WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            bits(0()) -> 0()
            bits(s(x)) -> s(bits(half(s(x))))
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            bits(0()) -> 0()
            bits(s(x)) -> s(bits(half(s(x))))
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          half(x){x -> s(s(x))} =
            half(s(s(x))) ->^+ s(half(x))
              = C[half(x) = half(x){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            bits(0()) -> 0()
            bits(s(x)) -> s(bits(half(s(x))))
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(bits) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {bits,half}
        TcT has computed the following interpretation:
             p(0) = 4     
          p(bits) = 4 + x1
          p(half) = x1    
             p(s) = x1    
        
        Following rules are strictly oriented:
        bits(0()) = 8  
                  > 4  
                  = 0()
        
        
        Following rules are (at-least) weakly oriented:
           bits(s(x)) =  4 + x              
                      >= 4 + x              
                      =  s(bits(half(s(x))))
        
            half(0()) =  4                  
                      >= 4                  
                      =  0()                
        
         half(s(0())) =  4                  
                      >= 4                  
                      =  0()                
        
        half(s(s(x))) =  x                  
                      >= x                  
                      =  s(half(x))         
        
** Step 1.b:2: Ara WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            bits(s(x)) -> s(bits(half(s(x))))
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
        - Weak TRS:
            bits(0()) -> 0()
        - Signature:
            {bits/1,half/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
    + Applied Processor:
        Ara {araHeuristics = Heuristics, minDegree = 1, maxDegree = 1, araTimeout = 3, araRuleShifting = Nothing}
    + Details:
        Signatures used:
        ----------------
          0 :: [] -(0)-> "A"(3)
          0 :: [] -(0)-> "A"(5)
          0 :: [] -(0)-> "A"(0)
          bits :: ["A"(5)] -(0)-> "A"(0)
          half :: ["A"(3)] -(1)-> "A"(5)
          s :: ["A"(5)] -(5)-> "A"(5)
          s :: ["A"(3)] -(3)-> "A"(3)
          s :: ["A"(0)] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        

WORST_CASE(Omega(n^1),O(n^1))