MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { from(X) -> cons(X, from(s(X))) , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , first(0(), Z) -> nil() , sel(s(X), cons(Y, Z)) -> sel(X, Z) , sel(0(), cons(X, Z)) -> 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)). [from](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x1 + [1] x2 + [0] [s](x1) = [0] [first](x1, x2) = [1] x2 + [0] [0] = [7] [nil] = [7] [sel](x1, x2) = [1] x2 + [4] The following symbols are considered usable {from, first, sel} The order satisfies the following ordering constraints: [from(X)] = [1] X + [0] >= [1] X + [0] = [cons(X, from(s(X)))] [first(s(X), cons(Y, Z))] = [1] Z + [1] Y + [0] >= [1] Z + [1] Y + [0] = [cons(Y, first(X, Z))] [first(0(), Z)] = [1] Z + [0] ? [7] = [nil()] [sel(s(X), cons(Y, Z))] = [1] Z + [1] Y + [4] >= [1] Z + [4] = [sel(X, Z)] [sel(0(), cons(X, Z))] = [1] X + [1] Z + [4] > [1] X + [0] = [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))) , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , first(0(), Z) -> nil() , sel(s(X), cons(Y, Z)) -> sel(X, Z) } Weak Trs: { sel(0(), cons(X, Z)) -> X } 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)). [from](x1) = [1] x1 + [7] [cons](x1, x2) = [1] x1 + [1] x2 + [7] [s](x1) = [0] [first](x1, x2) = [1] x2 + [7] [0] = [7] [nil] = [6] [sel](x1, x2) = [1] x2 + [0] The following symbols are considered usable {from, first, sel} The order satisfies the following ordering constraints: [from(X)] = [1] X + [7] ? [1] X + [14] = [cons(X, from(s(X)))] [first(s(X), cons(Y, Z))] = [1] Z + [1] Y + [14] >= [1] Z + [1] Y + [14] = [cons(Y, first(X, Z))] [first(0(), Z)] = [1] Z + [7] > [6] = [nil()] [sel(s(X), cons(Y, Z))] = [1] Z + [1] Y + [7] > [1] Z + [0] = [sel(X, Z)] [sel(0(), cons(X, Z))] = [1] X + [1] Z + [7] > [1] X + [0] = [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))) , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) } Weak Trs: { first(0(), Z) -> nil() , sel(s(X), cons(Y, Z)) -> sel(X, Z) , sel(0(), cons(X, Z)) -> 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: 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: 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)). [from](x1) = [0 1] x1 + [0] [0 0] [0] [cons](x1, x2) = [0 1] x1 + [1 0] x2 + [0] [1 0] [0 1] [0] [s](x1) = [1 0] x1 + [4] [0 0] [0] [first](x1, x2) = [1 0] x1 + [1 0] x2 + [4] [1 0] [0 1] [4] [0] = [0] [0] [nil] = [4] [4] [sel](x1, x2) = [1 0] x1 + [0 1] x2 + [7] [1 0] [1 0] [3] The following symbols are considered usable {from, first, sel} The order satisfies the following ordering constraints: [from(X)] = [0 1] X + [0] [0 0] [0] ? [0 1] X + [0] [1 0] [0] = [cons(X, from(s(X)))] [first(s(X), cons(Y, Z))] = [1 0] X + [1 0] Z + [0 1] Y + [8] [1 0] [0 1] [1 0] [8] > [1 0] X + [1 0] Z + [0 1] Y + [4] [1 0] [0 1] [1 0] [4] = [cons(Y, first(X, Z))] [first(0(), Z)] = [1 0] Z + [4] [0 1] [4] >= [4] [4] = [nil()] [sel(s(X), cons(Y, Z))] = [1 0] X + [0 1] Z + [1 0] Y + [11] [1 0] [1 0] [0 1] [7] > [1 0] X + [0 1] Z + [7] [1 0] [1 0] [3] = [sel(X, Z)] [sel(0(), cons(X, Z))] = [1 0] X + [0 1] Z + [7] [0 1] [1 0] [3] > [1 0] X + [0] [0 1] [0] = [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(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , first(0(), Z) -> nil() , sel(s(X), cons(Y, Z)) -> sel(X, Z) , sel(0(), cons(X, Z)) -> 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: 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: { from^#(X) -> c_1(from^#(s(X))) , first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) , first^#(0(), Z) -> c_3() , sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z)) , sel^#(0(), cons(X, Z)) -> c_5() } 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(from^#(s(X))) , first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) , first^#(0(), Z) -> c_3() , sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z)) , sel^#(0(), cons(X, Z)) -> c_5() } Strict Trs: { from(X) -> cons(X, from(s(X))) , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , first(0(), Z) -> nil() , sel(s(X), cons(Y, Z)) -> sel(X, Z) , sel(0(), cons(X, Z)) -> 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: { from^#(X) -> c_1(from^#(s(X))) , first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) , first^#(0(), Z) -> c_3() , sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z)) , sel^#(0(), cons(X, Z)) -> c_5() } 