WORST_CASE(?,O(n^1))
* Step 1: NaturalMI 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:
        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) = [0]         
          p(bits) = [1] x1 + [4]
          p(half) = [1] x1 + [0]
             p(s) = [1] x1 + [0]
        
        Following rules are strictly oriented:
           bits(0()) = [4]   
                     > [0]   
                     = 0()   
        
        bits(s(0())) = [4]   
                     > [0]   
                     = s(0())
        
        
        Following rules are (at-least) weakly oriented:
        bits(s(s(x))) =  [1] x + [4]        
                      >= [1] x + [4]        
                      =  s(bits(s(half(x))))
        
            half(0()) =  [0]                
                      >= [0]                
                      =  0()                
        
         half(s(0())) =  [0]                
                      >= [0]                
                      =  0()                
        
        half(s(s(x))) =  [1] x + [0]        
                      >= [1] x + [0]        
                      =  s(half(x))         
        
* Step 2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            bits(s(s(x))) -> s(bits(s(half(x))))
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
        - Weak TRS:
            bits(0()) -> 0()
            bits(s(0())) -> s(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:
        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) = 0       
          p(bits) = 9 + 8*x1
          p(half) = x1      
             p(s) = 1 + x1  
        
        Following rules are strictly oriented:
        bits(s(s(x))) = 25 + 8*x           
                      > 18 + 8*x           
                      = s(bits(s(half(x))))
        
         half(s(0())) = 1                  
                      > 0                  
                      = 0()                
        
        half(s(s(x))) = 2 + x              
                      > 1 + x              
                      = s(half(x))         
        
        
        Following rules are (at-least) weakly oriented:
           bits(0()) =  9     
                     >= 0     
                     =  0()   
        
        bits(s(0())) =  17    
                     >= 1     
                     =  s(0())
        
           half(0()) =  0     
                     >= 0     
                     =  0()   
        
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            half(0()) -> 0()
        - Weak TRS:
            bits(0()) -> 0()
            bits(s(0())) -> s(0())
            bits(s(s(x))) -> s(bits(s(half(x))))
            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) = [2]         
          p(bits) = [4] x1 + [6]
          p(half) = [1] x1 + [1]
             p(s) = [1] x1 + [3]
        
        Following rules are strictly oriented:
        half(0()) = [3]
                  > [2]
                  = 0()
        
        
        Following rules are (at-least) weakly oriented:
            bits(0()) =  [14]               
                      >= [2]                
                      =  0()                
        
         bits(s(0())) =  [26]               
                      >= [5]                
                      =  s(0())             
        
        bits(s(s(x))) =  [4] x + [30]       
                      >= [4] x + [25]       
                      =  s(bits(s(half(x))))
        
         half(s(0())) =  [6]                
                      >= [2]                
                      =  0()                
        
        half(s(s(x))) =  [1] x + [7]        
                      >= [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))