MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { and(tt(), X) -> X , plus(N, 0()) -> N , plus(N, s(M)) -> s(plus(N, M)) , x(N, 0()) -> 0() , x(N, s(M)) -> plus(x(N, M), N) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { and^#(tt(), X) -> c_1() , plus^#(N, 0()) -> c_2() , plus^#(N, s(M)) -> c_3(plus^#(N, M)) , x^#(N, 0()) -> c_4() , x^#(N, s(M)) -> c_5(plus^#(x(N, M), N), x^#(N, M)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { and^#(tt(), X) -> c_1() , plus^#(N, 0()) -> c_2() , plus^#(N, s(M)) -> c_3(plus^#(N, M)) , x^#(N, 0()) -> c_4() , x^#(N, s(M)) -> c_5(plus^#(x(N, M), N), x^#(N, M)) } Weak Trs: { and(tt(), X) -> X , plus(N, 0()) -> N , plus(N, s(M)) -> s(plus(N, M)) , x(N, 0()) -> 0() , x(N, s(M)) -> plus(x(N, M), N) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,2,4} by applications of Pre({1,2,4}) = {3,5}. Here rules are labeled as follows: DPs: { 1: and^#(tt(), X) -> c_1() , 2: plus^#(N, 0()) -> c_2() , 3: plus^#(N, s(M)) -> c_3(plus^#(N, M)) , 4: x^#(N, 0()) -> c_4() , 5: x^#(N, s(M)) -> c_5(plus^#(x(N, M), N), x^#(N, M)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { plus^#(N, s(M)) -> c_3(plus^#(N, M)) , x^#(N, s(M)) -> c_5(plus^#(x(N, M), N), x^#(N, M)) } Weak DPs: { and^#(tt(), X) -> c_1() , plus^#(N, 0()) -> c_2() , x^#(N, 0()) -> c_4() } Weak Trs: { and(tt(), X) -> X , plus(N, 0()) -> N , plus(N, s(M)) -> s(plus(N, M)) , x(N, 0()) -> 0() , x(N, s(M)) -> plus(x(N, 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. { and^#(tt(), X) -> c_1() , plus^#(N, 0()) -> c_2() , x^#(N, 0()) -> c_4() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { plus^#(N, s(M)) -> c_3(plus^#(N, M)) , x^#(N, s(M)) -> c_5(plus^#(x(N, M), N), x^#(N, M)) } Weak Trs: { and(tt(), X) -> X , plus(N, 0()) -> N , plus(N, s(M)) -> s(plus(N, M)) , x(N, 0()) -> 0() , x(N, s(M)) -> plus(x(N, M), N) } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { plus(N, 0()) -> N , plus(N, s(M)) -> s(plus(N, M)) , x(N, 0()) -> 0() , x(N, s(M)) -> plus(x(N, M), N) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { plus^#(N, s(M)) -> c_3(plus^#(N, M)) , x^#(N, s(M)) -> c_5(plus^#(x(N, M), N), x^#(N, M)) } Weak Trs: { plus(N, 0()) -> N , plus(N, s(M)) -> s(plus(N, M)) , x(N, 0()) -> 0() , x(N, s(M)) -> plus(x(N, 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..