MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { from(X) -> cons(X, from(s(X))) , head(cons(X, XS)) -> X , 2nd(cons(X, XS)) -> head(XS) , take(s(N), cons(X, XS)) -> cons(X, take(N, XS)) , take(0(), XS) -> nil() , sel(s(N), cons(X, XS)) -> sel(N, XS) , sel(0(), cons(X, XS)) -> 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 + [7] [cons](x1, x2) = [1] x1 + [1] x2 + [7] [s](x1) = [0] [head](x1) = [1] x1 + [7] [2nd](x1) = [1] x1 + [7] [take](x1, x2) = [1] x2 + [7] [0] = [7] [nil] = [6] [sel](x1, x2) = [1] x2 + [7] The following symbols are considered usable {from, head, 2nd, take, sel} The order satisfies the following ordering constraints: [from(X)] = [1] X + [7] ? [1] X + [14] = [cons(X, from(s(X)))] [head(cons(X, XS))] = [1] X + [1] XS + [14] > [1] X + [0] = [X] [2nd(cons(X, XS))] = [1] X + [1] XS + [14] > [1] XS + [7] = [head(XS)] [take(s(N), cons(X, XS))] = [1] X + [1] XS + [14] >= [1] X + [1] XS + [14] = [cons(X, take(N, XS))] [take(0(), XS)] = [1] XS + [7] > [6] = [nil()] [sel(s(N), cons(X, XS))] = [1] X + [1] XS + [14] > [1] XS + [7] = [sel(N, XS)] [sel(0(), cons(X, XS))] = [1] X + [1] XS + [14] > [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))) , take(s(N), cons(X, XS)) -> cons(X, take(N, XS)) } Weak Trs: { head(cons(X, XS)) -> X , 2nd(cons(X, XS)) -> head(XS) , take(0(), XS) -> nil() , sel(s(N), cons(X, XS)) -> sel(N, XS) , sel(0(), cons(X, XS)) -> 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) = [1 0] x1 + [0] [0 0] [0] [cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [0] [s](x1) = [0 0] x1 + [1] [1 1] [3] [head](x1) = [1 0] x1 + [7] [0 1] [7] [2nd](x1) = [1 0] x1 + [7] [0 1] [7] [take](x1, x2) = [1 1] x1 + [1 0] x2 + [4] [0 0] [0 1] [0] [0] = [0] [0] [nil] = [4] [0] [sel](x1, x2) = [1 1] x1 + [1 0] x2 + [7] [0 0] [0 1] [3] The following symbols are considered usable {from, head, 2nd, take, sel} The order satisfies the following ordering constraints: [from(X)] = [1 0] X + [0] [0 0] [0] ? [1 0] X + [1] [0 1] [0] = [cons(X, from(s(X)))] [head(cons(X, XS))] = [1 0] X + [1 0] XS + [7] [0 1] [0 1] [7] > [1 0] X + [0] [0 1] [0] = [X] [2nd(cons(X, XS))] = [1 0] X + [1 0] XS + [7] [0 1] [0 1] [7] >= [1 0] XS + [7] [0 1] [7] = [head(XS)] [take(s(N), cons(X, XS))] = [1 0] X + [1 0] XS + [1 1] N + [8] [0 1] [0 1] [0 0] [0] > [1 0] X + [1 0] XS + [1 1] N + [4] [0 1] [0 1] [0 0] [0] = [cons(X, take(N, XS))] [take(0(), XS)] = [1 0] XS + [4] [0 1] [0] >= [4] [0] = [nil()] [sel(s(N), cons(X, XS))] = [1 0] X + [1 0] XS + [1 1] N + [11] [0 1] [0 1] [0 0] [3] > [1 0] XS + [1 1] N + [7] [0 1] [0 0] [3] = [sel(N, XS)] [sel(0(), cons(X, XS))] = [1 0] X + [1 0] XS + [7] [0 1] [0 1] [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: { head(cons(X, XS)) -> X , 2nd(cons(X, XS)) -> head(XS) , take(s(N), cons(X, XS)) -> cons(X, take(N, XS)) , take(0(), XS) -> nil() , sel(s(N), cons(X, XS)) -> sel(N, XS) , sel(0(), cons(X, XS)) -> 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) '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 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))) , head^#(cons(X, XS)) -> c_2() , 2nd^#(cons(X, XS)) -> c_3(head^#(XS)) , take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS)) , take^#(0(), XS) -> c_5() , sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS)) , sel^#(0(), cons(X, XS)) -> c_7() } 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))) , head^#(cons(X, XS)) -> c_2() , 2nd^#(cons(X, XS)) -> c_3(head^#(XS)) , take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS)) , take^#(0(), XS) -> c_5() , sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS)) , sel^#(0(), cons(X, XS)) -> c_7() } Strict Trs: { from(X) -> cons(X, from(s(X))) , head(cons(X, XS)) -> X , 2nd(cons(X, XS)) -> head(XS) , take(s(N), cons(X, XS)) -> cons(X, take(N, XS)) , take(0(), XS) -> nil() , sel(s(N), cons(X, XS)) -> sel(N, XS) , sel(0(), cons(X, XS)) -> 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))) , head^#(cons(X, XS)) -> c_2() , 2nd^#(cons(X, XS)) -> c_3(head^#(XS)) , take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS)) , take^#(0(), XS) -> c_5() , sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS)) , sel^#(0(), cons(X, XS)) -> c_7() } 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_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {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] = [1] [0] [from^#](x1) = [1 