MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , dbls(nil()) -> nil() , dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) , sel(0(), cons(X, Y)) -> X , sel(s(X), cons(Y, Z)) -> sel(X, Z) , indx(nil(), X) -> nil() , indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) , from(X) -> cons(X, from(s(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: 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) '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) '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) 'WithProblem' 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) 'WithProblem' 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) '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. 2) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: We add the following innermost weak dependency pairs: Strict DPs: { dbl^#(0()) -> c_1() , dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(nil()) -> c_3() , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , sel^#(0(), cons(X, Y)) -> c_5() , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(nil(), X) -> c_7() , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { dbl^#(0()) -> c_1() , dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(nil()) -> c_3() , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , sel^#(0(), cons(X, Y)) -> c_5() , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(nil(), X) -> c_7() , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } Strict Trs: { dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , dbls(nil()) -> nil() , dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) , sel(0(), cons(X, Y)) -> X , sel(s(X), cons(Y, Z)) -> sel(X, Z) , indx(nil(), X) -> nil() , indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) , from(X) -> cons(X, from(s(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^#(0()) -> c_1() , dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(nil()) -> c_3() , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , sel^#(0(), cons(X, Y)) -> c_5() , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(nil(), X) -> c_7() , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } 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_4) = {1, 2}, Uargs(c_6) = {1}, Uargs(c_8) = {1, 2}, Uargs(c_9) = {1} TcT has computed the following constructor-restricted matrix interpretation. [0] = [0] [0] [s](x1) = [1 0] x1 + [0] [0 0] [0] [nil] = [0] [0] [cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] [dbl^#](x1) = [0] [0] [c_1] = [1] [0] [c_2](x1) = [1 0] x1 + [2] [0 1] [0] [dbls^#](x1) = [0] [0] [c_3] = [1] [0] [c_4](x1, x2) = [1 0] x1 + [1 0] x2 + [1] [0 1] [0 1] [2] [sel^#](x1, x2) = [0] [0] [c_5] = [1] [0] [c_6](x1) = [1 0] x1 + [2] [0 1] [0] [indx^#](x1, x2) = [2 2] x2 + [1] [2 2] [0] [c_7] = [0] [0] [c_8](x1, x2) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [0] [from^#](x1) = [0 1] x1 + [2] [1 1] [2] [c_9](x1) = [1 0] x1 + [2] [0 1] [1] The following symbols are considered usable {dbl^#, dbls^#, sel^#, indx^#, from^#} The order satisfies the following ordering constraints: [dbl^#(0())] = [0] [0] ? [1] [0] = [c_1()] [dbl^#(s(X))] = [0] [0] ? [2] [0] = [c_2(dbl^#(X))] [dbls^#(nil())] = [0] [0] ? [1] [0] = [c_3()] [dbls^#(cons(X, Y))] = [0] [0] ? [1] [2] = [c_4(dbl^#(X), dbls^#(Y))] [sel^#(0(), cons(X, Y))] = [0] [0] ? [1] [0] = [c_5()] [sel^#(s(X), cons(Y, Z))] = [0] [0] ? [2] [0] = [c_6(sel^#(X, Z))] [indx^#(nil(), X)] = [2 2] X + [1] [2 2] [0] > [0] [0] = [c_7()] [indx^#(cons(X, Y), Z)] = [2 2] Z + [1] [2 2] [0] >= [2 2] Z + [1] [2 2] [0] = [c_8(sel^#(X, Z), indx^#(Y, Z))] [from^#(X)] = [0 1] X + [2] [1 1] [2] ? [0 0] X + [4] [1 0] [3] = [c_9(from^#(s(X)))] 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: { dbl^#(0()) -> c_1() , dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(nil()) -> c_3() , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , sel^#(0(), cons(X, Y)) -> c_5() , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } Weak DPs: { indx^#(nil(), X) -> c_7() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,3,5} by applications of Pre({1,3,5}) = {2,4,6,7}. Here rules are labeled as follows: DPs: { 1: dbl^#(0()) -> c_1() , 2: dbl^#(s(X)) -> c_2(dbl^#(X)) , 3: dbls^#(nil()) -> c_3() , 4: dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , 5: sel^#(0(), cons(X, Y)) -> c_5() , 