MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { del(.(x, .(y, z))) -> f(=(x, y), x, y, z) , f(true(), x, y, z) -> del(.(y, z)) , f(false(), x, y, z) -> .(x, del(.(y, z))) , =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) , =(.(x, y), nil()) -> false() , =(nil(), .(y, z)) -> false() , =(nil(), nil()) -> true() } Obligation: innermost runtime complexity Answer: MAYBE We add following weak dependency pairs: Strict DPs: { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) , =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) , =^#(.(x, y), nil()) -> c_5() , =^#(nil(), .(y, z)) -> c_6() , =^#(nil(), nil()) -> c_7() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) , =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) , =^#(.(x, y), nil()) -> c_5() , =^#(nil(), .(y, z)) -> c_6() , =^#(nil(), nil()) -> c_7() } Strict Trs: { del(.(x, .(y, z))) -> f(=(x, y), x, y, z) , f(true(), x, y, z) -> del(.(y, z)) , f(false(), x, y, z) -> .(x, del(.(y, z))) , =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) , =(.(x, y), nil()) -> false() , =(nil(), .(y, z)) -> false() , =(nil(), nil()) -> true() } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Strict Usable Rules: { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) , =(.(x, y), nil()) -> false() , =(nil(), .(y, z)) -> false() , =(nil(), nil()) -> true() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) , =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) , =^#(.(x, y), nil()) -> c_5() , =^#(nil(), .(y, z)) -> c_6() , =^#(nil(), nil()) -> c_7() } Strict Trs: { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) , =(.(x, y), nil()) -> false() , =(nil(), .(y, z)) -> false() , =(nil(), nil()) -> true() } Obligation: innermost runtime complexity Answer: MAYBE The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(c_1) = {1}, Uargs(f^#) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1} TcT has computed following constructor-restricted matrix interpretation. [.](x1, x2) = [0] [=](x1, x2) = [1] [true] = [0] [false] = [0] [nil] = [0] [u] = [0] [v] = [0] [and](x1, x2) = [0] [del^#](x1) = [1] x1 + [2] [c_1](x1) = [1] x1 + [2] [f^#](x1, x2, x3, x4) = [2] x1 + [2] [c_2](x1) = [1] x1 + [2] [c_3](x1) = [1] x1 + [2] [=^#](x1, x2) = [0] [c_4](x1, x2) = [2] [c_5] = [1] [c_6] = [1] [c_7] = [1] This order satisfies following ordering constraints: [=(.(x, y), .(u(), v()))] = [1] > [0] = [and(=(x, u()), =(y, v()))] [=(.(x, y), nil())] = [1] > [0] = [false()] [=(nil(), .(y, z))] = [1] > [0] = [false()] [=(nil(), nil())] = [1] > [0] = [true()] 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 DPs: { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) , =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) , =^#(.(x, y), nil()) -> c_5() , =^#(nil(), .(y, z)) -> c_6() , =^#(nil(), nil()) -> c_7() } Weak Trs: { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) , =(.(x, y), nil()) -> false() , =(nil(), .(y, z)) -> false() , =(nil(), nil()) -> true() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {4,5,6,7} by applications of Pre({4,5,6,7}) = {}. Here rules are labeled as follows: DPs: { 1: del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) , 2: f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) , 3: f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) , 4: =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) , 5: =^#(.(x, y), nil()) -> c_5() , 6: =^#(nil(), .(y, z)) -> c_6() , 7: =^#(nil(), nil()) -> c_7() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) } Weak DPs: { =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) , =^#(.(x, y), nil()) -> c_5() , =^#(nil(), .(y, z)) -> c_6() , =^#(nil(), nil()) -> c_7() } Weak Trs: { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) , =(.(x, y), nil()) -> false() , =(nil(), .(y, z)) -> false() , =(nil(), nil()) -> true() } 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. { =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v())) , =^#(.(x, y), nil()) -> c_5() , =^#(nil(), .(y, z)) -> c_6() , =^#(nil(), nil()) -> c_7() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z)) , f^#(true(), x, y, z) -> c_2(del^#(.(y, z))) , f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) } Weak Trs: { =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v())) , =(.(x, y), nil()) -> false() , =(nil(), .(y, z)) -> false() , =(nil(), nil()) -> true() } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..