MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { eq(X, Y) -> false() , eq(0(), 0()) -> true() , eq(s(X), s(Y)) -> eq(X, Y) , inf(X) -> cons(X, inf(s(X))) , take(0(), X) -> nil() , take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) , length(cons(X, L)) -> s(length(L)) , length(nil()) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE We add following weak dependency pairs: Strict DPs: { eq^#(X, Y) -> c_1() , eq^#(0(), 0()) -> c_2() , eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) , inf^#(X) -> c_4(inf^#(s(X))) , take^#(0(), X) -> c_5() , take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , length^#(cons(X, L)) -> c_7(length^#(L)) , length^#(nil()) -> c_8() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { eq^#(X, Y) -> c_1() , eq^#(0(), 0()) -> c_2() , eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) , inf^#(X) -> c_4(inf^#(s(X))) , take^#(0(), X) -> c_5() , take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , length^#(cons(X, L)) -> c_7(length^#(L)) , length^#(nil()) -> c_8() } Strict Trs: { eq(X, Y) -> false() , eq(0(), 0()) -> true() , eq(s(X), s(Y)) -> eq(X, Y) , inf(X) -> cons(X, inf(s(X))) , take(0(), X) -> nil() , take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) , length(cons(X, L)) -> s(length(L)) , length(nil()) -> 0() } 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: { eq^#(X, Y) -> c_1() , eq^#(0(), 0()) -> c_2() , eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) , inf^#(X) -> c_4(inf^#(s(X))) , take^#(0(), X) -> c_5() , take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , length^#(cons(X, L)) -> c_7(length^#(L)) , length^#(nil()) -> c_8() } 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_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1} TcT has computed following constructor-restricted matrix interpretation. [0] = [1] [s](x1) = [1] x1 + [1] [cons](x1, x2) = [1] x2 + [1] [nil] = [1] [eq^#](x1, x2) = [2] x1 + [2] x2 + [2] [c_1] = [1] [c_2] = [1] [c_3](x1) = [1] x1 + [2] [inf^#](x1) = [1] x1 + [2] [c_4](x1) = [1] x1 + [2] [take^#](x1, x2) = [2] x1 + [1] x2 + [1] [c_5] = [2] [c_6](x1) = [1] x1 + [1] [length^#](x1) = [2] x1 + [2] [c_7](x1) = [1] x1 + [1] [c_8] = [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: { inf^#(X) -> c_4(inf^#(s(X))) } Weak DPs: { eq^#(X, Y) -> c_1() , eq^#(0(), 0()) -> c_2() , eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) , take^#(0(), X) -> c_5() , take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , length^#(cons(X, L)) -> c_7(length^#(L)) , length^#(nil()) -> c_8() } 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. { eq^#(X, Y) -> c_1() , eq^#(0(), 0()) -> c_2() , eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) , take^#(0(), X) -> c_5() , take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , length^#(cons(X, L)) -> c_7(length^#(L)) , length^#(nil()) -> c_8() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { inf^#(X) -> c_4(inf^#(s(X))) } 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..