MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , logarithm(x) -> ifa(lt(0(), x), x) , ifa(true(), x) -> help(x, 1()) , ifa(false(), x) -> logZeroError() , help(x, y) -> ifb(lt(y, x), x, y) , ifb(true(), x, y) -> help(half(x), s(y)) , ifb(false(), x, y) -> y , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , logarithm^#(x) -> c_4(ifa^#(lt(0(), x), x), lt^#(0(), x)) , ifa^#(true(), x) -> c_5(help^#(x, 1())) , ifa^#(false(), x) -> c_6() , help^#(x, y) -> c_7(ifb^#(lt(y, x), x, y), lt^#(y, x)) , ifb^#(true(), x, y) -> c_8(help^#(half(x), s(y)), half^#(x)) , ifb^#(false(), x, y) -> c_9() , half^#(0()) -> c_10() , half^#(s(0())) -> c_11() , half^#(s(s(x))) -> c_12(half^#(x)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , logarithm^#(x) -> c_4(ifa^#(lt(0(), x), x), lt^#(0(), x)) , ifa^#(true(), x) -> c_5(help^#(x, 1())) , ifa^#(false(), x) -> c_6() , help^#(x, y) -> c_7(ifb^#(lt(y, x), x, y), lt^#(y, x)) , ifb^#(true(), x, y) -> c_8(help^#(half(x), s(y)), half^#(x)) , ifb^#(false(), x, y) -> c_9() , half^#(0()) -> c_10() , half^#(s(0())) -> c_11() , half^#(s(s(x))) -> c_12(half^#(x)) } Weak Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , logarithm(x) -> ifa(lt(0(), x), x) , ifa(true(), x) -> help(x, 1()) , ifa(false(), x) -> logZeroError() , help(x, y) -> ifb(lt(y, x), x, y) , ifb(true(), x, y) -> help(half(x), s(y)) , ifb(false(), x, y) -> y , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,2,6,9,10,11} by applications of Pre({1,2,6,9,10,11}) = {3,4,7,8,12}. Here rules are labeled as follows: DPs: { 1: lt^#(x, 0()) -> c_1() , 2: lt^#(0(), s(x)) -> c_2() , 3: lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , 4: logarithm^#(x) -> c_4(ifa^#(lt(0(), x), x), lt^#(0(), x)) , 5: ifa^#(true(), x) -> c_5(help^#(x, 1())) , 6: ifa^#(false(), x) -> c_6() , 7: help^#(x, y) -> c_7(ifb^#(lt(y, x), x, y), lt^#(y, x)) , 8: ifb^#(true(), x, y) -> c_8(help^#(half(x), s(y)), half^#(x)) , 9: ifb^#(false(), x, y) -> c_9() , 10: half^#(0()) -> c_10() , 11: half^#(s(0())) -> c_11() , 12: half^#(s(s(x))) -> c_12(half^#(x)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , logarithm^#(x) -> c_4(ifa^#(lt(0(), x), x), lt^#(0(), x)) , ifa^#(true(), x) -> c_5(help^#(x, 1())) , help^#(x, y) -> c_7(ifb^#(lt(y, x), x, y), lt^#(y, x)) , ifb^#(true(), x, y) -> c_8(help^#(half(x), s(y)), half^#(x)) , half^#(s(s(x))) -> c_12(half^#(x)) } Weak DPs: { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , ifa^#(false(), x) -> c_6() , ifb^#(false(), x, y) -> c_9() , half^#(0()) -> c_10() , half^#(s(0())) -> c_11() } Weak Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , logarithm(x) -> ifa(lt(0(), x), x) , ifa(true(), x) -> help(x, 1()) , ifa(false(), x) -> logZeroError() , help(x, y) -> ifb(lt(y, x), x, y) , ifb(true(), x, y) -> help(half(x), s(y)) , ifb(false(), x, y) -> y , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(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. { lt^#(x, 0()) -> c_1() , lt^#(0(), s(x)) -> c_2() , ifa^#(false(), x) -> c_6() , ifb^#(false(), x, y) -> c_9() , half^#(0()) -> c_10() , half^#(s(0())) -> c_11() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { lt^#(s(x), s(y)) -> c_3(lt^#(x, y)) , logarithm^#(x) -> c_4(ifa^#(lt(0(), x), x), lt^#(0(), x)) , ifa^#(true(), x) -> c_5(help^#(x, 1())) , help^#(x, y) -> c_7(ifb^#(lt(y, x), x, y), lt^#(y, x)) , ifb^#(true(), x, y) -> c_8(help^#(half(x), s(y)), half^#(x)) , half^#(s(s(x))) -> c_12(half^#(x)) } Weak Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , logarithm(x) -> ifa(lt(0(), x), x) , ifa(true(), x) -> help(x, 1()) , ifa(false(), x) -> logZeroError() , help(x, y) -> ifb(lt(y, x), x, y) , ifb(true(), x, y) -> help(half(x), s(y)) , ifb(false(), x, y) -> y , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) } Obligation: innermost runtime complexity Answer: MAYBE Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { logarithm^#(x) -> c_4(ifa^#(lt(0(), x), x), lt^#(0(), x)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { lt^#(s(x), s(y)) -> c_1(lt^#(x, y)) , logarithm^#(x) -> c_2(ifa^#(lt(0(), x), x)) , ifa^#(true(), x) -> c_3(help^#(x, 1())) , help^#(x, y) -> c_4(ifb^#(lt(y, x), x, y), lt^#(y, x)) , ifb^#(true(), x, y) -> c_5(help^#(half(x), s(y)), half^#(x)) , half^#(s(s(x))) -> c_6(half^#(x)) } Weak Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , logarithm(x) -> ifa(lt(0(), x), x) , ifa(true(), x) -> help(x, 1()) , ifa(false(), x) -> logZeroError() , help(x, y) -> ifb(lt(y, x), x, y) , ifb(true(), x, y) -> help(half(x), s(y)) , ifb(false(), x, y) -> y , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(x)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { lt^#(s(x), s(y)) -> c_1(lt^#(x, y)) , logarithm^#(x) -> c_2(ifa^#(lt(0(), x), x)) , ifa^#(true(), x) -> c_3(help^#(x, 1())) , help^#(x, y) -> c_4(ifb^#(lt(y, x), x, y), lt^#(y, x)) , ifb^#(true(), x, y) -> c_5(help^#(half(x), s(y)), half^#(x)) , half^#(s(s(x))) -> c_6(half^#(x)) } Weak Trs: { lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) , half(0()) -> 0() , half(s(0())) -> 0() , half(s(s(x))) -> s(half(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: The input cannot be shown compatible 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..