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

The input cannot be shown compatible

Arrrr..