MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { gt(s(x), s(y)) -> gt(x, y) , gt(s(x), 0()) -> true() , gt(0(), y) -> false() , divides(x, y) -> div(x, y, y) , div(s(x), s(y), z) -> div(x, y, z) , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z)) , div(0(), s(x), z) -> false() , div(0(), 0(), z) -> true() , prime(x) -> test(x, s(s(0()))) , test(x, y) -> if1(gt(x, y), x, y) , if1(true(), x, y) -> if2(divides(x, y), x, y) , if1(false(), x, y) -> true() , if2(true(), x, y) -> false() , if2(false(), x, y) -> test(x, s(y)) } Obligation: runtime complexity Answer: MAYBE 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(if1) = {1}, Uargs(if2) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [gt](x1, x2) = [4] [s](x1) = [7] [0] = [7] [true] = [0] [false] = [0] [divides](x1, x2) = [0] [div](x1, x2, x3) = [0] [prime](x1) = [1] x1 + [7] [test](x1, x2) = [1] x1 + [0] [if1](x1, x2, x3) = [1] x1 + [1] x2 + [0] [if2](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {gt, divides, div, prime, test, if1, if2} The order satisfies the following ordering constraints: [gt(s(x), s(y))] = [4] >= [4] = [gt(x, y)] [gt(s(x), 0())] = [4] > [0] = [true()] [gt(0(), y)] = [4] > [0] = [false()] [divides(x, y)] = [0] >= [0] = [div(x, y, y)] [div(s(x), s(y), z)] = [0] >= [0] = [div(x, y, z)] [div(s(x), 0(), s(z))] = [0] >= [0] = [div(s(x), s(z), s(z))] [div(0(), s(x), z)] = [0] >= [0] = [false()] [div(0(), 0(), z)] = [0] >= [0] = [true()] [prime(x)] = [1] x + [7] > [1] x + [0] = [test(x, s(s(0())))] [test(x, y)] = [1] x + [0] ? [1] x + [4] = [if1(gt(x, y), x, y)] [if1(true(), x, y)] = [1] x + [0] >= [1] x + [0] = [if2(divides(x, y), x, y)] [if1(false(), x, y)] = [1] x + [0] >= [0] = [true()] [if2(true(), x, y)] = [1] x + [0] >= [0] = [false()] [if2(false(), x, y)] = [1] x + [0] >= [1] x + [0] = [test(x, s(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 Trs: { gt(s(x), s(y)) -> gt(x, y) , divides(x, y) -> div(x, y, y) , div(s(x), s(y), z) -> div(x, y, z) , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z)) , div(0(), s(x), z) -> false() , div(0(), 0(), z) -> true() , test(x, y) -> if1(gt(x, y), x, y) , if1(true(), x, y) -> if2(divides(x, y), x, y) , if1(false(), x, y) -> true() , if2(true(), x, y) -> false() , if2(false(), x, y) -> test(x, s(y)) } Weak Trs: { gt(s(x), 0()) -> true() , gt(0(), y) -> false() , prime(x) -> test(x, s(s(0()))) } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(if1) = {1}, Uargs(if2) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [gt](x1, x2) = [4] [s](x1) = [7] [0] = [7] [true] = [0] [false] = [0] [divides](x1, x2) = [1] [div](x1, x2, x3) = [0] [prime](x1) = [1] x1 + [7] [test](x1, x2) = [1] x1 + [0] [if1](x1, x2, x3) = [1] x1 + [1] x2 + [0] [if2](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {gt, divides, div, prime, test, if1, if2} The order satisfies the following ordering constraints: [gt(s(x), s(y))] = [4] >= [4] = [gt(x, y)] [gt(s(x), 0())] = [4] > [0] = [true()] [gt(0(), y)] = [4] > [0] = [false()] [divides(x, y)] = [1] > [0] = [div(x, y, y)] [div(s(x), s(y), z)] = [0] >= [0] = [div(x, y, z)] [div(s(x), 0(), s(z))] = [0] >= [0] = [div(s(x), s(z), s(z))] [div(0(), s(x), z)] = [0] >= [0] = [false()] [div(0(), 