MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , reverse(nil()) -> nil() , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) , shuffle(nil()) -> nil() , shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { app^#(nil(), y) -> c_1() , app^#(add(n, x), y) -> c_2(app^#(x, y)) , reverse^#(nil()) -> c_3() , reverse^#(add(n, x)) -> c_4(app^#(reverse(x), add(n, nil())), reverse^#(x)) , shuffle^#(nil()) -> c_5() , shuffle^#(add(n, x)) -> c_6(shuffle^#(reverse(x)), reverse^#(x)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { app^#(nil(), y) -> c_1() , app^#(add(n, x), y) -> c_2(app^#(x, y)) , reverse^#(nil()) -> c_3() , reverse^#(add(n, x)) -> c_4(app^#(reverse(x), add(n, nil())), reverse^#(x)) , shuffle^#(nil()) -> c_5() , shuffle^#(add(n, x)) -> c_6(shuffle^#(reverse(x)), reverse^#(x)) } Weak Trs: { app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , reverse(nil()) -> nil() , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) , shuffle(nil()) -> nil() , shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,3,5} by applications of Pre({1,3,5}) = {2,4,6}. Here rules are labeled as follows: DPs: { 1: app^#(nil(), y) -> c_1() , 2: app^#(add(n, x), y) -> c_2(app^#(x, y)) , 3: reverse^#(nil()) -> c_3() , 4: reverse^#(add(n, x)) -> c_4(app^#(reverse(x), add(n, nil())), reverse^#(x)) , 5: shuffle^#(nil()) -> c_5() , 6: shuffle^#(add(n, x)) -> c_6(shuffle^#(reverse(x)), reverse^#(x)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { app^#(add(n, x), y) -> c_2(app^#(x, y)) , reverse^#(add(n, x)) -> c_4(app^#(reverse(x), add(n, nil())), reverse^#(x)) , shuffle^#(add(n, x)) -> c_6(shuffle^#(reverse(x)), reverse^#(x)) } Weak DPs: { app^#(nil(), y) -> c_1() , reverse^#(nil()) -> c_3() , shuffle^#(nil()) -> c_5() } Weak Trs: { app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , reverse(nil()) -> nil() , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) , shuffle(nil()) -> nil() , shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) } 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. { app^#(nil(), y) -> c_1() , reverse^#(nil()) -> c_3() , shuffle^#(nil()) -> c_5() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { app^#(add(n, x), y) -> c_2(app^#(x, y)) , reverse^#(add(n, x)) -> c_4(app^#(reverse(x), add(n, nil())), reverse^#(x)) , shuffle^#(add(n, x)) -> c_6(shuffle^#(reverse(x)), reverse^#(x)) } Weak Trs: { app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , reverse(nil()) -> nil() , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) , shuffle(nil()) -> nil() , shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , reverse(nil()) -> nil() , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { app^#(add(n, x), y) -> c_2(app^#(x, y)) , reverse^#(add(n, x)) -> c_4(app^#(reverse(x), add(n, nil())), reverse^#(x)) , shuffle^#(add(n, x)) -> c_6(shuffle^#(reverse(x)), reverse^#(x)) } Weak Trs: { app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , reverse(nil()) -> nil() , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) } 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..