MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            bitIter(x,y) -> if(zero(x),x,inc(y))
            bits(x) -> bitIter(x,0())
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
            if(false(),x,y) -> bitIter(half(x),y)
            if(true(),x,y) -> p(y)
            inc(0()) -> 0()
            inc(s(x)) -> s(inc(x))
            p(0()) -> 0()
            p(s(x)) -> x
            zero(0()) -> true()
            zero(s(x)) -> false()
        - Signature:
            {bitIter/2,bits/1,half/1,if/3,inc/1,p/1,zero/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bitIter,bits,half,if,inc,p,zero} and constructors {0
            ,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y))
          bits#(x) -> c_2(bitIter#(x,0()))
          half#(0()) -> c_3()
          half#(s(0())) -> c_4()
          half#(s(s(x))) -> c_5(half#(x))
          if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
          if#(true(),x,y) -> c_7(p#(y))
          inc#(0()) -> c_8()
          inc#(s(x)) -> c_9(inc#(x))
          p#(0()) -> c_10()
          p#(s(x)) -> c_11()
          zero#(0()) -> c_12()
          zero#(s(x)) -> c_13()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y))
            bits#(x) -> c_2(bitIter#(x,0()))
            half#(0()) -> c_3()
            half#(s(0())) -> c_4()
            half#(s(s(x))) -> c_5(half#(x))
            if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
            if#(true(),x,y) -> c_7(p#(y))
            inc#(0()) -> c_8()
            inc#(s(x)) -> c_9(inc#(x))
            p#(0()) -> c_10()
            p#(s(x)) -> c_11()
            zero#(0()) -> c_12()
            zero#(s(x)) -> c_13()
        - Weak TRS:
            bitIter(x,y) -> if(zero(x),x,inc(y))
            bits(x) -> bitIter(x,0())
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
            if(false(),x,y) -> bitIter(half(x),y)
            if(true(),x,y) -> p(y)
            inc(0()) -> 0()
            inc(s(x)) -> s(inc(x))
            p(0()) -> 0()
            p(s(x)) -> x
            zero(0()) -> true()
            zero(s(x)) -> false()
        - Signature:
            {bitIter/2,bits/1,half/1,if/3,inc/1,p/1,zero/1,bitIter#/2,bits#/1,half#/1,if#/3,inc#/1,p#/1,zero#/1} / {0/0
            ,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bitIter#,bits#,half#,if#,inc#,p#
            ,zero#} and constructors {0,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by 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()
          bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y))
          bits#(x) -> c_2(bitIter#(x,0()))
          half#(0()) -> c_3()
          half#(s(0())) -> c_4()
          half#(s(s(x))) -> c_5(half#(x))
          if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
          if#(true(),x,y) -> c_7(p#(y))
          inc#(0()) -> c_8()
          inc#(s(x)) -> c_9(inc#(x))
          p#(0()) -> c_10()
          p#(s(x)) -> c_11()
          zero#(0()) -> c_12()
          zero#(s(x)) -> c_13()
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y))
            bits#(x) -> c_2(bitIter#(x,0()))
            half#(0()) -> c_3()
            half#(s(0())) -> c_4()
            half#(s(s(x))) -> c_5(half#(x))
            if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
            if#(true(),x,y) -> c_7(p#(y))
            inc#(0()) -> c_8()
            inc#(s(x)) -> c_9(inc#(x))
            p#(0()) -> c_10()
            p#(s(x)) -> c_11()
            zero#(0()) -> c_12()
            zero#(s(x)) -> c_13()
        - 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()
        - Signature:
            {bitIter/2,bits/1,half/1,if/3,inc/1,p/1,zero/1,bitIter#/2,bits#/1,half#/1,if#/3,inc#/1,p#/1,zero#/1} / {0/0
