MAYBE

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

Strict Trs:
  { -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , if(true(), x, y) -> x
  , if(true(), x, y) -> true()
  , if(false(), x, y) -> y
  , if(false(), x, y) -> false()
  , odd(0()) -> false()
  , odd(s(0())) -> true()
  , odd(s(s(x))) -> odd(x)
  , half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , pow(x, y) -> f(x, y, s(0()))
  , f(x, 0(), z) -> z
  , f(x, s(y), z) ->
    if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { -^#(x, 0()) -> c_1()
  , -^#(s(x), s(y)) -> c_2(-^#(x, y))
  , *^#(x, 0()) -> c_3()
  , *^#(x, s(y)) -> c_4(*^#(x, y))
  , if^#(true(), x, y) -> c_5()
  , if^#(true(), x, y) -> c_6()
  , if^#(false(), x, y) -> c_7()
  , if^#(false(), x, y) -> c_8()
  , odd^#(0()) -> c_9()
  , odd^#(s(0())) -> c_10()
  , odd^#(s(s(x))) -> c_11(odd^#(x))
  , half^#(0()) -> c_12()
  , half^#(s(0())) -> c_13()
  , half^#(s(s(x))) -> c_14(half^#(x))
  , pow^#(x, y) -> c_15(f^#(x, y, s(0())))
  , f^#(x, 0(), z) -> c_16()
  , f^#(x, s(y), z) ->
    c_17(if^#(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)),
         odd^#(s(y)),
         f^#(x, y, *(x, z)),
         *^#(x, z),
         f^#(*(x, x), half(s(y)), z),
         *^#(x, x),
         half^#(s(y))) }

and mark the set of starting terms.

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

Strict DPs:
  { -^#(x, 0()) -> c_1()
  , -^#(s(x), s(y)) -> c_2(-^#(x, y))
  , *^#(x, 0()) -> c_3()
  , *^#(x, s(y)) -> c_4(*^#(x, y))
  , if^#(true(), x, y) -> c_5()
  , if^#(true(), x, y) -> c_6()
  , if^#(false(), x, y) -> c_7()
  , if^#(false(), x, y) -> c_8()
  , odd^#(0()) -> c_9()
  , odd^#(s(0())) -> c_10()
  , odd^#(s(s(x))) -> c_11(odd^#(x))
  , half^#(0()) -> c_12()
  , half^#(s(0())) -> c_13()
  , half^#(s(s(x))) -> c_14(half^#(x))
  , pow^#(x, y) -> c_15(f^#(x, y, s(0())))
  , f^#(x, 0(), z) -> c_16()
  , f^#(x, s(y), z) ->
    c_17(if^#(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)),
         odd^#(s(y)),
         f^#(x, y, *(x, z)),
         *^#(x, z),
         f^#(*(x, x), half(s(y)), z),
         *^#(x, x),
         half^#(s(y))) }
Weak Trs:
  { -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , if(true(), x, y) -> x
  , if(true(), x, y) -> true()
  , if(false(), x, y) -> y
  , if(false(), x, y) -> false()
  , odd(0()) -> false()
  , odd(s(0())) -> true()
  , odd(s(s(x))) -> odd(x)
  , half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , pow(x, y) -> f(x, y, s(0()))
  , f(x, 0(), z) -> z
  , f(x, s(y), z) ->
    if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of
{1,3,5,6,7,8,9,10,12,13,16} by applications of
Pre({1,3,5,6,7,8,9,10,12,13,16}) = {2,4,11,14,15,17}. Here rules
are labeled as follows:

  DPs:
    { 1: -^#(x, 0()) -> c_1()
    , 2: -^#(s(x), s(y)) -> c_2(-^#(x, y))
    , 3: *^#(x, 0()) -> c_3()
    , 4: *^#(x, s(y)) -> c_4(*^#(x, y))
    , 5: if^#(true(), x, y) -> c_5()
    , 6: if^#(true(), x, y) -> c_6()
    , 7: if^#(false(), x, y) -> c_7()
    , 8: if^#(false(), x, y) -> c_8()
    , 9: odd^#(0()) -> c_9()
    , 10: odd^#(s(0())) -> c_10()
    , 11: odd^#(s(s(x))) -> c_11(odd^#(x))
    , 12: half^#(0()) -> c_12()
    , 13: half^#(s(0())) -> c_13()
    , 14: half^#(s(s(x))) -> c_14(half^#(x))
    , 15: pow^#(x, y) -> c_15(f^#(x, y, s(0())))
    , 16: f^#(x, 0(), z) -> c_16()
    , 17: f^#(x, s(y), z) ->
          c_17(if^#(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)),
               odd^#(s(y)),
               f^#(x, y, *(x, z)),
               *^#(x, z),
               f^#(*(x, x), half(s(y)), z),
               *^#(x, x),
               half^#(s(y))) }

