MAYBE

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

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

We add following dependency tuples:

Strict DPs:
  { half^#(0()) -> c_1()
  , half^#(s(0())) -> c_2()
  , half^#(s(s(x))) -> c_3(half^#(x))
  , inc^#(0()) -> c_4()
  , inc^#(s(x)) -> c_5(inc^#(x))
  , zero^#(0()) -> c_6()
  , zero^#(s(x)) -> c_7()
  , p^#(0()) -> c_8()
  , p^#(s(x)) -> c_9()
  , bits^#(x) -> c_10(bitIter^#(x, 0()))
  , bitIter^#(x, y) ->
    c_11(if^#(zero(x), x, inc(y)), zero^#(x), inc^#(y))
  , if^#(true(), x, y) -> c_12(p^#(y))
  , if^#(false(), x, y) -> c_13(bitIter^#(half(x), y), half^#(x)) }

and mark the set of starting terms.

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

Strict DPs:
  { half^#(0()) -> c_1()
  , half^#(s(0())) -> c_2()
  , half^#(s(s(x))) -> c_3(half^#(x))
  , inc^#(0()) -> c_4()
  , inc^#(s(x)) -> c_5(inc^#(x))
  , zero^#(0()) -> c_6()
  , zero^#(s(x)) -> c_7()
  , p^#(0()) -> c_8()
  , p^#(s(x)) -> c_9()
  , bits^#(x) -> c_10(bitIter^#(x, 0()))
  , bitIter^#(x, y) ->
    c_11(if^#(zero(x), x, inc(y)), zero^#(x), inc^#(y))
  , if^#(true(), x, y) -> c_12(p^#(y))
  , if^#(false(), x, y) -> c_13(bitIter^#(half(x), y), half^#(x)) }
Weak Trs:
  { half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , inc(0()) -> 0()
  , inc(s(x)) -> s(inc(x))
  , zero(0()) -> true()
  , zero(s(x)) -> false()
  , p(0()) -> 0()
  , p(s(x)) -> x
  , bits(x) -> bitIter(x, 0())
  , bitIter(x, y) -> if(zero(x), x, inc(y))
  , if(true(), x, y) -> p(y)
  , if(false(), x, y) -> bitIter(half(x), y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,4,6,7,8,9} by
applications of Pre({1,2,4,6,7,8,9}) = {3,5,11,12,13}. Here rules
are labeled as follows:

  DPs:
    { 1: half^#(0()) -> c_1()
    , 2: half^#(s(0())) -> c_2()
    , 3: half^#(s(s(x))) -> c_3(half^#(x))
    , 4: inc^#(0()) -> c_4()
    , 5: inc^#(s(x)) -> c_5(inc^#(x))
    , 6: zero^#(0()) -> c_6()
    , 7: zero^#(s(x)) -> c_7()
    , 8: p^#(0()) -> c_8()
    , 9: p^#(s(x)) -> c_9()
    , 10: bits^#(x) -> c_10(bitIter^#(x, 0()))
    , 11: bitIter^#(x, y) ->
          c_11(if^#(zero(x), x, inc(y)), zero^#(x), inc^#(y))
    , 12: if^#(true(), x, y) -> c_12(p^#(y))
    , 13: if^#(false(), x, y) ->
          c_13(bitIter^#(half(x), y), half^#(x)) }

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

Strict DPs:
  { half^#(s(s(x))) -> c_3(half^#(x))
  , inc^#(s(x)) -> c_5(inc^#(x))
  , bits^#(x) -> c_10(bitIter^#(x, 0()))
  , bitIter^#(x, y) ->
    c_11(if^#(zero(x), x, inc(y)), zero^#(x), inc^#(y))
  , if^#(true(), x, y) -> c_12(p^#(y))
  , if^#(false(), x, y) -> c_13(bitIter^#(half(x), y), half^#(x)) }
Weak DPs:
  { half^#(0()) -> c_1()
  , half^#(s(0())) -> c_2()
  , inc^#(0()) -> c_4()
  , zero^#(0()) -> c_6()
  , zero^#(s(x)) -> c_7()
  , p^#(0()) -> c_8()
  , p^#(s(x)) -> c_9() }
Weak Trs:
  { half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , inc(0()) -> 0()
  , inc(s(x)) -> s(inc(x))
  , zero(0()) -> true()
  , zero(s(x)) -> false()
  , p(0()) -> 0()
  , p(s(x)) -> x
  , bits(x) -> bitIter(x, 0())
  , bitIter(x, y) -> if(zero(x), x, inc(y))
  , if(true(), x, y) -> p(y)
  , if(false(), x, y) -> bitIter(half(x), y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {5} by applications of
Pre({5}) = {4}. Here rules are labeled as follows:

