MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(0()) -> 1() , f(s(x)) -> g(f(x)) , f(s(x)) -> +(f(x), s(f(x))) , g(x) -> +(x, s(x)) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { f^#(0()) -> c_1() , f^#(s(x)) -> c_2(g^#(f(x)), f^#(x)) , f^#(s(x)) -> c_3(f^#(x), f^#(x)) , g^#(x) -> c_4() } 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^#(s(x)) -> c_2(g^#(f(x)), f^#(x)) , f^#(s(x)) -> c_3(f^#(x), f^#(x)) , g^#(x) -> c_4() } Weak Trs: { f(0()) -> 1() , f(s(x)) -> g(f(x)) , f(s(x)) -> +(f(x), s(f(x))) , g(x) -> +(x, s(x)) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,4} by applications of Pre({1,4}) = {2,3}. Here rules are labeled as follows: DPs: { 1: f^#(0()) -> c_1() , 2: f^#(s(x)) -> c_2(g^#(f(x)), f^#(x)) , 3: f^#(s(x)) -> c_3(f^#(x), f^#(x)) , 4: g^#(x) -> c_4() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(x)) -> c_2(g^#(f(x)), f^#(x)) , f^#(s(x)) -> c_3(f^#(x), f^#(x)) } Weak DPs: { f^#(0()) -> c_1() , g^#(x) -> c_4() } Weak Trs: { f(0()) -> 1() , f(s(x)) -> g(f(x)) , f(s(x)) -> +(f(x), s(f(x))) , g(x) -> +(x, s(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. { f^#(0()) -> c_1() , g^#(x) -> c_4() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(x)) -> c_2(g^#(f(x)), f^#(x)) , f^#(s(x)) -> c_3(f^#(x), f^#(x)) } Weak Trs: { f(0()) -> 1() , f(s(x)) -> g(f(x)) , f(s(x)) -> +(f(x), s(f(x))) , g(x) -> +(x, s(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: { f^#(s(x)) -> c_2(g^#(f(x)), f^#(x)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(x)) -> c_1(f^#(x), f^#(x)) , f^#(s(x)) -> c_2(f^#(x)) } Weak Trs: { f(0()) -> 1() , f(s(x)) -> g(f(x)) , f(s(x)) -> +(f(x), s(f(x))) , g(x) -> +(x, s(x)) } Obligation: innermost runtime complexity Answer: MAYBE No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(x)) -> c_1(f^#(x), f^#(x)) , f^#(s(x)) -> c_2(f^#(x)) } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrices' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrix interpretation of dimension 4' failed due to the following reason: The input cannot be shown compatible 2) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 3) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 4) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 5) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 6) 'matrix interpretation of dimension 1' failed due to the following reason: The input cannot be shown compatible 2) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. Arrrr..