MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { app(X1, X2) -> n__app(X1, X2) , app(nil(), YS) -> YS , app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS)) , nil() -> n__nil() , activate(X) -> X , activate(n__app(X1, X2)) -> app(X1, X2) , activate(n__from(X)) -> from(X) , activate(n__nil()) -> nil() , activate(n__zWadr(X1, X2)) -> zWadr(X1, X2) , from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , zWadr(X1, X2) -> n__zWadr(X1, X2) , zWadr(XS, nil()) -> nil() , zWadr(nil(), YS) -> nil() , zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil())), n__zWadr(activate(XS), activate(YS))) , prefix(L) -> cons(nil(), n__zWadr(L, prefix(L))) } Obligation: innermost runtime complexity Answer: MAYBE Arguments of following rules are not normal-forms: { app(nil(), YS) -> YS , zWadr(XS, nil()) -> nil() , zWadr(nil(), YS) -> nil() } All above mentioned rules can be savely removed. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { app(X1, X2) -> n__app(X1, X2) , app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS)) , nil() -> n__nil() , activate(X) -> X , activate(n__app(X1, X2)) -> app(X1, X2) , activate(n__from(X)) -> from(X) , activate(n__nil()) -> nil() , activate(n__zWadr(X1, X2)) -> zWadr(X1, X2) , from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , zWadr(X1, X2) -> n__zWadr(X1, X2) , zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil())), n__zWadr(activate(XS), activate(YS))) , prefix(L) -> cons(nil(), n__zWadr(L, prefix(L))) } 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) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The input cannot be shown compatible 2) 'bsearch-popstar (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 perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with minimal-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: { app^#(X1, X2) -> c_1() , app^#(cons(X, XS), YS) -> c_2(activate^#(XS)) , activate^#(X) -> c_4() , activate^#(n__app(X1, X2)) -> c_5(app^#(X1, X2)) , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__nil()) -> c_7(nil^#()) , activate^#(n__zWadr(X1, X2)) -> c_8(zWadr^#(X1, X2)) , nil^#() -> c_3() , from^#(X) -> c_9() , from^#(X) -> c_10() , zWadr^#(X1, X2) -> c_11() , zWadr^#(cons(X, XS), cons(Y, YS)) -> c_12(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS)) , prefix^#(L) -> c_13(nil^#(), prefix^#(L)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { app^#(X1, X2) -> c_1() , app^#(cons(X, XS), YS) -> c_2(activate^#(XS)) , activate^#(X) -> c_4() , activate^#(n__app(X1, X2)) -> c_5(app^#(X1, X2)) , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__nil()) -> c_7(nil^#()) , activate^#(n__zWadr(X1, X2)) -> c_8(zWadr^#(X1, X2)) , nil^#() -> c_3() , from^#(X) -> c_9() , from^#(X) -> c_10() , zWadr^#(X1, X2) -> c_11() , zWadr^#(cons(X, XS), cons(Y, YS)) -> c_12(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS)) , prefix^#(L) -> c_13(nil^#(), prefix^#(L)) } Strict Trs: { app(X1, X2) -> n__app(X1, X2) , app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS)) , nil() -> n__nil() , activate(X) -> X , activate(n__app(X1, X2)) -> app(X1, X2) , activate(n__from(X)) -> from(X) , activate(n__nil()) -> nil() , activate(n__zWadr(X1, X2)) -> zWadr(X1, X2) , from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , zWadr(X1, X2) -> n__zWadr(X1, X2) , zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil())), n__zWadr(activate(XS), activate(YS))) , prefix(L) -> cons(nil(), n__zWadr(L, prefix(L))) } 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: { app^#(X1, X2) -> c_1() , app^#(cons(X, XS), YS) -> c_2(activate^#(XS)) , activate^#(X) -> c_4() , activate^#(n__app(X1, X2)) -> c_5(app^#(X1, X2)) , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__nil()) -> c_7(nil^#()) , activate^#(n__zWadr(X1, X2)) -> c_8(zWadr^#(X1, X2)) , nil^#() -> c_3() , from^#(X) -> c_9() , from^#(X) -> c_10() , zWadr^#(X1, X2) -> c_11() , zWadr^#(cons(X, XS), cons(Y, YS)) -> c_12(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS)) , prefix^#(L) -> c_13(nil^#(), prefix^#(L)) } 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_5) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_12) = {1, 2, 3}, Uargs(c_13) = {1, 2} TcT has computed the following constructor-restricted matrix interpretation. [cons](x1, x2) = [1 0] x1 + [0] [0 0] [0] [n__app](x1, x2) = [1 2] x1 + [1 1] x2 + [0] [0 1] [0 1] [0] [n__from](x1) = [1 2] x1 + [0] [0 1] [0] [n__nil] = [0] [0] [n__zWadr](x1, x2) = [1 1] x1 + [1 1] x2 + [0] [0 1] [0 1] [0] [app^#](x1, x2) = [0 0] x1 + [2] [1 1] [2] [c_1] = [1] [1] [c_2](x1) = [1 0] x1 + [2] [0 1] [2] [activate^#](x1) = [0] [0] [nil^#] = [1] [1] [c_3] = [0] [1] [c_4] = [1] [0] [c_5](x1) = [1 0] x1 + [1] [0 1] [2] [c_6](x1) = [1 0] x1 + [1] [0 1] [2] [from^#](x1) = [2] [2] [c_7](x1) = [1 0] x1 + [1] [0 1] [1] [c_8](x1) = [1 0] x1 + [1] [0 1] [2] [zWadr^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] [1 2] [1 2] [2] [c_9] = [1] [1] [c_10] = [1] [1] [c_11] = [1] [2] [c_12](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1] [0 1] [0 1] [0 1] [2] [prefix^#](x1) = [1 2] x1 + [2] [2 2] [2] [c_13](x1, x2) = [1 0] x1 + [1 0] x2 + [1] [0 1] [0 1] [1] The following symbols are considered usable {app^#, activate^#, nil^#, from^#, zWadr^#, prefix^#} The order satisfies the following ordering constraints: [app^#(X1, X2)] = [0 0] X1 + [2] [1 1] [2] > [1] [1] = [c_1()] [app^#(cons(X, XS), YS)] = [0 0] X + [2] [1 0] [2] >= [2] [2] = [c_2(activate^#(XS))] [activate^#(X)] = [0] [0] ? [1] [0] = [c_4()] [activate^#(n__app(X1, X2))] = [0] [0] ? [0 0] X1 + [3] [1 1] [4] = [c_5(app^#(X1, X2))] [activate^#(n__from(X))] = [0] [0] ? [3] [4] = [c_6(from^#(X))] [activate^#(n__nil())] = [0] [0] ? [2] [2] = [c_7(nil^#())] [activate^#(n__zWadr(X1, X2))] = [0] [0] ? [0 0] X1 + [0 0] X2 + [3] [1 2] [1 2] [4] = [c_8(zWadr^#(X1, X2))] [nil^#()] = [1] [1] > [0] [1] = [c_3()] [from^#(X)] = [2] [2] > [1] [1] = [c_9()] [from^#(X)] = [2] [2] > [1] [1] = [c_10()] [zWadr^#(X1, X2)] = [0 0] X1 + [0 0] X2 + [2] [1 2] [1 2] [2] > [1] [2] = [c_11()] [zWadr^#(cons(X, XS), cons(Y, YS))] = [0 0] X + [0 0] Y + [2] [1 0] [1 0] [2] ? [0 0] Y + [3] [1 1] [4] = [c_12(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS))] [prefix^#(L)] = [1 2] L + [2] [2 