  DPs:
    { 1: half^#(s(s(x))) -> c_3(half^#(x))
    , 2: inc^#(s(x)) -> c_5(inc^#(x))
    , 3: bits^#(x) -> c_10(bitIter^#(x, 0()))
    , 4: bitIter^#(x, y) ->
         c_11(if^#(zero(x), x, inc(y)), zero^#(x), inc^#(y))
    , 5: if^#(true(), x, y) -> c_12(p^#(y))
    , 6: if^#(false(), x, y) -> c_13(bitIter^#(half(x), y), half^#(x))
    , 7: half^#(0()) -> c_1()
    , 8: half^#(s(0())) -> c_2()
    , 9: inc^#(0()) -> c_4()
    , 10: zero^#(0()) -> c_6()
    , 11: zero^#(s(x)) -> c_7()
    , 12: p^#(0()) -> c_8()
    , 13: p^#(s(x)) -> c_9() }

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

Strict DPs:
  { half^#(s(s(x))) -> c_3(half^#(x))
  , inc^#(s(x)) -> c_5(inc^#(x))
  , bits^#(x) -> c_10(bitIter^#(x, 0()))
  , bitIter^#(x, y) ->
    c_11(if^#(zero(x), x, inc(y)), zero^#(x), inc^#(y))
  , if^#(false(), x, y) -> c_13(bitIter^#(half(x), y), half^#(x)) }
Weak DPs:
  { half^#(0()) -> c_1()
  , half^#(s(0())) -> c_2()
  , inc^#(0()) -> c_4()
  , zero^#(0()) -> c_6()
  , zero^#(s(x)) -> c_7()
  , p^#(0()) -> c_8()
  , p^#(s(x)) -> c_9()
  , if^#(true(), x, y) -> c_12(p^#(y)) }
Weak Trs:
  { half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , inc(0()) -> 0()
  , inc(s(x)) -> s(inc(x))
  , zero(0()) -> true()
  , zero(s(x)) -> false()
  , p(0()) -> 0()
  , p(s(x)) -> x
  , bits(x) -> bitIter(x, 0())
  , bitIter(x, y) -> if(zero(x), x, inc(y))
  , if(true(), x, y) -> p(y)
  , if(false(), x, y) -> bitIter(half(x), y) }
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.

{ half^#(0()) -> c_1()
, half^#(s(0())) -> c_2()
, inc^#(0()) -> c_4()
, zero^#(0()) -> c_6()
, zero^#(s(x)) -> c_7()
, p^#(0()) -> c_8()
, p^#(s(x)) -> c_9()
, if^#(true(), x, y) -> c_12(p^#(y)) }

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

Strict DPs:
  { half^#(s(s(x))) -> c_3(half^#(x))
  , inc^#(s(x)) -> c_5(inc^#(x))
  , bits^#(x) -> c_10(bitIter^#(x, 0()))
  , bitIter^#(x, y) ->
    c_11(if^#(zero(x), x, inc(y)), zero^#(x), inc^#(y))
  , if^#(false(), x, y) -> c_13(bitIter^#(half(x), y), half^#(x)) }
Weak Trs:
  { half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , inc(0()) -> 0()
  , inc(s(x)) -> s(inc(x))
  , zero(0()) -> true()
  , zero(s(x)) -> false()
  , p(0()) -> 0()
  , p(s(x)) -> x
  , bits(x) -> bitIter(x, 0())
  , bitIter(x, y) -> if(zero(x), x, inc(y))
  , if(true(), x, y) -> p(y)
  , if(false(), x, y) -> bitIter(half(x), y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  { bitIter^#(x, y) ->
    c_11(if^#(zero(x), x, inc(y)), zero^#(x), inc^#(y)) }

