MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { dbl(X) -> n__dbl(X) , dbl(0()) -> 0() , dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) , s(X) -> n__s(X) , activate(X) -> X , activate(n__s(X)) -> s(X) , activate(n__dbl(X)) -> dbl(X) , activate(n__dbls(X)) -> dbls(X) , activate(n__sel(X1, X2)) -> sel(X1, X2) , activate(n__indx(X1, X2)) -> indx(X1, X2) , activate(n__from(X)) -> from(X) , dbls(X) -> n__dbls(X) , dbls(nil()) -> nil() , dbls(cons(X, Y)) -> cons(n__dbl(activate(X)), n__dbls(activate(Y))) , sel(X1, X2) -> n__sel(X1, X2) , sel(0(), cons(X, Y)) -> activate(X) , sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z)) , indx(X1, X2) -> n__indx(X1, X2) , indx(nil(), X) -> nil() , indx(cons(X, Y), Z) -> cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), 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: { dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) , sel(s(X), cons(Y, Z)) -> sel(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: { dbl(X) -> n__dbl(X) , dbl(0()) -> 0() , s(X) -> n__s(X) , activate(X) -> X , activate(n__s(X)) -> s(X) , activate(n__dbl(X)) -> dbl(X) , activate(n__dbls(X)) -> dbls(X) , activate(n__sel(X1, X2)) -> sel(X1, X2) , activate(n__indx(X1, X2)) -> indx(X1, X2) , activate(n__from(X)) -> from(X) , dbls(X) -> n__dbls(X) , dbls(nil()) -> nil() , dbls(cons(X, Y)) -> cons(n__dbl(activate(X)), n__dbls(activate(Y))) , sel(X1, X2) -> n__sel(X1, X2) , sel(0(), cons(X, Y)) -> activate(X) , indx(X1, X2) -> n__indx(X1, X2) , indx(nil(), X) -> nil() , indx(cons(X, Y), Z) -> cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z))) , 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) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: We add the following innermost weak dependency pairs: Strict DPs: { dbl^#(X) -> c_1() , dbl^#(0()) -> c_2() , s^#(X) -> c_3() , activate^#(X) -> c_4() , activate^#(n__s(X)) -> c_5(s^#(X)) , activate^#(n__dbl(X)) -> c_6(dbl^#(X)) , activate^#(n__dbls(X)) -> c_7(dbls^#(X)) , activate^#(n__sel(X1, X2)) -> c_8(sel^#(X1, X2)) , activate^#(n__indx(X1, X2)) -> c_9(indx^#(X1, X2)) , activate^#(n__from(X)) -> c_10(from^#(X)) , dbls^#(X) -> c_11() , dbls^#(nil()) -> c_12() , dbls^#(cons(X, Y)) -> c_13(activate^#(X), activate^#(Y)) , sel^#(X1, X2) -> c_14() , sel^#(0(), cons(X, Y)) -> c_15(activate^#(X)) , indx^#(X1, X2) -> c_16() , indx^#(nil(), X) -> c_17() , indx^#(cons(X, Y), Z) -> c_18(activate^#(X), activate^#(Z), activate^#(Y), activate^#(Z)) , from^#(X) -> c_19(activate^#(X), activate^#(X)) , from^#(X) -> c_20() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { dbl^#(X) -> c_1() , dbl^#(0()) -> c_2() , s^#(X) -> c_3() , activate^#(X) -> c_4() , activate^#(n__s(X)) -> c_5(s^#(X)) , activate^#(n__dbl(X)) -> c_6(dbl^#(X)) , activate^#(n__dbls(X)) -> c_7(dbls^#(X)) , activate^#(n__sel(X1, X2)) -> c_8(sel^#(X1, X2)) , activate^#(n__indx(X1, X2)) -> c_9(indx^#(X1, X2)) , activate^#(n__from(X)) -> c_10(from^#(X)) , dbls^#(X) -> c_11() , dbls^#(nil()) -> c_12() , dbls^#(cons(X, Y)) -> c_13(activate^#(X), activate^#(Y)) , sel^#(X1, X2) -> c_14() , sel^#(0(), cons(X, Y)) -> c_15(activate^#(X)) , indx^#(X1, X2) -> c_16() , indx^#(nil(), X) -> c_17() , indx^#(cons(X, Y), Z) -> c_18(activate^#(X), activate^#(Z), activate^#(Y), activate^#(Z)) , from^#(X) -> c_19(activate^#(X), activate^#(X)) , from^#(X) -> c_20() } Strict