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