MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { nonZero(0()) -> false() , nonZero(s(x)) -> true() , p(0()) -> 0() , p(s(x)) -> x , id_inc(x) -> x , id_inc(x) -> s(x) , random(x) -> rand(x, 0()) , rand(x, y) -> if(nonZero(x), x, y) , if(false(), x, y) -> y , if(true(), x, y) -> rand(p(x), id_inc(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) '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(rand) = {1, 2}, Uargs(if) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [nonZero](x1) = [0] [0] = [0] [false] = [0] [s](x1) = [1] x1 + [0] [true] = [1] [p](x1) = [1] x1 + [0] [id_inc](x1) = [1] x1 + [0] [random](x1) = [1] x1 + [7] [rand](x1, x2) = [1] x1 + [1] x2 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {nonZero, p, id_inc, random, rand, if} The order satisfies the following ordering constraints: [nonZero(0())] = [0] >= [0] = [false()] [nonZero(s(x))] = [0] ? [1] = [true()] [p(0())] = [0] >= [0] = [0()] [p(s(x))] = [1] x + [0] >= [1] x + [0] = [x] [id_inc(x)] = [1] x + [0] >= [1] x + [0] = [x] [id_inc(x)] = [1] x + [0] >= [1] x + [0] = [s(x)] [random(x)] = [1] x + [7] > [1] x + [0] = [rand(x, 0())] [rand(x, y)] = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = [if(nonZero(x), x, y)] [if(false(), x, y)] = [1] x + [1] y + [0] >= [1] y + [0] = [y] [if(true(), x, y)] = [1] x + [1] y + [1] > [1] x + [1] y + [0] = [rand(p(x), id_inc(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: { nonZero(0()) -> false() , nonZero(s(x)) -> true() , p(0()) -> 0() , p(s(x)) -> x , id_inc(x) -> x , id_inc(x) -> s(x) , rand(x, y) -> if(nonZero(x), x, y) , if(false(), x, y) -> y } Weak Trs: { random(x) -> rand(x, 0()) , if(true(), x, y) -> rand(p(x), id_inc(y)) } Obligation: innermost runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(rand) = {1, 2}, Uargs(if) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [nonZero](x1) = [7] [0] = [3] [false] = [7] [s](x1) = [1] x1 + [7] [true] = [7] [p](x1) = [1] x1 + [7] [id_inc](x1) = [1] x1 + [6] [random](x1) = [1] x1 + [7] [rand](x1, x2) = [1] x1 + [1] x2 + [1] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [7] The following symbols are considered usable {nonZero, p, id_inc, random, rand, if} The order satisfies the following ordering constraints: [nonZero(0())] = [7] >= [7] = [false()] [nonZero(s(x))] = [7] >= [7] = [true()] [p(0())] = [10] > [3] = [0()] [p(s(x))] = [1] x + [14] > [1] x + [0] = [x] [id_inc(x)] = [1] x + [6] > [1] x + [0] = [x] [id_inc(x)] = [1] x + [6] ? [1] x + [7] = [s(x)] [random(x)] = [1] x + [7] > [1] x + [4] = [rand(x, 0())] [rand(x, y)] = [1] x + [1] y + [1] ? [1] x + [1] y + [14] = [if(nonZero(x), x, y)] [if(false(), x, y)] = [1] x + [1] y + [14] > [1] y + [0] = [y] [if(true(), x, y)] = [1] x + [1] y + [14] >= [1] x + [1] y + [14] = [rand(p(x), id_inc(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: { nonZero(0()) -> false() , nonZero(s(x)) -> true() , id_inc(x) -> s(x) , rand(x, y) -> if(nonZero(x), x, y) } Weak Trs: { p(0()) -> 0() , p(s(x)) -> x , id_inc(x) -> x , random(x) -> rand(x, 0()) , if(false(), x, y) -> y , if(true(), x, y) -> rand(p(x), id_inc(y)) } Obligation: innermost runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(rand) = {1, 2}, Uargs(if) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [nonZero](x1) = [7] [0] = [0] [false] = [5] [s](x1) = [1] x1 + [5] [true] = [7] [p](x1) = [1] x1 + [1] [id_inc](x1) = [1] x1 + [6] [random](x1) = [1] x1 + [7] [rand](x1, x2) = [1] x1 + [1] x2 + [7] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [7] The following symbols are considered usable {nonZero, p, id_inc, random, rand, if} The order satisfies the following ordering constraints: [nonZero(0())] = [7] > [5] = [false()] [nonZero(s(x))] = [7] >= [7] = [true()] [p(0())] = [1] > [0] = [0()] [p(s(x))] = [1] x + [6] > [1] x + [0] = [x] [id_inc(x)] = [1] x + [6] > [1] x + [0] = [x] [id_inc(x)] = [1] x + [6] > [1] x + [5] = [s(x)] [random(x)] = [1] x + [7] >= [1] x + [7] = [rand(x, 0())] [rand(x, y)] = [1] x + [1] y + [7] ? [1] x + [1] y + [14] = [if(nonZero(x), x, y)] [if(false(), x, y)] = [1] x + [1] y + [12] > [1] y + [0] = [y] [if(true(), x, y)] = [1] x + [1] y + [14] >= [1] x + [1] y + [14] = [rand(p(x), id_inc(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: { nonZero(s(x)) -> true() , rand(x, y) -> if(nonZero(x), x, y) } Weak Trs: { nonZero(0()) -> false() , p(0()) -> 0() , p(s(x)) -> x , id_inc(x) -> x , id_inc(x) -> s(x) , random(x) -> rand(x, 0()) , if(false(), x, y) -> y , if(true(), x, y) -> rand(p(x), id_inc(y)) } Obligation: innermost runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(rand) = {1, 2}, Uargs(if) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [nonZero](x1) = [4] [0] = [6] [false] = [3] [s](x1) = [1] x1 + [0] [true] = [1] [p](x1) = [1] x1 + [0] [id_inc](x1) = [1] x1 + [0] [random](x1) = [1] x1 + [7] [rand](x1, x2) = [1] x1 + [1] x2 + [1] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {nonZero, p, id_inc, random, rand, if} The order satisfies the following ordering constraints: [nonZero(0())] = [4] > [3] = [false()] [nonZero(s(x))] = [4] > [1] = [true()] [p(0())] = [6] >= [6] = [0()] [p(s(x))] = [1] x + [0] >= [1] x + [0] = [x] [id_inc(x)] = [1] x + [0] >= [1] x + [0] = [x] [id_inc(x)] = [1] x + [0] >= [1] x + [0] = [s(x)] [random(x)] = [1] x + [7] >= [1] x + [7] = [rand(x, 0())] [rand(x, y)] = [1] x + [1] y + [1] ? [1] x + [1] y + [4] = [if(nonZero(x), x, y)] [if(false(), x, y)] = [1] x + [1] y + [3] > [1] y + [0] = [y] [if(true(), x, y)] = [1] x + [1] y + [1] >= [1] x + [1] y + [1] = [rand(p(x), id_inc(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: { rand(x, y) -> if(nonZero(x), x, y) } Weak Trs: { nonZero(0()) -> false() , nonZero(s(x)) -> true() , p(0()) -> 0() , p(s(x)) -> x , id_inc(x) -> x , id_inc(x) -> s(x) , random(x) -> rand(x, 0()) , if(false(), x, y) -> y , if(true(), x, y) -> rand(p(x), id_inc(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: 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) '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 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. Arrrr..