Trs: { dbl(X) -> n__dbl(X) , dbl(0()) -> 0() , s(X) -> n__s(X) , activate(X) -> X , activate(n__s(X)) -> s(X) , activate(n__dbl(X)) -> dbl(X) , activate(n__dbls(X)) -> dbls(X) , activate(n__sel(X1, X2)) -> sel(X1, X2) , activate(n__indx(X1, X2)) -> indx(X1, X2) , activate(n__from(X)) -> from(X) , dbls(X) -> n__dbls(X) , dbls(nil()) -> nil() , dbls(cons(X, Y)) -> cons(n__dbl(activate(X)), n__dbls(activate(Y))) , sel(X1, X2) -> n__sel(X1, X2) , sel(0(), cons(X, Y)) -> activate(X) , indx(X1, X2) -> n__indx(X1, X2) , indx(nil(), X) -> nil() , indx(cons(X, Y), Z) -> cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z))) , 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: { dbl^#(X) -> c_1() , dbl^#(0()) -> c_2() , s^#(X) -> c_3() , activate^#(X) -> c_4() , activate^#(n__s(X)) -> c_5(s^#(X)) , activate^#(n__dbl(X)) -> c_6(dbl^#(X)) , activate^#(n__dbls(X)) -> c_7(dbls^#(X)) , activate^#(n__sel(X1, X2)) -> c_8(sel^#(X1, X2)) , activate^#(n__indx(X1, X2)) -> c_9(indx^#(X1, X2)) , activate^#(n__from(X)) -> c_10(from^#(X)) , dbls^#(X) -> c_11() , dbls^#(nil()) -> c_12() , dbls^#(cons(X, Y)) -> c_13(activate^#(X), activate^#(Y)) , sel^#(X1, X2) -> c_14() , sel^#(0(), cons(X, Y)) -> c_15(activate^#(X)) , indx^#(X1, X2) -> c_16() , indx^#(nil(), X) -> c_17() , indx^#(cons(X, Y), Z) -> c_18(activate^#(X), activate^#(Z), activate^#(Y), activate^#(Z)) , from^#(X) -> c_19(activate^#(X), activate^#(X)) , from^#(X) -> c_20() } 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_5) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(c_10) = {1}, Uargs(c_13) = {1, 2}, Uargs(c_15) = {1}, Uargs(c_18) = {1, 2, 3, 4}, Uargs(c_19) = {1, 2} TcT has computed the following constructor-restricted matrix interpretation. [0] = [1] [1] [n__s](x1) = [1 2] x1 + [1] [0 1] [2] [n__dbl](x1) = [1 2] x1 + [1] [0 1] [1] [nil] = [1] [1] [cons](x1, x2) = [1 2] x1 + [1 1] x2 + [1] [0 1] [0 1] [1] [n__dbls](x1) = [1 2] x1 + [1] [0 1] [1] [n__sel](x1, x2) = [1 2] x1 + [1 2] x2 + [1] [0 1] [0 1] [1] [n__indx](x1, x2) = [1 2] x1 + [1 2] x2 + [1] [0 1] [0 1] [2] [n__from](x1) = [1 2] x1 + [1] [0 1] [2] [dbl^#](x1) = [0 0] x1 + [1] [2 1] [1] [c_1] = [0] [1] [c_2] = [0] [0] [s^#](x1) = [0 0] x1 + [1] [1 1] [1] [c_3] = [0] [1] [activate^#](x1) = [0 0] x1 + [1] [1 1] [0] [c_4] = [0] [0] [c_5](x1) = [1 0] x1 + [2] [0 1] [2] [c_6](x1) = [1 0] x1 + [2] [0 1] [1] [c_7](x1) = [1 0] x1 + [2] [0 1] [1] [dbls^#](x1) = [0 0] x1 + [1] [2 2] [2] [c_8](x1) = [1 0] x1 + [2] [0 1] [2] [sel^#](x1, x2) = [0 0] x1 + [0 0] x2 + [1] [1 2] [2 2] [0] [c_9](x1) = [1 0] x1 + [2] [0 1] [1] [indx^#](x1, x2) = [0 0] x1 + [0 0] x2 + [1] [1 1] [2 1] [2] [c_10](x1) = [1 