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_1) = {1}, Uargs(c_2) = {1}, Uargs(c_4) = {1} TcT has computed the following constructor-restricted matrix interpretation. [cons](x1, x2) = [1 0] x2 + [0] [0 0] [0] [s](x1) = [1 0] x1 + [0] [0 0] [0] [0] = [0] [0] [from^#](x1) = [1 1] x1 + [2] [2 1] [2] [c_1](x1) = [1 0] x1 + [2] [0 1] [2] [first^#](x1, x2) = [0] [2] [c_2](x1) = [1 0] x1 + [2] [0 1] [0] [c_3] = [2] [1] [sel^#](x1, x2) = [2] [2] [c_4](x1) = [1 0] x1 + [2] [0 1] [0] [c_5] = [1] [1] The following symbols are considered usable {from^#, first^#, sel^#} The order satisfies the following ordering constraints: [from^#(X)] = [1 1] X + [2] [2 1] [2] ? [1 0] X + [4] [2 0] [4] = [c_1(from^#(s(X)))] [first^#(s(X), cons(Y, Z))] = [0] [2] ? [2] [2] = [c_2(first^#(X, Z))] [first^#(0(), Z)] = [0] [2] ? [2] [1] = [c_3()] [sel^#(s(X), cons(Y, Z))] = [2] [2] ? [4] [2] = [c_4(sel^#(X, Z))] [sel^#(0(), cons(X, Z))] = [2] [2] > [1] [1] = [c_5()] 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(from^#(s(X))) , first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) , first^#(0(), Z) -> c_3() , sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z)) } Weak DPs: { sel^#(0(), cons(X, Z)) -> c_5() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {3} by applications of Pre({3}) = {2}. Here rules are labeled as follows: DPs: { 1: from^#(X) -> c_1(from^#(s(X))) , 2: first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) , 3: first^#(0(), Z) -> c_3() , 4: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z)) , 5: sel^#(0(), cons(X, Z)) -> c_5() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_1(from^#(s(X))) , first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) , sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z)) } Weak DPs: { first^#(0(), Z) -> c_3() , sel^#(0(), cons(X, Z)) -> c_5() } 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(), Z) -> c_3() , sel^#(0(), cons(X, Z)) -> c_5() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_1(from^#(s(X))) , first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) , sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(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: We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 3: sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, 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] [first](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [0] [nil] = [0] [sel](x1, x2) = [7] x1 + [7] x2 + [0] [from^#](x1) = [0] [c_1](x1) = [1] x1 + [0] [first^#](x1, x2) = [0] [c_2](x1) = [4] x1 + [0] [c_3] = [0] [sel^#](x1, x2) = [2] x2 + [0] [c_4](x1) = [1] x1 + [1] [c_5] = [0] The following symbols are considered usable {from^#, first^#, sel^#} The order satisfies the following ordering constraints: [from^#(X)] = [0] >= [0] = [c_1(from^#(s(X)))] [first^#(s(X), cons(Y, Z))] = [0] >= [0] = [c_2(first^#(X, Z))] [sel^#(s(X), cons(Y, Z))] = [2] Z + [2] Y + [8] > [2] Z + [1] = [c_4(sel^#(X, Z))] 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(from^#(s(X))) , first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) } Weak DPs: { sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(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. { sel^#(s(X), cons(Y, Z)) -> c_4(sel^#(X, Z)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_1(from^#(s(X))) , first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 2: first^#(s(X), cons(Y, Z)) -> c_2(first^#(X, Z)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {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 + [7] [s](x1) = [1] x1 + [2] [first](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [0] [nil] = [0] [sel](x1, x2) = [7] x1 + [7] x2 + [0] [from^#](x1) = [0] [c_1](x1) = [1] x1 + [0] [first^#](x1, x2) = [4] x1 + [0] [c_2](x1) = [1] x1 + [5] [c_3] = [0] [sel^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_4](x1) = [7] x1 + [0] [c_5] = [0] The following symbols are considered usable {from^#, first^#} The order satisfies the following ordering constraints: [from^#(X)] = [0] >= [0] = [c_1(from^#(s(X)))] [first^#(s(X), cons(Y, Z))] = [4] X + [8] > [4] X + [5] = [c_2(first^#(X, Z))] 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(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_1(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..