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

Strict DPs:
  { -^#(s(x), s(y)) -> c_2(-^#(x, y))
  , *^#(x, s(y)) -> c_4(*^#(x, y))
  , odd^#(s(s(x))) -> c_11(odd^#(x))
  , half^#(s(s(x))) -> c_14(half^#(x))
  , pow^#(x, y) -> c_15(f^#(x, y, s(0())))
  , f^#(x, s(y), z) ->
    c_17(if^#(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)),
         odd^#(s(y)),
         f^#(x, y, *(x, z)),
         *^#(x, z),
         f^#(*(x, x), half(s(y)), z),
         *^#(x, x),
         half^#(s(y))) }
Weak DPs:
  { -^#(x, 0()) -> c_1()
  , *^#(x, 0()) -> c_3()
  , if^#(true(), x, y) -> c_5()
  , if^#(true(), x, y) -> c_6()
  , if^#(false(), x, y) -> c_7()
  , if^#(false(), x, y) -> c_8()
  , odd^#(0()) -> c_9()
  , odd^#(s(0())) -> c_10()
  , half^#(0()) -> c_12()
  , half^#(s(0())) -> c_13()
  , f^#(x, 0(), z) -> c_16() }
Weak Trs:
  { -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , if(true(), x, y) -> x
  , if(true(), x, y) -> true()
  , if(false(), x, y) -> y
  , if(false(), x, y) -> false()
  , odd(0()) -> false()
  , odd(s(0())) -> true()
  , odd(s(s(x))) -> odd(x)
  , half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , pow(x, y) -> f(x, y, s(0()))
  , f(x, 0(), z) -> z
  , f(x, s(y), z) ->
    if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)) }
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, 0()) -> c_1()
, *^#(x, 0()) -> c_3()
, if^#(true(), x, y) -> c_5()
, if^#(true(), x, y) -> c_6()
, if^#(false(), x, y) -> c_7()
, if^#(false(), x, y) -> c_8()
, odd^#(0()) -> c_9()
, odd^#(s(0())) -> c_10()
, half^#(0()) -> c_12()
, half^#(s(0())) -> c_13()
, f^#(x, 0(), z) -> c_16() }

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

Strict DPs:
  { -^#(s(x), s(y)) -> c_2(-^#(x, y))
  , *^#(x, s(y)) -> c_4(*^#(x, y))
  , odd^#(s(s(x))) -> c_11(odd^#(x))
  , half^#(s(s(x))) -> c_14(half^#(x))
  , pow^#(x, y) -> c_15(f^#(x, y, s(0())))
  , f^#(x, s(y), z) ->
    c_17(if^#(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)),
         odd^#(s(y)),
         f^#(x, y, *(x, z)),
         *^#(x, z),
         f^#(*(x, x), half(s(y)), z),
         *^#(x, x),
         half^#(s(y))) }
Weak Trs:
  { -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , if(true(), x, y) -> x
  , if(true(), x, y) -> true()
  , if(false(), x, y) -> y
  , if(false(), x, y) -> false()
  , odd(0()) -> false()
  , odd(s(0())) -> true()
  , odd(s(s(x))) -> odd(x)
  , half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , pow(x, y) -> f(x, y, s(0()))
  , f(x, 0(), z) -> z
  , f(x, s(y), z) ->
    if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  { f^#(x, s(y), z) ->
    c_17(if^#(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)),
         odd^#(s(y)),
         f^#(x, y, *(x, z)),
         *^#(x, z),
         f^#(*(x, x), half(s(y)), z),
         *^#(x, x),
         half^#(s(y))) }

