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))) , dbl1(0()) -> 01() , dbl1(s(X)) -> s1(s1(dbl1(X))) , sel1(0(), cons(X, Y)) -> X , sel1(s(X), cons(Y, Z)) -> sel1(X, Z) , quote(dbl(X)) -> dbl1(X) , quote(0()) -> 01() , quote(s(X)) -> s1(quote(X)) , quote(sel(X, Y)) -> sel1(X, Y) } 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 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: { 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))) , dbl1^#(0()) -> c_10() , dbl1^#(s(X)) -> c_11(dbl1^#(X)) , sel1^#(0(), cons(X, Y)) -> c_12() , sel1^#(s(X), cons(Y, Z)) -> c_13(sel1^#(X, Z)) , quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(0()) -> c_15() , quote^#(s(X)) -> c_16(quote^#(X)) , quote^#(sel(X, Y)) -> c_17(sel1^#(X, Y)) } 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))) , dbl1^#(0()) -> c_10() , dbl1^#(s(X)) -> c_11(dbl1^#(X)) , sel1^#(0(), cons(X, Y)) -> c_12() , sel1^#(s(X), cons(Y, Z)) -> c_13(sel1^#(X, Z)) , quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(0()) -> c_15() , quote^#(s(X)) -> c_16(quote^#(X)) , quote^#(sel(X, Y)) -> c_17(sel1^#(X, Y)) } 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))) , dbl1(0()) -> 01() , dbl1(s(X)) -> s1(s1(dbl1(X))) , sel1(0(), cons(X, Y)) -> X , sel1(s(X), cons(Y, Z)) -> sel1(X, Z) , quote(dbl(X)) -> dbl1(X) , quote(0()) -> 01() , quote(s(X)) -> s1(quote(X)) , quote(sel(X, Y)) -> sel1(X, Y) } 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))) , dbl1^#(0()) -> c_10() , dbl1^#(s(X)) -> c_11(dbl1^#(X)) , sel1^#(0(), cons(X, Y)) -> c_12() , sel1^#(s(X), cons(Y, Z)) -> c_13(sel1^#(X, Z)) , quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(0()) -> c_15() , quote^#(s(X)) -> c_16(quote^#(X)) , quote^#(sel(X, Y)) -> c_17(sel1^#(X, 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_2) = {1}, Uargs(c_4) = {1, 2}, Uargs(c_6) = {1}, Uargs(c_8) = {1, 2}, Uargs(c_9) = {1}, Uargs(c_11) = {1}, Uargs(c_13) = {1}, Uargs(c_14) = {1}, Uargs(c_16) = {1}, Uargs(c_17) = {1} TcT has computed the following constructor-restricted matrix interpretation. [dbl](x1) = [0] [0] [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] [sel](x1, x2) = [0] [0] [dbl^#](x1) = [0] [0] [c_1] = [2] [0] [c_2](x1) = [1 0] x1 + [2] [0 1] [0] [dbls^#](x1) = [0] [0] [c_3] = [2] [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 1] x2 + [0] [2 2] [0] [c_7] = [1] [0] [c_8](x1, x2) = [1 0] x1 + [1 0] x2 + [1] [0 1] [0 1] [2] [from^#](x1) = [2 1] x1 + [2] [2 1] [2] [c_9](x1) = [1 0] x1 + [2] [0 1] [2] [dbl1^#](x1) = [0] [0] [c_10] = [1] [0] [c_11](x1) = [1 0] x1 + [2] [0 1] [0] [sel1^#](x1, x2) = [1] [0] [c_12] = [0] [0] [c_13](x1) = [1 0] x1 + [1] [0 1] [2] [quote^#](x1) = [0] [0] [c_14](x1) = [1 0] x1 + [2] [0 1] [0] [c_15] = [1] [0] [c_16](x1) = [1 0] x1 + [2] [0 1] [0] [c_17](x1) = [1 0] x1 + [1] [0 1] [0] The following symbols are considered usable {dbl^#, dbls^#, sel^#, indx^#, from^#, dbl1^#, sel1^#, quote^#} The order satisfies the following ordering constraints: [dbl^#(0())] = [0] [0] ? [2] [0] = [c_1()] [dbl^#(s(X))] = [0] [0] ? [2] [0] = [c_2(dbl^#(X))] [dbls^#(nil())] = [0] [0] ? [2] [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 