MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(0()) -> true() , f(1()) -> false() , f(s(x)) -> f(x) , if(true(), x, y) -> x , if(false(), x, y) -> y , g(x, c(y)) -> g(x, g(s(c(y)), y)) , g(s(x), s(y)) -> if(f(x), s(x), s(y)) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { f^#(0()) -> c_1() , f^#(1()) -> c_2() , f^#(s(x)) -> c_3(f^#(x)) , if^#(true(), x, y) -> c_4() , if^#(false(), x, y) -> c_5() , g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y)) , g^#(s(x), s(y)) -> c_7(if^#(f(x), s(x), s(y)), f^#(x)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(0()) -> c_1() , f^#(1()) -> c_2() , f^#(s(x)) -> c_3(f^#(x)) , if^#(true(), x, y) -> c_4() , if^#(false(), x, y) -> c_5() , g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y)) , g^#(s(x), s(y)) -> c_7(if^#(f(x), s(x), s(y)), f^#(x)) } Weak Trs: { f(0()) -> true() , f(1()) -> false() , f(s(x)) -> f(x) , if(true(), x, y) -> x , if(false(), x, y) -> y , g(x, c(y)) -> g(x, g(s(c(y)), y)) , g(s(x), s(y)) -> if(f(x), s(x), s(y)) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,2,4,5} by applications of Pre({1,2,4,5}) = {3,7}. Here rules are labeled as follows: DPs: { 1: f^#(0()) -> c_1() , 2: f^#(1()) -> c_2() , 3: f^#(s(x)) -> c_3(f^#(x)) , 4: if^#(true(), x, y) -> c_4() , 5: if^#(false(), x, y) -> c_5() , 6: g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y)) , 7: g^#(s(x), s(y)) -> c_7(if^#(f(x), s(x), s(y)), f^#(x)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(x)) -> c_3(f^#(x)) , g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y)) , g^#(s(x), s(y)) -> c_7(if^#(f(x), s(x), s(y)), f^#(x)) } Weak DPs: { f^#(0()) -> c_1() , f^#(1()) -> c_2() , if^#(true(), x, y) -> c_4() , if^#(false(), x, y) -> c_5() } Weak Trs: { f(0()) -> true() , f(1()) -> false() , f(s(x)) -> f(x) , if(true(), x, y) -> x , if(false(), x, y) -> y , g(x, c(y)) -> g(x, g(s(c(y)), y)) , g(s(x), s(y)) -> if(f(x), s(x), s(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. { f^#(0()) -> c_1() , f^#(1()) -> c_2() , if^#(true(), x, y) -> c_4() , if^#(false(), x, y) -> c_5() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(x)) -> c_3(f^#(x)) , g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y)) , g^#(s(x), s(y)) -> c_7(if^#(f(x), s(x), s(y)), f^#(x)) } Weak Trs: { f(0()) -> true() , f(1()) -> false() , f(s(x)) -> f(x) , if(true(), x, y) -> x , if(false(), x, y) -> y , g(x, c(y)) -> g(x, g(s(c(y)), y)) , g(s(x), s(y)) -> if(f(x), s(x), s(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: { g^#(s(x), s(y)) -> c_7(if^#(f(x), s(x), s(y)), f^#(x)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(x)) -> c_1(f^#(x)) , g^#(x, c(y)) -> c_2(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y)) , g^#(s(x), s(y)) -> c_3(f^#(x)) } Weak Trs: { f(0()) -> true() , f(1()) -> false() , f(s(x)) -> f(x) , if(true(), x, y) -> x , if(false(), x, y) -> y , g(x, c(y)) -> g(x, g(s(c(y)), y)) , g(s(x), s(y)) -> if(f(x), s(x), s(y)) } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..