MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(x, 0()) -> s(0()) , f(s(x), s(y)) -> s(f(x, y)) , g(0(), x) -> g(f(x, x), x) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) '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: The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(g) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [f](x1, x2) = [4] [0] = [0] [s](x1) = [1] x1 + [0] [g](x1, x2) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {f, g} The order satisfies the following ordering constraints: [f(x, 0())] = [4] > [0] = [s(0())] [f(s(x), s(y))] = [4] >= [4] = [s(f(x, y))] [g(0(), x)] = [1] x + [0] ? [1] x + [4] = [g(f(x, x), x)] 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: { f(s(x), s(y)) -> s(f(x, y)) , g(0(), x) -> g(f(x, x), x) } Weak Trs: { f(x, 0()) -> s(0()) } Obligation: 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: We use the processor 'matrix interpretation of dimension 2' to orient following rules strictly. Trs: { g(0(), x) -> g(f(x, x), x) } 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}, Uargs(g) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA) and not(IDA(1)). [f](x1, x2) = [0] [1] [0] = [0] [2] [s](x1) = [1 0] x1 + [0] [0 0] [0] [g](x1, x2) = [4 7] x1 + [1] [0 0] [1] The following symbols are considered usable {f, g} The order satisfies the following ordering constraints: [f(x, 0())] = [0] [1] >= [0] [0] = [s(0())] [f(s(x), s(y))] = [0] [1] >= [0] [0] = [s(f(x, y))] [g(0(), x)] = [15] [1] > [8] [1] = [g(f(x, x), x)] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(s(x), s(y)) -> s(f(x, y)) } Weak Trs: { f(x, 0()) -> s(0()) , g(0(), x) -> g(f(x, x), x) } Obligation: 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) '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) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { f^#(x, 0()) -> c_1() , f^#(s(x), s(y)) -> c_2(f^#(x, y)) , g^#(0(), x) -> c_3(g^#(f(x, x), x)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(x, 0()) -> c_1() , f^#(s(x), s(y)) -> c_2(f^#(x, y)) , g^#(0(), x) -> c_3(g^#(f(x, x), x)) } Strict Trs: { f(x, 0()) -> s(0()) , f(s(x), s(y)) -> s(f(x, y)) , g(0(), x) -> g(f(x, x), x) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,3} by applications of Pre({1,3}) = {2}. Here rules are labeled as follows: DPs: { 1: f^#(x, 0()) -> c_1() , 2: f^#(s(x), s(y)) -> c_2(f^#(x, y)) , 3: g^#(0(), x) -> c_3(g^#(f(x, x), x)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(s(x), s(y)) -> c_2(f^#(x, y)) } Strict Trs: { f(x, 0()) -> s(0()) , f(s(x), s(y)) -> s(f(x, y)) , g(0(), x) -> g(f(x, x), x) } Weak DPs: { f^#(x, 0()) -> c_1() , g^#(0(), x) -> c_3(g^#(f(x, x), x)) } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..