Homework set 2: Formal program semantics: what's it all mean, anyway?

Ex i.j denotes exercise j at end of chapter i.
Ex i.j.k denotes exercise k at end of chapter i, section j.

1. Deduce Laws (7.5) and (7.6) using (7.3) and (7.4) and the basic laws in chapter 2. Do not use the definition of wp in (7.1). (This exercise is a slight modification of Ex 7.4.

   Suggestion: Since S is held fixed in all these laws, you can use this abbreviation: W(R) := wp(S, R).

   Hints:

   1. It will be helpful to first prove this meta-theorem: For any two predicates P and Q, (P => Q) holds if and only if (P = P ^ Q) holds. Then use substitution inferences from it (in both directions).
   2. Remember that our proof system only proves equalities, so the statements to prove are
      • (W(Q) => W(R)) = T, for Monotonicity, (7.5), and
      • (W(Q) v W(R) => W(Q v R)) = T, for Distributivity of Disjunction, (7.6).

      Thus, derive the left hand side, T, from the right hand side by a series of equivalences.

   3. In the case of Monotonicity, somewhere along the way, you use the hypothesis Q => R, or what is the same use Q = Q ^ R, just as you would an axiom.
2. Ex 7.5
3. Ex 8.2
4. Ex 8.5
5. Ex 9.2.1 (Hint: use section 4.4)
6. Ex 9.2.2