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

Strict DPs:
  { -^#(s(x), s(y)) -> c_1(-^#(x, y))
  , *^#(x, s(y)) -> c_2(*^#(x, y))
  , odd^#(s(s(x))) -> c_3(odd^#(x))
  , half^#(s(s(x))) -> c_4(half^#(x))
  , pow^#(x, y) -> c_5(f^#(x, y, s(0())))
  , f^#(x, s(y), z) ->
    c_6(odd^#(s(y)),
        f^#(x, y, *(x, z)),
        *^#(x, z),
        f^#(*(x, x), half(s(y)), z),
        *^#(x, x),
        half^#(s(y))) }
Weak Trs:
  { -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , if(true(), x, y) -> x
  , if(true(), x, y) -> true()
  , if(false(), x, y) -> y
  , if(false(), x, y) -> false()
  , odd(0()) -> false()
  , odd(s(0())) -> true()
  , odd(s(s(x))) -> odd(x)
  , half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , pow(x, y) -> f(x, y, s(0()))
  , f(x, 0(), z) -> z
  , f(x, s(y), z) ->
    if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { *(x, 0()) -> 0()
    , *(x, s(y)) -> +(*(x, y), x)
    , 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:
  { -^#(s(x), s(y)) -> c_1(-^#(x, y))
  , *^#(x, s(y)) -> c_2(*^#(x, y))
  , odd^#(s(s(x))) -> c_3(odd^#(x))
  , half^#(s(s(x))) -> c_4(half^#(x))
  , pow^#(x, y) -> c_5(f^#(x, y, s(0())))
  , f^#(x, s(y), z) ->
    c_6(odd^#(s(y)),
        f^#(x, y, *(x, z)),
        *^#(x, z),
        f^#(*(x, x), half(s(y)), z),
        *^#(x, x),
        half^#(s(y))) }
Weak Trs:
  { *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Consider the dependency graph

  1: -^#(s(x), s(y)) -> c_1(-^#(x, y))
     -->_1 -^#(s(x), s(y)) -> c_1(-^#(x, y)) :1
  
  2: *^#(x, s(y)) -> c_2(*^#(x, y))
     -->_1 *^#(x, s(y)) -> c_2(*^#(x, y)) :2
  
  3: odd^#(s(s(x))) -> c_3(odd^#(x))
     -->_1 odd^#(s(s(x))) -> c_3(odd^#(x)) :3
  
  4: half^#(s(s(x))) -> c_4(half^#(x))
     -->_1 half^#(s(s(x))) -> c_4(half^#(x)) :4
  
  5: pow^#(x, y) -> c_5(f^#(x, y, s(0())))
     -->_1 f^#(x, s(y), z) ->
           c_6(odd^#(s(y)),
               f^#(x, y, *(x, z)),
               *^#(x, z),
               f^#(*(x, x), half(s(y)), z),
               *^#(x, x),
               half^#(s(y))) :6
  
  6: f^#(x, s(y), z) ->
     c_6(odd^#(s(y)),
         f^#(x, y, *(x, z)),
         *^#(x, z),
         f^#(*(x, x), half(s(y)), z),
         *^#(x, x),
         half^#(s(y)))
     -->_4 f^#(x, s(y), z) ->
           c_6(odd^#(s(y)),
               f^#(x, y, *(x, z)),
               *^#(x, z),
               f^#(*(x, x), half(s(y)), z),
               *^#(x, x),
               half^#(s(y))) :6
     -->_2 f^#(x, s(y), z) ->
           c_6(odd^#(s(y)),
               f^#(x, y, *(x, z)),
               *^#(x, z),
               f^#(*(x, x), half(s(y)), z),
               *^#(x, x),
               half^#(s(y))) :6
     -->_6 half^#(s(s(x))) -> c_4(half^#(x)) :4
     -->_1 odd^#(s(s(x))) -> c_3(odd^#(x)) :3
     -->_5 *^#(x, s(y)) -> c_2(*^#(x, y)) :2
     -->_3 *^#(x, s(y)) -> c_2(*^#(x, y)) :2
  

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).

  { pow^#(x, y) -> c_5(f^#(x, y, s(0()))) }


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

Strict DPs:
  { -^#(s(x), s(y)) -> c_1(-^#(x, y))
  , *^#(x, s(y)) -> c_2(*^#(x, y))
  , odd^#(s(s(x))) -> c_3(odd^#(x))
  , half^#(s(s(x))) -> c_4(half^#(x))
  , f^#(x, s(y), z) ->
    c_6(odd^#(s(y)),
        f^#(x, y, *(x, z)),
        *^#(x, z),
        f^#(*(x, x), half(s(y)), z),
        *^#(x, x),
        half^#(s(y))) }
Weak Trs:
  { *(x, 0()) -> 0()
  , *(x, s(y)) -> +(*(x, y), x)
  , 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..