2] [2] ? [1 2] L + [4] [2 2] [4] = [c_13(nil^#(), prefix^#(L))] 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: { app^#(cons(X, XS), YS) -> c_2(activate^#(XS)) , activate^#(X) -> c_4() , activate^#(n__app(X1, X2)) -> c_5(app^#(X1, X2)) , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__nil()) -> c_7(nil^#()) , activate^#(n__zWadr(X1, X2)) -> c_8(zWadr^#(X1, X2)) , zWadr^#(cons(X, XS), cons(Y, YS)) -> c_12(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS)) , prefix^#(L) -> c_13(nil^#(), prefix^#(L)) } Weak DPs: { app^#(X1, X2) -> c_1() , nil^#() -> c_3() , from^#(X) -> c_9() , from^#(X) -> c_10() , zWadr^#(X1, X2) -> c_11() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {2,4,5} by applications of Pre({2,4,5}) = {1,7}. Here rules are labeled as follows: DPs: { 1: app^#(cons(X, XS), YS) -> c_2(activate^#(XS)) , 2: activate^#(X) -> c_4() , 3: activate^#(n__app(X1, X2)) -> c_5(app^#(X1, X2)) , 4: activate^#(n__from(X)) -> c_6(from^#(X)) , 5: activate^#(n__nil()) -> c_7(nil^#()) , 6: activate^#(n__zWadr(X1, X2)) -> c_8(zWadr^#(X1, X2)) , 7: zWadr^#(cons(X, XS), cons(Y, YS)) -> c_12(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS)) , 8: prefix^#(L) -> c_13(nil^#(), prefix^#(L)) , 9: app^#(X1, X2) -> c_1() , 10: nil^#() -> c_3() , 11: from^#(X) -> c_9() , 12: from^#(X) -> c_10() , 13: zWadr^#(X1, X2) -> c_11() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { app^#(cons(X, XS), YS) -> c_2(activate^#(XS)) , activate^#(n__app(X1, X2)) -> c_5(app^#(X1, X2)) , activate^#(n__zWadr(X1, X2)) -> c_8(zWadr^#(X1, X2)) , zWadr^#(cons(X, XS), cons(Y, YS)) -> c_12(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS)) , prefix^#(L) -> c_13(nil^#(), prefix^#(L)) } Weak DPs: { app^#(X1, X2) -> c_1() , activate^#(X) -> c_4() , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__nil()) -> c_7(nil^#()) , nil^#() -> c_3() , from^#(X) -> c_9() , from^#(X) -> c_10() , zWadr^#(X1, X2) -> c_11() } 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^#(X1, X2) -> c_1() , activate^#(X) -> c_4() , activate^#(n__from(X)) -> c_6(from^#(X)) , activate^#(n__nil()) -> c_7(nil^#()) , nil^#() -> c_3() , from^#(X) -> c_9() , from^#(X) -> c_10() , zWadr^#(X1, X2) -> c_11() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { app^#(cons(X, XS), YS) -> c_2(activate^#(XS)) , activate^#(n__app(X1, X2)) -> c_5(app^#(X1, X2)) , activate^#(n__zWadr(X1, X2)) -> c_8(zWadr^#(X1, X2)) , zWadr^#(cons(X, XS), cons(Y, YS)) -> c_12(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS)) , prefix^#(L) -> c_13(nil^#(), prefix^#(L)) } Obligation: innermost runtime complexity Answer: MAYBE Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { prefix^#(L) -> c_13(nil^#(), prefix^#(L)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { app^#(cons(X, XS), YS) -> c_1(activate^#(XS)) , activate^#(n__app(X1, X2)) -> c_2(app^#(X1, X2)) , activate^#(n__zWadr(X1, X2)) -> c_3(zWadr^#(X1, X2)) , zWadr^#(cons(X, XS), cons(Y, YS)) -> c_4(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS)) , prefix^#(L) -> c_5(prefix^#(L)) } 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: