MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { eq(X, Y) -> false() , eq(0(), 0()) -> true() , eq(s(X), s(Y)) -> eq(X, Y) , inf(X) -> cons(X, inf(s(X))) , take(0(), X) -> nil() , take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) , length(cons(X, L)) -> s(length(L)) , length(nil()) -> 0() } 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: The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(cons) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [eq](x1, x2) = [7] [0] = [7] [true] = [6] [s](x1) = [1] x1 + [7] [false] = [6] [inf](x1) = [1] x1 + [7] [cons](x1, x2) = [1] x2 + [1] [take](x1, x2) = [1] x1 + [1] x2 + [7] [nil] = [7] [length](x1) = [1] x1 + [7] The following symbols are considered usable {eq, inf, take, length} The order satisfies the following ordering constraints: [eq(X, Y)] = [7] > [6] = [false()] [eq(0(), 0())] = [7] > [6] = [true()] [eq(s(X), s(Y))] = [7] >= [7] = [eq(X, Y)] [inf(X)] = [1] X + [7] ? [1] X + [15] = [cons(X, inf(s(X)))] [take(0(), X)] = [1] X + [14] > [7] = [nil()] [take(s(X), cons(Y, L))] = [1] X + [1] L + [15] > [1] X + [1] L + [8] = [cons(Y, take(X, L))] [length(cons(X, L))] = [1] L + [8] ? [1] L + [14] = [s(length(L))] [length(nil())] = [14] > [7] = [0()] 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 Trs: { eq(s(X), s(Y)) -> eq(X, Y) , inf(X) -> cons(X, inf(s(X))) , length(cons(X, L)) -> s(length(L)) } Weak Trs: { eq(X, Y) -> false() , eq(0(), 0()) -> true() , take(0(), X) -> nil() , take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) , length(nil()) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(cons) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [eq](x1, x2) = [1] [0] = [0] [true] = [1] [s](x1) = [1] x1 + [0] [false] = [1] [inf](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x2 + [1] [take](x1, x2) = [1] x2 + [0] [nil] = [0] [length](x1) = [1] x1 + [0] The following symbols are considered usable {eq, inf, take, length} The order satisfies the following ordering constraints: [eq(X, Y)] = [1] >= [1] = [false()] [eq(0(), 0())] = [1] >= [1] = [true()] [eq(s(X), s(Y))] = [1] >= [1] = [eq(X, Y)] [inf(X)] = [1] X + [0] ? [1] X + [1] = [cons(X, inf(s(X)))] [take(0(), X)] = [1] X + [0] >= [0] = [nil()] [take(s(X), cons(Y, L))] = [1] L + [1] >= [1] L + [1] = [cons(Y, take(X, L))] [length(cons(X, L))] = [1] L + [1] > [1] L + [0] = [s(length(L))] [length(nil())] = [0] >= [0] = [0()] 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 Trs: { eq(s(X), s(Y)) -> eq(X, Y) , inf(X) -> cons(X, inf(s(X))) } Weak Trs: { eq(X, Y) -> false() , eq(0(), 0()) -> true() , take(0(), X) -> nil() , take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) , length(cons(X, L)) -> s(length(L)) , length(nil()) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(cons) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [eq](x1, x2) = [1] x2 + [7] [0] = [0] [true] = [7] [s](x1) = [1] x1 + [1] [false] = [7] [inf](x1) = [0] [cons](x1, x2) = [1] x2 + [1] [take](x1, x2) = [1] x2 + [0] [nil] = [0] [length](x1) = [1] x1 + [0] The following symbols are considered usable {eq, inf, take, length} The order satisfies the following ordering constraints: [eq(X, Y)] = [1] Y + [7] >= [7] = [false()] [eq(0(), 0())] = [7] >= [7] = [true()] [eq(s(X), s(Y))] = [1] Y + [8] > [1] Y + [7] = [eq(X, Y)] [inf(X)] = [0] ? [1] = [cons(X, inf(s(X)))] [take(0(), X)] = [1] X + [0] >= [0] = [nil()] [take(s(X), cons(Y, L))] = [1] L + [1] >= [1] L + [1] = [cons(Y, take(X, L))] [length(cons(X, L))] = [1] L + [1] >= [1] L + [1] = [s(length(L))] [length(nil())] = [0] >= [0] = [0()] 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 Trs: { inf(X) -> cons(X, inf(s(X))) } Weak Trs: { eq(X, Y) -> false() , eq(0(), 0()) -> true() , eq(s(X), s(Y)) -> eq(X, Y) , take(0(), X) -> nil() , take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) , length(cons(X, L)) -> s(length(L)) , length(nil()) -> 0() } 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) '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: { eq^#(X, Y) -> c_1() , eq^#(0(), 0()) -> c_2() , eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) , inf^#(X) -> c_4(inf^#(s(X))) , take^#(0(), X) -> c_5() , take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , length^#(cons(X, L)) -> c_7(length^#(L)) , length^#(nil()) -> c_8() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { eq^#(X, Y) -> c_1() , eq^#(0(), 0()) -> c_2() , eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) , inf^#(X) -> c_4(inf^#(s(X))) , take^#(0(), X) -> c_5() , take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , length^#(cons(X, L)) -> c_7(length^#(L)) , length^#(nil()) -> c_8() } Strict Trs: { eq(X, Y) -> false() , eq(0(), 0()) -> true() , eq(s(X), s(Y)) -> eq(X, Y) , inf(X) -> cons(X, inf(s(X))) , take(0(), X) -> nil() , take(s(X), cons(Y, L)) -> cons(Y, take(X, L)) , length(cons(X, L)) -> s(length(L)) , length(nil()) -> 0() } 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: { eq^#(X, Y) -> c_1() , eq^#(0(), 0()) -> c_2() , eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) , inf^#(X) -> c_4(inf^#(s(X))) , take^#(0(), X) -> c_5() , take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , length^#(cons(X, L)) -> c_7(length^#(L)) , length^#(nil()) -> c_8() } 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_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1} TcT has computed the following constructor-restricted matrix interpretation. [0] = [0] [0] [s](x1) = [1 0] x1 + [0] [0 0] [0] [cons](x1, x2) = [1 2] x2 + [0] [0 1] [0] [nil] = [2] [1] [eq^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] [2 1] [2 2] [2] [c_1] = [1] [1] [c_2] = [1] [1] [c_3](x1) = [1 0] x1 + [2] [0 1] [0] [inf^#](x1) = [1 2] x1 + [2] [2 1] [2] [c_4](x1) = [1 0] x1 + [1] [0 1] [1] [take^#](x1, x2) = [0 0] x1 + [1 1] x2 + [2] [1 2] [1 1] [2] [c_5] = [1] [1] [c_6](x1) = [1 0] x1 + [2] [0 1] [0] [length^#](x1) = [1 2] x1 + [2] [2 2] [1] [c_7](x1) = [1 0] x1 + [2] [0 1] [1] [c_8] = [1] [1] The following symbols are considered usable {eq^#, inf^#, take^#, length^#} The order satisfies the following ordering constraints: [eq^#(X, Y)] = [0 0] X + [0 0] Y + [2] [2 1] [2 2] [2] > [1] [1] = [c_1()] [eq^#(0(), 0())] = [2] [2] > [1] [1] = [c_2()] [eq^#(s(X), s(Y))] = [0 0] X + [0 0] Y + [2] [2 0] [2 0] [2] ? [0 0] X + [0 0] Y + [4] [2 1] [2 2] [2] = [c_3(eq^#(X, Y))] [inf^#(X)] = [1 2] X + [2] [2 1] [2] ? [1 0] X + [3] [2 0] [3] = [c_4(inf^#(s(X)))] [take^#(0(), X)] = [1 1] X + [2] [1 1] [2] > [1] [1] = [c_5()] [take^#(s(X), cons(Y, L))] = [0 0] X + [1 3] L + [2] [1 0] [1 3] [2] ? [0 0] X + [1 1] L + [4] [1 2] [1 1] [2] = [c_6(take^#(X, L))] [length^#(cons(X, L))] = [1 4] L + [2] [2 6] [1] ? [1 2] L + [4] [2 2] [2] = [c_7(length^#(L))] [length^#(nil())] = [6] [7] > [1] [1] = [c_8()] 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: { eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) , inf^#(X) -> c_4(inf^#(s(X))) , take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , length^#(cons(X, L)) -> c_7(length^#(L)) } Weak DPs: { eq^#(X, Y) -> c_1() , eq^#(0(), 0()) -> c_2() , take^#(0(), X) -> c_5() , length^#(nil()) -> c_8() } 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. { eq^#(X, Y) -> c_1() , eq^#(0(), 0()) -> c_2() , take^#(0(), X) -> c_5() , length^#(nil()) -> c_8() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) , inf^#(X) -> c_4(inf^#(s(X))) , take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , length^#(cons(X, L)) -> c_7(length^#(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: { 3: take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , 4: length^#(cons(X, L)) -> c_7(length^#(L)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [eq](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [0] [true] = [0] [s](x1) = [1] x1 + [4] [false] = [0] [inf](x1) = [7] x1 + [0] [cons](x1, x2) = [1] x1 + [1] x2 + [4] [take](x1, x2) = [7] x1 + [7] x2 + [0] [nil] = [0] [length](x1) = [7] x1 + [0] [eq^#](x1, x2) = [0] [c_1] = [0] [c_2] = [0] [c_3](x1) = [4] x1 + [0] [inf^#](x1) = [0] [c_4](x1) = [4] x1 + [0] [take^#](x1, x2) = [1] x1 + [1] x2 + [1] [c_5] = [0] [c_6](x1) = [1] x1 + [1] [length^#](x1) = [2] x1 + [0] [c_7](x1) = [1] x1 + [3] [c_8] = [0] The following symbols are considered usable {eq^#, inf^#, take^#, length^#} The order satisfies the following ordering constraints: [eq^#(s(X), s(Y))] = [0] >= [0] = [c_3(eq^#(X, Y))] [inf^#(X)] = [0] >= [0] = [c_4(inf^#(s(X)))] [take^#(s(X), cons(Y, L))] = [1] X + [1] Y + [1] L + [9] > [1] X + [1] L + [2] = [c_6(take^#(X, L))] [length^#(cons(X, L))] = [2] X + [2] L + [8] > [2] L + [3] = [c_7(length^#(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: { eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) , inf^#(X) -> c_4(inf^#(s(X))) } Weak DPs: { take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , length^#(cons(X, L)) -> c_7(length^#(L)) } 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. { take^#(s(X), cons(Y, L)) -> c_6(take^#(X, L)) , length^#(cons(X, L)) -> c_7(length^#(L)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) , inf^#(X) -> c_4(inf^#(s(X))) } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_4) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [eq](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [0] [true] = [0] [s](x1) = [1] x1 + [2] [false] = [0] [inf](x1) = [7] x1 + [0] [cons](x1, x2) = [1] x1 + [1] x2 + [0] [take](x1, x2) = [7] x1 + [7] x2 + [0] [nil] = [0] [length](x1) = [7] x1 + [0] [eq^#](x1, x2) = [1] x1 + [3] x2 + [0] [c_1] = [0] [c_2] = [0] [c_3](x1) = [1] x1 + [5] [inf^#](x1) = [0] [c_4](x1) = [4] x1 + [0] [take^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_5] = [0] [c_6](x1) = [7] x1 + [0] [length^#](x1) = [7] x1 + [0] [c_7](x1) = [7] x1 + [0] [c_8] = [0] The following symbols are considered usable {eq^#, inf^#} The order satisfies the following ordering constraints: [eq^#(s(X), s(Y))] = [1] X + [3] Y + [8] > [1] X + [3] Y + [5] = [c_3(eq^#(X, Y))] [inf^#(X)] = [0] >= [0] = [c_4(inf^#(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: { inf^#(X) -> c_4(inf^#(s(X))) } Weak DPs: { eq^#(s(X), s(Y)) -> c_3(eq^#(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. { eq^#(s(X), s(Y)) -> c_3(eq^#(X, Y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { inf^#(X) -> c_4(inf^#(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..