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