Consider the TRS R consisting of the rewrite rules 1: not(true) -> false 2: not(false) -> true 3: odd(0) -> false 4: odd(s(x)) -> not(odd(x)) 5: x + 0 -> x 6: x + s(y) -> s(x + y) 7: s(x) + y -> s(x + y) There are 4 dependency pairs: 8: ODD(s(x)) -> NOT(odd(x)) 9: ODD(s(x)) -> ODD(x) 10: x +# s(y) -> x +# y 11: s(x) +# y -> x +# y The approximated dependency graph contains 2 SCCs: {10,11} and {9}. - Consider the SCC {10,11}. There are no usable rules. By taking the polynomial interpretation [s](x) = x + 1 and [+#](x,y) = x + y + 1, the rules in {10,11} are strictly decreasing. - Consider the SCC {9}. There are no usable rules. By taking the polynomial interpretation [ODD](x) = [s](x) = x + 1, rule 9 is strictly decreasing. Hence the TRS is terminating.