MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { and(true(), X) -> activate(X) , and(false(), Y) -> false() , activate(X) -> X , activate(n__add(X1, X2)) -> add(X1, X2) , activate(n__first(X1, X2)) -> first(X1, X2) , activate(n__from(X)) -> from(X) , activate(n__s(X)) -> s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) , add(X1, X2) -> n__add(X1, X2) , add(0(), X) -> activate(X) , add(s(X), Y) -> s(n__add(activate(X), activate(Y))) , s(X) -> n__s(X) , first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(activate(Y), n__first(activate(X), activate(Z))) , from(X) -> cons(activate(X), n__from(n__s(activate(X)))) , from(X) -> n__from(X) } Obligation: innermost runtime complexity Answer: MAYBE Arguments of following rules are not normal-forms: { add(s(X), Y) -> s(n__add(activate(X), activate(Y))) , first(s(X), cons(Y, Z)) -> cons(activate(Y), n__first(activate(X), activate(Z))) } All above mentioned rules can be savely removed. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { and(true(), X) -> activate(X) , and(false(), Y) -> false() , activate(X) -> X , activate(n__add(X1, X2)) -> add(X1, X2) , activate(n__first(X1, X2)) -> first(X1, X2) , activate(n__from(X)) -> from(X) , activate(n__s(X)) -> s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) , add(X1, X2) -> n__add(X1, X2) , add(0(), X) -> activate(X) , s(X) -> n__s(X) , first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , from(X) -> cons(activate(X), n__from(n__s(activate(X)))) , from(X) -> n__from(X) } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The input cannot be shown compatible 2) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The input cannot be shown compatible 3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: We add the following innermost weak dependency pairs: Strict DPs: { and^#(true(), X) -> c_1(activate^#(X)) , and^#(false(), Y) -> c_2() , activate^#(X) -> c_3() , activate^#(n__add(X1, X2)) -> c_4(add^#(X1, X2)) , activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2)) , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__s(X)) -> c_7(s^#(X)) , add^#(X1, X2) -> c_10() , add^#(0(), X) -> c_11(activate^#(X)) , first^#(X1, X2) -> c_13() , first^#(0(), X) -> c_14() , from^#(X) -> c_15(activate^#(X), activate^#(X)) , from^#(X) -> c_16() , s^#(X) -> c_12() , if^#(true(), X, Y) -> c_8(activate^#(X)) , if^#(false(), X, Y) -> c_9(activate^#(Y)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { and^#(true(), X) -> c_1(activate^#(X)) , and^#(false(), Y) -> c_2() , activate^#(X) -> c_3() , activate^#(n__add(X1, X2)) -> c_4(add^#(X1, X2)) , activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2)) , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__s(X)) -> c_7(s^#(X)) , add^#(X1, X2) -> c_10() , add^#(0(), X) -> c_11(activate^#(X)) , first^#(X1, X2) -> c_13() , first^#(0(), X) -> c_14() , from^#(X) -> c_15(activate^#(X), activate^#(X)) , from^#(X) -> c_16() , s^#(X) -> c_12() , if^#(true(), X, Y) -> c_8(activate^#(X)) , if^#(false(), X, Y) -> c_9(activate^#(Y)) } Strict Trs: { and(true(), X) -> activate(X) , and(false(), Y) -> false() , activate(X) -> X , activate(n__add(X1, X2)) -> add(X1, X2) , activate(n__first(X1, X2)) -> first(X1, X2) , activate(n__from(X)) -> from(X) , activate(n__s(X)) -> s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) , add(X1, X2) -> n__add(X1, X2) , add(0(), X) -> activate(X) , s(X) -> n__s(X) , first(X1, X2) -> n__first(X1, X2) , first(0(), X) -> nil() , from(X) -> cons(activate(X), n__from(n__s(activate(X)))) , from(X) -> n__from(X) } 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: { and^#(true(), X) -> c_1(activate^#(X)) , and^#(false(), Y) -> c_2() , activate^#(X) -> c_3() , activate^#(n__add(X1, X2)) -> c_4(add^#(X1, X2)) , activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2)) , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__s(X)) -> c_7(s^#(X)) , add^#(X1, X2) -> c_10() , add^#(0(), X) -> c_11(activate^#(X)) , first^#(X1, X2) -> c_13() , first^#(0(), X) -> c_14() , from^#(X) -> c_15(activate^#(X), activate^#(X)) , from^#(X) -> c_16() , s^#(X) -> c_12() , if^#(true(), X, Y) -> c_8(activate^#(X)) , if^#(false(), X, Y) -> c_9(activate^#(Y)) } 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_1) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(c_11) = {1}, Uargs(c_15) = {1, 2} TcT has computed the following constructor-restricted matrix interpretation. [true] = [1] [1] [false] = [2] [1] [0] = [2] [2] [n__add](x1, x2) = [1 2] x1 + [1 2] x2 + [1] [0 1] [0 1] [2] [n__first](x1, x2) = [1 2] x1 + [1 2] x2 + [2] [0 1] [0 1] [1] [n__from](x1) = [1 2] x1 + [2] [0 1] [1] [n__s](x1) = [1 2] x1 + [1] [0 1] [2] [and^#](x1, x2) = [2 2] x1 + [2 2] x2 + [1] [1 2] [1 2] [1] [c_1](x1) = [1 0] x1 + [1] [0 1] [1] [activate^#](x1) = [0 0] x1 + [2] [1 1] [1] [c_2] = [2] [1] [c_3] = [1] [1] [c_4](x1) = [1 0] x1 + [2] [0 1] [1] [add^#](x1, x2) = [0 0] x1 + [0 0] x2 + [1] [1 0] [2 1] [2] [c_5](x1) = [1 0] x1 + [2] [0 1] [1] [first^#](x1, x2) = [0 0] x1 + [0 0] x2 + [1] [2 1] [1 1] [1] [c_6](x1) = [1 0] x1 + [2] [0 1] [2] [from^#](x1) = [0 0] x1 + [1] [1 1] [1] [c_7](x1) = [1 0] x1 + [2] [0 1] [2] [s^#](x1) = [0 0] x1 + [1] [1 1] [1] [if^#](x1, x2, x3) = [1 2] x1 + [1 1] x2 + [1 2] x3 + [1] [2 2] [1 2] [2 1] [1] [c_8](x1) = [1 0] x1 + [1] [0 1] [1] [c_9](x1) = [1 0] x1 + [1] [0 1] [1] [c_10] = [0] [2] [c_11](x1) = [1 0] x1 + [1] [0 1] [1] [c_12] = [0] [1] [c_13] = [0] [1] [c_14] = [0] [2] [c_15](x1, x2) = [1 0] x1 + [1 0] x2 + [2] [0 1] [0 1] [1] [c_16] = [0] [1] The following symbols are considered usable {and^#, activate^#, add^#, first^#, from^#, s^#, if^#} The order satisfies the following ordering constraints: [and^#(true(), X)] = [2 2] X + [5] [1 2] [4] > [0 0] X + [3] [1 1] [2] = [c_1(activate^#(X))] [and^#(false(), Y)] = [2 2] Y + [7] [1 2] [5] > [2] [1] = [c_2()] [activate^#(X)] = [0 0] X + [2] [1 1] [1] > [1] [1] = [c_3()] [activate^#(n__add(X1, X2))] = [0 0] X1 + [0 0] X2 + [2] [1 3] [1 3] [4] ? [0 0] X1 + [0 0] X2 + [3] [1 0] [2 1] [3] = [c_4(add^#(X1, X2))] [activate^#(n__first(X1, X2))] = [0 0] X1 + [0 0] X2 + [2] [1 3] [1 3] [4] ? [0 0] X1 + [0 0] X2 + [3] [2 1] [1 1] [2] = [c_5(first^#(X1, X2))] [activate^#(n__from(X))] = [0 0] X + [2] [1 3] [4] ? [0 0] X + [3] [1 1] [3] = [c_6(from^#(X))] [activate^#(n__s(X))] = [0 0] X + [2] [1 3] [4] ? [0 0] X + [3] [1 