MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { zeros() -> cons(0(), zeros()) , U11(tt(), L) -> U12(tt(), L) , U12(tt(), L) -> s(length(L)) , length(cons(N, L)) -> U11(tt(), L) , length(nil()) -> 0() , U21(tt(), IL, M, N) -> U22(tt(), IL, M, N) , U22(tt(), IL, M, N) -> U23(tt(), IL, M, N) , U23(tt(), IL, M, N) -> cons(N, take(M, IL)) , take(0(), IL) -> nil() , take(s(M), cons(N, IL)) -> U21(tt(), IL, M, N) } Obligation: innermost runtime complexity Answer: MAYBE We add following weak dependency pairs: Strict DPs: { zeros^#() -> c_1(zeros^#()) , U11^#(tt(), L) -> c_2(U12^#(tt(), L)) , U12^#(tt(), L) -> c_3(length^#(L)) , length^#(cons(N, L)) -> c_4(U11^#(tt(), L)) , length^#(nil()) -> c_5() , U21^#(tt(), IL, M, N) -> c_6(U22^#(tt(), IL, M, N)) , U22^#(tt(), IL, M, N) -> c_7(U23^#(tt(), IL, M, N)) , U23^#(tt(), IL, M, N) -> c_8(take^#(M, IL)) , take^#(0(), IL) -> c_9() , take^#(s(M), cons(N, IL)) -> c_10(U21^#(tt(), IL, M, N)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { zeros^#() -> c_1(zeros^#()) , U11^#(tt(), L) -> c_2(U12^#(tt(), L)) , U12^#(tt(), L) -> c_3(length^#(L)) , length^#(cons(N, L)) -> c_4(U11^#(tt(), L)) , length^#(nil()) -> c_5() , U21^#(tt(), IL, M, N) -> c_6(U22^#(tt(), IL, M, N)) , U22^#(tt(), IL, M, N) -> c_7(U23^#(tt(), IL, M, N)) , U23^#(tt(), IL, M, N) -> c_8(take^#(M, IL)) , take^#(0(), IL) -> c_9() , take^#(s(M), cons(N, IL)) -> c_10(U21^#(tt(), IL, M, N)) } Strict Trs: { zeros() -> cons(0(), zeros()) , U11(tt(), L) -> U12(tt(), L) , U12(tt(), L) -> s(length(L)) , length(cons(N, L)) -> U11(tt(), L) , length(nil()) -> 0() , U21(tt(), IL, M, N) -> U22(tt(), IL, M, N) , U22(tt(), IL, M, N) -> U23(tt(), IL, M, N) , U23(tt(), IL, M, N) -> cons(N, take(M, IL)) , take(0(), IL) -> nil() , take(s(M), cons(N, IL)) -> U21(tt(), IL, M, N) } 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: { zeros^#() -> c_1(zeros^#()) , U11^#(tt(), L) -> c_2(U12^#(tt(), L)) , U12^#(tt(), L) -> c_3(length^#(L)) , length^#(cons(N, L)) -> c_4(U11^#(tt(), L)) , length^#(nil()) -> c_5() , U21^#(tt(), IL, M, N) -> c_6(U22^#(tt(), IL, M, N)) , U22^#(tt(), IL, M, N) -> c_7(U23^#(tt(), IL, M, N)) , U23^#(tt(), IL, M, N) -> c_8(take^#(M, IL)) , take^#(0(), IL) -> c_9() , take^#(s(M), cons(N, IL)) -> c_10(U21^#(tt(), IL, M, N)) } 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_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_10) = {1} TcT has computed following constructor-restricted matrix interpretation. [cons](x1, x2) = [1] x1 + [1] x2 + [0] [0] = [2] [tt] = [2] [s](x1) = [1] x1 + [2] [nil] = [2] [zeros^#] = [2] [c_1](x1) = [1] x1 + [1] [U11^#](x1, x2) = [0] [c_2](x1) = [1] x1 + [1] [U12^#](x1, x2) = [0] [c_3](x1) = [1] x1 + [2] [length^#](x1) = [0] [c_4](x1) = [1] x1 + [1] [c_5] = [1] [U21^#](x1, x2, x3, x4) = [2] x1 + [1] x2 + [1] x4 + [0] [c_6](x1) = [1] x1 + [1] [U22^#](x1, x2, x3, x4) = [1] x2 + [1] x4 + [0] [c_7](x1) = [1] x1 + [1] [U23^#](x1, x2, x3, x4) = [1] x2 + [1] x4 + [0] [c_8](x1) = [1] x1 + [1] [take^#](x1, x2) = [1] x2 + [0] [c_9] = [1] [c_10](x1) = [1] x1 + [1] This order satisfies following ordering constraints: 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: { zeros^#() -> c_1(zeros^#()) , U11^#(tt(), L) -> c_2(U12^#(tt(), L)) , U12^#(tt(), L) -> c_3(length^#(L)) , length^#(cons(N, L)) -> c_4(U11^#(tt(), L)) , length^#(nil()) -> c_5() , U22^#(tt(), IL, M, N) -> c_7(U23^#(tt(), IL, M, N)) , U23^#(tt(), IL, M, N) -> c_8(take^#(M, IL)) , take^#(0(), IL) -> c_9() , take^#(s(M), cons(N, IL)) -> c_10(U21^#(tt(), IL, M, N)) } Weak