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

Strict DPs:
  { half^#(s(s(x))) -> c_1(half^#(x))
  , inc^#(s(x)) -> c_2(inc^#(x))
  , bits^#(x) -> c_3(bitIter^#(x, 0()))
  , bitIter^#(x, y) -> c_4(if^#(zero(x), x, inc(y)), inc^#(y))
  , if^#(false(), x, y) -> c_5(bitIter^#(half(x), y), half^#(x)) }
Weak Trs:
  { half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , inc(0()) -> 0()
  , inc(s(x)) -> s(inc(x))
  , zero(0()) -> true()
  , zero(s(x)) -> false()
  , p(0()) -> 0()
  , p(s(x)) -> x
  , bits(x) -> bitIter(x, 0())
  , bitIter(x, y) -> if(zero(x), x, inc(y))
  , if(true(), x, y) -> p(y)
  , if(false(), x, y) -> bitIter(half(x), y) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { half(0()) -> 0()
    , half(s(0())) -> 0()
    , half(s(s(x))) -> s(half(x))
    , inc(0()) -> 0()
    , inc(s(x)) -> s(inc(x))
    , zero(0()) -> true()
    , zero(s(x)) -> false() }

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

Strict DPs:
  { half^#(s(s(x))) -> c_1(half^#(x))
  , inc^#(s(x)) -> c_2(inc^#(x))
  , bits^#(x) -> c_3(bitIter^#(x, 0()))
  , bitIter^#(x, y) -> c_4(if^#(zero(x), x, inc(y)), inc^#(y))
  , if^#(false(), x, y) -> c_5(bitIter^#(half(x), y), half^#(x)) }
Weak Trs:
  { half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , inc(0()) -> 0()
  , inc(s(x)) -> s(inc(x))
  , zero(0()) -> true()
  , zero(s(x)) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Consider the dependency graph

  1: half^#(s(s(x))) -> c_1(half^#(x))
     -->_1 half^#(s(s(x))) -> c_1(half^#(x)) :1
  
  2: inc^#(s(x)) -> c_2(inc^#(x))
     -->_1 inc^#(s(x)) -> c_2(inc^#(x)) :2
  
  3: bits^#(x) -> c_3(bitIter^#(x, 0()))
     -->_1 bitIter^#(x, y) -> c_4(if^#(zero(x), x, inc(y)), inc^#(y)) :4
  
  4: bitIter^#(x, y) -> c_4(if^#(zero(x), x, inc(y)), inc^#(y))
     -->_1 if^#(false(), x, y) ->
           c_5(bitIter^#(half(x), y), half^#(x)) :5
     -->_2 inc^#(s(x)) -> c_2(inc^#(x)) :2
  
  5: if^#(false(), x, y) -> c_5(bitIter^#(half(x), y), half^#(x))
     -->_1 bitIter^#(x, y) -> c_4(if^#(zero(x), x, inc(y)), inc^#(y)) :4
     -->_2 half^#(s(s(x))) -> c_1(half^#(x)) :1
  

Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).

  { bits^#(x) -> c_3(bitIter^#(x, 0())) }


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

Strict DPs:
  { half^#(s(s(x))) -> c_1(half^#(x))
  , inc^#(s(x)) -> c_2(inc^#(x))
  , bitIter^#(x, y) -> c_4(if^#(zero(x), x, inc(y)), inc^#(y))
  , if^#(false(), x, y) -> c_5(bitIter^#(half(x), y), half^#(x)) }
Weak Trs:
  { half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , inc(0()) -> 0()
  , inc(s(x)) -> s(inc(x))
  , zero(0()) -> true()
  , zero(s(x)) -> false() }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..