0] x1 + [2] [0 1] [2] [from^#](x1) = [0 0] x1 + [1] [1 1] [1] [c_11] = [0] [1] [c_12] = [0] [1] [c_13](x1, x2) = [1 0] x1 + [1 0] x2 + [1] [0 1] [0 1] [2] [c_14] = [0] [0] [c_15](x1) = [1 0] x1 + [1] [0 1] [2] [c_16] = [0] [1] [c_17] = [0] [0] [c_18](x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1 0] x4 + [2] [0 1] [0 1] [0 1] [0 1] [2] [c_19](x1, x2) = [1 0] x1 + [1 0] x2 + [1] [0 1] [0 1] [2] [c_20] = [0] [1] The following symbols are considered usable {dbl^#, s^#, activate^#, dbls^#, sel^#, indx^#, from^#} The order satisfies the following ordering constraints: [dbl^#(X)] = [0 0] X + [1] [2 1] [1] > [0] [1] = [c_1()] [dbl^#(0())] = [1] [4] > [0] [0] = [c_2()] [s^#(X)] = [0 0] X + [1] [1 1] [1] > [0] [1] = [c_3()] [activate^#(X)] = [0 0] X + [1] [1 1] [0] > [0] [0] = [c_4()] [activate^#(n__s(X))] = [0 0] X + [1] [1 3] [3] ? [0 0] X + [3] [1 1] [3] = [c_5(s^#(X))] [activate^#(n__dbl(X))] = [0 0] X + [1] [1 3] [2] ? [0 0] X + [3] [2 1] [2] = [c_6(dbl^#(X))] [activate^#(n__dbls(X))] = [0 0] X + [1] [1 3] [2] ? [0 0] X + [3] [2 2] [3] = [c_7(dbls^#(X))] [activate^#(n__sel(X1, X2))] = [0 0] X1 + [0 0] X2 + [1] [1 3] [1 3] [2] ? [0 0] X1 + [0 0] X2 + [3] [1 2] [2 2] [2] = [c_8(sel^#(X1, X2))] [activate^#(n__indx(X1, X2))] = [0 0] X1 + [0 0] X2 + [1] [1 3] [1 3] [3] ? [0 0] X1 + [0 0] X2 + [3] [1 1] [2 1] [3] = [c_9(indx^#(X1, X2))] [activate^#(n__from(X))] = [0 0] X + [1] [1 3] [3] ? [0 0] X + [3] [1 1] [3] = [c_10(from^#(X))] [dbls^#(X)] = [0 0] X + [1] [2 2] [2] > [0] [1] = [c_11()] [dbls^#(nil())] = [1] [6] > [0] [1] = [c_12()] [dbls^#(cons(X, Y))] = [0 0] X + [0 0] Y + [1] [2 6] [2 4] [6] ? [0 0] X + [0 0] Y + [3] [1 1] [1 1] [2] = [c_13(activate^#(X), activate^#(Y))] [sel^#(X1, X2)] = [0 0] X1 + [0 0] X2 + [1] [1 2] [2 2] [0] > [0] [0] = [c_14()] [sel^#(0(), cons(X, Y))] = [0 0] X + [0 0] Y + [1] [2 6] [2 4] [7] ? [0 0] X + [2] [1 1] [2] = [c_15(activate^#(X))] [indx^#(X1, X2)] = [0 0] X1 + [0 0] X2 + [1] [1 1] [2 1] [2] > [0] [1] = [c_16()] [indx^#(nil(), X)] = [0 0] X + [1] [2 1] [4] > [0] [0] = [c_17()] [indx^#(cons(X, Y), Z)] = [0 0] X + [0 0] Y + [0 0] Z + [1] [1 3] [1 2] [2 1] [4] ? [0 0] X + [0 0] Y + [0 0] Z + [6] [1 1] [1 1] [2 2] [2] = [c_18(activate^#(X), activate^#(Z), activate^#(Y), activate^#(Z))] [from^#(X)] = [0 0] X + [1] [1 1] [1] ? [0 0] X + [3] [2 2] [2] = [c_19(activate^#(X), activate^#(X))] [from^#(X)] = [0 0] X + [1] [1 1] [1] > [0] [1] = [c_20()] 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__s(X)) -> c_5(s^#(X)) , activate^#(n__dbl(X)) -> c_6(dbl^#(X)) , activate^#(n__dbls(X)) -> c_7(dbls^#(X)) , activate^#(n__sel(X1, X2)) -> c_8(sel^#(X1, X2)) , activate^#(n__indx(X1, X2)) -> c_9(indx^#(X1, X2)) , activate^#(n__from(X)) -> c_10(from^#(X)) , dbls^#(cons(X, Y)) -> c_13(activate^#(X), activate^#(Y)) , sel^#(0(), cons(X, Y)) -> c_15(activate^#(X)) , indx^#(cons(X, Y), Z) -> c_18(activate^#(X), activate^#(Z), activate^#(Y), activate^#(Z)) , from^#(X) -> c_19(activate^#(X), activate^#(X)) } Weak DPs: { dbl^#(X) -> c_1() , dbl^#(0()) -> c_2() , s^#(X) -> c_3() , activate^#(X) -> c_4() , dbls^#(X) -> c_11() , dbls^#(nil()) -> c_12() , sel^#(X1, X2) -> c_14() , indx^#(X1, X2) -> c_16() , indx^#(nil(), X) -> c_17() , from^#(X) -> c_20() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,2} by applications of Pre({1,2}) = {7,8,9,10}. Here rules are labeled as follows: DPs: { 1: activate^#(n__s(X)) -> c_5(s^#(X)) , 2: activate^#(n__dbl(X)) -> c_6(dbl^#(X)) , 3: activate^#(n__dbls(X)) -> c_7(dbls^#(X)) , 4: activate^#(n__sel(X1, X2)) -> c_8(sel^#(X1, X2)) , 5: activate^#(n__indx(X1, X2)) -> c_9(indx^#(X1, X2)) , 6: activate^#(n__from(X)) -> c_10(from^#(X)) , 7: dbls^#(cons(X, Y)) -> c_13(activate^#(X), activate^#(Y)) , 8: sel^#(0(), cons(X, Y)) -> c_15(activate^#(X)) , 9: indx^#(cons(X, Y), Z) -> c_18(activate^#(X), activate^#(Z), activate^#(Y), activate^#(Z)) , 10: from^#(X) -> c_19(activate^#(X), activate^#(X)) , 11: dbl^#(X) -> c_1() , 12: dbl^#(0()) -> c_2() , 13: s^#(X) -> c_3() , 14: activate^#(X) -> c_4() , 15: dbls^#(X) -> c_11() , 16: dbls^#(nil()) -> c_12() , 17: sel^#(X1, X2) -> c_14() , 18: indx^#(X1, X2) -> c_16() , 19: indx^#(nil(), X) -> c_17() , 20: from^#(X) -> c_20() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { activate^#(n__dbls(X)) -> c_7(dbls^#(X)) , activate^#(n__sel(X1, X2)) -> c_8(sel^#(X1, X2)) , activate^#(n__indx(X1, X2)) -> c_9(indx^#(X1, X2)) , activate^#(n__from(X)) -> c_10(from^#(X)) , dbls^#(cons(X, Y)) -> c_13(activate^#(X), activate^#(Y)) , sel^#(0(), cons(X, Y)) -> c_15(activate^#(X)) , indx^#(cons(X, Y), Z) -> c_18(activate^#(X), activate^#(Z), activate^#(Y), activate^#(Z)) , from^#(X) -> c_19(activate^#(X), activate^#(X)) } Weak DPs: { dbl^#(X) -> c_1() , dbl^#(0()) -> c_2() , s^#(X) -> c_3() , activate^#(X) -> c_4() , activate^#(n__s(X)) -> c_5(s^#(X)) , activate^#(n__dbl(X)) -> c_6(dbl^#(X)) , dbls^#(X) -> c_11() , dbls^#(nil()) -> c_12() , sel^#(X1, X2) -> c_14() , indx^#(X1, X2) -> c_16() , indx^#(nil(), X) -> c_17() , from^#(X) -> c_20() } 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. { dbl^#(X) -> c_1() , dbl^#(0()) -> c_2() , s^#(X) -> c_3() , activate^#(X) -> c_4() , activate^#(n__s(X)) -> c_5(s^#(X)) , activate^#(n__dbl(X)) -> c_6(dbl^#(X)) , dbls^#(X) -> c_11() , dbls^#(nil()) -> c_12() , sel^#(X1, X2) -> c_14() , indx^#(X1, X2) -> c_16() , indx^#(nil(), X) -> c_17() , from^#(X) -> c_20() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { activate^#(n__dbls(X)) -> c_7(dbls^#(X)) , activate^#(n__sel(X1, X2)) -> c_8(sel^#(X1, X2)) , activate^#(n__indx(X1, X2)) -> c_9(indx^#(X1, X2)) , activate^#(n__from(X)) -> c_10(from^#(X)) , dbls^#(cons(X, Y)) -> c_13(activate^#(X), activate^#(Y)) , sel^#(0(), cons(X, Y)) -> c_15(activate^#(X)) , indx^#(cons(X, Y), Z) -> c_18(activate^#(X), activate^#(Z), activate^#(Y), activate^#(Z)) , from^#(X) -> c_19(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. 2) '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 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..