Accumulators, Examples of proving w/accumulators

Plans

Rest of Sem. Sched.

Outline next project, due dates

Project presentations

Next class? Next, next class!

Discussed earlier plans. Those plans upon us. I can’t imagine enjoying a final exam more.

So meeting w/folk re: these.

Accumulators + Reasoning w/Accumulators

Additional documentation

Recap what we saw w/accumulators on Last Mon.

Defining functions w/accs

1. Start with a function f.
2. Define ft, a tail-recursive version of f with an accumulator.
3. Define f*, a non-recursive function that calls ft and is logically equivalent to f, i.e., the following is a theorem hyps ⇒ (f* ...) = (f ...)

Proofs w/, wrt/ accumulator fns.

1. Identify a lemma that relates ft to f. It should have the following form: hyps ⇒ (ft ... acc) = ... (f ...) ... Remember that you have to generalize, so all arguments to ft should be variables (no constants). The RHS should include acc.

2. Assuming that the lemma in 4 is true, and using only equational reasoning, prove the main theorem hyps ⇒ (f* ...) = (f ... )

   If you have to prove lemmas, prove them later. (Like an adult!)

3. Prove the lemma in 4. Use the induction scheme of ft.

4. Prove any remaining lemmas. (Like an adult!)

Example

(definec powof3 (b)
  (if (zp b) 1
    (* 3 (powof3 (1- b)))))

(definec powof32h (b)
  (powof32h b 1))

(definec powof32h (b c)
  (if (zp b) c
   (powof32h (1- b) (* 3 c))))

The lemma of 4

(pow32 b c) = (* (pow3 b) c)

The main theorem