Consider the TRS R consisting of the rewrite rules 1: and(x,or(y,z)) -> or(and(x,y),and(x,z)) 2: and(x,and(y,y)) -> and(x,y) 3: or(or(x,y),and(y,z)) -> or(x,y) 4: or(x,and(x,y)) -> x 5: or(true,y) -> true 6: or(x,false) -> x 7: or(x,x) -> x 8: or(x,or(y,y)) -> or(x,y) 9: and(x,true) -> x 10: and(false,y) -> false 11: and(x,x) -> x There are 5 dependency pairs: 12: AND(x,or(y,z)) -> OR(and(x,y),and(x,z)) 13: AND(x,or(y,z)) -> AND(x,y) 14: AND(x,or(y,z)) -> AND(x,z) 15: AND(x,and(y,y)) -> AND(x,y) 16: OR(x,or(y,y)) -> OR(x,y) The approximated dependency graph contains 2 SCCs: {16} and {13-15}. - Consider the SCC {16}. There are no usable rules. By taking the polynomial interpretation [or](x,y) = [OR](x,y) = x + y + 1, rule 16 is strictly decreasing. - Consider the SCC {13-15}. There are no usable rules. By taking the polynomial interpretation [and](x,y) = [AND](x,y) = [or](x,y) = x + y + 1, the rules in {13-15} are strictly decreasing. Hence the TRS is terminating.