6: sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , 7: indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , 8: from^#(X) -> c_9(from^#(s(X))) , 9: indx^#(nil(), X) -> c_7() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } Weak DPs: { dbl^#(0()) -> c_1() , dbls^#(nil()) -> c_3() , sel^#(0(), cons(X, Y)) -> c_5() , indx^#(nil(), X) -> c_7() } 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^#(0()) -> c_1() , dbls^#(nil()) -> c_3() , sel^#(0(), cons(X, Y)) -> c_5() , indx^#(nil(), X) -> c_7() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(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: We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: dbl^#(s(X)) -> c_2(dbl^#(X)) , 3: sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_2) = {1}, Uargs(c_4) = {1, 2}, Uargs(c_6) = {1}, Uargs(c_8) = {1, 2}, Uargs(c_9) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [dbl](x1) = [7] x1 + [0] [0] = [0] [s](x1) = [1] x1 + [2] [dbls](x1) = [7] x1 + [0] [nil] = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [0] [sel](x1, x2) = [7] x1 + [7] x2 + [0] [indx](x1, x2) = [7] x1 + [7] x2 + [0] [from](x1) = [7] x1 + [0] [dbl^#](x1) = [4] x1 + [0] [c_1] = [0] [c_2](x1) = [1] x1 + [1] [dbls^#](x1) = [4] x1 + [0] [c_3] = [0] [c_4](x1, x2) = [1] x1 + [1] x2 + [0] [sel^#](x1, x2) = [4] x1 + [0] [c_5] = [0] [c_6](x1) = [1] x1 + [7] [indx^#](x1, x2) = [4] x1 + [0] [c_7] = [0] [c_8](x1, x2) = [1] x1 + [1] x2 + [0] [from^#](x1) = [0] [c_9](x1) = [1] x1 + [0] The following symbols are considered usable {dbl^#, dbls^#, sel^#, indx^#, from^#} The order satisfies the following ordering constraints: [dbl^#(s(X))] = [4] X + [8] > [4] X + [1] = [c_2(dbl^#(X))] [dbls^#(cons(X, Y))] = [4] X + [4] Y + [0] >= [4] X + [4] Y + [0] = [c_4(dbl^#(X), dbls^#(Y))] [sel^#(s(X), cons(Y, Z))] = [4] X + [8] > [4] X + [7] = [c_6(sel^#(X, Z))] [indx^#(cons(X, Y), Z)] = [4] X + [4] Y + [0] >= [4] X + [4] Y + [0] = [c_8(sel^#(X, Z), indx^#(Y, Z))] [from^#(X)] = [0] >= [0] = [c_9(from^#(s(X)))] 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: { dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } Weak DPs: { dbl^#(s(X)) -> c_2(dbl^#(X)) , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) } 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^#(s(X)) -> c_2(dbl^#(X)) , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } Obligation: innermost runtime complexity Answer: MAYBE Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { dbls^#(cons(X, Y)) -> c_1(dbls^#(Y)) , indx^#(cons(X, Y), Z) -> c_2(indx^#(Y, Z)) , from^#(X) -> c_3(from^#(s(X))) } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: dbls^#(cons(X, Y)) -> c_1(dbls^#(Y)) , 2: indx^#(cons(X, Y), Z) -> c_2(indx^#(Y, Z)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [dbl](x1) = [7] x1 + [0] [0] = [0] [s](x1) = [1] x1 + [4] [dbls](x1) = [7] x1 + [0] [nil] = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [4] [sel](x1, x2) = [7] x1 + [7] x2 + [0] [indx](x1, x2) = [7] x1 + [7] x2 + [0] [from](x1) = [7] x1 + [0] [dbl^#](x1) = [7] x1 + [0] [c_1] = [0] [c_2](x1) = [7] x1 + [0] [dbls^#](x1) = [2] x1 + [0] [c_3] = [0] [c_4](x1, x2) = [7] x1 + [7] x2 + [0] [sel^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_5] = [0] [c_6](x1) = [7] x1 + [0] [indx^#](x1, x2) = [2] x1 + [7] x2 + [0] [c_7] = [0] [c_8](x1, x2) = [7] x1 + [7] x2 + [0] [from^#](x1) = [0] [c_9](x1) = [7] x1 + [0] [c] = [0] [c_1](x1) = [1] x1 + [1] [c_2](x1) = [1] x1 + [7] [c_3](x1) = [4] x1 + [0] The following symbols are considered usable {dbls^#, indx^#, from^#} The order satisfies the following ordering constraints: [dbls^#(cons(X, Y))] = [2] X + [2] Y + [8] > [2] Y + [1] = [c_1(dbls^#(Y))] [indx^#(cons(X, Y), Z)] = [2] X + [2] Y + [7] Z + [8] > [2] Y + [7] Z + [7] = [c_2(indx^#(Y, Z))] [from^#(X)] = [0] >= [0] = [c_3(from^#(s(X)))] 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: { from^#(X) -> c_3(from^#(s(X))) } Weak DPs: { dbls^#(cons(X, Y)) -> c_1(dbls^#(Y)) , indx^#(cons(X, Y), Z) -> c_2(indx^#(Y, Z)) } 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. { dbls^#(cons(X, Y)) -> c_1(dbls^#(Y)) , indx^#(cons(X, Y), Z) -> c_2(indx^#(Y, Z)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_3(from^#(s(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) '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..