Monomial orderings, rewriting systems, and Gröbner bases for the ...

the well-founded ordering L(c mj b j+1 , w ′ ).
Similarly, applying a rule of the form (5 j ), j ≤ J leaves α(w) = α(w ′ ),
L(c mJ b J+1 , w) = L(c mJ b J+1 , w ′ ), R(c mJ b J , w) = R(c mJ b J , w ′ ),..., L(c mj+1 b j+2 , w) =
L(c mj+1 b j+2 , w ′ ), and R(c mj b j , w) > R(c mj b j , w ′ ). In each case, then, the rules
(4 j ), (5 j ) decrease Ψ, and if one of these rules is applied to rewrite w to w ′ ,
then w ′ ≺r w.
Finally, if rule (1) or (3) is applied to w, then α(w ′ ) = α(w). For each
index j, applying rule (1) or (3) either does not move past a barrier used to
compute L(c mj b j+1 , ·) or R(c mj b j , ·), or else it moves an a past a barrier in a
way that will decrease the corresponding variable. Therefore Ψ(w ′ ) ≤ Ψ(w).
However, rewriting by (1) and (3) decreases w lexicographically, since b ≺r
a ≺r c, so w ′ ≺r w. Thus, we have shown that the reduction process always
decreases the well-founded ordering ≺r no matter what rule is applied, so
this process will always terminate after finitely many steps.
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