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