MAYBE

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

Strict Trs:
  { even(0()) -> true()
  , even(s(0())) -> false()
  , even(s(s(x))) -> even(x)
  , half(0()) -> 0()
  , half(s(s(x))) -> s(half(x))
  , plus(0(), y) -> y
  , plus(s(x), y) -> s(plus(x, y))
  , times(0(), y) -> 0()
  , times(s(x), y) -> if_times(even(s(x)), s(x), y)
  , if_times(true(), s(x), y) ->
    plus(times(half(s(x)), y), times(half(s(x)), y))
  , if_times(false(), s(x), y) -> plus(y, times(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { even^#(0()) -> c_1()
  , even^#(s(0())) -> c_2()
  , even^#(s(s(x))) -> c_3(even^#(x))
  , half^#(0()) -> c_4()
  , half^#(s(s(x))) -> c_5(half^#(x))
  , plus^#(0(), y) -> c_6()
  , plus^#(s(x), y) -> c_7(plus^#(x, y))
  , times^#(0(), y) -> c_8()
  , times^#(s(x), y) ->
    c_9(if_times^#(even(s(x)), s(x), y), even^#(s(x)))
  , if_times^#(true(), s(x), y) ->
    c_10(plus^#(times(half(s(x)), y), times(half(s(x)), y)),
         times^#(half(s(x)), y),
         half^#(s(x)),
         times^#(half(s(x)), y),
         half^#(s(x)))
  , if_times^#(false(), s(x), y) ->
    c_11(plus^#(y, times(x, y)), times^#(x, y)) }

and mark the set of starting terms.

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

Strict DPs:
  { even^#(0()) -> c_1()
  , even^#(s(0())) -> c_2()
  , even^#(s(s(x))) -> c_3(even^#(x))
  , half^#(0()) -> c_4()
  , half^#(s(s(x))) -> c_5(half^#(x))
  , plus^#(0(), y) -> c_6()
  , plus^#(s(x), y) -> c_7(plus^#(x, y))
  , times^#(0(), y) -> c_8()
  , times^#(s(x), y) ->
    c_9(if_times^#(even(s(x)), s(x), y), even^#(s(x)))
  , if_times^#(true(), s(x), y) ->
    c_10(plus^#(times(half(s(x)), y), times(half(s(x)), y)),
         times^#(half(s(x)), y),
         half^#(s(x)),
         times^#(half(s(x)), y),
         half^#(s(x)))
  , if_times^#(false(), s(x), y) ->
    c_11(plus^#(y, times(x, y)), times^#(x, y)) }
Weak Trs:
  { even(0()) -> true()
  , even(s(0())) -> false()
  , even(s(s(x))) -> even(x)
  , half(0()) -> 0()
  , half(s(s(x))) -> s(half(x))
  , plus(0(), y) -> y
  , plus(s(x), y) -> s(plus(x, y))
  , times(0(), y) -> 0()
  , times(s(x), y) -> if_times(even(s(x)), s(x), y)
  , if_times(true(), s(x), y) ->
    plus(times(half(s(x)), y), times(half(s(x)), y))
  , if_times(false(), s(x), y) -> plus(y, times(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

  DPs:
    { 1: even^#(0()) -> c_1()
    , 2: even^#(s(0())) -> c_2()
    , 3: even^#(s(s(x))) -> c_3(even^#(x))
    , 4: half^#(0()) -> c_4()
    , 5: half^#(s(s(x))) -> c_5(half^#(x))
    , 6: plus^#(0(), y) -> c_6()
    , 7: plus^#(s(x), y) -> c_7(plus^#(x, y))
    , 8: times^#(0(), y) -> c_8()
    , 9: times^#(s(x), y) ->
         c_9(if_times^#(even(s(x)), s(x), y), even^#(s(x)))
    , 10: if_times^#(true(), s(x), y) ->
          c_10(plus^#(times(half(s(x)), y), times(half(s(x)), y)),
               times^#(half(s(x)), y),
               half^#(s(x)),
               times^#(half(s(x)), y),
               half^#(s(x)))
    , 11: if_times^#(false(), s(x), y) ->
          c_11(plus^#(y, times(x, y)), times^#(x, y)) }

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

Strict DPs:
  { even^#(s(s(x))) -> c_3(even^#(x))
  , half^#(s(s(x))) -> c_5(half^#(x))
  , plus^#(s(x), y) -> c_7(plus^#(x, y))
  , times^#(s(x), y) ->
    c_9(if_times^#(even(s(x)), s(x), y), even^#(s(x)))
  , if_times^#(true(), s(x), y) ->
    c_10(plus^#(times(half(s(x)), y), times(half(s(x)), y)),
         times^#(half(s(x)), y),
         half^#(s(x)),
         times^#(half(s(x)), y),
         half^#(s(x)))
  , if_times^#(false(), s(x), y) ->
    c_11(plus^#(y, times(x, y)), times^#(x, y)) }
Weak DPs:
  { even^#(0()) -> c_1()
  , even^#(s(0())) -> c_2()
  , half^#(0()) -> c_4()
  , plus^#(0(), y) -> c_6()
  , times^#(0(), y) -> c_8() }
Weak Trs:
  { even(0()) -> true()
  , even(s(0())) -> false()
  , even(s(s(x))) -> even(x)
  , half(0()) -> 0()
  , half(s(s(x))) -> s(half(x))
  , plus(0(), y) -> y
  , plus(s(x), y) -> s(plus(x, y))
  , times(0(), y) -> 0()
  , times(s(x), y) -> if_times(even(s(x)), s(x), y)
  , if_times(true(), s(x), y) ->
    plus(times(half(s(x)), y), times(half(s(x)), y))
  , if_times(false(), s(x), y) -> plus(y, times(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.

{ even^#(0()) -> c_1()
, even^#(s(0())) -> c_2()
, half^#(0()) -> c_4()
, plus^#(0(), y) -> c_6()
, times^#(0(), y) -> c_8() }

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

Strict DPs:
  { even^#(s(s(x))) -> c_3(even^#(x))
  , half^#(s(s(x))) -> c_5(half^#(x))
  , plus^#(s(x), y) -> c_7(plus^#(x, y))
  , times^#(s(x), y) ->
    c_9(if_times^#(even(s(x)), s(x), y), even^#(s(x)))
  , if_times^#(true(), s(x), y) ->
    c_10(plus^#(times(half(s(x)), y), times(half(s(x)), y)),
         times^#(half(s(x)), y),
         half^#(s(x)),
         times^#(half(s(x)), y),
         half^#(s(x)))
  , if_times^#(false(), s(x), y) ->
    c_11(plus^#(y, times(x, y)), times^#(x, y)) }
Weak Trs:
  { even(0()) -> true()
  , even(s(0())) -> false()
  , even(s(s(x))) -> even(x)
  , half(0()) -> 0()
  , half(s(s(x))) -> s(half(x))
  , plus(0(), y) -> y
  , plus(s(x), y) -> s(plus(x, y))
  , times(0(), y) -> 0()
  , times(s(x), y) -> if_times(even(s(x)), s(x), y)
  , if_times(true(), s(x), y) ->
    plus(times(half(s(x)), y), times(half(s(x)), y))
  , if_times(false(), s(x), y) -> plus(y, times(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..