0(), z)] = [0] >= [0] = [true()] [prime(x)] = [1] x + [7] > [1] x + [0] = [test(x, s(s(0())))] [test(x, y)] = [1] x + [0] ? [1] x + [4] = [if1(gt(x, y), x, y)] [if1(true(), x, y)] = [1] x + [0] ? [1] x + [1] = [if2(divides(x, y), x, y)] [if1(false(), x, y)] = [1] x + [0] >= [0] = [true()] [if2(true(), x, y)] = [1] x + [0] >= [0] = [false()] [if2(false(), x, y)] = [1] x + [0] >= [1] x + [0] = [test(x, s(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 Trs: { gt(s(x), s(y)) -> gt(x, y) , div(s(x), s(y), z) -> div(x, y, z) , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z)) , div(0(), s(x), z) -> false() , div(0(), 0(), z) -> true() , test(x, y) -> if1(gt(x, y), x, y) , if1(true(), x, y) -> if2(divides(x, y), x, y) , if1(false(), x, y) -> true() , if2(true(), x, y) -> false() , if2(false(), x, y) -> test(x, s(y)) } Weak Trs: { gt(s(x), 0()) -> true() , gt(0(), y) -> false() , divides(x, y) -> div(x, y, y) , prime(x) -> test(x, s(s(0()))) } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(if1) = {1}, Uargs(if2) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [gt](x1, x2) = [4] [s](x1) = [7] [0] = [7] [true] = [0] [false] = [0] [divides](x1, x2) = [4] [div](x1, x2, x3) = [0] [prime](x1) = [1] x1 + [7] [test](x1, x2) = [1] x1 + [5] [if1](x1, x2, x3) = [1] x1 + [1] x2 + [0] [if2](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {gt, divides, div, prime, test, if1, if2} The order satisfies the following ordering constraints: [gt(s(x), s(y))] = [4] >= [4] = [gt(x, y)] [gt(s(x), 0())] = [4] > [0] = [true()] [gt(0(), y)] = [4] > [0] = [false()] [divides(x, y)] = [4] > [0] = [div(x, y, y)] [div(s(x), s(y), z)] = [0] >= [0] = [div(x, y, z)] [div(s(x), 0(), s(z))] = [0] >= [0] = [div(s(x), s(z), s(z))] [div(0(), s(x), z)] = [0] >= [0] = [false()] [div(0(), 0(), z)] = [0] >= [0] = [true()] [prime(x)] = [1] x + [7] > [1] x + [5] = [test(x, s(s(0())))] [test(x, y)] = [1] x + [5] > [1] x + [4] = [if1(gt(x, y), x, y)] [if1(true(), x, y)] = [1] x + [0] ? [1] x + [4] = [if2(divides(x, y), x, y)] [if1(false(), x, y)] = [1] x + [0] >= [0] = [true()] [if2(true(), x, y)] = [1] x + [0] >= [0] = [false()] [if2(false(), x, y)] = [1] x + [0] ? [1] x + [5] = [test(x, s(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 Trs: { gt(s(x), s(y)) -> gt(x, y) , div(s(x), s(y), z) -> div(x, y, z) , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z)) , div(0(), s(x), z) -> false() , div(0(), 0(), z) -> true() , if1(true(), x, y) -> if2(divides(x, y), x, y) , if1(false(), x, y) -> true() , if2(true(), x, y) -> false() , if2(false(), x, y) -> test(x, s(y)) } Weak Trs: { gt(s(x), 0()) -> true() , gt(0(), y) -> false() , divides(x, y) -> div(x, y, y) , prime(x) -> test(x, s(s(0()))) , test(x, y) -> if1(gt(x, y), x, y) } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(if1) = {1}, Uargs(if2) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [gt](x1, x2) = [1] [s](x1) = [7] [0] = [7] [true] = [0] [false] = [1] [divides](x1, x2) = [4] [div](x1, x2, x3) = [0] [prime](x1) = [1] x1 + [7] [test](x1, x2) = [1] x1 + [4] [if1](x1, x2, x3) = [1] x1 + [1] x2 + [0] [if2](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {gt, divides, div, prime, test, if1, if2} The order satisfies the following ordering constraints: [gt(s(x), s(y))] = [1] >= [1] = [gt(x, y)] [gt(s(x), 0())] = [1] > [0] = [true()] [gt(0(), y)] = [1] >= [1] = [false()] [divides(x, y)] = [4] > [0] = [div(x, y, y)] [div(s(x), s(y), z)] = [0] >= [0] = [div(x, y, z)] [div(s(x), 0(), s(z))] = [0] >= [0] = [div(s(x), s(z), s(z))] [div(0(), s(x), z)] = [0] ? [1] = [false()] [div(0(), 0(), z)] = [0] >= [0] = [true()] [prime(x)] = [1] x + [7] > [1] x + [4] = [test(x, s(s(0())))] [test(x, y)] = [1] x + [4] > [1] x + [1] = [if1(gt(x, y), x, y)] [if1(true(), x, y)] = [1] x + [0] ? [1] x + [4] = [if2(divides(x, y), x, y)] [if1(false(), x, y)] = [1] x + [1] > [0] = [true()] [if2(true(), x, y)] = [1] x + [0] ? [1] = [false()] [if2(false(), x, y)] = [1] x + [1] ? [1] x + [4] = [test(x, s(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 Trs: { gt(s(x), s(y)) -> gt(x, y) , div(s(x), s(y), z) -> div(x, y, z) , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z)) , div(0(), s(x), z) -> false() , div(0(), 0(), z) -> true() , if1(true(), x, y) -> if2(divides(x, y), x, y) , if2(true(), x, y) -> false() , if2(false(), x, y) -> test(x, s(y)) } Weak Trs: { gt(s(x), 0()) -> true() , gt(0(), y) -> false() , divides(x, y) -> div(x, y, y) , prime(x) -> test(x, s(s(0()))) , test(x, y) -> if1(gt(x, y), x, y) , if1(false(), x, y) -> true() } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(if1) = {1}, Uargs(if2) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [gt](x1, x2) = [0] [s](x1) = [3] [0] = [7] [true] = [0] [false] = [0] [divides](x1, x2) = [4] [div](x1, x2, x3) = [4] [prime](x1) = [1] x1 + [7] [test](x1, x2) = [1] x1 + [4] [if1](x1, x2, x3) = [1] x1 + [1] x2 + [0] [if2](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {gt, divides, div, prime, test, if1, if2} The order satisfies the following ordering constraints: [gt(s(x), s(y))] = [0] >= [0] = [gt(x, y)] [gt(s(x), 0())] = [0] >= [0] = [true()] [gt(0(), y)] = [0] >= [0] = [false()] [divides(x, y)] = [4] >= [4] = [div(x, y, y)] [div(s(x), s(y), z)] = [4] >= [4] = [div(x, y, z)] [div(s(x), 0(), s(z))] = [4] >= [4] = [div(s(x), s(z), s(z))] [div(0(), s(x), z)] = [4] > [0] = [false()] [div(0(), 0(), z)] = [4] > [0] = [true()] [prime(x)] = [1] x + [7] > [1] x + [4] = [test(x, s(s(0())))] [test(x, y)] = [1] x + [4] > [1] x + [0] = [if1(gt(x, y), x, y)] [if1(true(), x, y)] = [1] x + [0] ? [1] x + [4] = [if2(divides(x, y), x, y)] [if1(false(), x, y)] = [1] x + [0] >= [0] = [true()] [if2(true(), x, y)] = [1] x + [0] >= [0] = [false()] [if2(false(), x, y)] = [1] x + [0] ? [1] x + [4] = [test(x, s(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 Trs: { gt(s(x), s(y)) -> gt(x, y) , div(s(x), s(y), z) -> div(x, y, z) , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z)) , if1(true(), x, y) -> if2(divides(x, y), x, y) , if2(true(), x, y) -> false() , if2(false(), x, y) -> test(x, s(y)) } Weak Trs: { gt(s(x), 0()) -> true() , gt(0(), y) -> false() , divides(x, y) -> div(x, y, y) , div(0(), s(x), z) -> false() , div(0(), 