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