DPs: { U21^#(tt(), IL, M, N) -> c_6(U22^#(tt(), IL, M, N)) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {5,8,9} by applications of Pre({5,8,9}) = {3,7}. Here rules are labeled as follows: DPs: { 1: zeros^#() -> c_1(zeros^#()) , 2: U11^#(tt(), L) -> c_2(U12^#(tt(), L)) , 3: U12^#(tt(), L) -> c_3(length^#(L)) , 4: length^#(cons(N, L)) -> c_4(U11^#(tt(), L)) , 5: length^#(nil()) -> c_5() , 6: U22^#(tt(), IL, M, N) -> c_7(U23^#(tt(), IL, M, N)) , 7: U23^#(tt(), IL, M, N) -> c_8(take^#(M, IL)) , 8: take^#(0(), IL) -> c_9() , 9: take^#(s(M), cons(N, IL)) -> c_10(U21^#(tt(), IL, M, N)) , 10: U21^#(tt(), IL, M, N) -> c_6(U22^#(tt(), IL, M, N)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { zeros^#() -> c_1(zeros^#()) , U11^#(tt(), L) -> c_2(U12^#(tt(), L)) , U12^#(tt(), L) -> c_3(length^#(L)) , length^#(cons(N, L)) -> c_4(U11^#(tt(), L)) , U22^#(tt(), IL, M, N) -> c_7(U23^#(tt(), IL, M, N)) , U23^#(tt(), IL, M, N) -> c_8(take^#(M, IL)) } Weak DPs: { length^#(nil()) -> c_5() , U21^#(tt(), IL, M, N) -> c_6(U22^#(tt(), IL, M, N)) , take^#(0(), IL) -> c_9() , take^#(s(M), cons(N, IL)) -> c_10(U21^#(tt(), IL, M, N)) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {6} by applications of Pre({6}) = {5}. Here rules are labeled as follows: DPs: { 1: zeros^#() -> c_1(zeros^#()) , 2: U11^#(tt(), L) -> c_2(U12^#(tt(), L)) , 3: U12^#(tt(), L) -> c_3(length^#(L)) , 4: length^#(cons(N, L)) -> c_4(U11^#(tt(), L)) , 5: U22^#(tt(), IL, M, N) -> c_7(U23^#(tt(), IL, M, N)) , 6: U23^#(tt(), IL, M, N) -> c_8(take^#(M, IL)) , 7: length^#(nil()) -> c_5() , 8: U21^#(tt(), IL, M, N) -> c_6(U22^#(tt(), IL, M, N)) , 9: take^#(0(), IL) -> c_9() , 10: take^#(s(M), cons(N, IL)) -> c_10(U21^#(tt(), IL, M, N)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { zeros^#() -> c_1(zeros^#()) , U11^#(tt(), L) -> c_2(U12^#(tt(), L)) , U12^#(tt(), L) -> c_3(length^#(L)) , length^#(cons(N, L)) -> c_4(U11^#(tt(), L)) , U22^#(tt(), IL, M, N) -> c_7(U23^#(tt(), IL, M, N)) } Weak DPs: { length^#(nil()) -> c_5() , U21^#(tt(), IL, M, N) -> c_6(U22^#(tt(), IL, M, N)) , U23^#(tt(), IL, M, N) -> c_8(take^#(M, IL)) , take^#(0(), IL) -> c_9() , take^#(s(M), cons(N, IL)) -> c_10(U21^#(tt(), IL, M, N)) } 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. { length^#(nil()) -> c_5() , take^#(0(), IL) -> c_9() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { zeros^#() -> c_1(zeros^#()) , U11^#(tt(), L) -> c_2(U12^#(tt(), L)) , U12^#(tt(), L) -> c_3(length^#(L)) , length^#(cons(N, L)) -> c_4(U11^#(tt(), L)) , U22^#(tt(), IL, M, N) -> c_7(U23^#(tt(), IL, M, N)) } Weak DPs: { U21^#(tt(), IL, M, N) -> c_6(U22^#(tt(), IL, M, N)) , U23^#(tt(), IL, M, N) -> c_8(take^#(M, IL)) , take^#(s(M), cons(N, IL)) -> c_10(U21^#(tt(), IL, M, N)) } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrices' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrix interpretation of dimension 4' failed due to the following reason: The input cannot be shown compatible 2) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 3) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 4) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 5) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 6) 'matrix interpretation of dimension 1' failed due to the following reason: The input cannot be shown compatible 2) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. Arrrr..