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