MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , d(0()) -> 0() , d(s(x)) -> s(s(d(x))) , q(0()) -> 0() , q(s(x)) -> s(plus(q(x), d(x))) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { plus^#(x, 0()) -> c_1() , plus^#(x, s(y)) -> c_2(plus^#(x, y)) , d^#(0()) -> c_3() , d^#(s(x)) -> c_4(d^#(x)) , q^#(0()) -> c_5() , q^#(s(x)) -> c_6(plus^#(q(x), d(x)), q^#(x), d^#(x)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { plus^#(x, 0()) -> c_1() , plus^#(x, s(y)) -> c_2(plus^#(x, y)) , d^#(0()) -> c_3() , d^#(s(x)) -> c_4(d^#(x)) , q^#(0()) -> c_5() , q^#(s(x)) -> c_6(plus^#(q(x), d(x)), q^#(x), d^#(x)) } Weak Trs: { plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , d(0()) -> 0() , d(s(x)) -> s(s(d(x))) , q(0()) -> 0() , q(s(x)) -> s(plus(q(x), d(x))) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,3,5} by applications of Pre({1,3,5}) = {2,4,6}. Here rules are labeled as follows: DPs: { 1: plus^#(x, 0()) -> c_1() , 2: plus^#(x, s(y)) -> c_2(plus^#(x, y)) , 3: d^#(0()) -> c_3() , 4: d^#(s(x)) -> c_4(d^#(x)) , 5: q^#(0()) -> c_5() , 6: q^#(s(x)) -> c_6(plus^#(q(x), d(x)), q^#(x), d^#(x)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { plus^#(x, s(y)) -> c_2(plus^#(x, y)) , d^#(s(x)) -> c_4(d^#(x)) , q^#(s(x)) -> c_6(plus^#(q(x), d(x)), q^#(x), d^#(x)) } Weak DPs: { plus^#(x, 0()) -> c_1() , d^#(0()) -> c_3() , q^#(0()) -> c_5() } Weak Trs: { plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , d(0()) -> 0() , d(s(x)) -> s(s(d(x))) , q(0()) -> 0() , q(s(x)) -> s(plus(q(x), d(x))) } Obligation: innermost runtime complexity Answer: MAYBE The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { plus^#(x, 0()) -> c_1() , d^#(0()) -> c_3() , q^#(0()) -> c_5() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { plus^#(x, s(y)) -> c_2(plus^#(x, y)) , d^#(s(x)) -> c_4(d^#(x)) , q^#(s(x)) -> c_6(plus^#(q(x), d(x)), q^#(x), d^#(x)) } Weak Trs: { plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , d(0()) -> 0() , d(s(x)) -> s(s(d(x))) , q(0()) -> 0() , q(s(x)) -> s(plus(q(x), d(x))) } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..