MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { fac(0()) -> 1() , fac(0()) -> s(0()) , fac(s(x)) -> *(s(x), fac(x)) , 1() -> s(0()) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , floop(0(), y) -> y , floop(s(x), y) -> floop(x, *(s(x), y)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 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: The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(*) = {2}, Uargs(floop) = {2}, Uargs(+) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [fac](x1) = [0] [0] = [0] [1] = [7] [s](x1) = [1] x1 + [0] [*](x1, x2) = [1] x2 + [0] [floop](x1, x2) = [1] x2 + [0] [+](x1, x2) = [1] x1 + [0] The following symbols are considered usable {fac, 1, *, floop, +} The order satisfies the following ordering constraints: [fac(0())] = [0] ? [7] = [1()] [fac(0())] = [0] >= [0] = [s(0())] [fac(s(x))] = [0] >= [0] = [*(s(x), fac(x))] [1()] = [7] > [0] = [s(0())] [*(x, 0())] = [0] >= [0] = [0()] [*(x, s(y))] = [1] y + [0] >= [1] y + [0] = [+(*(x, y), x)] [floop(0(), y)] = [1] y + [0] >= [1] y + [0] = [y] [floop(s(x), y)] = [1] y + [0] >= [1] y + [0] = [floop(x, *(s(x), y))] [+(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [+(x, s(y))] = [1] x + [0] >= [1] x + [0] = [s(+(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 Trs: { fac(0()) -> 1() , fac(0()) -> s(0()) , fac(s(x)) -> *(s(x), fac(x)) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , floop(0(), y) -> y , floop(s(x), y) -> floop(x, *(s(x), y)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) } Weak Trs: { 1() -> 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(s) = {1}, Uargs(*) = {2}, Uargs(floop) = {2}, Uargs(+) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [fac](x1) = [0] [0] = [1] [1] = [7] [s](x1) = [1] x1 + [0] [*](x1, x2) = [1] x2 + [0] [floop](x1, x2) = [1] x1 + [1] x2 + [0] [+](x1, x2) = [1] x1 + [0] The following symbols are considered usable {fac, 1, *, floop, +} The order satisfies the following ordering constraints: [fac(0())] = [0] ? [7] = [1()] [fac(0())] = [0] ? [1] = [s(0())] [fac(s(x))] = [0] >= [0] = [*(s(x), fac(x))] [1()] = [7] > [1] = [s(0())] [*(x, 0())] = [1] >= [1] = [0()] [*(x, s(y))] = [1] y + [0] >= [1] y + [0] = [+(*(x, y), x)] [floop(0(), y)] = [1] y + [1] > [1] y + [0] = [y] [floop(s(x), y)] = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = [floop(x, *(s(x), y))] [+(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [+(x, s(y))] = [1] x + [0] >= [1] x + [0] = [s(+(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 Trs: { fac(0()) -> 1() , fac(0()) -> s(0()) , fac(s(x)) -> *(s(x), fac(x)) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , floop(s(x), y) -> floop(x, *(s(x), y)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) } Weak Trs: { 1() -> s(0()) , floop(0(), y) -> y } Obligation: 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(*) = {2}, Uargs(floop) = {2}, Uargs(+) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [fac](x1) = [1] x1 + [7] [0] = [0] [1] = [7] [s](x1) = [1] x1 + [7] [*](x1, x2) = [1] x2 + [7] [floop](x1, x2) = [1] x1 + [1] x2 + [7] [+](x1, x2) = [1] x1 + [3] The