WORST_CASE(?,O(n^1))
* Step 1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            bits(0()) -> 0()
            bits(s(0())) -> s(0())
            bits(s(s(x))) -> s(bits(s(half(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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(bits) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [0]          
            p(bits) = [1] x1 + [1] 
            p(half) = [1] x1 + [10]
               p(s) = [1] x1 + [0] 
          
          Following rules are strictly oriented:
             bits(0()) = [1]   
                       > [0]   
                       = 0()   
          
          bits(s(0())) = [1]   
                       > [0]   
                       = s(0())
          
             half(0()) = [10]  
                       > [0]   
                       = 0()   
          
          half(s(0())) = [10]  
                       > [0]   
                       = 0()   
          
          
          Following rules are (at-least) weakly oriented:
          bits(s(s(x))) =  [1] x + [1]        
                        >= [1] x + [11]       
                        =  s(bits(s(half(x))))
          
          half(s(s(x))) =  [1] x + [10]       
                        >= [1] x + [10]       
                        =  s(half(x))         
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            bits(s(s(x))) -> s(bits(s(half(x))))
            half(s(s(x))) -> s(half(x))
        - Weak TRS:
            bits(0()) -> 0()
            bits(s(0())) -> s(0())
            half(0()) -> 0()
            half(s(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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(bits) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [2]         
            p(bits) = [1] x1 + [2]
            p(half) = [1] x1 + [0]
               p(s) = [1] x1 + [2]
          
          Following rules are strictly oriented:
          half(s(s(x))) = [1] x + [4]
                        > [1] x + [2]
                        = s(half(x)) 
          
          
          Following rules are (at-least) weakly oriented:
              bits(0()) =  [4]                
                        >= [2]                
                        =  0()                
          
           bits(s(0())) =  [6]                
                        >= [4]                
                        =  s(0())             
          
          bits(s(s(x))) =  [1] x + [6]        
                        >= [1] x + [6]        
                        =  s(bits(s(half(x))))
          
              half(0()) =  [2]                
                        >= [2]                
                        =  0()                
          
           half(s(0())) =  [4]                
                        >= [2]                
                        =  0()                
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            bits(s(s(x))) -> s(bits(s(half(x))))
        - Weak TRS:
            bits(0()) -> 0()
            bits(s(0())) -> s(0())
            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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        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) = [2] x1 + [0]
          p(half) = [1] x1 + [0]
             p(s) = [1] x1 + [4]
        
        Following rules are strictly oriented:
        bits(s(s(x))) = [2] x + [16]       
                      > [2] x + [12]       
                      = s(bits(s(half(x))))
        
        
        Following rules are (at-least) weakly oriented:
            bits(0()) =  [8]        
                      >= [4]        
                      =  0()        
        
         bits(s(0())) =  [16]       
                      >= [8]        
                      =  s(0())     
        
            half(0()) =  [4]        
                      >= [4]        
                      =  0()        
        
         half(s(0())) =  [8]        
                      >= [4]        
                      =  0()        
        
        half(s(s(x))) =  [1] x + [8]
                      >= [1] x + [4]
                      =  s(half(x)) 
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            bits(0()) -> 0()
            bits(s(0())) -> s(0())
            bits(s(s(x))) -> s(bits(s(half(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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))