CS 641 meeting -*- Outline -*-

* developing loops, an introduction (Cohen's chapter 8)

	The key idea is figuring out what the predicates are
		so can go back to calculating the loopless parts

** before and after
*** other forms of postconditions
	Note that the Invariance theorem requires postconditions to have
		a special form.

	Can use loops when have postconditions in other forms,
		by rule of consequence
	I.e., have to choose P and B to satisfy  P /\ ~B  ==>  R
		(this requires creativity/experience,
			both to see to use a loop,
			and to choose appropriate P and /\ ~B)
*** establishing the invariant
	The precondition may not be the same as the invariant.
	Can use the rule of consequence again,
		but in general need some intitialization code.

*** summary
-----------------
  FORM OF TYPICAL LOOP {PROOF}

 {Q}
 S0
 {invariant: P}
 ; do B -> S od
 {P /\ ~B, hence R}

  STEPS IN DEVELOPMENT OF LOOP

0. choose invariant P and guard B
1. find way to establish P		(develop {Q} S0 {P})
---------------------------
	0. is creative, but we'll give heuristics
	1. we covered earlier, if can do it with loopless program

** the loop body
	idea: use invariance theorem

	Recall:
---------------------
Invariance Thm:
 {P} do B -> S od {P /\ ~B}
  <==  {P /\ B} S {P}	      (Invariance)
    /\ {P /\ B /\ t=T} S {t<T}  (Progress)
    /\ P /\ t<=0 ==> ~B      (Boundedness)
----------------------
	so far we've chosen P and B, so spec of S is complete
	need to choose t.  How?  So that repeated decrease will falsify B

	example:
		if P is n >= 10 and guard B is ~(n = 10),
			can falsify B by choosing t = (n - 10),
			because decreasing t will approach making B false

	Q: what would you choose for t if P is 0 <= i and B is i > 1?

	So last steps are:
--------------------
   FINISHING STEPS

2. choose bound function, t, such that
	P /\ (t <= 0) ==> ~B
3. develop loop body S to satisfy
	invariance and progress
--------------------
	2. is creative in general
	3. is a development again

** remarks
	Invariant is crucial, and the step requiring the most
		creativity and experience
	But will try to codify this based on the shape of the postcondition
		will see heuristics "deleting a conjunct"
				and "replaacing constants by fresh variables"