Consider the TRS R consisting of the rewrite rules 1: exp(x,0) -> s(0) 2: exp(x,s(y)) -> x * exp(x,y) 3: 0 * y -> 0 4: s(x) * y -> y + (x * y) 5: 0 - y -> 0 6: x - 0 -> x 7: s(x) - s(y) -> x - y There are 4 dependency pairs: 8: EXP(x,s(y)) -> x *# exp(x,y) 9: EXP(x,s(y)) -> EXP(x,y) 10: s(x) *# y -> x *# y 11: s(x) -# s(y) -> x -# y The approximated dependency graph contains 3 SCCs: {10}, {11} and {9}. - Consider the SCC {10}. There are no usable rules. By taking the polynomial interpretation [s](x) = x + 1 and [*#](x,y) = x + y + 1, rule 10 is strictly decreasing. - Consider the SCC {11}. There are no usable rules. By taking the polynomial interpretation [s](x) = x + 1 and [-#](x,y) = x + y + 1, rule 11 is strictly decreasing. - Consider the SCC {9}. There are no usable rules. By taking the polynomial interpretation [s](x) = x + 1 and [EXP](x,y) = x + y + 1, rule 9 is strictly decreasing. Hence the TRS is terminating.