1] X + [0] [2 2] [0] ? [1] [0] = [c_7()] [indx^#(cons(X, Y), Z)] = [2 1] Z + [0] [2 2] [0] ? [2 1] Z + [1] [2 2] [2] = [c_8(sel^#(X, Z), indx^#(Y, Z))] [from^#(X)] = [2 1] X + [2] [2 1] [2] ? [2 0] X + [4] [2 0] [4] = [c_9(from^#(s(X)))] [dbl1^#(0())] = [0] [0] ? [1] [0] = [c_10()] [dbl1^#(s(X))] = [0] [0] ? [2] [0] = [c_11(dbl1^#(X))] [sel1^#(0(), cons(X, Y))] = [1] [0] > [0] [0] = [c_12()] [sel1^#(s(X), cons(Y, Z))] = [1] [0] ? [2] [2] = [c_13(sel1^#(X, Z))] [quote^#(dbl(X))] = [0] [0] ? [2] [0] = [c_14(dbl1^#(X))] [quote^#(0())] = [0] [0] ? [1] [0] = [c_15()] [quote^#(s(X))] = [0] [0] ? [2] [0] = [c_16(quote^#(X))] [quote^#(sel(X, Y))] = [0] [0] ? [2] [0] = [c_17(sel1^#(X, 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: { 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))) , dbl1^#(0()) -> c_10() , dbl1^#(s(X)) -> c_11(dbl1^#(X)) , sel1^#(s(X), cons(Y, Z)) -> c_13(sel1^#(X, Z)) , quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(0()) -> c_15() , quote^#(s(X)) -> c_16(quote^#(X)) , quote^#(sel(X, Y)) -> c_17(sel1^#(X, Y)) } Weak DPs: { sel1^#(0(), cons(X, Y)) -> c_12() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,3,5,7,10,14} by applications of Pre({1,3,5,7,10,14}) = {2,4,6,8,11,13,15}. 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^#(nil(), X) -> c_7() , 8: indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , 9: from^#(X) -> c_9(from^#(s(X))) , 10: dbl1^#(0()) -> c_10() , 11: dbl1^#(s(X)) -> c_11(dbl1^#(X)) , 12: sel1^#(s(X), cons(Y, Z)) -> c_13(sel1^#(X, Z)) , 13: quote^#(dbl(X)) -> c_14(dbl1^#(X)) , 14: quote^#(0()) -> c_15() , 15: quote^#(s(X)) -> c_16(quote^#(X)) , 16: quote^#(sel(X, Y)) -> c_17(sel1^#(X, Y)) , 17: sel1^#(0(), cons(X, Y)) -> c_12() } 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))) , dbl1^#(s(X)) -> c_11(dbl1^#(X)) , sel1^#(s(X), cons(Y, Z)) -> c_13(sel1^#(X, Z)) , quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(s(X)) -> c_16(quote^#(X)) , quote^#(sel(X, Y)) -> c_17(sel1^#(X, Y)) } Weak DPs: { dbl^#(0()) -> c_1() , dbls^#(nil()) -> c_3() , sel^#(0(), cons(X, Y)) -> c_5() , indx^#(nil(), X) -> c_7() , dbl1^#(0()) -> c_10() , sel1^#(0(), cons(X, Y)) -> c_12() , quote^#(0()) -> c_15() } 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() , dbl1^#(0()) -> c_10() , sel1^#(0(), cons(X, Y)) -> c_12() , quote^#(0()) -> c_15() } 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))) , dbl1^#(s(X)) -> c_11(dbl1^#(X)) , sel1^#(s(X), cons(Y, Z)) -> c_13(sel1^#(X, Z)) , quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(s(X)) -> c_16(quote^#(X)) , quote^#(sel(X, Y)) -> c_17(sel1^#(X, Y)) } 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: { 7: sel1^#(s(X), cons(Y, Z)) -> c_13(sel1^#(X, Z)) , 10: quote^#(sel(X, Y)) -> c_17(sel1^#(X, Y)) } 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}, Uargs(c_11) = {1}, Uargs(c_13) = {1}, Uargs(c_14) = {1}, Uargs(c_16) = {1}, Uargs(c_17) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [dbl](x1) = [4] x1 + [0] [0] = [0] [s](x1) = [1] x1 + [0] [dbls](x1) = [7] x1 + [0] [nil] = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [2] [sel](x1, x2) = [4] x2 + [4] [indx](x1, x2) = [7] x1 + [7] x2 + [0] [from](x1) = [7] x1 + [0] [dbl1](x1) = [7] x1 + [0] [01] = [0] [s1](x1) = [1] x1 + [0] [sel1](x1, x2) = [7] x1 + [7] x2 + [0] [quote](x1) = [7] x1 + [0] [dbl^#](x1) = [0] [c_1] = [0] [c_2](x1) = [4] x1 + [0] [dbls^#](x1) = [0] [c_3] = [0] [c_4](x1, x2) = [2] x1 + [1] x2 + [0] [sel^#](x1, x2) = [0] [c_5] = [0] [c_6](x1) = [4] x1 + [0] [indx^#](x1, x2) = [0] [c_7] = [0] [c_8](x1, x2) = [1] x1 + [4] x2 + [0] [from^#](x1) = [0] [c_9](x1) = [4] x1 + [0] [dbl1^#](x1) = [0] [c_10] = [0] [c_11](x1) = [4] x1 + [0] [sel1^#](x1, x2) = [2] x2 + [0] [c_12] = [0] [c_13](x1) = [1] x1 + [1] [quote^#](x1) = [2] x1 + [0] [c_14](x1) = [1] x1 + [0] [c_15] = [0] [c_16](x1) = [1] x1 + [0] [c_17](x1) = [4] x1 + [5] The following symbols are considered usable {dbl^#, dbls^#, sel^#, indx^#, from^#, dbl1^#, sel1^#, quote^#} The order satisfies the following ordering constraints: [dbl^#(s(X))] = [0] >= [0] = [c_2(dbl^#(X))] [dbls^#(cons(X, Y))] = [0] >= [0] = [c_4(dbl^#(X), dbls^#(Y))] [sel^#(s(X), cons(Y, Z))] = [0] >= [0] = [c_6(sel^#(X, Z))] [indx^#(cons(X, Y), Z)] = [0] >= [0] = [c_8(sel^#(X, Z), indx^#(Y, Z))] [from^#(X)] = [0] >= [0] = [c_9(from^#(s(X)))] [dbl1^#(s(X))] = [0] >= [0] = [c_11(dbl1^#(X))] [sel1^#(s(X), cons(Y, Z))] = [2] Y + [2] Z + [4] > [2] Z + [1] = [c_13(sel1^#(X, Z))] [quote^#(dbl(X))] = [8] X + [0] >= [0] = [c_14(dbl1^#(X))] [quote^#(s(X))] = [2] X + [0] >= [2] X + [0] = [c_16(quote^#(X))] [quote^#(sel(X, Y))] = [8] Y + [8] > [8] Y + [5] = [c_17(sel1^#(X, Y))] 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: { 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))) , dbl1^#(s(X)) -> c_11(dbl1^#(X)) , quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(s(X)) -> c_16(quote^#(X)) } Weak DPs: { sel1^#(s(X), cons(Y, Z)) -> c_13(sel1^#(X, Z)) , quote^#(sel(X, Y)) -> c_17(sel1^#(X, 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. { sel1^#(s(X), cons(Y, Z)) -> c_13(sel1^#(X, Z)) , quote^#(sel(X, Y)) -> c_17(sel1^#(X, Y)) } 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))) , dbl1^#(s(X)) -> c_11(dbl1^#(X)) , quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(s(X)) -> c_16(quote^#(X)) } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 3: sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , 4: indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , 7: quote^#(dbl(X)) -> c_14(dbl1^#(X)) , 8: quote^#(s(X)) -> c_16(quote^#(X)) } 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}, Uargs(c_11) = {1}, Uargs(c_14) = {1}, Uargs(c_16) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [dbl](x1) = [4] x1 + [4] [0] = [0] [s](x1) = [1] x1 + [4] [dbls](x1) = [7] x1 + [0] [nil] = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [2] [sel](x1, x2) = [7] x1 + [7] x2 + [0] [indx](x1, x2) = [7] x1 + [7] x2 + [0] [from](x1) = [7] x1 + [0] [dbl1](x1) = [7] x1 + [0] [01] = [0] [s1](x1) = [1] x1 + [0] [sel1](x1, x2) = [7] x1 + [7] x2 + [0] [quote](x1) = [7] x1 + [0] [dbl^#](x1) = [0] [c_1] = [0] [c_2](x1) = [4] x1 + [0] [dbls^#](x1) = [0] [c_3] = [0] [c_4](x1, x2) = [4] x1 + [1] x2 + [0] [sel^#](x1, x2) = [1] x1 + [0] [c_5] = [0] [c_6](x1) = [1] x1 + [1] [indx^#](x1, x2) = [4] x1 + [0] [c_7] = [0] [c_8](x1, x2) = [4] x1 + [1] x2 + [3] [from^#](x1) = [0] [c_9](x1) = [2] x1 + [0] [dbl1^#](x1) = [0] [c_10] = [0] [c_11](x1) = [4] x1 + [0] [sel1^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_12] = [0] [c_13](x1) = [7] x1 + [0] [quote^#](x1) = [2] x1 + [0] [c_14](x1) = [2] x1 + [1] [c_15] = [0] [c_16](x1) = [1] x1 + [5] [c_17](x1) = [7] x1 + [0] The following symbols are considered usable {dbl^#, dbls^#, sel^#, indx^#, from^#, dbl1^#, quote^#} The order satisfies the following ordering constraints: [dbl^#(s(X))] = [0] >= [0] = [c_2(dbl^#(X))] [dbls^#(cons(X, Y))] = [0] >= [0] = [c_4(dbl^#(X), dbls^#(Y))] [sel^#(s(X), cons(Y, Z))] = [1] X + [4] > [1] X + [1] = [c_6(sel^#(X, Z))] [indx^#(cons(X, Y), Z)] = [4] X + [4] Y + [8] > [4] X + [4] Y + [3] = [c_8(sel^#(X, Z), indx^#(Y, Z))] [from^#(X)] = [0] >= [0] = [c_9(from^#(s(X)))] [dbl1^#(s(X))] = [0] >= [0] = [c_11(dbl1^#(X))] [quote^#(dbl(X))] = [8] X + [8] > [1] = [c_14(dbl1^#(X))] [quote^#(s(X))] = [2] X + [8] > [2] X + [5] = [c_16(quote^#(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: { dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , from^#(X) -> c_9(from^#(s(X))) , dbl1^#(s(X)) -> c_11(dbl1^#(X)) } Weak DPs: { sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(s(X)) -> c_16(quote^#(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. { sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , 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: { dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , from^#(X) -> c_9(from^#(s(X))) , dbl1^#(s(X)) -> c_11(dbl1^#(X)) } Weak DPs: { quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(s(X)) -> c_16(quote^#(X)) } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: dbl^#(s(X)) -> c_2(dbl^#(X)) , 2: dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , 5: quote^#(dbl(X)) -> c_14(dbl1^#(X)) , 6: quote^#(s(X)) -> c_16(quote^#(X)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_2) = {1}, Uargs(c_4) = {1, 2}, Uargs(c_9) = {1}, Uargs(c_11) = {1}, Uargs(c_14) = {1}, Uargs(c_16) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [dbl](x1) = [4] x1 + [4] [0] = [0] [s](x1) = [1] x1 + [4] [dbls](x1) = [7] x1 + [0] [nil] = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [2] [sel](x1, x2) = [7] x1 + [7] x2 + [0] [indx](x1, x2) = [7] x1 + [7] x2 + [0] [from](x1) = [7] x1 + [0] [dbl1](x1) = [7] x1 + [0] [01] = [0] [s1](x1) = [1] x1 + [0] [sel1](x1, x2) = [7] x1 + [7] x2 + [0] [quote](x1) = [7] x1 + [0] [dbl^#](x1) = [2] 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 + [7] [sel^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_5] = [0] [c_6](x1) = [7] x1 + [0] [indx^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_7] = [0] [c_8](x1, x2) = [7] x1 + [7] x2 + [0] [from^#](x1) = [0] [c_9](x1) = [1] x1 + [0] [dbl1^#](x1) = [0] [c_10] = [0] [c_11](x1) = [4] x1 + [0] [sel1^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_12] = [0] [c_13](x1) = [7] x1 + [0] [quote^#](x1) = [2] x1 + [0] [c_14](x1) = [4] x1 + [5] [c_15] = [0] [c_16](x1) = [1] x1 + [5] [c_17](x1) = [7] x1 + [0] The following symbols are considered usable {dbl^#, dbls^#, from^#, dbl1^#, quote^#} The order satisfies the following ordering constraints: [dbl^#(s(X))] = [2] X + [8] > [2] X + [1] = [c_2(dbl^#(X))] [dbls^#(cons(X, Y))] = [4] X + [4] Y + [8] > [2] X + [4] Y + [7] = [c_4(dbl^#(X), dbls^#(Y))] [from^#(X)] = [0] >= [0] = [c_9(from^#(s(X)))] [dbl1^#(s(X))] = [0] >= [0] = [c_11(dbl1^#(X))] [quote^#(dbl(X))] = [8] X + [8] > [5] = [c_14(dbl1^#(X))] [quote^#(s(X))] = [2] X + [8] > [2] X + [5] = [c_16(quote^#(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_9(from^#(s(X))) , dbl1^#(s(X)) -> c_11(dbl1^#(X)) } Weak DPs: { dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(s(X)) -> c_16(quote^#(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. { dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_9(from^#(s(X))) , dbl1^#(s(X)) -> c_11(dbl1^#(X)) } Weak DPs: { quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(s(X)) -> c_16(quote^#(X)) } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 2: dbl1^#(s(X)) -> c_11(dbl1^#(X)) , 3: quote^#(dbl(X)) -> c_14(dbl1^#(X)) , 4: quote^#(s(X)) -> c_16(quote^#(X)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_9) = {1}, Uargs(c_11) = {1}, Uargs(c_14) = {1}, Uargs(c_16) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [dbl](x1) = [4] x1 + [4] [0] = [0] [s](x1) = [1] x1 + [4] [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] [dbl1](x1) = [7] x1 + [0] [01] = [0] [s1](x1) = [1] x1 + [0] [sel1](x1, x2) = [7] x1 + [7] x2 + [0] [quote](x1) = [7] x1 + [0] [dbl^#](x1) = [7] x1 + [0] [c_1] = [0] [c_2](x1) = [7] x1 + [0] [dbls^#](x1) = [7] 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) = [7] x1 + [7] x2 + [0] [c_7] = [0] [c_8](x1, x2) = [7] x1 + [7] x2 + [0] [from^#](x1) = [0] [c_9](x1) = [1] x1 + [0] [dbl1^#](x1) = [2] x1 + [0] [c_10] = [0] [c_11](x1) = [1] x1 + [1] [sel1^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_12] = [0] [c_13](x1) = [7] x1 + [0] [quote^#](x1) = [2] x1 + [0] [c_14](x1) = [1] x1 + [5] [c_15] = [0] [c_16](x1) = [1] x1 + [1] [c_17](x1) = [7] x1 + [0] The following symbols are considered usable {from^#, dbl1^#, quote^#} The order satisfies the following ordering constraints: [from^#(X)] = [0] >= [0] = [c_9(from^#(s(X)))] [dbl1^#(s(X))] = [2] X + [8] > [2] X + [1] = [c_11(dbl1^#(X))] [quote^#(dbl(X))] = [8] X + [8] > [2] X + [5] = [c_14(dbl1^#(X))] [quote^#(s(X))] = [2] X + [8] > [2] X + [1] = [c_16(quote^#(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_9(from^#(s(X))) } Weak DPs: { dbl1^#(s(X)) -> c_11(dbl1^#(X)) , quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(s(X)) -> c_16(quote^#(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. { dbl1^#(s(X)) -> c_11(dbl1^#(X)) , quote^#(dbl(X)) -> c_14(dbl1^#(X)) , quote^#(s(X)) -> c_16(quote^#(X)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { 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) '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..