loopinvariant

A loop invariant is a condition that holds true before, during, and after the execution of each iteration of a loop. It is a property that is established before the loop begins, and the loop's logic is designed to maintain this property with each iteration. When the loop terminates, the loop invariant, combined with the termination condition, can be used to prove the correctness of the algorithm.

Loop invariants are a fundamental concept in program verification and algorithm design. They help to reason about the behavior of loops and ensure that they produce the desired results. For example, in a sorting algorithm that uses a loop to place elements in their correct positions, a loop invariant might state that all elements before the current index are sorted, and all elements after the current index are yet to be considered.

To establish a loop invariant, one must prove three things: initialization, maintenance, and termination. Initialization means the invariant holds before the first iteration. Maintenance means if the invariant holds at the start of an iteration, it will also hold at the end of that iteration. Termination means that when the loop finishes, the invariant, along with the loop's termination condition, implies that the algorithm has achieved its goal.

Programmers often implicitly rely on loop invariants when writing loops, but explicitly defining and verifying them can catch subtle bugs and lead to more robust code. Formal methods use loop invariants extensively for proving program correctness.