Partial correctness

Partial correctness means that every terminating execution begun in a program state satisfying the precondition $P$ finishes in a program state satisfying the postcondition $Q$. For a sequential program $C$, the usual Hoare triple notation isHere the notation expresses partial correctness; it makes no promise that $C$ finishes. The precondition and postcondition describe permitted starting circumstances and required final circumstances, rather than tests performed automatically by the program.^[ref]

For a sharp example, take the precondition $n=4$ and the following sequential program over mathematical integers:The condition remains true, so execution never reaches the instruction after the loop. Consequently its partial correctness claim holds for the postcondition $2s=n(n+1)$, and even for the impossible postcondition $0=1$: there is no terminating execution with a wrong final result. This is vacuous truth, not evidence that the loop computes a sum. The same program fails total correctness for this permitted input.

The distinction is useful in a proof: first show that an answer, whenever produced, is right; then establish algorithm termination. The summation loop verification supplies both arguments for a loop that actually progresses. Its finite trace illustrates one execution, while its loop invariant covers every permitted input.