#### Reading material

Pages 211-214 (Section 5.4) and pages 109-110 (Section 3.4.3)
#### Loop invariants

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.

Consider the following Java snippet.

int i = 1;
int x = 1;
while (i < N)
{
x *= i + 1;
i++;
}

For the above loop, the assertion x = i! i <= N is a
loop invariant.
**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*

*i* 0

**while** *i* < *length* **do**

*j* 0

**while** *j* < *length* - 1 **do**

**if** *j*th element of *sequence* > (*j*+1)th element of
*sequence* **then**

swap *j*th and (*j*+1)th element of *sequence*

The outer loop has loop invariant

(A *k* : *length* - *i* <= *k* < *length* :
(A *l* : *k* < *l* < *length* :
*k*th element of *sequence* <= *l*th element of *sequence*))

and the inner loop has loop invariant

(A *k* : *length* - *i* <= *k* < *length* :
(A *l* : *k* < *l* < *length* :
*k*th element of *sequence* <= *l*th element of *sequence*))

(A *k* : 0 <= *k* < *j* :
*k*th element of *sequence* <= *j*th element of *sequence*)

#### Question

Consider the following main method.
public static void main(String[] args)
{
int lo = 1;
int hi = 1;
int n = Integer.parseInt(args[0]);
int i = 1;
while (i < n)
{
int temp = hi;
hi = lo + hi;
lo = temp;
i++;
}
System.out.println(hi);
}

Try to come up with a loop invariant.