# Write a loop invariant definition

Review: Check that the Loop Invariant is setup true by the initialization of nextToCheck and smallestSoFar; Check that each time through the loop, assuming the Loop Invariant true going into the loop, it is true going out of the loop; Check that headway towards termination is being made in this case, nextToCheck is always incremented in the loop body.

That is, the above statement is an invariant assuming Blue follows the specified strategy with respect to the playing of the game.

The combination of the loop invariant and the termination condition gives that smallestSoFar is the index of the smallest value in the array.

Since we are running a loop, we are gaining information step by step. Such a proof is broken down into the following parts: Initialization: It is true in a limited sense before the loop runs.

### Loop invariant bubble sort

You can often use a similar process to demonstrate the correctness of many algorithms. Two players take alternating moves. In effect, this amounts to a proof by mathematical induction on the number of loop iterations executed. This breaks the complicated overall behavior of the loop into small simple steps, each which can be considered separately. This maybe sounds complicated, but is actually quite simple. The key to identifying it is to first identify an invariant of the loop having to do with the number of BLUE marbles in the jar. Red and Blue take alternating turns, with Red going first. In Insertion Sort 2 we saw this by printing the array each time an element was properly inserted.

Challenge In the InsertionSort code below, there is an error. The aim of such a code segment is to ensure that, after it has executed, the loop invariant will have been truthified i.

Think about: What you want to know at the end? But then the same property holds with respect to any number of iterations of the loop.

### Loop invariant proof

Of course, there is nothing special about upper right corners. The game ends when either Red has formed a closed curve or there are no more line segments to draw. Now what about what I said about the loop conditional necessarily being false when the loop terminates? Sometimes one choice works better than the other; sometimes it makes little difference. Use of loop invariants[ edit ] A loop invariant can serve one of the following purposes: purely documentary to be checked within in the code by an assertion call to be verified based on the Floyd-Hoare approach For 1. One idea is to set it to negative infinity, which works in the abstract, but which at least in many circumstances has no obvious counterpart in a computer program. Loop Invariant Condition with Examples of Sorting Algorithms 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. This breaks the complicated overall behavior of the loop into small simple steps, each which can be considered separately. The biggest take from it is the purpose of having this loop invariant is to help prove the correctness of the algorithm. Answering this question in some way shouldn't be difficult, because we wrote the algorithm to solve a specific computational problem. Question: Does either Red or Blue have a "winning strategy", i. How do you show that an invariant really is one? The fundamental difference is that each clause of the loop invariant identifies the result as being correct up to the current element i, whilst the postconditions identify the result as being correct for all elements. The other answers only focus on what is a loop invariant! But how would you prove that the code is correct?

About Termination: We have formulated a loop invariant, that -if true after stepping out of the loop- gives us a statement with which we can prove correctness. The most obvious way to increase m's value is to increment it.

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