            ,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bitIter#,bits#,half#,if#,inc#,p#
            ,zero#} and constructors {0,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,4,8,10,11,12,13}
        by application of
          Pre({3,4,8,10,11,12,13}) = {1,5,6,7,9}.
        Here rules are labelled as follows:
          1: bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y))
          2: bits#(x) -> c_2(bitIter#(x,0()))
          3: half#(0()) -> c_3()
          4: half#(s(0())) -> c_4()
          5: half#(s(s(x))) -> c_5(half#(x))
          6: if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
          7: if#(true(),x,y) -> c_7(p#(y))
          8: inc#(0()) -> c_8()
          9: inc#(s(x)) -> c_9(inc#(x))
          10: p#(0()) -> c_10()
          11: p#(s(x)) -> c_11()
          12: zero#(0()) -> c_12()
          13: zero#(s(x)) -> c_13()
* Step 4: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y))
            bits#(x) -> c_2(bitIter#(x,0()))
            half#(s(s(x))) -> c_5(half#(x))
            if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
            if#(true(),x,y) -> c_7(p#(y))
            inc#(s(x)) -> c_9(inc#(x))
        - Weak DPs:
            half#(0()) -> c_3()
            half#(s(0())) -> c_4()
            inc#(0()) -> c_8()
            p#(0()) -> c_10()
            p#(s(x)) -> c_11()
            zero#(0()) -> c_12()
            zero#(s(x)) -> c_13()
        - 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()
        - Signature:
            {bitIter/2,bits/1,half/1,if/3,inc/1,p/1,zero/1,bitIter#/2,bits#/1,half#/1,if#/3,inc#/1,p#/1,zero#/1} / {0/0
            ,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bitIter#,bits#,half#,if#,inc#,p#
            ,zero#} and constructors {0,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {5}
        by application of
          Pre({5}) = {1}.
        Here rules are labelled as follows:
          1: bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y))
          2: bits#(x) -> c_2(bitIter#(x,0()))
          3: half#(s(s(x))) -> c_5(half#(x))
          4: if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
          5: if#(true(),x,y) -> c_7(p#(y))
          6: inc#(s(x)) -> c_9(inc#(x))
          7: half#(0()) -> c_3()
          8: half#(s(0())) -> c_4()
          9: inc#(0()) -> c_8()
          10: p#(0()) -> c_10()
          11: p#(s(x)) -> c_11()
          12: zero#(0()) -> c_12()
          13: zero#(s(x)) -> c_13()
* Step 5: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y))
            bits#(x) -> c_2(bitIter#(x,0()))
            half#(s(s(x))) -> c_5(half#(x))
            if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
            inc#(s(x)) -> c_9(inc#(x))
        - Weak DPs:
            half#(0()) -> c_3()
            half#(s(0())) -> c_4()
            if#(true(),x,y) -> c_7(p#(y))
            inc#(0()) -> c_8()
            p#(0()) -> c_10()
            p#(s(x)) -> c_11()
            zero#(0()) -> c_12()
            zero#(s(x)) -> c_13()
        - 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()
        - Signature:
            {bitIter/2,bits/1,half/1,if/3,inc/1,p/1,zero/1,bitIter#/2,bits#/1,half#/1,if#/3,inc#/1,p#/1,zero#/1} / {0/0
            ,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bitIter#,bits#,half#,if#,inc#,p#
            ,zero#} and constructors {0,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y))
             -->_1 if#(true(),x,y) -> c_7(p#(y)):8
             -->_3 inc#(s(x)) -> c_9(inc#(x)):5
             -->_1 if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x)):4
             -->_2 zero#(s(x)) -> c_13():13
             -->_2 zero#(0()) -> c_12():12
             -->_3 inc#(0()) -> c_8():9
          
