Pre and Post Condition Rules

1
Pre and Post Condition Rules

• Definition If R and S are two assertions, then
  R is said to be stronger than S if R gt S (R
  implies S).
• Example
• the assertion i lt 0 is stronger than the
  assertion i lt 1 because i lt 0 implies that i lt
  1
• Note that if R is stronger than S, then all
  states that satisfy R will satisfy S. But there
  is at least one state that satisfies S which will
  not satisfy R. So the number of states that
  satisfies S is larger than that of R. Thus one
  may view the notion of stronger as more
  selective because less states will satisfy the
  stronger condition.
• Of course if R is stronger than S, then S is
  weaker than R

2
Precondition Strengthening

• If P is stronger than P and if PCQ is
  correct, then with the strengthened precondition
  assertion the PCQ is correct
• Example
• if PCQ is correct, for P which is (xgt0), then
  for P, which asserts (xgt2), the triple PCQ
  will also be correct.
• More formally
• P gt P
• PCQ
• PCQ
• Example if x lt5 x x1 x lt 6 is correct,
  then strengthening P to x lt 3 should also
  give us xlt3 xx1 xlt6 as also correct
  because
• xlt3 gt xlt5
• xlt5 x x1 xlt6
• xlt3 x x1 xlt6

3
Post Condition Weakening

• If Q gt Q and PCQ is correct then PCQ
  is correct
• Formally we have
• PCQ
• Q gt Q
• PCQ
• Example
• if max b maxb then show maxb
  max gt b
• max b max b
• max b gt max gt b
• maxb max gtb

4
Conjunction (AND) and Disjunction (OR) Rules

• If C is a piece of code, PCQ AND PCQ
  (note that both of the conditions have to be True
  simlutaneously), then P
  AND P C Q AND Q
• Formally
• P C Q
• P C Q
• P AND P C Q AND Q
• If C is a piece of code, PCQ AND PCQ,
  then P OR P C Q OR Q
• Formally
• P C Q
• P C Q
• P OR P C Q OR Q

5
Example with Conjunction Rule

• Problem given the following Hoare Triples
• x x1 x x1 and
• xgt0 x x 1 x gt 0
• show that xgt0 xx1 xgt-1
• Proof (a little more detailed than needed)
• a) using conjunction rule, we get xgt0 x x
  1xx1 AND xgt0
• b) using the weakening the post-condition rule,
  we have xx1 and xgt0 gt xgt0, and from second
  triple xgt0 xx1 xx1 AND xgt0 also imply
  xgt0 xx1 xgt0 by weakening the post
  condition
• Furthermore x gt 0 gt x gt - 1
• Therefore we have xgt0 x x1 xgt-1
• Alternatively note that the weakening of post
  condition can be achieved through dropping xx1
  and also directly weakening xgt0 to xgt-1.