1] [3] = [c_7(s^#(X))] [add^#(X1, X2)] = [0 0] X1 + [0 0] X2 + [1] [1 0] [2 1] [2] > [0] [2] = [c_10()] [add^#(0(), X)] = [0 0] X + [1] [2 1] [4] ? [0 0] X + [3] [1 1] [2] = [c_11(activate^#(X))] [first^#(X1, X2)] = [0 0] X1 + [0 0] X2 + [1] [2 1] [1 1] [1] > [0] [1] = [c_13()] [first^#(0(), X)] = [0 0] X + [1] [1 1] [7] > [0] [2] = [c_14()] [from^#(X)] = [0 0] X + [1] [1 1] [1] ? [0 0] X + [6] [2 2] [3] = [c_15(activate^#(X), activate^#(X))] [from^#(X)] = [0 0] X + [1] [1 1] [1] > [0] [1] = [c_16()] [s^#(X)] = [0 0] X + [1] [1 1] [1] > [0] [1] = [c_12()] [if^#(true(), X, Y)] = [1 1] X + [1 2] Y + [4] [1 2] [2 1] [5] > [0 0] X + [3] [1 1] [2] = [c_8(activate^#(X))] [if^#(false(), X, Y)] = [1 1] X + [1 2] Y + [5] [1 2] [2 1] [7] > [0 0] Y + [3] [1 1] [2] = [c_9(activate^#(Y))] 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: { activate^#(n__add(X1, X2)) -> c_4(add^#(X1, X2)) , activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2)) , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__s(X)) -> c_7(s^#(X)) , add^#(0(), X) -> c_11(activate^#(X)) , from^#(X) -> c_15(activate^#(X), activate^#(X)) } Weak DPs: { and^#(true(), X) -> c_1(activate^#(X)) , and^#(false(), Y) -> c_2() , activate^#(X) -> c_3() , add^#(X1, X2) -> c_10() , first^#(X1, X2) -> c_13() , first^#(0(), X) -> c_14() , from^#(X) -> c_16() , s^#(X) -> c_12() , if^#(true(), X, Y) -> c_8(activate^#(X)) , if^#(false(), X, Y) -> c_9(activate^#(Y)) } 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^#(false(), Y) -> c_2() , activate^#(X) -> c_3() , add^#(X1, X2) -> c_10() , first^#(X1, X2) -> c_13() , first^#(0(), X) -> c_14() , from^#(X) -> c_16() , s^#(X) -> c_12() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { activate^#(n__add(X1, X2)) -> c_4(add^#(X1, X2)) , activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2)) , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__s(X)) -> c_7(s^#(X)) , add^#(0(), X) -> c_11(activate^#(X)) , from^#(X) -> c_15(activate^#(X), activate^#(X)) } Weak DPs: { and^#(true(), X) -> c_1(activate^#(X)) , if^#(true(), X, Y) -> c_8(activate^#(X)) , if^#(false(), X, Y) -> c_9(activate^#(Y)) } Obligation: innermost runtime complexity Answer: MAYBE Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2)) , activate^#(n__s(X)) -> c_7(s^#(X)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { activate^#(n__add(X1, X2)) -> c_1(add^#(X1, X2)) , activate^#(n__first(X1, X2)) -> c_2() , activate^#(n__from(X)) -> c_3(from^#(X)) , activate^#(n__s(X)) -> c_4() , add^#(0(), X) -> c_5(activate^#(X)) , from^#(X) -> c_6(activate^#(X), activate^#(X)) } Weak DPs: { and^#(true(), X) -> c_7(activate^#(X)) , if^#(true(), X, Y) -> c_8(activate^#(X)) , if^#(false(), X, Y) -> c_9(activate^#(Y)) } Obligation: innermost runtime complexity Answer: MAYBE Consider the dependency graph 1: activate^#(n__add(X1, X2)) -> c_1(add^#(X1, X2)) -->_1 add^#(0(), X) -> c_5(activate^#(X)) :5 2: activate^#(n__first(X1, X2)) -> c_2() 3: activate^#(n__from(X)) -> c_3(from^#(X)) -->_1 from^#(X) -> c_6(activate^#(X), activate^#(X)) :6 4: activate^#(n__s(X)) -> c_4() 5: add^#(0(), X) -> c_5(activate^#(X)) -->_1 activate^#(n__s(X)) -> c_4() :4 -->_1 activate^#(n__from(X)) -> c_3(from^#(X)) :3 -->_1 activate^#(n__first(X1, X2)) -> c_2() :2 -->_1 activate^#(n__add(X1, X2)) -> c_1(add^#(X1, X2)) :1 6: from^#(X) -> c_6(activate^#(X), activate^#(X)) -->_2 activate^#(n__s(X)) -> c_4() :4 -->_1 activate^#(n__s(X)) -> c_4() :4 -->_2 activate^#(n__from(X)) -> c_3(from^#(X)) :3 -->_1 activate^#(n__from(X)) -> c_3(from^#(X)) :3 -->_2 activate^#(n__first(X1, X2)) -> c_2() :2 -->_1 activate^#(n__first(X1, X2)) -> c_2() :2 -->_2 activate^#(n__add(X1, X2)) -> c_1(add^#(X1, X2)) :1 -->_1 activate^#(n__add(X1, X2)) -> c_1(add^#(X1, X2)) :1 7: and^#(true(), X) -> c_7(activate^#(X)) -->_1 activate^#(n__s(X)) -> c_4() :4 -->_1 activate^#(n__from(X)) -> c_3(from^#(X)) :3 -->_1 activate^#(n__first(X1, X2)) -> c_2() :2 -->_1 activate^#(n__add(X1, X2)) -> c_1(add^#(X1, X2)) :1 8: if^#(true(), X, Y) -> c_8(activate^#(X)) -->_1 activate^#(n__s(X)) -> c_4() :4 -->_1 activate^#(n__from(X)) -> c_3(from^#(X)) :3 -->_1 activate^#(n__first(X1, X2)) -> c_2() :2 -->_1 activate^#(n__add(X1, X2)) -> c_1(add^#(X1, X2)) :1 9: if^#(false(), X, Y) -> c_9(activate^#(Y)) -->_1 activate^#(n__s(X)) -> c_4() :4 -->_1 activate^#(n__from(X)) -> c_3(from^#(X)) :3 -->_1 activate^#(n__first(X1, X2)) -> c_2() :2 -->_1 activate^#(n__add(X1, X2)) -> c_1(add^#(X1, X2)) :1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). { and^#(true(), X) -> c_7(activate^#(X)) , if^#(true(), X, Y) -> c_8(activate^#(X)) , if^#(false(), X, Y) -> c_9(activate^#(Y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { activate^#(n__add(X1, X2)) -> c_1(add^#(X1, X2)) , activate^#(n__first(X1, X2)) -> c_2() , activate^#(n__from(X)) -> c_3(from^#(X)) , activate^#(n__s(X)) -> c_4() , add^#(0(), X) -> c_5(activate^#(X)) , from^#(X) -> c_6(activate^#(X), activate^#(X)) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {2,4} by applications of Pre({2,4}) = {5,6}. Here rules are labeled as follows: DPs: { 1: activate^#(n__add(X1, X2)) -> c_1(add^#(X1, X2)) , 2: activate^#(n__first(X1, X2)) -> c_2() , 3: activate^#(n__from(X)) -> c_3(from^#(X)) , 4: activate^#(n__s(X)) -> c_4() , 5: add^#(0(), X) -> c_5(activate^#(X)) , 6: from^#(X) -> c_6(activate^#(X), activate^#(X)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { activate^#(n__add(X1, X2)) -> c_1(add^#(X1, X2)) , activate^#(n__from(X)) -> c_3(from^#(X)) , add^#(0(), X) -> c_5(activate^#(X)) , from^#(X) -> c_6(activate^#(X), activate^#(X)) } Weak DPs: { activate^#(n__first(X1, X2)) -> c_2() , activate^#(n__s(X)) -> c_4() } 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. { activate^#(n__first(X1, X2)) -> c_2() , activate^#(n__s(X)) -> c_4() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { activate^#(n__add(X1, X2)) -> c_1(add^#(X1, X2)) , activate^#(n__from(X)) -> c_3(from^#(X)) , add^#(0(), X) -> c_5(activate^#(X)) , from^#(X) -> c_6(activate^#(X), activate^#(X)) } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Polynomial Path Order (PS)' failed due to the following reason: The input cannot be shown compatible 2) 'Polynomial Path Order (PS)' failed due to the following reason: The input cannot be shown compatible 3) 'Fastest (timeout of 5 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. Arrrr..