We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: app^#(cons(X, XS), YS) -> c_1(activate^#(XS)) , 2: activate^#(n__app(X1, X2)) -> c_2(app^#(X1, X2)) , 3: activate^#(n__zWadr(X1, X2)) -> c_3(zWadr^#(X1, X2)) , 4: zWadr^#(cons(X, XS), cons(Y, YS)) -> c_4(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1, 2, 3}, Uargs(c_5) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [app](x1, x2) = [7] x1 + [7] x2 + [0] [nil] = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [5] [n__app](x1, x2) = [1] x1 + [1] x2 + [4] [activate](x1) = [7] x1 + [0] [from](x1) = [7] x1 + [0] [n__from](x1) = [1] x1 + [0] [s](x1) = [1] x1 + [0] [zWadr](x1, x2) = [7] x1 + [7] x2 + [0] [n__nil] = [4] [n__zWadr](x1, x2) = [1] x1 + [1] x2 + [4] [prefix](x1) = [7] x1 + [0] [app^#](x1, x2) = [1] x1 + [3] [c_1] = [0] [c_2](x1) = [7] x1 + [0] [activate^#](x1) = [1] x1 + [0] [nil^#] = [0] [c_3] = [0] [c_4] = [0] [c_5](x1) = [7] x1 + [0] [c_6](x1) = [7] x1 + [0] [from^#](x1) = [7] x1 + [0] [c_7](x1) = [7] x1 + [0] [c_8](x1) = [7] x1 + [0] [zWadr^#](x1, x2) = [1] x1 + [1] x2 + [0] [c_9] = [0] [c_10] = [0] [c_11] = [0] [c_12](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [prefix^#](x1) = [0] [c_13](x1, x2) = [7] x1 + [7] x2 + [0] [c] = [0] [c_1](x1) = [1] x1 + [5] [c_2](x1) = [1] x1 + [0] [c_3](x1) = [1] x1 + [3] [c_4](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] [c_5](x1) = [2] x1 + [0] The following symbols are considered usable {app^#, activate^#, zWadr^#, prefix^#} The order satisfies the following ordering constraints: [app^#(cons(X, XS), YS)] = [1] X + [1] XS + [8] > [1] XS + [5] = [c_1(activate^#(XS))] [activate^#(n__app(X1, X2))] = [1] X1 + [1] X2 + [4] > [1] X1 + [3] = [c_2(app^#(X1, X2))] [activate^#(n__zWadr(X1, X2))] = [1] X1 + [1] X2 + [4] > [1] X1 + [1] X2 + [3] = [c_3(zWadr^#(X1, X2))] [zWadr^#(cons(X, XS), cons(Y, YS))] = [1] YS + [1] X + [1] XS + [1] Y + [10] > [1] YS + [1] XS + [1] Y + [3] = [c_4(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS))] [prefix^#(L)] = [0] >= [0] = [c_5(prefix^#(L))] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { prefix^#(L) -> c_5(prefix^#(L)) } Weak DPs: { app^#(cons(X, XS), YS) -> c_1(activate^#(XS)) , activate^#(n__app(X1, X2)) -> c_2(app^#(X1, X2)) , activate^#(n__zWadr(X1, X2)) -> c_3(zWadr^#(X1, X2)) , zWadr^#(cons(X, XS), cons(Y, YS)) -> c_4(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS)) } 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^#(cons(X, XS), YS) -> c_1(activate^#(XS)) , activate^#(n__app(X1, X2)) -> c_2(app^#(X1, X2)) , activate^#(n__zWadr(X1, X2)) -> c_3(zWadr^#(X1, X2)) , zWadr^#(cons(X, XS), cons(Y, YS)) -> c_4(app^#(Y, cons(X, n__nil())), activate^#(XS), activate^#(YS)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { prefix^#(L) -> c_5(prefix^#(L)) } 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) '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) 'Fastest (timeout of 5 seconds)' failed due to the following reason: Computation stopped due to timeout after 5.0 seconds. 3) 'Polynomial Path Order (PS)' failed due to the following reason: The input cannot be shown compatible Arrrr..