1] x1 + [2] [1 1] [2] [c_1](x1) = [1 0] x1 + [2] [0 1] [2] [head^#](x1) = [2 0] x1 + [1] [1 0] [1] [c_2] = [0] [1] [2nd^#](x1) = [2 2] x1 + [2] [1 1] [2] [c_3](x1) = [1 0] x1 + [0] [0 1] [1] [take^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] [2 1] [2 2] [2] [c_4](x1) = [1 0] x1 + [2] [0 1] [0] [c_5] = [1] [0] [sel^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] [2 2] [1 2] [2] [c_6](x1) = [1 0] x1 + [2] [0 1] [0] [c_7] = [1] [0] The following symbols are considered usable {from^#, head^#, 2nd^#, take^#, sel^#} The order satisfies the following ordering constraints: [from^#(X)] = [1 1] X + [2] [1 1] [2] ? [1 0] X + [4] [1 0] [4] = [c_1(from^#(s(X)))] [head^#(cons(X, XS))] = [2 0] XS + [1] [1 0] [1] > [0] [1] = [c_2()] [2nd^#(cons(X, XS))] = [2 0] XS + [2] [1 0] [2] > [2 0] XS + [1] [1 0] [2] = [c_3(head^#(XS))] [take^#(s(N), cons(X, XS))] = [0 0] XS + [0 0] N + [2] [2 0] [2 0] [2] ? [0 0] XS + [0 0] N + [4] [2 2] [2 1] [2] = [c_4(take^#(N, XS))] [take^#(0(), XS)] = [0 0] XS + [2] [2 2] [4] > [1] [0] = [c_5()] [sel^#(s(N), cons(X, XS))] = [0 0] XS + [0 0] N + [2] [1 0] [2 0] [2] ? [0 0] XS + [0 0] N + [4] [1 2] [2 2] [2] = [c_6(sel^#(N, XS))] [sel^#(0(), cons(X, XS))] = [0 0] XS + [2] [1 0] [4] > [1] [0] = [c_7()] 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))) , take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS)) , sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS)) } Weak DPs: { head^#(cons(X, XS)) -> c_2() , 2nd^#(cons(X, XS)) -> c_3(head^#(XS)) , take^#(0(), XS) -> c_5() , sel^#(0(), cons(X, XS)) -> c_7() } 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. { head^#(cons(X, XS)) -> c_2() , 2nd^#(cons(X, XS)) -> c_3(head^#(XS)) , take^#(0(), XS) -> c_5() , sel^#(0(), cons(X, XS)) -> c_7() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_1(from^#(s(X))) , take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS)) , sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS)) } 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: { 2: take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {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] [head](x1) = [7] x1 + [0] [2nd](x1) = [7] x1 + [0] [take](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) = [4] x1 + [0] [head^#](x1) = [7] x1 + [0] [c_2] = [0] [2nd^#](x1) = [7] x1 + [0] [c_3](x1) = [7] x1 + [0] [take^#](x1, x2) = [2] x2 + [0] [c_4](x1) = [1] x1 + [5] [c_5] = [0] [sel^#](x1, x2) = [0] [c_6](x1) = [4] x1 + [0] [c_7] = [0] The following symbols are considered usable {from^#, take^#, sel^#} The order satisfies the following ordering constraints: [from^#(X)] = [0] >= [0] = [c_1(from^#(s(X)))] [take^#(s(N), cons(X, XS))] = [2] X + [2] XS + [8] > [2] XS + [5] = [c_4(take^#(N, XS))] [sel^#(s(N), cons(X, XS))] = [0] >= [0] = [c_6(sel^#(N, XS))] 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))) , sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS)) } Weak DPs: { take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS)) } 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. { take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_1(from^#(s(X))) , sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS)) } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 2: sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_6) = {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] [head](x1) = [7] x1 + [0] [2nd](x1) = [7] x1 + [0] [take](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) = [2] x1 + [0] [head^#](x1) = [7] x1 + [0] [c_2] = [0] [2nd^#](x1) = [7] x1 + [0] [c_3](x1) = [7] x1 + [0] [take^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_4](x1) = [7] x1 + [0] [c_5] = [0] [sel^#](x1, x2) = [2] x2 + [0] [c_6](x1) = [1] x1 + [1] [c_7] = [0] The following symbols are considered usable {from^#, sel^#} The order satisfies the following ordering constraints: [from^#(X)] = [0] >= [0] = [c_1(from^#(s(X)))] [sel^#(s(N), cons(X, XS))] = [2] X + [2] XS + [8] > [2] XS + [1] = [c_6(sel^#(N, XS))] 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: { sel^#(s(N), cons(X, XS)) -> c_6(sel^#(N, XS)) } 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(N), cons(X, XS)) -> c_6(sel^#(N, XS)) } 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..