The nature of loops in programming

In program semantics and verification, reasoning about loops is complicated by the need to produce two separate mathematical arguments: an invariant, for functional properties (ignoring termination); and a variant, for termination (ignoring functional properties). A single and simple definition is possible, removing this split. A loop is just the limit (a variant of the reflexive transitive closure) of a Noetherian (well-founded) relation. To prove the loop correct there is no need to devise an invariant and a variant; it suffices to identify the relation, yielding both partial correctness and termination. The present note develops the (small) theory and applies it to standard loop examples and proofs of their correctness.