## 3 Answers

**Loop invariant** is some condition which holds at the end of each iteration of the loop. i.e. it is* "invariant"* => does not vary (or change). It might change inside one iteration, but it will be true at the end of every iteration.

We often use loop invariants to prove that our algorithm works correctly.

In the given program, a loop invariant is:

i.e. **i is even after every iteration**.

One can verify this as follows:

- Before the execution of first iteration the loop invariant is true, because of this line of code:
if i mod 2 = 0 then

- In every iteration, we divide i by $2$, so now i will be either odd or even.
- If odd, we subtract 1 from i
if i mod 2 < > 0 then i := i - 1;

- so it's now even.
- otherwise, if even, we subtract 2 from i
else i := i – 2;

- so, it remains even

- If odd, we subtract 1 from i
- So, at the end of every iteration
*i remains even.*

https://stackoverflow.com/questions/3221577/what-is-a-loop-invariant

### 3 Comments

A loop invariant is a condition [among program variables] that is necessarily true immediately before and immediately after each iteration of a loop. (Note that this says nothing about its truth or falsity part way through an iteration.)

Source:http://www.cs.uofs.edu/~mccloske/courses/cmps144/invariants_lec.html

In simple words, *a loop invariant is some predicate (condition) that holds for every iteration of the loop.*

**For example**, let's look at a simple`for`

loop that looks like this:

```
int j = 9;
for(int i=0; i<10; i++)
j--;
```

In this example it is true (for every iteration) that `i + j == 9`

. A weaker invariant that is also true is that` i >= 0 && i < 10`

(because this is the continuation condition!) or that `j <= 9 && j >= 0`

.

- The loop invariants will be true on entry into a loop and following each iteration,
*so that on exit from the loop both the loop invariants and the loop termination condition can be guaranteed.*

**Solution :**An appropriate loop-invariant for the while-loop is

*i mod 2 = 0*(i.e.

**' i ' must be even**, else the

**loop breaks**when

**' i ' is odd**)

### 6 Comments

How can i mod2==0 be loop invariant...

For i=6, 6%2==0[ In while loop as i>=0 ]

Then i=3, 3%2!=0[ In while loop as i>=0 ]

Then i=i-1, making i=2

For i=2, i%2==0[ In while loop as i>=0 ]

Then i=i-2, making i=0

For i=0, i%2==0[ In while loop as i==0 ]

Then i=i-2, making i=-2[OUT OF LOOP]

SO loop invariant should be **i>=0.**