CS 641 meeting -*- Outline -*-

* calculational proofs and notation (Cohen, Ch. 1)

	See Also Dijkstra & Scholten, chapter 4, and Cohen's chapter 6

** shapes of calculations

-------------------
  BASIC SHAPE OF CALCULATIONAL PROOF

  X
=   < hint why X = true >
  true
--------------------

	Other formats
------------------
  TO PROVE X = Z

  X
=   < hint why X = Y >
  Y
=   < hint why Y = Z >
  Z
------------------
	this is valid because = is transitive: (X=Y) /\ (Y=Z)  ==> X=Z

	format saves writing compared to
		  X = Z
		=   < hint why Y = Z >
		  X = Y
		=   < hint why X = Y >
		  X = X
		=   < = is reflexive >
		  true
	as don't have to write X out each time
	note also that you get more control over the order

-------------------
  TO PROVE A ==> C

    A
=     < hint why A = B >
    B
==>   < hint why B ==> C >
    C
-------------------
	this is valid because (A=B) /\ (B ==> C) ==> (A ==> C)

	format valid for any transtive relation,
		for example with implication (==>) and consequence (<==),
		provided they all go in the same direction!

--------------------
  PROOFS UNDER ASSUMPTIONS
   e.g., to prove E ==> (A = D)

  A
=   < hint why A = B >
  B
=   < hint why E ==> (B = C) >
  C
=   < hint why C = D >
  D
--------------------
	valid because (E ==> (A=B /\ B=C /\ C=D)) ==> (E ==> A=D)

	this is not being sneaky, but is an abbreviation.
		often useful when E is some property
		that allows some manipulation

	Also ok to prove universals by skolemization.

-----------------------
   EXAMPLE CALCULATIONAL PROOF

def: f is conjuctive =
 (all X,Y:: f.(X /\ Y) = (f.X /\ f.Y))
def: f is monotonic =
 (all P,Q:: (P ==> Q) ==> (f.P ==> f.Q))

Thm.  a conjuctive f is monotonic.
Proof: observe for any f, P, and Q

   f.P ==> f.Q
=    < predicate calculus >
   (f.P /\ f.Q) = f.P
=    < f is conjunctive >
   f.(P /\ Q) = f.P
<==  < Leibniz >
   P /\ Q = P
=    < Predicate calculus >
   P ==> Q
-----------------------
	Note this uses the consequence (<==) to start with the more
		complex side.
	we'll see the equivalence between P /\ Q = P  and P ==> Q later.

** heuristic (p. 15)
	start with the more complex side and simplify
	less rabbits generally are pulled out of the hat that way
** calculations vs. appeals to intuition (p. 19)
	calculations are ideally syntactic manipulations
		based on syntactic laws not "meaning" of formulas
	Q: does this mean that formal models can't be used in proofs?
		no, but then it isn't what Cohen would call a calculation
		certainly don't want to appeal to intuition

** notation
*** three dots "..." (section 1.6)
	Q: Why does Cohen say to do away with the 3 dots?
		ambiguous, verbose, doesn't allow substitution
*** naturals (1.7)
	start with 0
*** function (1.8)
	application indicated by infix dot; functions curried
		f.x.y instead of f(x,y)
	domain of partial function
		f(i:1<=i<9)
			here i is a dummy or bound variable
			variables that aren't dummies are free
			need to avoid capture of free variables...

** exercises
	have students show their work on exercises from Cohen's chapter 1