MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { half(x) -> if(ge(x, s(s(0()))), x)
  , if(false(), x) -> 0()
  , if(true(), x) -> s(half(p(p(x))))
  , ge(x, 0()) -> true()
  , ge(s(x), s(y)) -> ge(x, y)
  , ge(0(), s(x)) -> false()
  , p(s(x)) -> x
  , p(0()) -> 0()
  , log(s(x)) -> s(log(half(s(x))))
  , log(0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { half^#(x) -> c_1(if^#(ge(x, s(s(0()))), x), ge^#(x, s(s(0()))))
  , if^#(false(), x) -> c_2()
  , if^#(true(), x) -> c_3(half^#(p(p(x))), p^#(p(x)), p^#(x))
  , ge^#(x, 0()) -> c_4()
  , ge^#(s(x), s(y)) -> c_5(ge^#(x, y))
  , ge^#(0(), s(x)) -> c_6()
  , p^#(s(x)) -> c_7()
  , p^#(0()) -> c_8()
  , log^#(s(x)) -> c_9(log^#(half(s(x))), half^#(s(x)))
  , log^#(0()) -> c_10() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { half^#(x) -> c_1(if^#(ge(x, s(s(0()))), x), ge^#(x, s(s(0()))))
  , if^#(false(), x) -> c_2()
  , if^#(true(), x) -> c_3(half^#(p(p(x))), p^#(p(x)), p^#(x))
  , ge^#(x, 0()) -> c_4()
  , ge^#(s(x), s(y)) -> c_5(ge^#(x, y))
  , ge^#(0(), s(x)) -> c_6()
  , p^#(s(x)) -> c_7()
  , p^#(0()) -> c_8()
  , log^#(s(x)) -> c_9(log^#(half(s(x))), half^#(s(x)))
  , log^#(0()) -> c_10() }
Weak Trs:
  { half(x) -> if(ge(x, s(s(0()))), x)
  , if(false(), x) -> 0()
  , if(true(), x) -> s(half(p(p(x))))
  , ge(x, 0()) -> true()
  , ge(s(x), s(y)) -> ge(x, y)
  , ge(0(), s(x)) -> false()
  , p(s(x)) -> x
  , p(0()) -> 0()
  , log(s(x)) -> s(log(half(s(x))))
  , log(0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {2,4,6,7,8,10} by
applications of Pre({2,4,6,7,8,10}) = {1,3,5,9}. Here rules are
labeled as follows:

  DPs:
    { 1: half^#(x) ->
         c_1(if^#(ge(x, s(s(0()))), x), ge^#(x, s(s(0()))))
    , 2: if^#(false(), x) -> c_2()
    , 3: if^#(true(), x) -> c_3(half^#(p(p(x))), p^#(p(x)), p^#(x))
    , 4: ge^#(x, 0()) -> c_4()
    , 5: ge^#(s(x), s(y)) -> c_5(ge^#(x, y))
    , 6: ge^#(0(), s(x)) -> c_6()
    , 7: p^#(s(x)) -> c_7()
    , 8: p^#(0()) -> c_8()
    , 9: log^#(s(x)) -> c_9(log^#(half(s(x))), half^#(s(x)))
    , 10: log^#(0()) -> c_10() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { half^#(x) -> c_1(if^#(ge(x, s(s(0()))), x), ge^#(x, s(s(0()))))
  , if^#(true(), x) -> c_3(half^#(p(p(x))), p^#(p(x)), p^#(x))
  , ge^#(s(x), s(y)) -> c_5(ge^#(x, y))
  , log^#(s(x)) -> c_9(log^#(half(s(x))), half^#(s(x))) }
Weak DPs:
  { if^#(false(), x) -> c_2()
  , ge^#(x, 0()) -> c_4()
  , ge^#(0(), s(x)) -> c_6()
  , p^#(s(x)) -> c_7()
  , p^#(0()) -> c_8()
  , log^#(0()) -> c_10() }
Weak Trs:
  { half(x) -> if(ge(x, s(s(0()))), x)
  , if(false(), x) -> 0()
  , if(true(), x) -> s(half(p(p(x))))
  , ge(x, 0()) -> true()
  , ge(s(x), s(y)) -> ge(x, y)
  , ge(0(), s(x)) -> false()
  , p(s(x)) -> x
  , p(0()) -> 0()
  , log(s(x)) -> s(log(half(s(x))))
  , log(0()) -> 0() }
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.

{ if^#(false(), x) -> c_2()
, ge^#(x, 0()) -> c_4()
, ge^#(0(), s(x)) -> c_6()
, p^#(s(x)) -> c_7()
, p^#(0()) -> c_8()
, log^#(0()) -> c_10() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { half^#(x) -> c_1(if^#(ge(x, s(s(0()))), x), ge^#(x, s(s(0()))))
  , if^#(true(), x) -> c_3(half^#(p(p(x))), p^#(p(x)), p^#(x))
  , ge^#(s(x), s(y)) -> c_5(ge^#(x, y))
  , log^#(s(x)) -> c_9(log^#(half(s(x))), half^#(s(x))) }
Weak Trs:
  { half(x) -> if(ge(x, s(s(0()))), x)
  , if(false(), x) -> 0()
  , if(true(), x) -> s(half(p(p(x))))
  , ge(x, 0()) -> true()
  , ge(s(x), s(y)) -> ge(x, y)
  , ge(0(), s(x)) -> false()
  , p(s(x)) -> x
  , p(0()) -> 0()
  , log(s(x)) -> s(log(half(s(x))))
  , log(0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { if^#(true(), x) -> c_3(half^#(p(p(x))), p^#(p(x)), p^#(x)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { half^#(x) -> c_1(if^#(ge(x, s(s(0()))), x), ge^#(x, s(s(0()))))
  , if^#(true(), x) -> c_2(half^#(p(p(x))))
  , ge^#(s(x), s(y)) -> c_3(ge^#(x, y))
  , log^#(s(x)) -> c_4(log^#(half(s(x))), half^#(s(x))) }
Weak Trs:
  { half(x) -> if(ge(x, s(s(0()))), x)
  , if(false(), x) -> 0()
  , if(true(), x) -> s(half(p(p(x))))
  , ge(x, 0()) -> true()
  , ge(s(x), s(y)) -> ge(x, y)
  , ge(0(), s(x)) -> false()
  , p(s(x)) -> x
  , p(0()) -> 0()
  , log(s(x)) -> s(log(half(s(x))))
  , log(0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { half(x) -> if(ge(x, s(s(0()))), x)
    , if(false(), x) -> 0()
    , if(true(), x) -> s(half(p(p(x))))
    , ge(x, 0()) -> true()
    , ge(s(x), s(y)) -> ge(x, y)
    , ge(0(), s(x)) -> false()
    , p(s(x)) -> x
    , p(0()) -> 0() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { half^#(x) -> c_1(if^#(ge(x, s(s(0()))), x), ge^#(x, s(s(0()))))
  , if^#(true(), x) -> c_2(half^#(p(p(x))))
  , ge^#(s(x), s(y)) -> c_3(ge^#(x, y))
  , log^#(s(x)) -> c_4(log^#(half(s(x))), half^#(s(x))) }
Weak Trs:
  { half(x) -> if(ge(x, s(s(0()))), x)
  , if(false(), x) -> 0()
  , if(true(), x) -> s(half(p(p(x))))
  , ge(x, 0()) -> true()
  , ge(s(x), s(y)) -> ge(x, y)
  , ge(0(), s(x)) -> false()
  , p(s(x)) -> x
  , p(0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..