MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , <=(0(), y) -> true() , <=(s(x), 0()) -> false() , <=(s(x), s(y)) -> <=(x, y) , if(true(), x, y) -> x , if(false(), x, y) -> y , perfectp(0()) -> false() , perfectp(s(x)) -> f(x, s(0()), s(x), s(x)) , f(0(), y, 0(), u) -> true() , f(0(), y, s(z), u) -> false() , f(s(x), 0(), z, u) -> f(x, u, -(z, s(x)), u) , f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u)) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { -^#(x, 0()) -> c_1() , -^#(s(x), s(y)) -> c_2(-^#(x, y)) , <=^#(0(), y) -> c_3() , <=^#(s(x), 0()) -> c_4() , <=^#(s(x), s(y)) -> c_5(<=^#(x, y)) , if^#(true(), x, y) -> c_6() , if^#(false(), x, y) -> c_7() , perfectp^#(0()) -> c_8() , perfectp^#(s(x)) -> c_9(f^#(x, s(0()), s(x), s(x))) , f^#(0(), y, 0(), u) -> c_10() , f^#(0(), y, s(z), u) -> c_11() , f^#(s(x), 0(), z, u) -> c_12(f^#(x, u, -(z, s(x)), u), -^#(z, s(x))) , f^#(s(x), s(y), z, u) -> c_13(if^#(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u)), <=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { -^#(x, 0()) -> c_1() , -^#(s(x), s(y)) -> c_2(-^#(x, y)) , <=^#(0(), y) -> c_3() , <=^#(s(x), 0()) -> c_4() , <=^#(s(x), s(y)) -> c_5(<=^#(x, y)) , if^#(true(), x, y) -> c_6() , if^#(false(), x, y) -> c_7() , perfectp^#(0()) -> c_8() , perfectp^#(s(x)) -> c_9(f^#(x, s(0()), s(x), s(x))) , f^#(0(), y, 0(), u) -> c_10() , f^#(0(), y, s(z), u) -> c_11() , f^#(s(x), 0(), z, u) -> c_12(f^#(x, u, -(z, s(x)), u), -^#(z, s(x))) , f^#(s(x), s(y), z, u) -> c_13(if^#(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u)), <=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) } Weak Trs: { -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , <=(0(), y) -> true() , <=(s(x), 0()) -> false() , <=(s(x), s(y)) -> <=(x, y) , if(true(), x, y) -> x , if(false(), x, y) -> y , perfectp(0()) -> false() , perfectp(s(x)) -> f(x, s(0()), s(x), s(x)) , f(0(), y, 0(), u) -> true() , f(0(), y, s(z), u) -> false() , f(s(x), 0(), z, u) -> f(x, u, -(z, s(x)), u) , f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u)) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,3,4,6,7,8,10,11} by applications of Pre({1,3,4,6,7,8,10,11}) = {2,5,9,12,13}. Here rules are labeled as follows: DPs: { 1: -^#(x, 0()) -> c_1() , 2: -^#(s(x), s(y)) -> c_2(-^#(x, y)) , 3: <=^#(0(), y) -> c_3() , 4: <=^#(s(x), 0()) -> c_4() , 5: <=^#(s(x), s(y)) -> c_5(<=^#(x, y)) , 6: if^#(true(), x, y) -> c_6() , 7: if^#(false(), x, y) -> c_7() , 8: perfectp^#(0()) -> c_8() , 9: perfectp^#(s(x)) -> c_9(f^#(x, s(0()), s(x), s(x))) , 10: f^#(0(), y, 0(), u) -> c_10() , 11: f^#(0(), y, s(z), u) -> c_11() , 12: f^#(s(x), 0(), z, u) -> c_12(f^#(x, u, -(z, s(x)), u), -^#(z, s(x))) , 13: f^#(s(x), s(y), z, u) -> c_13(if^#(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u)), <=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { -^#(s(x), s(y)) -> c_2(-^#(x, y)) , <=^#(s(x), s(y)) -> c_5(<=^#(x, y)) , perfectp^#(s(x)) -> c_9(f^#(x, s(0()), s(x), s(x))) , f^#(s(x), 0(), z, u) -> c_12(f^#(x, u, -(z, s(x)), u), -^#(z, s(x))) , f^#(s(x), s(y), z, u) -> c_13(if^#(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u)), <=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) } Weak DPs: { -^#(x, 0()) -> c_1() , <=^#(0(), y) -> c_3() , <=^#(s(x), 0()) -> c_4() , if^#(true(), x, y) -> c_6() , if^#(false(), x, y) -> c_7() , perfectp^#(0()) -> c_8() , f^#(0(), y, 0(), u) -> c_10() , f^#(0(), y, s(z), u) -> c_11() } Weak Trs: { -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , <=(0(), y) -> true() , <=(s(x), 0()) -> false() , <=(s(x), s(y)) -> <=(x, y) , if(true(), x, y) -> x , if(false(), x, y) -> y , perfectp(0()) -> false() , perfectp(s(x)) -> f(x, s(0()), s(x), s(x)) , f(0(), y, 0(), u) -> true() , f(0(), y, s(z), u) -> false() , f(s(x), 0(), z, u) -> f(x, u, -(z, s(x)), u) , f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u)) } 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. { -^#(x, 0()) -> c_1() , <=^#(0(), y) -> c_3() , <=^#(s(x), 