following symbols are considered usable {fac, 1, *, floop, +} The order satisfies the following ordering constraints: [fac(0())] = [7] >= [7] = [1()] [fac(0())] = [7] >= [7] = [s(0())] [fac(s(x))] = [1] x + [14] >= [1] x + [14] = [*(s(x), fac(x))] [1()] = [7] >= [7] = [s(0())] [*(x, 0())] = [7] > [0] = [0()] [*(x, s(y))] = [1] y + [14] > [1] y + [10] = [+(*(x, y), x)] [floop(0(), y)] = [1] y + [7] > [1] y + [0] = [y] [floop(s(x), y)] = [1] x + [1] y + [14] >= [1] x + [1] y + [14] = [floop(x, *(s(x), y))] [+(x, 0())] = [1] x + [3] > [1] x + [0] = [x] [+(x, s(y))] = [1] x + [3] ? [1] x + [10] = [s(+(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 Trs: { fac(0()) -> 1() , fac(0()) -> s(0()) , fac(s(x)) -> *(s(x), fac(x)) , floop(s(x), y) -> floop(x, *(s(x), y)) , +(x, s(y)) -> s(+(x, y)) } Weak Trs: { 1() -> s(0()) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , floop(0(), y) -> y , +(x, 0()) -> x } Obligation: 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(*) = {2}, Uargs(floop) = {2}, Uargs(+) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [fac](x1) = [0] [0] = [0] [1] = [7] [s](x1) = [1] x1 + [4] [*](x1, x2) = [1] x2 + [0] [floop](x1, x2) = [1] x1 + [1] x2 + [0] [+](x1, x2) = [1] x1 + [0] The following symbols are considered usable {fac, 1, *, floop, +} The order satisfies the following ordering constraints: [fac(0())] = [0] ? [7] = [1()] [fac(0())] = [0] ? [4] = [s(0())] [fac(s(x))] = [0] >= [0] = [*(s(x), fac(x))] [1()] = [7] > [4] = [s(0())] [*(x, 0())] = [0] >= [0] = [0()] [*(x, s(y))] = [1] y + [4] > [1] y + [0] = [+(*(x, y), x)] [floop(0(), y)] = [1] y + [0] >= [1] y + [0] = [y] [floop(s(x), y)] = [1] x + [1] y + [4] > [1] x + [1] y + [0] = [floop(x, *(s(x), y))] [+(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [+(x, s(y))] = [1] x + [0] ? [1] x + [4] = [s(+(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 Trs: { fac(0()) -> 1() , fac(0()) -> s(0()) , fac(s(x)) -> *(s(x), fac(x)) , +(x, s(y)) -> s(+(x, y)) } Weak Trs: { 1() -> s(0()) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , floop(0(), y) -> y , floop(s(x), y) -> floop(x, *(s(x), y)) , +(x, 0()) -> x } Obligation: 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(*) = {2}, Uargs(floop) = {2}, Uargs(+) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [fac](x1) = [1] x1 + [0] [0] = [0] [1] = [7] [s](x1) = [1] x1 + [4] [*](x1, x2) = [1] x2 + [0] [floop](x1, x2) = [1] x1 + [1] x2 + [5] [+](x1, x2) = [1] x1 + [0] The following symbols are considered usable {fac, 1, *, floop, +} The order satisfies the following ordering constraints: [fac(0())] = [0] ? [7] = [1()] [fac(0())] = [0] ? [4] = [s(0())] [fac(s(x))] = [1] x + [4] > [1] x + [0] = [*(s(x), fac(x))] [1()] = [7] > [4] = [s(0())] [*(x, 0())] = [0] >= [0] = [0()] [*(x, s(y))] = [1] y + [4] > [1] y + [0] = [+(*(x, y), x)] [floop(0(), y)] = [1] y + [5] > [1] y + [0] = [y] [floop(s(x), y)] = [1] x + [1] y + [9] > [1] x + [1] y + [5] = [floop(x, *(s(x), y))] [+(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [+(x, s(y))] = [1] x + [0] ? [1] x + [4] = [s(+(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 