# Write a loop invariant for selection

Programming language support[ edit ] Eiffel[ edit ] The Eiffel programming language provides native support for loop invariants. For this to be defined, the array must contain at least one element. The postconditions of max require that the returned value is: This again can be weakened to give us the invariant. If c is not present in s, then -1 is returned. For this reason, we must pause our study of programming logic for a study of symbolic logic, that is, the algebra of logical assertions.

The phrase reads better like this: The specification of the previous example now looks like this: To understand how it might be used, consider this variation of the previous example. For string, s, s[: If c is not in s, a copy of s is returned, unchanged. You are welcome to deduce that the program meets its postcondition.

Reuse the loop invariant from find in the previous exercise. Worse yet, its presence can lead to false deductions at the point where the function is called!

We do it like this: No answer is returned because the precondition is violated and the loop is unable to terminate. The deduction law for loops guarantees, if the loop terminates, then the postcondition must hold true.

There can be silly applications of the loop law. Consider this faulty program: But the loop body preserves the invariant only because its body, pass, is too timid to make any progress at all towards the goal.

So, the loop never terminates. Now, if the loop would terminate, then the proof shows we will achieve the goal. But, for every argument but 0, the loop will not terminate.

Because of this limitation of the loop law, it is called a partial correctness law. To ensure total correctness, that is, to prove the loop must terminate and satisfies its goal, we must use additional reasoning. The value of the measure must always compute to a nonnegative integer, and after each iteration of the loop, the measure must decrease by at least 1.

This means, after some finite number of iterations, the measure hits 0 and the loop stops.

## Loop Invariants

It is called mathematical induction. We will introduce mathematical induction by means of examples from programming. Testing and induction Go back to the first section in this chapter. A typical computer programmer would pretend that the definition of!

For example, we can test 5! Before we grind out 4! There is an important principle hiding in here, and once we understand it we will realize there are only two test cases that matter: This will save us a lot of time testing the definition of!.

Now, go back to the beginning of this chapter, where there is a loop that computes factorial by repeated multiplications.Loop invariant of Selection Sort. From there, we move to invariant of statement 1: the loop starts at i=1 and will ensure that (I2) is true, so in particular that a1mathematical induction: (I3): every number in the array is smaller than its .

A loop invariant is a condition 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.).

Loop Invariant Condition: Loop invariant condition is a condition about the relationship between the variables of our program which is definitely true immediately before and immediately after each iteration of the loop.

For example: Consider an array A{7, 5, 3, 10, 2, 6} with 6 elements and we have to find maximum element max in the array. Another loop invariant you probably need in order to show the algorithm is correct is probably that the items of A are a permutation of the original items of A.

## Algorithm - What is a loop invariant? - Stack Overflow

If you show only swaps change A and you prove this for the swap once it will be usable for the loop invariant as well. Exercise Consider sorting \$n\$ numbers in an array \$A\$ by first finding the smallest element of \$A\$ and exchanging it with the element in \$A\$. A loop invariant is a condition 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.).

Loop Invariants