MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { from(X) -> cons(X) , length() -> 0() , length() -> s(length1()) , length1() -> length() } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { from(X) -> cons(X) , length() -> 0() } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(s) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [from](x1) = [7] x1 + [7] [cons](x1) = [1] x1 + [3] [length] = [7] [0] = [6] [s](x1) = [1] x1 + [0] [length1] = [7] The following symbols are considered usable {from, length, length1} The order satisfies the following ordering constraints: [from(X)] = [7] X + [7] > [1] X + [3] = [cons(X)] [length()] = [7] > [6] = [0()] [length()] = [7] >= [7] = [s(length1())] [length1()] = [7] >= [7] = [length()] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { length() -> s(length1()) , length1() -> length() } Weak Trs: { from(X) -> cons(X) , length() -> 0() } Obligation: innermost runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [from](x1) = [1] x1 + [7] [cons](x1) = [1] x1 + [7] [length] = [6] [0] = [5] [s](x1) = [1] x1 + [7] [length1] = [7] The following symbols are considered usable {from, length, length1} The order satisfies the following ordering constraints: [from(X)] = [1] X + [7] >= [1] X + [7] = [cons(X)] [length()] = [6] > [5] = [0()] [length()] = [6] ? [14] = [s(length1())] [length1()] = [7] > [6] = [length()] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { length() -> s(length1()) } Weak Trs: { from(X) -> cons(X) , length() -> 0() , length1() -> length() } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 5.0 seconds. 3) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The input cannot be shown compatible 2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The input cannot be shown compatible 2) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: We add the following innermost weak dependency pairs: Strict DPs: { from^#(X) -> c_1() , length^#() -> c_2() , length^#() -> c_3(length1^#()) , length1^#() -> c_4(length^#()) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_1() , length^#() -> c_2() , length^#() -> c_3(length1^#()) , length1^#() -> c_4(length^#()) } Strict Trs: { from(X) -> cons(X) , length() -> 0() , length() -> s(length1()) , length1() -> length() } 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: { from^#(X) -> c_1() , length^#() -> c_2() , length^#() -> c_3(length1^#()) , length1^#() -> c_4(length^#()) } Obligation: innermost runtime complexity Answer: MAYBE The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_4) = {1} TcT has computed the following constructor-restricted matrix interpretation. [from^#](x1) = [2] [1] [c_1] = [1] [1] [length^#] = [1] [1] [c_2] = [0] [1] [c_3](x1) = [1 0] x1 + [2] [0 1] [2] [length1^#] = [1] [1] [c_4](x1) = [1 0] x1 + [2] [0 1] [2] The following symbols are considered usable {from^#, length^#, length1^#} The order satisfies the following ordering constraints: [from^#(X)] = [2] [1] > [1] [1] = [c_1()] [length^#()] = [1] [1] > [0] [1] = [c_2()] [length^#()] = [1] [1] ? [3] [3] = [c_3(length1^#())] [length1^#()] = [1] [1] ? [3] [3] = [c_4(length^#())] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { length^#() -> c_3(length1^#()) , length1^#() -> c_4(length^#()) } Weak DPs: { from^#(X) -> c_1() , length^#() -> c_2() } 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. { from^#(X) -> c_1() , length^#() -> c_2() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { length^#() -> c_3(length1^#()) , length1^#() -> c_4(length^#()) } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Polynomial Path Order (PS)' failed due to the following reason: The input cannot be shown compatible 2) 'Fastest (timeout of 5 seconds)' failed due to the following reason: Computation stopped due to timeout after 5.0 seconds. 3) 'Polynomial Path Order (PS)' failed due to the following reason: The input cannot be shown compatible Arrrr..