MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , fac(x) -> help(x, 0()) , help(x, c) -> if(lt(c, x), x, c) , if(true(), x, c) -> times(s(c), help(x, s(c))) , if(false(), x, c) -> s(0()) } 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(if) = {1}, Uargs(times) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [lt](x1, x2) = [4] [0] = [7] [s](x1) = [0] [true] = [1] [false] = [1] [fac](x1) = [1] x1 + [7] [help](x1, x2) = [0] [if](x1, x2, x3) = [1] x1 + [0] [times](x1, x2) = [1] x2 + [7] The following symbols are considered usable {lt, fac, help, if} The order satisfies the following ordering constraints: [lt(x, 0())] = [4] > [1] = [false()] [lt(0(), s(x))] = [4] > [1] = [true()] [lt(s(x), s(y))] = [4] >= [4] = [lt(x, y)] [fac(x)] = [1] x + [7] > [0] = [help(x, 0())] [help(x, c)] = [0] ? [4] = [if(lt(c, x), x, c)] [if(true(), x, c)] = [1] ? [7] = [times(s(c), help(x, s(c)))] [if(false(), x, c)] = [1] > [0] = [s(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: { lt(s(x), s(y)) -> lt(x, y) , help(x, c) -> if(lt(c, x), x, c) , if(true(), x, c) -> times(s(c), help(x, s(c))) } Weak Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , fac(x) -> help(x, 0()) , if(false(), x, c) -> 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(if) = {1}, Uargs(times) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [lt](x1, x2) = [4] [0] = [7] [s](x1) = [0] [true] = [0] [false] = [0] [fac](x1) = [1] x1 + [7] [help](x1, x2) = [1] x1 + [5] [if](x1, x2, x3) = [1] x1 + [1] x2 + [0] [times](x1, x2) = [1] x2 + [6] The following symbols are considered usable {lt, fac, help, if} The order satisfies the following ordering constraints: [lt(x, 0())] = [4] > [0] = [false()] [lt(0(), s(x))] = [4] > [0] = [true()] [lt(s(x), s(y))] = [4] >= [4] = [lt(x, y)] [fac(x)] = [1] x + [7] > [1] x + [5] = [help(x, 0())] [help(x, c)] = [1] x + [5] > [1] x + [4] = [if(lt(c, x), x, c)] [if(true(), x, c)] = [1] x + [0] ? [1] x + [11] = [times(s(c), help(x, s(c)))] [if(false(), x, c)] = [1] x + [0] >= [0] = [s(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: { lt(s(x), s(y)) -> lt(x, y) , if(true(), x, c) -> times(s(c), help(x, s(c))) } Weak Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , fac(x) -> help(x, 0()) , help(x, c) -> if(lt(c, x), x, c) , if(false(), x, c) -> s(0()) } 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(if) = {1}, Uargs(times) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [lt](x1, x2) = [0 0] x1 + [4] [0 1] [0] [0] = [0] [1] [s](x1) = [0] [0] [true] = [3] [1] [false] = [4] [0] [fac](x1) = [0 4] x1 + [7] [0 0] [7] [help](x1, x2) = [0 2] x2 + [4] [0 0] [4] [if](x1, x2, x3) = [1 2] x1 + [0] [0 0] [0] [times](x1, x2) = [1 0] x2 + [0] [0 0] [0] The following symbols are considered usable {lt, fac, help, if} The order satisfies the following ordering constraints: [lt(x, 0())] = [0 0] x + [4] [0 1] [0] >= [4] [0] = [false()] [lt(0(), s(x))] = [4] [1] > [3] [1] = [true()] [lt(s(x), s(y))] = [4] [0] ? [0 0] x + [4] [0 1] [0] = [lt(x, y)] [fac(x)] = [0 4] x + [7] [0 0] [7] > [6] [4] = [help(x, 0())] [help(x, c)] = [0 2] c + [4] [0 0] [4] >= [0 2] c + [4] [0 0] [0] = [if(lt(c, x), x, c)] [if(true(), x, c)] = [5] [0] > [4] [0] = [times(s(c), help(x, s(c)))] [if(false(), x, c)] = [4] [0] > [0] [0] = [s(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: { lt(s(x), s(y)) -> lt(x, y) } Weak Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , fac(x) -> help(x, 0()) , help(x, c) -> if(lt(c, x), x, c) , if(true(), x, c) -> times(s(c), help(x, s(c))) , if(false(), x, c) -> s(0()) } 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) '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. 2) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , fac^#(x) -> c_4(help^#(x, 0())) , help^#(x, c) -> c_5(if^#(lt(c, x), x, c)) , if^#(true(), x, c) -> c_6(c, help^#(x, s(c))) , if^#(false(), x, c) -> c_7() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , fac^#(x) -> c_4(help^#(x, 0())) , help^#(x, c) -> c_5(if^#(lt(c, x), x, c)) , if^#(true(), x, c) -> c_6(c, help^#(x, s(c))) , if^#(false(), x, c) -> c_7() } Strict Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , fac(x) -> help(x, 0()) , help(x, c) -> if(lt(c, x), x, c) , if(true(), x, c) -> times(s(c), help(x, s(c))) , if(false(), x, c) -> s(0()) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,7} by applications of Pre({1,2,7}) = {3,5,6}. Here rules are labeled as follows: DPs: { 1: lt^#(x, 0()) -> c_1() , 2: lt^#(0(), s(x)) -> c_2() , 3: lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , 4: fac^#(x) -> c_4(help^#(x, 0())) , 5: help^#(x, c) -> c_5(if^#(lt(c, x), x, c)) , 6: if^#(true(), x, c) -> c_6(c, help^#(x, s(c))) , 7: if^#(false(), x, c) -> c_7() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , fac^#(x) -> c_4(help^#(x, 0())) , help^#(x, c) -> c_5(if^#(lt(c, x), x, c)) , if^#(true(), x, c) -> c_6(c, help^#(x, s(c))) } Strict Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , fac(x) -> help(x, 0()) , help(x, c) -> if(lt(c, x), x, c) , if(true(), x, c) -> times(s(c), help(x, s(c))) , if(false(), x, c) -> s(0()) } Weak DPs: { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , if^#(false(), x, c) -> c_7() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..