MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(x, y) -> cond(lt(x, y), x, y) , cond(tt(), x, y) -> f(s(x), s(y)) , lt(s(x), s(y)) -> lt(x, y) , lt(0(), y) -> tt() } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { f^#(x, y) -> c_1(cond^#(lt(x, y), x, y), lt^#(x, y)) , cond^#(tt(), x, y) -> c_2(f^#(s(x), s(y))) , lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , lt^#(0(), y) -> c_4() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(x, y) -> c_1(cond^#(lt(x, y), x, y), lt^#(x, y)) , cond^#(tt(), x, y) -> c_2(f^#(s(x), s(y))) , lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , lt^#(0(), y) -> c_4() } Weak Trs: { f(x, y) -> cond(lt(x, y), x, y) , cond(tt(), x, y) -> f(s(x), s(y)) , lt(s(x), s(y)) -> lt(x, y) , lt(0(), y) -> tt() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {4} by applications of Pre({4}) = {1,3}. Here rules are labeled as follows: DPs: { 1: f^#(x, y) -> c_1(cond^#(lt(x, y), x, y), lt^#(x, y)) , 2: cond^#(tt(), x, y) -> c_2(f^#(s(x), s(y))) , 3: lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , 4: lt^#(0(), y) -> c_4() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(x, y) -> c_1(cond^#(lt(x, y), x, y), lt^#(x, y)) , cond^#(tt(), x, y) -> c_2(f^#(s(x), s(y))) , lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) } Weak DPs: { lt^#(0(), y) -> c_4() } Weak Trs: { f(x, y) -> cond(lt(x, y), x, y) , cond(tt(), x, y) -> f(s(x), s(y)) , lt(s(x), s(y)) -> lt(x, y) , lt(0(), y) -> tt() } 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. { lt^#(0(), y) -> c_4() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(x, y) -> c_1(cond^#(lt(x, y), x, y), lt^#(x, y)) , cond^#(tt(), x, y) -> c_2(f^#(s(x), s(y))) , lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) } Weak Trs: { f(x, y) -> cond(lt(x, y), x, y) , cond(tt(), x, y) -> f(s(x), s(y)) , lt(s(x), s(y)) -> lt(x, y) , lt(0(), y) -> tt() } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { lt(s(x), s(y)) -> lt(x, y) , lt(0(), y) -> tt() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(x, y) -> c_1(cond^#(lt(x, y), x, y), lt^#(x, y)) , cond^#(tt(), x, y) -> c_2(f^#(s(x), s(y))) , lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) } Weak Trs: { lt(s(x), s(y)) -> lt(x, y) , lt(0(), y) -> tt() } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..