0()) -> c_4() , if^#(true(), x, y) -> c_6() , if^#(false(), x, y) -> c_7() , perfectp^#(0()) -> c_8() , f^#(0(), y, 0(), u) -> c_10() , f^#(0(), y, s(z), u) -> c_11() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { -^#(s(x), s(y)) -> c_2(-^#(x, y)) , <=^#(s(x), s(y)) -> c_5(<=^#(x, y)) , perfectp^#(s(x)) -> c_9(f^#(x, s(0()), s(x), s(x))) , f^#(s(x), 0(), z, u) -> c_12(f^#(x, u, -(z, s(x)), u), -^#(z, s(x))) , f^#(s(x), s(y), z, u) -> c_13(if^#(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u)), <=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) } Weak Trs: { -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , <=(0(), y) -> true() , <=(s(x), 0()) -> false() , <=(s(x), s(y)) -> <=(x, y) , if(true(), x, y) -> x , if(false(), x, y) -> y , perfectp(0()) -> false() , perfectp(s(x)) -> f(x, s(0()), s(x), s(x)) , f(0(), y, 0(), u) -> true() , f(0(), y, s(z), u) -> false() , f(s(x), 0(), z, u) -> f(x, u, -(z, s(x)), u) , f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u)) } 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), s(y), z, u) -> c_13(if^#(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u)), <=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { -^#(s(x), s(y)) -> c_1(-^#(x, y)) , <=^#(s(x), s(y)) -> c_2(<=^#(x, y)) , perfectp^#(s(x)) -> c_3(f^#(x, s(0()), s(x), s(x))) , f^#(s(x), 0(), z, u) -> c_4(f^#(x, u, -(z, s(x)), u), -^#(z, s(x))) , f^#(s(x), s(y), z, u) -> c_5(<=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) } Weak Trs: { -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) , <=(0(), y) -> true() , <=(s(x), 0()) -> false() , <=(s(x), s(y)) -> <=(x, y) , if(true(), x, y) -> x , if(false(), x, y) -> y , perfectp(0()) -> false() , perfectp(s(x)) -> f(x, s(0()), s(x), s(x)) , f(0(), y, 0(), u) -> true() , f(0(), y, s(z), u) -> false() , f(s(x), 0(), z, u) -> f(x, u, -(z, s(x)), u) , f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u)) } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { -^#(s(x), s(y)) -> c_1(-^#(x, y)) , <=^#(s(x), s(y)) -> c_2(<=^#(x, y)) , perfectp^#(s(x)) -> c_3(f^#(x, s(0()), s(x), s(x))) , f^#(s(x), 0(), z, u) -> c_4(f^#(x, u, -(z, s(x)), u), -^#(z, s(x))) , f^#(s(x), s(y), z, u) -> c_5(<=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) } Weak Trs: { -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) } Obligation: innermost runtime complexity Answer: MAYBE Consider the dependency graph 1: -^#(s(x), s(y)) -> c_1(-^#(x, y)) -->_1 -^#(s(x), s(y)) -> c_1(-^#(x, y)) :1 2: <=^#(s(x), s(y)) -> c_2(<=^#(x, y)) -->_1 <=^#(s(x), s(y)) -> c_2(<=^#(x, y)) :2 3: perfectp^#(s(x)) -> c_3(f^#(x, s(0()), s(x), s(x))) -->_1 f^#(s(x), s(y), z, u) -> c_5(<=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) :5 4: f^#(s(x), 0(), z, u) -> c_4(f^#(x, u, -(z, s(x)), u), -^#(z, s(x))) -->_1 f^#(s(x), s(y), z, u) -> c_5(<=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) :5 -->_1 f^#(s(x), 0(), z, u) -> c_4(f^#(x, u, -(z, s(x)), u), -^#(z, s(x))) :4 -->_2 -^#(s(x), s(y)) -> c_1(-^#(x, y)) :1 5: f^#(s(x), s(y), z, u) -> c_5(<=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) -->_4 f^#(s(x), s(y), z, u) -> c_5(<=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) :5 -->_2 f^#(s(x), s(y), z, u) -> c_5(<=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) :5 -->_4 f^#(s(x), 0(), z, u) -> c_4(f^#(x, u, -(z, s(x)), u), -^#(z, s(x))) :4 -->_2 f^#(s(x), 0(), z, u) -> c_4(f^#(x, u, -(z, s(x)), u), -^#(z, s(x))) :4 -->_1 <=^#(s(x), s(y)) -> c_2(<=^#(x, y)) :2 -->_3 -^#(s(x), s(y)) -> c_1(-^#(x, y)) :1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). { perfectp^#(s(x)) -> c_3(f^#(x, s(0()), s(x), s(x))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { -^#(s(x), s(y)) -> c_1(-^#(x, y)) , <=^#(s(x), s(y)) -> c_2(<=^#(x, y)) , f^#(s(x), 0(), z, u) -> c_4(f^#(x, u, -(z, s(x)), u), -^#(z, s(x))) , f^#(s(x), s(y), z, u) -> c_5(<=^#(x, y), f^#(s(x), -(y, x), z, u), -^#(y, x), f^#(x, u, z, u)) } Weak Trs: { -(x, 0()) -> x , -(s(x), s(y)) -> -(x, y) } 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..