Consider the TRS R consisting of the rewrite rules 1: 0 + y -> y 2: s(x) + y -> s(x + y) 3: 0 - y -> 0 4: x - 0 -> x 5: s(x) - s(y) -> x - y There are 2 dependency pairs: 6: s(x) +# y -> x +# y 7: s(x) -# s(y) -> x -# y The approximated dependency graph contains 2 SCCs: {6} and {7}. - Consider the SCC {6}. There are no usable rules. By taking the polynomial interpretation [s](x) = x + 1 and [+#](x,y) = x + y + 1, rule 6 is strictly decreasing. - Consider the SCC {7}. There are no usable rules. By taking the polynomial interpretation [s](x) = x + 1 and [-#](x,y) = x + y + 1, rule 7 is strictly decreasing. Hence the TRS is terminating.