MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { le(0(), y, z) -> greater(y, z) , le(s(x), 0(), z) -> false() , le(s(x), s(y), 0()) -> false() , le(s(x), s(y), s(z)) -> le(x, y, z) , greater(x, 0()) -> first() , greater(0(), s(y)) -> second() , greater(s(x), s(y)) -> greater(x, y) , double(0()) -> 0() , double(s(x)) -> s(s(double(x))) , triple(x) -> if(le(x, x, double(x)), x, 0(), 0()) , if(false(), x, y, z) -> true() , if(first(), x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) , if(second(), x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { le^#(0(), y, z) -> c_1(greater^#(y, z)) , le^#(s(x), 0(), z) -> c_2() , le^#(s(x), s(y), 0()) -> c_3() , le^#(s(x), s(y), s(z)) -> c_4(le^#(x, y, z)) , greater^#(x, 0()) -> c_5() , greater^#(0(), s(y)) -> c_6() , greater^#(s(x), s(y)) -> c_7(greater^#(x, y)) , double^#(0()) -> c_8() , double^#(s(x)) -> c_9(double^#(x)) , triple^#(x) -> c_10(if^#(le(x, x, double(x)), x, 0(), 0()), le^#(x, x, double(x)), double^#(x)) , if^#(false(), x, y, z) -> c_11() , if^#(first(), x, y, z) -> c_12(if^#(le(s(x), y, s(z)), s(x), y, s(z)), le^#(s(x), y, s(z))) , if^#(second(), x, y, z) -> c_13(if^#(le(s(x), s(y), z), s(x), s(y), z), le^#(s(x), s(y), z)) } 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, z) -> c_1(greater^#(y, z)) , le^#(s(x), 0(), z) -> c_2() , le^#(s(x), s(y), 0()) -> c_3() , le^#(s(x), s(y), s(z)) -> c_4(le^#(x, y, z)) , greater^#(x, 0()) -> c_5() , greater^#(0(), s(y)) -> c_6() , greater^#(s(x), s(y)) -> c_7(greater^#(x, y)) , double^#(0()) -> c_8() , double^#(s(x)) -> c_9(double^#(x)) , triple^#(x) -> c_10(if^#(le(x, x, double(x)), x, 0(), 0()), le^#(x, x, double(x)), double^#(x)) , if^#(false(), x, y, z) -> c_11() , if^#(first(), x, y, z) -> c_12(if^#(le(s(x), y, s(z)), s(x), y, s(z)), le^#(s(x), y, s(z))) , if^#(second(), x, y, z) -> c_13(if^#(le(s(x), s(y), z), s(x), s(y), z), le^#(s(x), s(y), z)) } Weak Trs: { le(0(), y, z) -> greater(y, z) , le(s(x), 0(), z) -> false() , le(s(x), s(y), 0()) -> false() , le(s(x), s(y), s(z)) -> le(x, y, z) , greater(x, 0()) -> first() , greater(0(), s(y)) -> second() , greater(s(x), s(y)) -> greater(x, y) , double(0()) -> 0() , double(s(x)) -> s(s(double(x))) , triple(x) -> if(le(x, x, double(x)), x, 0(), 0()) , if(false(), x, y, z) -> true() , if(first(), x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) , if(second(), x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {2,3,5,6,8,11} by applications of Pre({2,3,5,6,8,11}) = {1,4,7,9,10,12,13}. Here rules are labeled as follows: DPs: { 1: le^#(0(), y, z) -> c_1(greater^#(y, z)) , 2: le^#(s(x), 0(), z) -> c_2() , 3: le^#(s(x), s(y), 0()) -> c_3() , 4: le^#(s(x), s(y), s(z)) -> c_4(le^#(x, y, z)) , 5: greater^#(x, 0()) -> c_5() , 6: greater^#(0(), s(y)) -> c_6() , 7: greater^#(s(x), s(y)) -> c_7(greater^#(x, y)) , 8: double^#(0()) -> c_8() , 9: double^#(s(x)) -> c_9(double^#(x)) , 10: triple^#(x) -> c_10(if^#(le(x, x, double(x)), x, 0(), 0()), le^#(x, x, double(x)), double^#(x)) , 11: if^#(false(), x, y, z) -> c_11() , 12: if^#(first(), x, y, z) -> c_12(if^#(le(s(x), y, s(z)), s(x), y, s(z)), le^#(s(x), y, s(z))) , 13: if^#(second(), x, y, z) -> c_13(if^#(le(s(x), s(y), z), s(x), s(y), z), le^#(s(x), s(y), z)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(0(), y, z) -> c_1(greater^#(y, z)) , le^#(s(x), s(y), s(z)) -> c_4(le^#(x, y, z)) , greater^#(s(x), s(y)) -> c_7(greater^#(x, y)) , double^#(s(x)) -> c_9(double^#(x)) , triple^#(x) -> c_10(if^#(le(x, x, double(x)), x, 0(), 0()), le^#(x, x, double(x)), double^#(x)) , if^#(first(), x, y, z) -> c_12(if^#(le(s(x), y, s(z)), s(x), y, s(z)), le^#(s(x), y, s(z))) , if^#(second(), x, y, z) -> c_13(if^#(le(s(x), s(y), z), s(x), s(y), z), le^#(s(x), s(y), z)) } Weak DPs: { le^#(s(x), 