Trs: { fac(0()) -> 1() , fac(0()) -> s(0()) , +(x, s(y)) -> s(+(x, y)) } Weak Trs: { fac(s(x)) -> *(s(x), fac(x)) , 1() -> s(0()) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , floop(0(), y) -> y , floop(s(x), y) -> floop(x, *(s(x), y)) , +(x, 0()) -> x } Obligation: 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(*) = {2}, Uargs(floop) = {2}, Uargs(+) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [fac](x1) = [1] x1 + [6] [0] = [0] [1] = [7] [s](x1) = [1] x1 + [2] [*](x1, x2) = [1] x2 + [0] [floop](x1, x2) = [1] x2 + [5] [+](x1, x2) = [1] x1 + [2] The following symbols are considered usable {fac, 1, *, floop, +} The order satisfies the following ordering constraints: [fac(0())] = [6] ? [7] = [1()] [fac(0())] = [6] > [2] = [s(0())] [fac(s(x))] = [1] x + [8] > [1] x + [6] = [*(s(x), fac(x))] [1()] = [7] > [2] = [s(0())] [*(x, 0())] = [0] >= [0] = [0()] [*(x, s(y))] = [1] y + [2] >= [1] y + [2] = [+(*(x, y), x)] [floop(0(), y)] = [1] y + [5] > [1] y + [0] = [y] [floop(s(x), y)] = [1] y + [5] >= [1] y + [5] = [floop(x, *(s(x), y))] [+(x, 0())] = [1] x + [2] > [1] x + [0] = [x] [+(x, s(y))] = [1] x + [2] ? [1] x + [4] = [s(+(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 Trs: { fac(0()) -> 1() , +(x, s(y)) -> s(+(x, y)) } Weak Trs: { fac(0()) -> s(0()) , fac(s(x)) -> *(s(x), fac(x)) , 1() -> s(0()) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , floop(0(), y) -> y , floop(s(x), y) -> floop(x, *(s(x), y)) , +(x, 0()) -> x } Obligation: 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(*) = {2}, Uargs(floop) = {2}, Uargs(+) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [fac](x1) = [1] x1 + [4] [0] = [4] [1] = [7] [s](x1) = [1] x1 + [0] [*](x1, x2) = [1] x2 + [0] [floop](x1, x2) = [1] x1 + [1] x2 + [5] [+](x1, x2) = [1] x1 + [0] The following symbols are considered usable {fac, 1, *, floop, +} The order satisfies the following ordering constraints: [fac(0())] = [8] > [7] = [1()] [fac(0())] = [8] > [4] = [s(0())] [fac(s(x))] = [1] x + [4] >= [1] x + [4] = [*(s(x), fac(x))] [1()] = [7] > [4] = [s(0())] [*(x, 0())] = [4] >= [4] = [0()] [*(x, s(y))] = [1] y + [0] >= [1] y + [0] = [+(*(x, y), x)] [floop(0(), y)] = [1] y + [9] > [1] y + [0] = [y] [floop(s(x), y)] = [1] x + [1] y + [5] >= [1] x + [1] y + [5] = [floop(x, *(s(x), y))] [+(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [+(x, s(y))] = [1] x + [0] >= [1] x + [0] = [s(+(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 Trs: { +(x, s(y)) -> s(+(x, y)) } Weak Trs: { fac(0()) -> 1() , fac(0()) -> s(0()) , fac(s(x)) -> *(s(x), fac(x)) , 1() -> s(0()) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , floop(0(), y) -> y , floop(s(x), y) -> floop(x, *(s(x), y)) , +(x, 0()) -> x } 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: 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: 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) '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) '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 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. 