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