0(), z) -> true() , prime(x) -> test(x, s(s(0()))) , test(x, y) -> if1(gt(x, y), x, y) , if1(false(), x, y) -> true() } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(if1) = {1}, Uargs(if2) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [gt](x1, x2) = [2] [s](x1) = [7] [0] = [7] [true] = [0] [false] = [0] [divides](x1, x2) = [0] [div](x1, x2, x3) = [0] [prime](x1) = [1] x1 + [7] [test](x1, x2) = [4] [if1](x1, x2, x3) = [1] x1 + [2] [if2](x1, x2, x3) = [1] x1 + [0] The following symbols are considered usable {gt, divides, div, prime, test, if1, if2} The order satisfies the following ordering constraints: [gt(s(x), s(y))] = [2] >= [2] = [gt(x, y)] [gt(s(x), 0())] = [2] > [0] = [true()] [gt(0(), y)] = [2] > [0] = [false()] [divides(x, y)] = [0] >= [0] = [div(x, y, y)] [div(s(x), s(y), z)] = [0] >= [0] = [div(x, y, z)] [div(s(x), 0(), s(z))] = [0] >= [0] = [div(s(x), s(z), s(z))] [div(0(), s(x), z)] = [0] >= [0] = [false()] [div(0(), 0(), z)] = [0] >= [0] = [true()] [prime(x)] = [1] x + [7] > [4] = [test(x, s(s(0())))] [test(x, y)] = [4] >= [4] = [if1(gt(x, y), x, y)] [if1(true(), x, y)] = [2] > [0] = [if2(divides(x, y), x, y)] [if1(false(), x, y)] = [2] > [0] = [true()] [if2(true(), x, y)] = [0] >= [0] = [false()] [if2(false(), x, y)] = [0] ? [4] = [test(x, s(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 Trs: { gt(s(x), s(y)) -> gt(x, y) , div(s(x), s(y), z) -> div(x, y, z) , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z)) , if2(true(), x, y) -> false() , if2(false(), x, y) -> test(x, s(y)) } Weak Trs: { gt(s(x), 0()) -> true() , gt(0(), y) -> false() , divides(x, y) -> div(x, y, y) , div(0(), s(x), z) -> false() , div(0(), 0(), z) -> true() , prime(x) -> test(x, s(s(0()))) , test(x, y) -> if1(gt(x, y), x, y) , if1(true(), x, y) -> if2(divides(x, y), x, y) , if1(false(), x, y) -> true() } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(if1) = {1}, Uargs(if2) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [gt](x1, x2) = [1] [s](x1) = [7] [0] = [7] [true] = [1] [false] = [0] [divides](x1, x2) = [1] [div](x1, x2, x3) = [1] [prime](x1) = [1] x1 + [7] [test](x1, x2) = [1] x1 + [5] [if1](x1, x2, x3) = [1] x1 + [1] x2 + [4] [if2](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {gt, divides, div, prime, test, if1, if2} The order satisfies the following ordering constraints: [gt(s(x), s(y))] = [1] >= [1] = [gt(x, y)] [gt(s(x), 0())] = [1] >= [1] = [true()] [gt(0(), y)] = [1] > [0] = [false()] [divides(x, y)] = [1] >= [1] = [div(x, y, y)] [div(s(x), s(y), z)] = [1] >= [1] = [div(x, y, z)] [div(s(x), 0(), s(z))] = [1] >= [1] = [div(s(x), s(z), s(z))] [div(0(), s(x), z)] = [1] > [0] = [false()] [div(0(), 0(), z)] = [1] >= [1] = [true()] [prime(x)] = [1] x + [7] > [1] x + [5] = [test(x, s(s(0())))] [test(x, y)] = [1] x + [5] >= [1] x + [5] = [if1(gt(x, y), x, y)] [if1(true(), x, y)] = [1] x + [5] > [1] x + [1] = [if2(divides(x, y), x, y)] [if1(false(), x, y)] = [1] x + [4] > [1] = [true()] [if2(true(), x, y)] = [1] x + [1] > [0] = [false()] [if2(false(), x, y)] = [1] x + [0] ? [1] x + [5] = [test(x, s(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 Trs: { gt(s(x), s(y)) -> gt(x, y) , div(s(x), s(y), z) -> div(x, y, z) , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z)) , if2(false(), x, y) -> test(x, s(y)) } Weak