MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(tt(), X) -> mark(X) , mark(tt()) -> tt() , mark(0()) -> 0() , mark(s(X)) -> s(mark(X)) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, 0()) -> mark(N) , a__plus(N, s(M)) -> s(a__plus(mark(N), mark(M))) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { a__and^#(X1, X2) -> c_1() , a__and^#(tt(), X) -> c_2(mark^#(X)) , mark^#(tt()) -> c_3() , mark^#(0()) -> c_4() , mark^#(s(X)) -> c_5(mark^#(X)) , mark^#(and(X1, X2)) -> c_6(a__and^#(mark(X1), X2), mark^#(X1)) , mark^#(plus(X1, X2)) -> c_7(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , a__plus^#(X1, X2) -> c_8() , a__plus^#(N, 0()) -> c_9(mark^#(N)) , a__plus^#(N, s(M)) -> c_10(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__and^#(X1, X2) -> c_1() , a__and^#(tt(), X) -> c_2(mark^#(X)) , mark^#(tt()) -> c_3() , mark^#(0()) -> c_4() , mark^#(s(X)) -> c_5(mark^#(X)) , mark^#(and(X1, X2)) -> c_6(a__and^#(mark(X1), X2), mark^#(X1)) , mark^#(plus(X1, X2)) -> c_7(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , a__plus^#(X1, X2) -> c_8() , a__plus^#(N, 0()) -> c_9(mark^#(N)) , a__plus^#(N, s(M)) -> c_10(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(tt(), X) -> mark(X) , mark(tt()) -> tt() , mark(0()) -> 0() , mark(s(X)) -> s(mark(X)) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, 0()) -> mark(N) , a__plus(N, s(M)) -> s(a__plus(mark(N), mark(M))) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,3,4,8} by applications of Pre({1,3,4,8}) = {2,5,6,7,9,10}. Here rules are labeled as follows: DPs: { 1: a__and^#(X1, X2) -> c_1() , 2: a__and^#(tt(), X) -> c_2(mark^#(X)) , 3: mark^#(tt()) -> c_3() , 4: mark^#(0()) -> c_4() , 5: mark^#(s(X)) -> c_5(mark^#(X)) , 6: mark^#(and(X1, X2)) -> c_6(a__and^#(mark(X1), X2), mark^#(X1)) , 7: mark^#(plus(X1, X2)) -> c_7(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , 8: a__plus^#(X1, X2) -> c_8() , 9: a__plus^#(N, 0()) -> c_9(mark^#(N)) , 10: a__plus^#(N, s(M)) -> c_10(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__and^#(tt(), X) -> c_2(mark^#(X)) , mark^#(s(X)) -> c_5(mark^#(X)) , mark^#(and(X1, X2)) -> c_6(a__and^#(mark(X1), X2), mark^#(X1)) , mark^#(plus(X1, X2)) -> c_7(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , a__plus^#(N, 0()) -> c_9(mark^#(N)) , a__plus^#(N, s(M)) -> c_10(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) } Weak DPs: { a__and^#(X1, X2) -> c_1() , mark^#(tt()) -> c_3() , mark^#(0()) -> c_4() , a__plus^#(X1, X2) -> c_8() } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(tt(), X) -> mark(X) , mark(tt()) -> tt() , mark(0()) -> 0() , mark(s(X)) -> s(mark(X)) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, 0()) -> mark(N) , a__plus(N, s(M)) -> s(a__plus(mark(N), mark(M))) } 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. { a__and^#(X1, X2) -> c_1() , mark^#(tt()) -> c_3() , mark^#(0()) -> c_4() , a__plus^#(X1, X2) -> c_8() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__and^#(tt(), X) -> c_2(mark^#(X)) , mark^#(s(X)) -> c_5(mark^#(X)) , mark^#(and(X1, X2)) -> c_6(a__and^#(mark(X1), X2), mark^#(X1)) , mark^#(plus(X1, X2)) -> c_7(a__plus^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2)) , a__plus^#(N, 0()) -> c_9(mark^#(N)) , a__plus^#(N, s(M)) -> c_10(a__plus^#(mark(N), mark(M)), mark^#(N), mark^#(M)) } Weak Trs: { a__and(X1, X2) -> and(X1, X2) , a__and(tt(), X) -> mark(X) , mark(tt()) -> tt() , mark(0()) -> 0() , mark(s(X)) -> s(mark(X)) , mark(and(X1, X2)) -> a__and(mark(X1), X2) , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(N, 0()) -> mark(N) , a__plus(N, s(M)) -> s(a__plus(mark(N), mark(M))) } 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..