MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { from(X) -> cons(X, from(s(X))) , length(cons(X, Y)) -> s(length1(Y)) , length(nil()) -> 0() , length1(X) -> length(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: We add the following weak dependency pairs: Strict DPs: { from^#(X) -> c_1(X, from^#(s(X))) , length^#(cons(X, Y)) -> c_2(length1^#(Y)) , length^#(nil()) -> c_3() , length1^#(X) -> c_4(length^#(X)) } 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(X, from^#(s(X))) , length^#(cons(X, Y)) -> c_2(length1^#(Y)) , length^#(nil()) -> c_3() , length1^#(X) -> c_4(length^#(X)) } Strict Trs: { from(X) -> cons(X, from(s(X))) , length(cons(X, Y)) -> s(length1(Y)) , length(nil()) -> 0() , length1(X) -> length(X) } Obligation: 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(X, from^#(s(X))) , length^#(cons(X, Y)) -> c_2(length1^#(Y)) , length^#(nil()) -> c_3() , length1^#(X) -> c_4(length^#(X)) } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(c_1) = {2}, Uargs(c_2) = {1}, Uargs(c_4) = {1} TcT has computed the following constructor-restricted matrix interpretation. [cons](x1, x2) = [1 1] x2 + [2] [0 1] [1] [s](x1) = [1 1] x1 + [2] [0 0] [2] [nil] = [2] [1] [from^#](x1) = [1 1] x1 + [2] [1 1] [2] [c_1](x1, x2) = [0 0] x1 + [1 0] x2 + [1] [1 2] [0 1] [1] [length^#](x1) = [1 2] x1 + [2] [1 2] [2] [c_2](x1) = [1 0] x1 + [2] [0 1] [0] [length1^#](x1) = [1 2] x1 + [2] [1 2] [2] [c_3] = [1] [1] [c_4](x1) = [1 0] x1 + [2] [0 1] [1] The following symbols are considered usable {from^#, length^#, length1^#} The order satisfies the following ordering constraints: [from^#(X)] = [1 1] X + [2] [1 1] [2] ? [1 1] X + [7] [2 3] [7] = [c_1(X, from^#(s(X)))] [length^#(cons(X, Y))] = [1 3] Y + [6] [1 3] [6] > [1 2] Y + [4] [1 2] [2] = [c_2(length1^#(Y))] [length^#(nil())] = [6] [6] > [1] [1] = [c_3()] [length1^#(X)] = [1 2] X + [2] [1 2] [2] ? [1 2] X + [4] [1 2] [3] = [c_4(length^#(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 DPs: { from^#(X) -> c_1(X, from^#(s(X))) , length1^#(X) -> c_4(length^#(X)) } Weak DPs: { length^#(cons(X, Y)) -> c_2(length1^#(Y)) , length^#(nil()) -> c_3() } Obligation: runtime complexity Answer: MAYBE The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { length^#(nil()) -> c_3() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_1(X, from^#(s(X))) , length1^#(X) -> c_4(length^#(X)) } Weak DPs: { length^#(cons(X, Y)) -> c_2(length1^#(Y)) } 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) 'Inspecting Problem...' failed due to the following reason: We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 2: length1^#(X) -> c_4(length^#(X)) , 3: length^#(cons(X, Y)) -> c_2(length1^#(Y)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {2}, Uargs(c_2) = {1}, Uargs(c_4) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [from](x1) = [7] x1 + [0] [cons](x1, x2) = [1] x1 + [1] x2 + [4] [s](x1) = [0] [length](x1) = [7] x1 + [0] [nil] = [0] [0] = [0] [length1](x1) = [7] x1 + [0] [from^#](x1) = [4] x1 + [0] [c_1](x1, x2) = [3] x1 + [2] x2 + [0] [length^#](x1) = [2] x1 + [0] [c_2](x1) = [1] x1 + [3] [length1^#](x1) = [2] x1 + [4] [c_3] = [0] [c_4](x1) = [1] x1 + [1] The following symbols are considered usable {from^#, length^#, length1^#} The order satisfies the following ordering constraints: [from^#(X)] = [4] X + [0] >= [3] X + [0] = [c_1(X, from^#(s(X)))] [length^#(cons(X, Y))] = [2] X + [2] Y + [8] > [2] Y + [7] = [c_2(length1^#(Y))] [length1^#(X)] = [2] X + [4] > [2] X + [1] = [c_4(length^#(X))] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_1(X, from^#(s(X))) } Weak DPs: { length^#(cons(X, Y)) -> c_2(length1^#(Y)) , length1^#(X) -> c_4(length^#(X)) } Obligation: runtime complexity Answer: MAYBE The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { length^#(cons(X, Y)) -> c_2(length1^#(Y)) , length1^#(X) -> c_4(length^#(X)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_1(X, from^#(s(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) '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 processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'Fastest (timeout of 5 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) 'Polynomial Path Order (PS)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 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(cons) = {2}, Uargs(s) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [from](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x2 + [0] [s](x1) = [1] x1 + [0] [length](x1) = [1] x1 + [0] [nil] = [1] [0] = [0] [length1](x1) = [1] x1 + [0] The following symbols are considered usable {from, length, length1} The order satisfies the following ordering constraints: [from(X)] = [1] X + [0] >= [1] X + [0] = [cons(X, from(s(X)))] [length(cons(X, Y))] = [1] Y + [0] >= [1] Y + [0] = [s(length1(Y))] [length(nil())] = [1] > [0] = [0()] [length1(X)] = [1] X + [0] >= [1] X + [0] = [length(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: { from(X) -> cons(X, from(s(X))) , length(cons(X, Y)) -> s(length1(Y)) , length1(X) -> length(X) } Weak Trs: { length(nil()) -> 0() } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(cons) = {2}, Uargs(s) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [from](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x2 + [0] [s](x1) = [1] x1 + [0] [length](x1) = [1] x1 + [1] [nil] = [7] [0] = [0] [length1](x1) = [1] x1 + [0] The following symbols are considered usable {from, length, length1} The order satisfies the following ordering constraints: [from(X)] = [1] X + [0] >= [1] X + [0] = [cons(X, from(s(X)))] [length(cons(X, Y))] = [1] Y + [1] > [1] Y + [0] = [s(length1(Y))] [length(nil())] = [8] > [0] = [0()] [length1(X)] = [1] X + [0] ? [1] X + [1] = [length(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: { from(X) -> cons(X, from(s(X))) , length1(X) -> length(X) } Weak Trs: { length(cons(X, Y)) -> s(length1(Y)) , length(nil()) -> 0() } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(cons) = {2}, Uargs(s) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [from](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x2 + [4] [s](x1) = [1] x1 + [0] [length](x1) = [1] x1 + [0] [nil] = [7] [0] = [7] [length1](x1) = [1] x1 + [1] The following symbols are considered usable {from, length, length1} The order satisfies the following ordering constraints: [from(X)] = [1] X + [0] ? [1] X + [4] = [cons(X, from(s(X)))] [length(cons(X, Y))] = [1] Y + [4] > [1] Y + [1] = [s(length1(Y))] [length(nil())] = [7] >= [7] = [0()] [length1(X)] = [1] X + [1] > [1] X + [0] = [length(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: { from(X) -> cons(X, from(s(X))) } Weak Trs: { length(cons(X, Y)) -> s(length1(Y)) , length(nil()) -> 0() , length1(X) -> length(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) '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) '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 minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-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: { from^#(X) -> c_1(X, from^#(s(X))) , length^#(cons(X, Y)) -> c_2(length1^#(Y)) , length^#(nil()) -> c_3() , length1^#(X) -> c_4(length^#(X)) } 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(X, from^#(s(X))) , length^#(cons(X, Y)) -> c_2(length1^#(Y)) , length^#(nil()) -> c_3() , length1^#(X) -> c_4(length^#(X)) } Strict Trs: { from(X) -> cons(X, from(s(X))) , length(cons(X, Y)) -> s(length1(Y)) , length(nil()) -> 0() , length1(X) -> length(X) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3} by applications of Pre({3}) = {1,4}. Here rules are labeled as follows: DPs: { 1: from^#(X) -> c_1(X, from^#(s(X))) , 2: length^#(cons(X, Y)) -> c_2(length1^#(Y)) , 3: length^#(nil()) -> c_3() , 4: length1^#(X) -> c_4(length^#(X)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_1(X, from^#(s(X))) , length^#(cons(X, Y)) -> c_2(length1^#(Y)) , length1^#(X) -> c_4(length^#(X)) } Strict Trs: { from(X) -> cons(X, from(s(X))) , length(cons(X, Y)) -> s(length1(Y)) , length(nil()) -> 0() , length1(X) -> length(X) } Weak DPs: { length^#(nil()) -> c_3() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..