Trs: { gt(s(x), 0()) -> true() , gt(0(), y) -> false() , divides(x, y) -> div(x, y, y) , div(0(), s(x), z) -> false() , div(0(), 0(), z) -> true() , prime(x) -> test(x, s(s(0()))) , test(x, y) -> if1(gt(x, y), x, y) , if1(true(), x, y) -> if2(divides(x, y), x, y) , if1(false(), x, y) -> true() , if2(true(), x, y) -> false() } Obligation: 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: The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(if1) = {1}, Uargs(if2) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [gt](x1, x2) = [0 0] x2 + [0] [0 1] [0] [s](x1) = [0] [0] [0] = [0] [2] [true] = [0] [2] [false] = [0] [0] [divides](x1, x2) = [0] [4] [div](x1, x2, x3) = [0] [4] [prime](x1) = [7] [7] [test](x1, x2) = [0 1] x2 + [6] [0 0] [4] [if1](x1, x2, x3) = [1 1] x1 + [6] [0 0] [4] [if2](x1, x2, x3) = [1 0] x1 + [7] [0 0] [4] The following symbols are considered usable {gt, divides, div, prime, test, if1, if2} The order satisfies the following ordering constraints: [gt(s(x), s(y))] = [0] [0] ? [0 0] y + [0] [0 1] [0] = [gt(x, y)] [gt(s(x), 0())] = [0] [2] >= [0] [2] = [true()] [gt(0(), y)] = [0 0] y + [0] [0 1] [0] >= [0] [0] = [false()] [divides(x, y)] = [0] [4] >= [0] [4] = [div(x, y, y)] [div(s(x), s(y), z)] = [0] [4] >= [0] [4] = [div(x, y, z)] [div(s(x), 0(), s(z))] = [0] [4] >= [0] [4] = [div(s(x), s(z), s(z))] [div(0(), s(x), z)] = [0] [4] >= [0] [0] = [false()] [div(0(), 0(), z)] = [0] [4] >= [0] [2] = [true()] [prime(x)] = [7] [7] > [6] [4] = [test(x, s(s(0())))] [test(x, y)] = [0 1] y + [6] [0 0] [4] >= [0 1] y + [6] [0 0] [4] = [if1(gt(x, y), x, y)] [if1(true(), x, y)] = [8] [4] > [7] [4] = [if2(divides(x, y), x, y)] [if1(false(), x, y)] = [6] [4] > [0] [2] = [true()] [if2(true(), x, y)] = [7] [4] > [0] [0] = [false()] [if2(false(), x, y)] = [7] [4] > [6] [4] = [test(x, s(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 Trs: { gt(s(x), s(y)) -> gt(x, y) , div(s(x), s(y), z) -> div(x, y, z) , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z)) } Weak Trs: { gt(s(x), 0()) -> true() , gt(0(), y) -> false() , divides(x, y) -> div(x, y, y) , div(0(), s(x), z) -> false() , div(0(), 0(), z) -> true() , prime(x) -> test(x, s(s(0()))) , test(x, y) -> if1(gt(x, y), x, y) , if1(true(), x, y) -> if2(divides(x, y), x, y) , if1(false(), x, y) -> true() , if2(true(), x, y) -> false() , if2(false(), x, y) -> test(x, s(y)) } Obligation: 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) '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) '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. 3) '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 processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { gt^#(s(x), s(y)) -> c_1(gt^#(x, y)) , gt^#(s(x), 0()) -> c_2() , gt^#(0(), y) -> c_3() , divides^#(x, y) -> c_4(div^#(x, y, y)) , div^#(s(x), s(y), z) -> c_5(div^#(x, y, z)) , div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z))) , div^#(0(), s(x), z) -> c_7() , div^#(0(), 