          2:S:bits#(x) -> c_2(bitIter#(x,0()))
             -->_1 bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y)):1
          
          3:S:half#(s(s(x))) -> c_5(half#(x))
             -->_1 half#(s(0())) -> c_4():7
             -->_1 half#(0()) -> c_3():6
             -->_1 half#(s(s(x))) -> c_5(half#(x)):3
          
          4:S:if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
             -->_2 half#(s(0())) -> c_4():7
             -->_2 half#(0()) -> c_3():6
             -->_2 half#(s(s(x))) -> c_5(half#(x)):3
             -->_1 bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y)):1
          
          5:S:inc#(s(x)) -> c_9(inc#(x))
             -->_1 inc#(0()) -> c_8():9
             -->_1 inc#(s(x)) -> c_9(inc#(x)):5
          
          6:W:half#(0()) -> c_3()
             
          
          7:W:half#(s(0())) -> c_4()
             
          
          8:W:if#(true(),x,y) -> c_7(p#(y))
             -->_1 p#(s(x)) -> c_11():11
             -->_1 p#(0()) -> c_10():10
          
          9:W:inc#(0()) -> c_8()
             
          
          10:W:p#(0()) -> c_10()
             
          
          11:W:p#(s(x)) -> c_11()
             
          
          12:W:zero#(0()) -> c_12()
             
          
          13:W:zero#(s(x)) -> c_13()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          12: zero#(0()) -> c_12()
          13: zero#(s(x)) -> c_13()
          6: half#(0()) -> c_3()
          7: half#(s(0())) -> c_4()
          9: inc#(0()) -> c_8()
          8: if#(true(),x,y) -> c_7(p#(y))
          10: p#(0()) -> c_10()
          11: p#(s(x)) -> c_11()
* Step 6: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y))
            bits#(x) -> c_2(bitIter#(x,0()))
            half#(s(s(x))) -> c_5(half#(x))
            if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
            inc#(s(x)) -> c_9(inc#(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()
        - Signature:
            {bitIter/2,bits/1,half/1,if/3,inc/1,p/1,zero/1,bitIter#/2,bits#/1,half#/1,if#/3,inc#/1,p#/1,zero#/1} / {0/0
            ,false/0,s/1,true/0,c_1/3,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bitIter#,bits#,half#,if#,inc#,p#
            ,zero#} and constructors {0,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y))
             -->_3 inc#(s(x)) -> c_9(inc#(x)):5
             -->_1 if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x)):4
          
          2:S:bits#(x) -> c_2(bitIter#(x,0()))
             -->_1 bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y)):1
          
          3:S:half#(s(s(x))) -> c_5(half#(x))
             -->_1 half#(s(s(x))) -> c_5(half#(x)):3
          
          4:S:if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
             -->_2 half#(s(s(x))) -> c_5(half#(x)):3
             -->_1 bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),zero#(x),inc#(y)):1
          
          5:S:inc#(s(x)) -> c_9(inc#(x))
             -->_1 inc#(s(x)) -> c_9(inc#(x)):5
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),inc#(y))
* Step 7: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),inc#(y))
            bits#(x) -> c_2(bitIter#(x,0()))
            half#(s(s(x))) -> c_5(half#(x))
            if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
            inc#(s(x)) -> c_9(inc#(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()
        - Signature:
            {bitIter/2,bits/1,half/1,if/3,inc/1,p/1,zero/1,bitIter#/2,bits#/1,half#/1,if#/3,inc#/1,p#/1,zero#/1} / {0/0
            ,false/0,s/1,true/0,c_1/2,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bitIter#,bits#,half#,if#,inc#,p#
            ,zero#} and constructors {0,false,s,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),inc#(y))
           -->_2 inc#(s(x)) -> c_9(inc#(x)):5
           -->_1 if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x)):4
        
        2:S:bits#(x) -> c_2(bitIter#(x,0()))
           -->_1 bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),inc#(y)):1
        
        3:S:half#(s(s(x))) -> c_5(half#(x))
           -->_1 half#(s(s(x))) -> c_5(half#(x)):3
        
        4:S:if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
           -->_2 half#(s(s(x))) -> c_5(half#(x)):3
           -->_1 bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),inc#(y)):1
        
        5:S:inc#(s(x)) -> c_9(inc#(x))
           -->_1 inc#(s(x)) -> c_9(inc#(x)):5
        
        
        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).
        
        [(2,bits#(x) -> c_2(bitIter#(x,0())))]
* Step 8: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          bitIter#(x,y) -> c_1(if#(zero(x),x,inc(y)),inc#(y))
          half#(s(s(x))) -> c_5(half#(x))
          if#(false(),x,y) -> c_6(bitIter#(half(x),y),half#(x))
          inc#(s(x)) -> c_9(inc#(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()
      - Signature:
          {bitIter/2,bits/1,half/1,if/3,inc/1,p/1,zero/1,bitIter#/2,bits#/1,half#/1,if#/3,inc#/1,p#/1,zero#/1} / {0/0
          ,false/0,s/1,true/0,c_1/2,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {bitIter#,bits#,half#,if#,inc#,p#
          ,zero#} and constructors {0,false,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE