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))
  , lastbit(0()) -> 0()
  , lastbit(s(0())) -> s(0())
  , lastbit(s(s(x))) -> lastbit(x)
  , conv(0()) -> cons(nil(), 0())
  , conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following weak dependency pairs:

Strict DPs:
  { half^#(0()) -> c_1()
  , half^#(s(0())) -> c_2()
  , half^#(s(s(x))) -> c_3(half^#(x))
  , lastbit^#(0()) -> c_4()
  , lastbit^#(s(0())) -> c_5()
  , lastbit^#(s(s(x))) -> c_6(lastbit^#(x))
  , conv^#(0()) -> c_7()
  , conv^#(s(x)) -> c_8(conv^#(half(s(x))), lastbit^#(s(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))
  , lastbit^#(0()) -> c_4()
  , lastbit^#(s(0())) -> c_5()
  , lastbit^#(s(s(x))) -> c_6(lastbit^#(x))
  , conv^#(0()) -> c_7()
  , conv^#(s(x)) -> c_8(conv^#(half(s(x))), lastbit^#(s(x))) }
Strict Trs:
  { half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x))
  , lastbit(0()) -> 0()
  , lastbit(s(0())) -> s(0())
  , lastbit(s(s(x))) -> lastbit(x)
  , conv(0()) -> cons(nil(), 0())
  , conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Strict Usable Rules:
    { 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:
  { half^#(0()) -> c_1()
  , half^#(s(0())) -> c_2()
  , half^#(s(s(x))) -> c_3(half^#(x))
  , lastbit^#(0()) -> c_4()
  , lastbit^#(s(0())) -> c_5()
  , lastbit^#(s(s(x))) -> c_6(lastbit^#(x))
  , conv^#(0()) -> c_7()
  , conv^#(s(x)) -> c_8(conv^#(half(s(x))), lastbit^#(s(x))) }
Strict Trs:
  { half(0()) -> 0()
  , half(s(0())) -> 0()
  , half(s(s(x))) -> s(half(x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(s) = {1}, Uargs(c_3) = {1}, Uargs(c_6) = {1},
  Uargs(conv^#) = {1}, Uargs(c_8) = {1, 2}

TcT has computed following constructor-restricted matrix
interpretation.

       [half](x1) = [1] x1 + [2]         
                                         
              [0] = [1]                  
                                         
          [s](x1) = [1] x1 + [1]         
                                         
     [half^#](x1) = [1] x1 + [2]         
                                         
            [c_1] = [1]                  
                                         
            [c_2] = [1]                  
                                         
        [c_3](x1) = [1] x1 + [1]         
                                         
  [lastbit^#](x1) = [0]                  
                                         
            [c_4] = [1]                  
                                         
            [c_5] = [1]                  
                                         
        [c_6](x1) = [1] x1 + [2]         
                                         
     [conv^#](x1) = [2] x1 + [1]         
                                         
            [c_7] = [2]                  
                                         
    [c_8](x1, x2) = [1] x1 + [1] x2 + [0]

This order satisfies following ordering constraints:

      [half(0())] = [3]         
                  > [1]         
                  = [0()]       
                                
   [half(s(0()))] = [4]         
                  > [1]         
                  = [0()]       
                                
  [half(s(s(x)))] = [1] x + [4] 
                  > [1] x + [3] 
                  = [s(half(x))]
                                

Further, it can be verified that all rules not oriented are covered by the weightgap condition.

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

Strict DPs:
  { lastbit^#(0()) -> c_4()
  , lastbit^#(s(0())) -> c_5()
  , lastbit^#(s(s(x))) -> c_6(lastbit^#(x))
  , conv^#(s(x)) -> c_8(conv^#(half(s(x))), lastbit^#(s(x))) }
Weak DPs:
  { half^#(0()) -> c_1()
  , half^#(s(0())) -> c_2()
  , half^#(s(s(x))) -> c_3(half^#(x))
  , conv^#(0()) -> c_7() }
Weak Trs:
  { 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} by applications of
Pre({1,2}) = {3,4}. Here rules are labeled as follows:

  DPs:
    { 1: lastbit^#(0()) -> c_4()
    , 2: lastbit^#(s(0())) -> c_5()
    , 3: lastbit^#(s(s(x))) -> c_6(lastbit^#(x))
    , 4: conv^#(s(x)) -> c_8(conv^#(half(s(x))), lastbit^#(s(x)))
    , 5: half^#(0()) -> c_1()
    , 6: half^#(s(0())) -> c_2()
    , 7: half^#(s(s(x))) -> c_3(half^#(x))
    , 8: conv^#(0()) -> c_7() }

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

Strict DPs:
  { lastbit^#(s(s(x))) -> c_6(lastbit^#(x))
  , conv^#(s(x)) -> c_8(conv^#(half(s(x))), lastbit^#(s(x))) }
Weak DPs:
  { half^#(0()) -> c_1()
  , half^#(s(0())) -> c_2()
  , half^#(s(s(x))) -> c_3(half^#(x))
  , lastbit^#(0()) -> c_4()
  , lastbit^#(s(0())) -> c_5()
  , conv^#(0()) -> c_7() }
Weak Trs:
  { 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.

{ half^#(0()) -> c_1()
, half^#(s(0())) -> c_2()
, half^#(s(s(x))) -> c_3(half^#(x))
, lastbit^#(0()) -> c_4()
, lastbit^#(s(0())) -> c_5()
, conv^#(0()) -> c_7() }

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

Strict DPs:
  { lastbit^#(s(s(x))) -> c_6(lastbit^#(x))
  , conv^#(s(x)) -> c_8(conv^#(half(s(x))), lastbit^#(s(x))) }
Weak Trs:
  { 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..