MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { p(p(s(x))) -> p(x) , p(0()) -> s(s(0())) , p(s(x)) -> x , le(p(s(x)), x) -> le(x, x) , le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(x, y) -> if(le(x, y), x, y) , if(true(), x, y) -> 0() , if(false(), x, y) -> s(minus(p(x), y)) } Obligation: innermost runtime complexity Answer: MAYBE Arguments of following rules are not normal-forms: { p(p(s(x))) -> p(x) , le(p(s(x)), x) -> le(x, x) } All above mentioned rules can be savely removed. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { p(0()) -> s(s(0())) , p(s(x)) -> x , le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(x, y) -> if(le(x, y), x, y) , if(true(), x, y) -> 0() , if(false(), x, y) -> s(minus(p(x), 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(s) = {1}, Uargs(minus) = {1}, Uargs(if) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [p](x1) = [1] x1 + [0] [0] = [0] [s](x1) = [1] x1 + [0] [le](x1, x2) = [1] [true] = [0] [false] = [1] [minus](x1, x2) = [1] x1 + [1] x2 + [0] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {p, le, minus, if} The order satisfies the following ordering constraints: [p(0())] = [0] >= [0] = [s(s(0()))] [p(s(x))] = [1] x + [0] >= [1] x + [0] = [x] [le(0(), y)] = [1] > [0] = [true()] [le(s(x), 0())] = [1] >= [1] = [false()] [le(s(x), s(y))] = [1] >= [1] = [le(x, y)] [minus(x, y)] = [1] x + [1] y + [0] ? [1] x + [1] y + [1] = [if(le(x, y), x, y)] [if(true(), x, y)] = [1] x + [1] y + [0] >= [0] = [0()] [if(false(), x, y)] = [1] x + [1] y + [1] > [1] x + [1] y + [0] = [s(minus(p(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: { p(0()) -> s(s(0())) , p(s(x)) -> x , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(x, y) -> if(le(x, y), x, y) , if(true(), x, y) -> 0() } Weak Trs: { le(0(), y) -> true() , if(false(), x, y) -> s(minus(p(x), 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(s) = {1}, Uargs(minus) = {1}, Uargs(if) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [p](x1) = [1] x1 + [0] [0] = [0] [s](x1) = [1] x1 + [0] [le](x1, x2) = [0] [true] = [0] [false] = [4] [minus](x1, x2) = [1] x1 + [1] [if](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {p, le, minus, if} The order satisfies the following ordering constraints: [p(0())] = [0] >= [0] = [s(s(0()))] [p(s(x))] = [1] x + [0] >= [1] x + [0] = [x] [le(0(), y)] = [0] >= [0] = [true()] [le(s(x), 0())] = [0] ? [4] = [false()] [le(s(x), s(y))] = [0] >= [0] = [le(x, y)] [minus(x, y)] = [1] x + [1] > [1] x + [0] = [if(le(x, y), x, y)] [if(true(), x, y)] = [1] x + [0] >= [0] = [0()] [if(false(), x, y)] = [1] x + [4] > [1] x + [1] = [s(minus(p(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: { p(0()) -> s(s(0())) , p(s(x)) -> x , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , if(true(), x, y) -> 0() } Weak Trs: { le(0(), y) -> true() , minus(x, y) -> if(le(x, y), x, y) , if(false(), x, y) -> s(minus(p(x), 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(s) = {1}, Uargs(minus) = {1}, Uargs(if) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [p](x1) = [1] x1 + [1] [0] = [0] [s](x1) = [1] x1 + [0] [le](x1, x2) = [0] [true] = [0] [false] = [5] [minus](x1, x2) = [1] x1 + [1] x2 + [4] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {p, le, minus, if} The order satisfies the following ordering constraints: [p(0())] = [1] > [0] = [s(s(0()))] [p(s(x))] = [1] x + [1] > [1] x + [0] = [x] [le(0(), y)] = [0] >= [0] = [true()] [le(s(x), 0())] = [0] ? [5] = [false()] [le(s(x), s(y))] = [0] >= [0] = [le(x, y)] [minus(x, y)] = [1] x + [1] y + [4] > [1] x + [1] y + [0] = [if(le(x, y), x, y)] [if(true(), x, y)] = [1] x + [1] y + [0] >= [0] = [0()] [if(false(), x, y)] = [1] x + [1] y + [5] >= [1] x + [1] y + [5] = [s(minus(p(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: { le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , if(true(), x, y) -> 0() } Weak Trs: { p(0()) -> s(s(0())) , p(s(x)) -> x , le(0(), y) -> true() , minus(x, y) -> if(le(x, y), x, y) , if(false(), x, y) -> s(minus(p(x), 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(s) = {1}, Uargs(minus) = {1}, Uargs(if) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [p](x1) = [1] x1 + [0] [0] = [0] [s](x1) = [1] x1 + [0] [le](x1, x2) = [1] [true] = [1] [false] = [1] [minus](x1, x2) = [1] x1 + [1] x2 + [1] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {p, le, minus, if} The order satisfies the following ordering constraints: [p(0())] = [0] >= [0] = [s(s(0()))] [p(s(x))] = [1] x + [0] >= [1] x + [0] = [x] [le(0(), y)] = [1] >= [1] = [true()] [le(s(x), 0())] = [1] >= [1] = [false()] [le(s(x), s(y))] = [1] >= [1] = [le(x, y)] [minus(x, y)] = [1] x + [1] y + [1] >= [1] x + [1] y + [1] = [if(le(x, y), x, y)] [if(true(), x, y)] = [1] x + [1] y + [1] > [0] = [0()] [if(false(), x, y)] = [1] x + [1] y + [1] >= [1] x + [1] y + [1] = [s(minus(p(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: { le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) } Weak Trs: { p(0()) -> s(s(0())) , p(s(x)) -> x , le(0(), y) -> true() , minus(x, y) -> if(le(x, y), x, y) , if(true(), x, y) -> 0() , if(false(), x, y) -> s(minus(p(x), 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) '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. Arrrr..