0(), z) -> c_8() , prime^#(x) -> c_9(test^#(x, s(s(0())))) , test^#(x, y) -> c_10(if1^#(gt(x, y), x, y)) , if1^#(true(), x, y) -> c_11(if2^#(divides(x, y), x, y)) , if1^#(false(), x, y) -> c_12() , if2^#(true(), x, y) -> c_13() , if2^#(false(), x, y) -> c_14(test^#(x, s(y))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { gt^#(s(x), s(y)) -> c_1(gt^#(x, y)) , gt^#(s(x), 0()) -> c_2() , gt^#(0(), y) -> c_3() , divides^#(x, y) -> c_4(div^#(x, y, y)) , div^#(s(x), s(y), z) -> c_5(div^#(x, y, z)) , div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z))) , div^#(0(), s(x), z) -> c_7() , div^#(0(), 0(), z) -> c_8() , prime^#(x) -> c_9(test^#(x, s(s(0())))) , test^#(x, y) -> c_10(if1^#(gt(x, y), x, y)) , if1^#(true(), x, y) -> c_11(if2^#(divides(x, y), x, y)) , if1^#(false(), x, y) -> c_12() , if2^#(true(), x, y) -> c_13() , if2^#(false(), x, y) -> c_14(test^#(x, s(y))) } Strict Trs: { gt(s(x), s(y)) -> gt(x, y) , gt(s(x), 0()) -> true() , gt(0(), y) -> false() , divides(x, y) -> div(x, y, y) , div(s(x), s(y), z) -> div(x, y, z) , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z)) , div(0(), s(x), z) -> false() , div(0(), 0(), z) -> true() , prime(x) -> test(x, s(s(0()))) , test(x, y) -> if1(gt(x, y), x, y) , if1(true(), x, y) -> if2(divides(x, y), x, y) , if1(false(), x, y) -> true() , if2(true(), x, y) -> false() , if2(false(), x, y) -> test(x, s(y)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,3,7,8,12,13} by applications of Pre({2,3,7,8,12,13}) = {1,4,5,10,11}. Here rules are labeled as follows: DPs: { 1: gt^#(s(x), s(y)) -> c_1(gt^#(x, y)) , 2: gt^#(s(x), 0()) -> c_2() , 3: gt^#(0(), y) -> c_3() , 4: divides^#(x, y) -> c_4(div^#(x, y, y)) , 5: div^#(s(x), s(y), z) -> c_5(div^#(x, y, z)) , 6: div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z))) , 7: div^#(0(), s(x), z) -> c_7() , 8: div^#(0(), 0(), z) -> c_8() , 9: prime^#(x) -> c_9(test^#(x, s(s(0())))) , 10: test^#(x, y) -> c_10(if1^#(gt(x, y), x, y)) , 11: if1^#(true(), x, y) -> c_11(if2^#(divides(x, y), x, y)) , 12: if1^#(false(), x, y) -> c_12() , 13: if2^#(true(), x, y) -> c_13() , 14: if2^#(false(), x, y) -> c_14(test^#(x, s(y))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { gt^#(s(x), s(y)) -> c_1(gt^#(x, y)) , divides^#(x, y) -> c_4(div^#(x, y, y)) , div^#(s(x), s(y), z) -> c_5(div^#(x, y, z)) , div^#(s(x), 0(), s(z)) -> c_6(div^#(s(x), s(z), s(z))) , prime^#(x) -> c_9(test^#(x, s(s(0())))) , test^#(x, y) -> c_10(if1^#(gt(x, y), x, y)) , if1^#(true(), x, y) -> c_11(if2^#(divides(x, y), x, y)) , if2^#(false(), x, y) -> c_14(test^#(x, s(y))) } Strict Trs: { gt(s(x), s(y)) -> gt(x, y) , gt(s(x), 0()) -> true() , gt(0(), y) -> false() , divides(x, y) -> div(x, y, y) , div(s(x), s(y), z) -> div(x, y, z) , div(s(x), 0(), s(z)) -> div(s(x), s(z), s(z)) , div(0(), s(x), z) -> false() , div(0(), 0(), z) -> true() , prime(x) -> test(x, s(s(0()))) , test(x, y) -> if1(gt(x, y), x, y) , if1(true(), x, y) -> if2(divides(x, y), x, y) , if1(false(), x, y) -> true() , if2(true(), x, y) -> false() , if2(false(), x, y) -> test(x, s(y)) } Weak DPs: { gt^#(s(x), 0()) -> c_2() , gt^#(0(), y) -> c_3() , div^#(0(), s(x), z) -> c_7() , div^#(0(), 0(), z) -> c_8() , if1^#(false(), x, y) -> c_12() , if2^#(true(), x, y) -> c_13() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..