CS 641 meeting -*- Outline -*-

* semantics of recursion (4.4)

	We now will give the main definition of the book!

** the setup
	Let A = disjoint union of S and H.

*** simple commands (S)
	S is set of simple commands, assumed to contain all guards and
		assignments.
	wp and wlp defined on them as before.

	notation:
		ws_0: S -> MT,	 for wp
		ws_1: S -> MT,	 for wlp

*** procedure names (H)
	H is set of procedure names.

	body: H -> A(.)

*** the problem
--------------------------
      THE PROBLEM

Want to construct
	wp, wlp: A(.) -> MT
such that
	wp|S = ws_0
	wlp|S = ws_1,
and for all h \in H,
	wp.h = wp.(body.h)
	wlp.h = wlp.(body.h)
---------------------------
	Then we would also want to verify that:
		the homomorphism properties are satisfied,
		and the healthiness laws hold

** construction
*** overview
-----------------------
   CONSTRUCTION OVERVIEW

for e \in {0,1},
 ws_e is extended to A -> MT
 as (ws_e \cup v)
 for a v \in H -> MT to be selected.

Then use its language extension:
	(ws_e \cup v)(.) \in A(.) -> MT
-----------------------
	It remains to select v

*** fixpoints
	we find the v above as a fixpoints (least and greatest)
	want to satisfy wg.h = wg.(body.h), so we calculate
		for e \in {0,1}

	   (all h \in H :: (ws_e \cup v)(.).h = (ws_e \cup v)(.).(body.h))
	=	{def of language extension}
	   (all h \in H :: v.h = (ws_e \cup v)(.).(body.h))
	=	{def of composition}
	   (all h \in H :: v.h = ((ws_e \cup v)(.) o body).h)
	=	{Liebniz}
	   v = (ws_e \cup v)(.) o body

	Now we abstract out v from the rhs, by defining
		D_e: (H -> MT) -> (H -> MT)
		D_e.v = (ws_e \cup v)(.) o body

	and this lets us write a fixpoint equation:
		v:  v = D_e.v

	Q: Can we find fixpoints of D_e?
		Yes, MT is a complete lattice, and so (H -> MT) is a complete
			lattice in the induced order.
		Remains to show that D_e is monotone...

	Thm: Function D_e: (H -> MT) -> (H -> MT) defined above
		is monotone, and has a least fixpoint wa_e
		and greatest fixpoint wb_e in (H -> MT).

	Pf: By Knaster-Tarski theorem, suffies to show D_e is monotone.
		Let v, w \in (H -> MT).

	    D_e.v <= D_e.w
	=	{induced order on H -> MT, definition of D_e}
	    (all h \in H ::
		(ws_e \cup v)(.).(body.h) <= (ws_e \cup w)(.).(body.h))
	<==	{r must be in A(.), definition of language extension}
	    (all r \in A(.) ::
		(inf s \in r :: (ws_e \cup v)*.s)
			<= (inf s \in r :: (ws_e \cup w)*.s))
	<==	{if s \in r and r \in A(.), then s \in A*; inf is monotone}
	    (all s \in A* :: (ws_e \cup v)*.s <= (ws_e \cup w)*.s)

	It remains to show that this last is implied by v <= w.
	This is done by induction on length of strings s.
	QED

-------------------
   DEFINITIONS OF WP AND WLP

Using the theorem.

def: wp = (ws_0 \cup wa_0)(.)
     wlp = (ws_1 \cup wb_1)(.)
-------------------
	so wp is language extension of ws_0 by least fixpoint of D_0
	and wlp is language extension of ws_1 by greatest fixpoint of D_1

	By the first calculation, these satisfy the requirements.

*** auxiliary definitions
--------------------
    SOME NOTATION

def: Let v : (H -> MT), and e \in {0,1}.
     Then v^e : A(.) -> MT is defined by
	v^e = (ws_e \cup v)(.)

Thus
	D_e.v = v^e o body
	wp = (wa_0)^0
	wlp = (wb_1)^1
--------------------

** healthiness laws
*** wlp is universally conjunctive (4.5)
	want this to be a theorem now

	Q: how would you go about that?

	Assume: ws_1 \in S -> MU

	Thm: Suppose ws_1 \in S -> MU.  Then wlp \in A(.) -> MU.
	Pf:  using the Knaster-Tarski theorem it suffices to prove:
		1. (H -> MU) is inf-closed in (H -> MT).
		2. (H -> MU) is D_1-invariant.

	Q: can you prove (1)?

	For (2) Let v \in (H -> MT).
	Need to show: v \in (H -> MU) ==> D_1.v \in (H -> MU).

	    D_1.v \in MU
	=	{def of D_1}
	    (ws_1 \cup v)(.) o body \in (H -> MU)
	<==	{types of language extension and body}
	    (ws_1 \cup v)(.) \in A(.) -> MU
	<==	{lemma 17 in section 4.3}
	    (ws_1 \cup v) \in A -> MU
	<==	{A is disjoint union of S and H, assumption ws_1 \in S -> MU}
	    v \in (H -> MU)

	So by Knaster-Tarski theorem, the greatest fixpoint, wb_1,
	of D_1 in (H -> MT) is an element of (H -> MU):
		wb_1 \in (H -> MU)

	By the above calculation, note that,
		v \in (H -> MU)  ==> v^1 \in (A(.) -> MU)

	Since wlp = (wb_1)^1, the result follows.
	QED

	Q: why doesn't this proof work to show wp is universally conjunctive?
		You'd have to use "MU is sup-closed" for the analogous argument
		but MU isn't sup-closed in MT.

*** termination law (4.6)

	Thm: Suppose
		(all s \in S, p \in |P :: ws_0.s.p = ws_0.s.true /\ ws_1.s.p).
	Then for all r \in A(.) and p \in |P,
		wp.r.p = wp.r.true /\ wlp.r.p.

	Pf: Let WT \subseteq (H -> MT) be defined by
		v \in WT equiv (all h \in H, p \in |P ::
				v.h.p = v.h.true /\ wlp.h.p)

	To prove wp \in WT, it suffices to show that
		1. (all v \in WT, r \in A(.), p \in |P ::
				v^0.r.p = v^0.r.true /\ wlp.r.p)
		2. WT is D_0-invariant
		3. WT is sup-closed in (H -> MT).
	Once this is done, by (2,3) and the Knaster-Tarski theorem,
		the least fixpoint of D_0 in (H -> MT), wa_0,
		is such that
			wa_0 \in WT
	and thus by (1) wp = (wa_0)^0 satisfies the termination law

	Q: how would you prove (1)?
		let v be given, proceed by induction on structure of A(.)
		Hesselink defines a subset of A(.) called K that satisfies
			the result, then shows this is all of K
			by showing that A(.) is a subset of K.

	Q: how would you prove (2)?
		calculate
	Q: how would you prove (3)?
		What does it mean to be sup-closed?

	QED