  The F5 algorithm is generally believed as one of the fastest algorithms for
computing Gr\"{o}bner bases. However, its termination problem is still unclear.
Recently, an algorithm GVW and its variant GVWHS have been proposed, and their
efficiency are comparable to the F5 algorithm. In the paper, we clarify the
concept of an admissible module order. For the first time, the connection
between the reducible and rewritable check is discussed here. We show that the
top-reduced S-Gr\"{o}bner basis must be finite if the admissible monomial order
and the admissible module order are compatible. This paper presents a complete
proof of the termination and correctness of the GVWHS algorithm. What is more,
it can be seen that the GVWHS is in fact an F5-like algorithm. Different from
the GVWHS algorithm, the F5B algorithm may generate redundant sig-polynomials.
Taking into account this situation, we prove the termination and correctness of
the F5B algorithm. And we notice that the original F5 algorithm slightly
differs from the F5B algorithm in the insertion strategy on which the
F5-rewritten criterion is based. Exploring the potential ordering of
sig-polynomials computed by the original F5 algorithm, we propose an F5GEN
algorithm with a generalized insertion strategy, and prove the termination and
correctness of it. Therefore, we have a positive answer to the long standing
problem of proving the termination of the original F5 algorithm.