0(), z) -> c_2() , le^#(s(x), s(y), 0()) -> c_3() , greater^#(x, 0()) -> c_5() , greater^#(0(), s(y)) -> c_6() , double^#(0()) -> c_8() , if^#(false(), x, y, z) -> c_11() } Weak Trs: { le(0(), y, z) -> greater(y, z) , le(s(x), 0(), z) -> false() , le(s(x), s(y), 0()) -> false() , le(s(x), s(y), s(z)) -> le(x, y, z) , greater(x, 0()) -> first() , greater(0(), s(y)) -> second() , greater(s(x), s(y)) -> greater(x, y) , double(0()) -> 0() , double(s(x)) -> s(s(double(x))) , triple(x) -> if(le(x, x, double(x)), x, 0(), 0()) , if(false(), x, y, z) -> true() , if(first(), x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) , if(second(), x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) } Obligation: innermost runtime complexity Answer: MAYBE The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { le^#(s(x), 0(), z) -> c_2() , le^#(s(x), s(y), 0()) -> c_3() , greater^#(x, 0()) -> c_5() , greater^#(0(), s(y)) -> c_6() , double^#(0()) -> c_8() , if^#(false(), x, y, z) -> c_11() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(0(), y, z) -> c_1(greater^#(y, z)) , le^#(s(x), s(y), s(z)) -> c_4(le^#(x, y, z)) , greater^#(s(x), s(y)) -> c_7(greater^#(x, y)) , double^#(s(x)) -> c_9(double^#(x)) , triple^#(x) -> c_10(if^#(le(x, x, double(x)), x, 0(), 0()), le^#(x, x, double(x)), double^#(x)) , if^#(first(), x, y, z) -> c_12(if^#(le(s(x), y, s(z)), s(x), y, s(z)), le^#(s(x), y, s(z))) , if^#(second(), x, y, z) -> c_13(if^#(le(s(x), s(y), z), s(x), s(y), z), le^#(s(x), s(y), z)) } Weak Trs: { le(0(), y, z) -> greater(y, z) , le(s(x), 0(), z) -> false() , le(s(x), s(y), 0()) -> false() , le(s(x), s(y), s(z)) -> le(x, y, z) , greater(x, 0()) -> first() , greater(0(), s(y)) -> second() , greater(s(x), s(y)) -> greater(x, y) , double(0()) -> 0() , double(s(x)) -> s(s(double(x))) , triple(x) -> if(le(x, x, double(x)), x, 0(), 0()) , if(false(), x, y, z) -> true() , if(first(), x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) , if(second(), x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { le(0(), y, z) -> greater(y, z) , le(s(x), 0(), z) -> false() , le(s(x), s(y), 0()) -> false() , le(s(x), s(y), s(z)) -> le(x, y, z) , greater(x, 0()) -> first() , greater(0(), s(y)) -> second() , greater(s(x), s(y)) -> greater(x, y) , double(0()) -> 0() , double(s(x)) -> s(s(double(x))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(0(), y, z) -> c_1(greater^#(y, z)) , le^#(s(x), s(y), s(z)) -> c_4(le^#(x, y, z)) , greater^#(s(x), s(y)) -> c_7(greater^#(x, y)) , double^#(s(x)) -> c_9(double^#(x)) , triple^#(x) -> c_10(if^#(le(x, x, double(x)), x, 0(), 0()), le^#(x, x, double(x)), double^#(x)) , if^#(first(), x, y, z) -> c_12(if^#(le(s(x), y, s(z)), s(x), y, s(z)), le^#(s(x), y, s(z))) , if^#(second(), x, y, z) -> c_13(if^#(le(s(x), s(y), z), s(x), s(y), z), le^#(s(x), s(y), z)) } Weak Trs: { le(0(), y, z) -> greater(y, z) , le(s(x), 0(), z) -> false() , le(s(x), s(y), 0()) -> false() , le(s(x), s(y), s(z)) -> le(x, y, z) , greater(x, 0()) -> first() , greater(0(), s(y)) -> second() , greater(s(x), s(y)) -> greater(x, y) , double(0()) -> 0() , double(s(x)) -> s(s(double(x))) } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrices' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrix interpretation of dimension 4' failed due to the following reason: Following exception was raised: stack overflow 2) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 3) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 4) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 5) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 6) 'matrix interpretation of dimension 1' failed due to the following reason: The input cannot be shown compatible 2) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. Arrrr..