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { fac^#(0()) -> c_1(1^#()) , fac^#(0()) -> c_2() , fac^#(s(x)) -> c_3(*^#(s(x), fac(x))) , 1^#() -> c_4() , *^#(x, 0()) -> c_5() , *^#(x, s(y)) -> c_6(+^#(*(x, y), x)) , +^#(x, 0()) -> c_9(x) , +^#(x, s(y)) -> c_10(+^#(x, y)) , floop^#(0(), y) -> c_7(y) , floop^#(s(x), y) -> c_8(floop^#(x, *(s(x), y))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { fac^#(0()) -> c_1(1^#()) , fac^#(0()) -> c_2() , fac^#(s(x)) -> c_3(*^#(s(x), fac(x))) , 1^#() -> c_4() , *^#(x, 0()) -> c_5() , *^#(x, s(y)) -> c_6(+^#(*(x, y), x)) , +^#(x, 0()) -> c_9(x) , +^#(x, s(y)) -> c_10(+^#(x, y)) , floop^#(0(), y) -> c_7(y) , floop^#(s(x), y) -> c_8(floop^#(x, *(s(x), y))) } Strict Trs: { fac(0()) -> 1() , fac(0()) -> s(0()) , fac(s(x)) -> *(s(x), fac(x)) , 1() -> s(0()) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , floop(0(), y) -> y , floop(s(x), y) -> floop(x, *(s(x), y)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,4,5} by applications of Pre({2,4,5}) = {1,3,7,9}. Here rules are labeled as follows: DPs: { 1: fac^#(0()) -> c_1(1^#()) , 2: fac^#(0()) -> c_2() , 3: fac^#(s(x)) -> c_3(*^#(s(x), fac(x))) , 4: 1^#() -> c_4() , 5: *^#(x, 0()) -> c_5() , 6: *^#(x, s(y)) -> c_6(+^#(*(x, y), x)) , 7: +^#(x, 0()) -> c_9(x) , 8: +^#(x, s(y)) -> c_10(+^#(x, y)) , 9: floop^#(0(), y) -> c_7(y) , 10: floop^#(s(x), y) -> c_8(floop^#(x, *(s(x), y))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { fac^#(0()) -> c_1(1^#()) , fac^#(s(x)) -> c_3(*^#(s(x), fac(x))) , *^#(x, s(y)) -> c_6(+^#(*(x, y), x)) , +^#(x, 0()) -> c_9(x) , +^#(x, s(y)) -> c_10(+^#(x, y)) , floop^#(0(), y) -> c_7(y) , floop^#(s(x), y) -> c_8(floop^#(x, *(s(x), y))) } Strict Trs: { fac(0()) -> 1() , fac(0()) -> s(0()) , fac(s(x)) -> *(s(x), fac(x)) , 1() -> s(0()) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , floop(0(), y) -> y , floop(s(x), y) -> floop(x, *(s(x), y)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) } Weak DPs: { fac^#(0()) -> c_2() , 1^#() -> c_4() , *^#(x, 0()) -> c_5() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1} by applications of Pre({1}) = {4,6}. Here rules are labeled as follows: DPs: { 1: fac^#(0()) -> c_1(1^#()) , 2: fac^#(s(x)) -> c_3(*^#(s(x), fac(x))) , 3: *^#(x, s(y)) -> c_6(+^#(*(x, y), x)) , 4: +^#(x, 0()) -> c_9(x) , 5: +^#(x, s(y)) -> c_10(+^#(x, y)) , 6: floop^#(0(), y) -> c_7(y) , 7: floop^#(s(x), y) -> c_8(floop^#(x, *(s(x), y))) , 8: fac^#(0()) -> c_2() , 9: 1^#() -> c_4() , 10: *^#(x, 0()) -> c_5() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { fac^#(s(x)) -> c_3(*^#(s(x), fac(x))) , *^#(x, s(y)) -> c_6(+^#(*(x, y), x)) , +^#(x, 0()) -> c_9(x) , +^#(x, s(y)) -> c_10(+^#(x, y)) , floop^#(0(), y) -> c_7(y) , floop^#(s(x), y) -> c_8(floop^#(x, *(s(x), y))) } Strict Trs: { fac(0()) -> 1() , fac(0()) -> s(0()) , fac(s(x)) -> *(s(x), fac(x)) , 1() -> s(0()) , *(x, 0()) -> 0() , *(x, s(y)) -> +(*(x, y), x) , floop(0(), y) -> y , floop(s(x), y) -> floop(x, *(s(x), y)) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) } Weak DPs: { fac^#(0()) -> c_1(1^#()) , fac^#(0()) -> c_2() , 1^#() -> c_4() , *^#(x, 0()) -> c_5() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..