one point rule with j = s in the first conjunct and i - s in the second conjunct. } A <Vj:i<j<AT:a[5]<a[j]> { transitivity of < } < V i j : 0 ^ i < 5 A I ^ j <N : a[i] <



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<class namespace="Foo" name="Bar" access="family"> <property name="Height" type="SystemString" access="public" modifier="sealed" /> </class>

expression within the for loop corresponds to a statement (It is easy to remember that the separation character between expressions is a semicolon and not a comma, because each expression could be a statement) You write a for loop generically as follows:

iteration In Listing 343, it declares and initializes the variable count The initial expression does not have to be a declaration of a new variable It is possible, for example, to declare the variable beforehand and simply initialize it in the for loop Variables declared here, however, are bound within the scope of the for statement

Solution 14.3. An appropriate invariant property is A {ViJ : 0 ^ i < 5 A 5^ j <N : A (Vij : 0^i<iA The assignment

The boolean-expression portion of the for loop specifies an end con-





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Here s a matching XSLT stylesheet to transform the class design DSL into C# code:

dition The loop exits when this condition is false in a manner similar to the while loop s termination The for loop will repeat only as long as boolean-expression evaluates to true In the preceding example, the loop exits when count increments to 64

Solutions to Exercises initializes the 'small' and 'large' segments to the empty set and, so, the invariant is vacuously true. The bound function we use is l-s, the size of the 'medium' segment. The termination condition is s = K so that the loop body is executed when 5 < K. In order to make progress to the termination condition whilst maintaining the invariant, we again choose a[K-l] as borderline value, recording it in some local variable, X. (The justification for this choice is unchanged from the earlier algorithm.) The algorithm we are aiming to develop thus has the basic structure shown below:

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example, count++ executes after the right shift of the mask (mask >>= 1), but before the Boolean expression is evaluated During the sixtyfourth iteration, count increments to 64, causing boolean-expression to be false and, therefore, terminating the loop Because each expression may be thought of as a separate statement, each expression in the for loop is separated by a semicolon

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the conditional expression remains true If you wrote out each for loop execution step in pseudocode without using a for loop expression, it would look like this: 1 Declare and initialize count to 0 2 Verify that count is less than 64

,l := 0,N ;

3 Calculate bit and display it 4 Shift the mask 5 Increment count by one 6 If count<64, then jump back to line 3 The for statement doesn t require any of the elements between parentheses for(;;){ } is perfectly valid; although there still needs to be a means to escape from the loop to avoid executing infinitely Similarly, the initial and loop expressions can be a complex expression involving multiple subexpressions, as shown in Listing 344

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