Linear search correctness

The loop invariant just before inspecting index $i$ is: $0\le i\le n$, and no entry at an index $k<i$ matches $x$. Initially $i=0$, so there are no earlier entries. A failed equality test extends the excluded region by one position and preserves the loop invariant. A successful equality test at index $i$ establishes a match; the loop invariant establishes that it is the first one. At exhaustion, $i=n$ and every entry has failed, so the absent result is correct. The number $n-i$ of remaining entries decreases after each failed equality test, proving termination for a finite sequence. Together these establish algorithm correctness under the assumptions that the input is stable and each equality test terminates. No step of the argument compares the relative order of values.