Pre- and postconditions are predicates. Both tell us something about the
values of the variables of our programs. For example, `n = 1' and
`list is nonempty' are such predicates. We use
{ P } S { Q }
to specify that
if the variables have values such that the predicate
P is satisfied at the start of the execution of
statement S
and the execution of the statement S terminates
then at termination the variables have values such that the predicate
Q is satisfied.
For example,
{ n = 0 } n = n + 1; { n = 1 }
and
{ true } list.insertLast("x");
{ list is nonempty }
A loop invariant is also a predicate. It tells us something about the
values of the variables while executing a loop. This predicate should be
satisfied at the beginning of every iteration of the loop. Consider the
loop
while B do S
with loop invariant LI. If
{ LI 
B }
S { LI },
that is, statement S preserves the loop invariant LI
within the loop, then we can conclude that
{ LI } while B do S
{ LI
B },
that is, if the loop invariant LI is satisfied at
the start of the execution of the loop and the loop terminates then the
loop invariant LI (and of course also
B) is satisfied at termination.
Bubble-sort
Algorithm bubbleSort(sequence):
Input: sequence of integers sequence
Postcondition: sequence is sorted and
contains the same integers as the original sequence
length 
length of sequence
for i 0 to length - 1
do
for j 0 to length -
i - 2 do
if jth element of sequence > (j+1)th element of
sequence then
swap jth and (j+1)th element of sequence
A loop invariant for the outer loop is: the last i elements of sequence are sorted and are all greater than or equal to the other elements of sequence. A loop invariant for the inner loop is: the last i elements of sequence are sorted and are all greater than or equal to the other elements of sequence and the jth element of sequence is greater than or equal to the first j elements of sequence.