MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) , f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z) , and(x, true()) -> x , and(x, false()) -> false() , gt(s(u), s(v)) -> gt(u, v) , gt(s(u), 0()) -> true() , gt(0(), v) -> false() } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { f^#(true(), x, y, z) -> c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)), and^#(gt(x, y), gt(x, z)), gt^#(x, y), gt^#(x, z)) , f^#(true(), x, y, z) -> c_2(f^#(and(gt(x, y), gt(x, z)), x, s(y), z), and^#(gt(x, y), gt(x, z)), gt^#(x, y), gt^#(x, z)) , and^#(x, true()) -> c_3() , and^#(x, false()) -> c_4() , gt^#(s(u), s(v)) -> c_5(gt^#(u, v)) , gt^#(s(u), 0()) -> c_6() , gt^#(0(), v) -> c_7() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(true(), x, y, z) -> c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)), and^#(gt(x, y), gt(x, z)), gt^#(x, y), gt^#(x, z)) , f^#(true(), x, y, z) -> c_2(f^#(and(gt(x, y), gt(x, z)), x, s(y), z), and^#(gt(x, y), gt(x, z)), gt^#(x, y), gt^#(x, z)) , and^#(x, true()) -> c_3() , and^#(x, false()) -> c_4() , gt^#(s(u), s(v)) -> c_5(gt^#(u, v)) , gt^#(s(u), 0()) -> c_6() , gt^#(0(), v) -> c_7() } Weak Trs: { f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) , f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z) , and(x, true()) -> x , and(x, false()) -> false() , gt(s(u), s(v)) -> gt(u, v) , gt(s(u), 0()) -> true() , gt(0(), v) -> false() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {3,4,6,7} by applications of Pre({3,4,6,7}) = {1,2,5}. Here rules are labeled as follows: DPs: { 1: f^#(true(), x, y, z) -> c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)), and^#(gt(x, y), gt(x, z)), gt^#(x, y), gt^#(x, z)) , 2: f^#(true(), x, y, z) -> c_2(f^#(and(gt(x, y), gt(x, z)), x, s(y), z), and^#(gt(x, y), gt(x, z)), gt^#(x, y), gt^#(x, z)) , 3: and^#(x, true()) -> c_3() , 4: and^#(x, false()) -> c_4() , 5: gt^#(s(u), s(v)) -> c_5(gt^#(u, v)) , 6: gt^#(s(u), 0()) -> c_6() , 7: gt^#(0(), v) -> c_7() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(true(), x, y, z) -> c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)), and^#(gt(x, y), gt(x, z)), gt^#(x, y), gt^#(x, z)) , f^#(true(), x, y, z) -> c_2(f^#(and(gt(x, y), gt(x, z)), x, s(y), z), and^#(gt(x, y), gt(x, z)), gt^#(x, y), gt^#(x, z)) , gt^#(s(u), s(v)) -> c_5(gt^#(u, v)) } Weak DPs: { and^#(x, true()) -> c_3() , and^#(x, false()) -> c_4() , gt^#(s(u), 0()) -> c_6() , gt^#(0(), v) -> c_7() } Weak Trs: { f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) , f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z) , and(x, true()) -> x , and(x, false()) -> false() , gt(s(u), s(v)) -> gt(u, v) , gt(s(u), 0()) -> true() , gt(0(), v) -> false() } 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, true()) -> c_3() , and^#(x, false()) -> c_4() , gt^#(s(u), 0()) -> c_6() , gt^#(0(), v) -> c_7() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(true(), x, y, z) -> c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)), and^#(gt(x, y), gt(x, z)), gt^#(x, y), gt^#(x, z)) , f^#(true(), x, y, z) -> c_2(f^#(and(gt(x, y), gt(x, z)), x, s(y), z), and^#(gt(x, y), gt(x, z)), gt^#(x, y), gt^#(x, z)) , gt^#(s(u), s(v)) -> c_5(gt^#(u, v)) } Weak Trs: { f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) , f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z) , and(x, true()) -> x , and(x, false()) -> false() , gt(s(u), s(v)) -> gt(u, v) , gt(s(u), 0()) -> true() , gt(0(), v) -> false() } 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^#(true(), x, y, z) -> c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)), and^#(gt(x, y), gt(x, z)), gt^#(x, y), gt^#(x, z)) , f^#(true(), x, y, z) -> c_2(f^#(and(gt(x, y), gt(x, z)), x, s(y), z), and^#(gt(x, y), gt(x, z)), gt^#(x, y), gt^#(x, z)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(true(), x, y, z) -> c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)), gt^#(x, y), gt^#(x, z)) , f^#(true(), x, y, z) -> c_2(f^#(and(gt(x, y), gt(x, z)), x, s(y), z), gt^#(x, y), gt^#(x, z)) , gt^#(s(u), s(v)) -> c_3(gt^#(u, v)) } Weak Trs: { f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) , f(true(), x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z) , and(x, true()) -> x , and(x, false()) -> false() , gt(s(u), s(v)) -> gt(u, v) , gt(s(u), 0()) -> true() , gt(0(), v) -> false() } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { and(x, true()) -> x , and(x, false()) -> false() , gt(s(u), s(v)) -> gt(u, v) , gt(s(u), 0()) -> true() , gt(0(), v) -> false() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(true(), x, y, z) -> c_1(f^#(and(gt(x, y), gt(x, z)), x, y, s(z)), gt^#(x, y), gt^#(x, z)) , f^#(true(), x, y, z) -> c_2(f^#(and(gt(x, y), gt(x, z)), x, s(y), z), gt^#(x, y), gt^#(x, z)) , gt^#(s(u), s(v)) -> c_3(gt^#(u, v)) } Weak Trs: { and(x, true()) -> x , and(x, false()) -> false() , gt(s(u), s(v)) -> gt(u, v) , gt(s(u), 0()) -> true() , gt(0(), v) -> false() } 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: Following exception was raised: stack overflow 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..