MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , from(X) -> cons(X, from(s(X))) } 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: The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(cons) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [first](x1, x2) = [1] x1 + [1] x2 + [0] [0] = [1] [nil] = [0] [s](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x2 + [0] [from](x1) = [1] x1 + [0] The following symbols are considered usable {first, from} The order satisfies the following ordering constraints: [first(0(), X)] = [1] X + [1] > [0] = [nil()] [first(s(X), cons(Y, Z))] = [1] X + [1] Z + [0] >= [1] X + [1] Z + [0] = [cons(Y, first(X, Z))] [from(X)] = [1] X + [0] >= [1] X + [0] = [cons(X, from(s(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: { first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , from(X) -> cons(X, from(s(X))) } Weak Trs: { first(0(), X) -> nil() } Obligation: innermost runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(cons) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [first](x1, x2) = [1] x1 + [1] x2 + [0] [0] = [7] [nil] = [7] [s](x1) = [1] x1 + [4] [cons](x1, x2) = [1] x2 + [0] [from](x1) = [0] The following symbols are considered usable {first, from} The order satisfies the following ordering constraints: [first(0(), X)] = [1] X + [7] >= [7] = [nil()] [first(s(X), cons(Y, Z))] = [1] X + [1] Z + [4] > [1] X + [1] Z + [0] = [cons(Y, first(X, Z))] [from(X)] = [0] >= [0] = [cons(X, from(s(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: { first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) } 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) '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 input cannot be shown compatible 2) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The input cannot be shown compatible 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. 2) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: We add the following innermost weak dependency pairs: Strict DPs: { first^#(0(), X) -> c_1() , first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) , from^#(X) -> c_3(from^#(s(X))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { first^#(0(), X) -> c_1() , first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) , from^#(X) -> c_3(from^#(s(X))) } Strict Trs: { first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , from(X) -> cons(X, from(s(X))) } 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: { first^#(0(), X) -> c_1() , first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) , from^#(X) -> c_3(from^#(s(X))) } 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_2) = {1}, Uargs(c_3) = {1} TcT has computed the following constructor-restricted matrix interpretation. [0] = [2] [1] [s](x1) = [1 0] x1 + [0] [0 0] [0] [cons](x1, x2) = [1 0] x2 + [0] [0 0] [0] [first^#](x1, x2) = [2] [2] [c_1] = [1] [1] [c_2](x1) = [1 0] x1 + [2] [0 1] [0] [from^#](x1) = [0 0] x1 + [1] [2 0] [2] [c_3](x1) = [1 0] x1 + [1] [0 1] [1] The following symbols are considered usable {first^#, from^#} The order satisfies the following ordering constraints: [first^#(0(), X)] = [2] [2] > [1] [1] = [c_1()] [first^#(s(X), cons(Y, Z))] = [2] [2] ? [4] [2] = [c_2(first^#(X, Z))] [from^#(X)] = [0 0] X + [1] [2 0] [2] ? [0 0] X + [2] [2 0] [3] = [c_3(from^#(s(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: { first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) , from^#(X) -> c_3(from^#(s(X))) } Weak DPs: { first^#(0(), X) -> c_1() } 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. { first^#(0(), X) -> c_1() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) , from^#(X) -> c_3(from^#(s(X))) } 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: We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_2) = {1}, Uargs(c_3) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [first](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [0] [nil] = [0] [s](x1) = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [2] [from](x1) = [7] x1 + [0] [first^#](x1, x2) = [4] x2 + [0] [c_1] = [0] [c_2](x1) = [1] x1 + [1] [from^#](x1) = [0] [c_3](x1) = [1] x1 + [0] The following symbols are considered usable {first^#, from^#} The order satisfies the following ordering constraints: [first^#(s(X), cons(Y, Z))] = [4] Y + [4] Z + [8] > [4] Z + [1] = [c_2(first^#(X, Z))] [from^#(X)] = [0] >= [0] = [c_3(from^#(s(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_3(from^#(s(X))) } Weak DPs: { first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) } 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. { first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_3(from^#(s(X))) } 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) '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..