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: innermost runtime complexity Answer: MAYBE We add following dependency tuples: 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), le^#(x, y)) , if^#(true(), x, y) -> c_5(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), le^#(x, y)) , if^#(true(), x, y) -> c_5(int^#(s(x), y)) , if^#(false(), x, y) -> c_6() } Weak 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: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,2,6} by applications of Pre({1,2,6}) = {3,4}. 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), le^#(x, y)) , 5: if^#(true(), x, y) -> c_5(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), le^#(x, y)) , if^#(true(), x, y) -> c_5(int^#(s(x), y)) } Weak DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , if^#(false(), x, y) -> c_6() } Weak 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: 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^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , 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), le^#(x, y)) , if^#(true(), x, y) -> c_5(int^#(s(x), y)) } Weak 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: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) } 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), le^#(x, y)) , if^#(true(), x, y) -> c_5(int^#(s(x), y)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..