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) , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), x, y) -> mod(minus(x, y), y) , if_mod(false(), s(x), s(y)) -> s(x) } 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)) , minus^#(x, 0()) -> c_4() , minus^#(s(x), s(y)) -> c_5(minus^#(x, y)) , mod^#(0(), y) -> c_6() , mod^#(s(x), 0()) -> c_7() , mod^#(s(x), s(y)) -> c_8(if_mod^#(le(y, x), s(x), s(y)), le^#(y, x)) , if_mod^#(true(), x, y) -> c_9(mod^#(minus(x, y), y), minus^#(x, y)) , if_mod^#(false(), s(x), s(y)) -> c_10() } 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)) , minus^#(x, 0()) -> c_4() , minus^#(s(x), s(y)) -> c_5(minus^#(x, y)) , mod^#(0(), y) -> c_6() , mod^#(s(x), 0()) -> c_7() , mod^#(s(x), s(y)) -> c_8(if_mod^#(le(y, x), s(x), s(y)), le^#(y, x)) , if_mod^#(true(), x, y) -> c_9(mod^#(minus(x, y), y), minus^#(x, y)) , if_mod^#(false(), s(x), s(y)) -> c_10() } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), x, y) -> mod(minus(x, y), y) , if_mod(false(), s(x), s(y)) -> s(x) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,2,4,6,7,10} by applications of Pre({1,2,4,6,7,10}) = {3,5,8,9}. 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: minus^#(x, 0()) -> c_4() , 5: minus^#(s(x), s(y)) -> c_5(minus^#(x, y)) , 6: mod^#(0(), y) -> c_6() , 7: mod^#(s(x), 0()) -> c_7() , 8: mod^#(s(x), s(y)) -> c_8(if_mod^#(le(y, x), s(x), s(y)), le^#(y, x)) , 9: if_mod^#(true(), x, y) -> c_9(mod^#(minus(x, y), y), minus^#(x, y)) , 10: if_mod^#(false(), s(x), s(y)) -> c_10() } 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)) , minus^#(s(x), s(y)) -> c_5(minus^#(x, y)) , mod^#(s(x), s(y)) -> c_8(if_mod^#(le(y, x), s(x), s(y)), le^#(y, x)) , if_mod^#(true(), x, y) -> c_9(mod^#(minus(x, y), y), minus^#(x, y)) } Weak DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , minus^#(x, 0()) -> c_4() , mod^#(0(), y) -> c_6() , mod^#(s(x), 0()) -> c_7() , if_mod^#(false(), s(x), s(y)) -> c_10() } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), x, y) -> mod(minus(x, y), y) , if_mod(false(), s(x), s(y)) -> s(x) } 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() , minus^#(x, 0()) -> c_4() , mod^#(0(), y) -> c_6() , mod^#(s(x), 0()) -> c_7() , if_mod^#(false(), s(x), s(y)) -> c_10() } 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)) , minus^#(s(x), s(y)) -> c_5(minus^#(x, y)) , mod^#(s(x), s(y)) -> c_8(if_mod^#(le(y, x), s(x), s(y)), le^#(y, x)) , if_mod^#(true(), x, y) -> c_9(mod^#(minus(x, y), y), minus^#(x, y)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), x, y) -> mod(minus(x, y), y) , if_mod(false(), s(x), s(y)) -> s(x) } 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) , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(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)) , minus^#(s(x), s(y)) -> c_5(minus^#(x, y)) , mod^#(s(x), s(y)) -> c_8(if_mod^#(le(y, x), s(x), s(y)), le^#(y, x)) , if_mod^#(true(), x, y) -> c_9(mod^#(minus(x, y), y), minus^#(x, y)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..