MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { cond(true(), x, y, z) -> cond(and(gr(x, z), gr(y, z)), p(x), p(y), z) , and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , gr(0(), x) -> false() , gr(0(), 0()) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { cond^#(true(), x, y, z) -> c_1(cond^#(and(gr(x, z), gr(y, z)), p(x), p(y), z), and^#(gr(x, z), gr(y, z)), gr^#(x, z), gr^#(y, z), p^#(x), p^#(y)) , and^#(x, false()) -> c_2() , and^#(true(), true()) -> c_3() , and^#(false(), x) -> c_4() , gr^#(0(), x) -> c_5() , gr^#(0(), 0()) -> c_6() , gr^#(s(x), 0()) -> c_7() , gr^#(s(x), s(y)) -> c_8(gr^#(x, y)) , p^#(0()) -> c_9() , p^#(s(x)) -> c_10() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { cond^#(true(), x, y, z) -> c_1(cond^#(and(gr(x, z), gr(y, z)), p(x), p(y), z), and^#(gr(x, z), gr(y, z)), gr^#(x, z), gr^#(y, z), p^#(x), p^#(y)) , and^#(x, false()) -> c_2() , and^#(true(), true()) -> c_3() , and^#(false(), x) -> c_4() , gr^#(0(), x) -> c_5() , gr^#(0(), 0()) -> c_6() , gr^#(s(x), 0()) -> c_7() , gr^#(s(x), s(y)) -> c_8(gr^#(x, y)) , p^#(0()) -> c_9() , p^#(s(x)) -> c_10() } Weak Trs: { cond(true(), x, y, z) -> cond(and(gr(x, z), gr(y, z)), p(x), p(y), z) , and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , gr(0(), x) -> false() , gr(0(), 0()) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {2,3,4,5,6,7,9,10} by applications of Pre({2,3,4,5,6,7,9,10}) = {1,8}. Here rules are labeled as follows: DPs: { 1: cond^#(true(), x, y, z) -> c_1(cond^#(and(gr(x, z), gr(y, z)), p(x), p(y), z), and^#(gr(x, z), gr(y, z)), gr^#(x, z), gr^#(y, z), p^#(x), p^#(y)) , 2: and^#(x, false()) -> c_2() , 3: and^#(true(), true()) -> c_3() , 4: and^#(false(), x) -> c_4() , 5: gr^#(0(), x) -> c_5() , 6: gr^#(0(), 0()) -> c_6() , 7: gr^#(s(x), 0()) -> c_7() , 8: gr^#(s(x), s(y)) -> c_8(gr^#(x, y)) , 9: p^#(0()) -> c_9() , 10: p^#(s(x)) -> c_10() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { cond^#(true(), x, y, z) -> c_1(cond^#(and(gr(x, z), gr(y, z)), p(x), p(y), z), and^#(gr(x, z), gr(y, z)), gr^#(x, z), gr^#(y, z), p^#(x), p^#(y)) , gr^#(s(x), s(y)) -> c_8(gr^#(x, y)) } Weak DPs: { and^#(x, false()) -> c_2() , and^#(true(), true()) -> c_3() , and^#(false(), x) -> c_4() , gr^#(0(), x) -> c_5() , gr^#(0(), 0()) -> c_6() , gr^#(s(x), 0()) -> c_7() , p^#(0()) -> c_9() , p^#(s(x)) -> c_10() } Weak Trs: { cond(true(), x, y, z) -> cond(and(gr(x, z), gr(y, z)), p(x), p(y), z) , and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , gr(0(), x) -> false() , gr(0(), 0()) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> 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. { and^#(x, false()) -> c_2() , and^#(true(), true()) -> c_3() , and^#(false(), x) -> c_4() , gr^#(0(), x) -> c_5() , gr^#(0(), 0()) -> c_6() , gr^#(s(x), 0()) -> c_7() , p^#(0()) -> c_9() , p^#(s(x)) -> c_10() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { cond^#(true(), x, y, z) -> c_1(cond^#(and(gr(x, z), gr(y, z)), p(x), p(y), z), and^#(gr(x, z), gr(y, z)), gr^#(x, z), gr^#(y, z), p^#(x), p^#(y)) , gr^#(s(x), s(y)) -> c_8(gr^#(x, y)) } Weak Trs: { cond(true(), x, y, z) -> cond(and(gr(x, z), gr(y, z)), p(x), p(y), z) , and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , gr(0(), x) -> false() , gr(0(), 0()) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x } Obligation: innermost runtime complexity Answer: MAYBE Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { cond^#(true(), x, y, z) -> c_1(cond^#(and(gr(x, z), gr(y, z)), p(x), p(y), z), and^#(gr(x, z), gr(y, z)), gr^#(x, z), gr^#(y, z), p^#(x), p^#(y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { cond^#(true(), x, y, z) -> c_1(cond^#(and(gr(x, z), gr(y, z)), p(x), p(y), z), gr^#(x, z), gr^#(y, z)) , gr^#(s(x), s(y)) -> c_2(gr^#(x, y)) } Weak Trs: { cond(true(), x, y, z) -> cond(and(gr(x, z), gr(y, z)), p(x), p(y), z) , and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , gr(0(), x) -> false() , gr(0(), 0()) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , gr(0(), x) -> false() , gr(0(), 0()) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { cond^#(true(), x, y, z) -> c_1(cond^#(and(gr(x, z), gr(y, z)), p(x), p(y), z), gr^#(x, z), gr^#(y, z)) , gr^#(s(x), s(y)) -> c_2(gr^#(x, y)) } Weak Trs: { and(x, false()) -> false() , and(true(), true()) -> true() , and(false(), x) -> false() , gr(0(), x) -> false() , gr(0(), 0()) -> false() , gr(s(x), 0()) -> true() , gr(s(x), s(y)) -> gr(x, y) , p(0()) -> 0() , p(s(x)) -> x } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..