/export/starexec/sandbox2/solver/bin/starexec_run_tct_dc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(?,O(n^2))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^2))
    + 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:
             derivational complexity wrt. signature {0,bits,half,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
             p(0) = [1]         
          p(bits) = [1] x1 + [2]
          p(half) = [1] x1 + [0]
             p(s) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        bits(0()) = [3]
                  > [1]
                  = 0()
        
        
        Following rules are (at-least) weakly oriented:
           bits(s(x)) =  [1] x + [2]        
                      >= [1] x + [2]        
                      =  s(bits(half(s(x))))
        
            half(0()) =  [1]                
                      >= [1]                
                      =  0()                
        
         half(s(0())) =  [1]                
                      >= [1]                
                      =  0()                
        
        half(s(s(x))) =  [1] x + [0]        
                      >= [1] x + [0]        
                      =  s(half(x))         
        
* Step 2: WeightGap.  WORST_CASE(?,O(n^2))
    + 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:
             derivational complexity wrt. signature {0,bits,half,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind triangular matrix interpretation:
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [0]          
            p(bits) = [1] x1 + [0] 
            p(half) = [1] x1 + [13]
               p(s) = [1] x1 + [1] 
          
          Following rules are strictly oriented:
              half(0()) = [13]        
                        > [0]         
                        = 0()         
          
           half(s(0())) = [14]        
                        > [0]         
                        = 0()         
          
          half(s(s(x))) = [1] x + [15]
                        > [1] x + [14]
                        = s(half(x))  
          
          
          Following rules are (at-least) weakly oriented:
           bits(0()) =  [0]                
                     >= [0]                
                     =  0()                
          
          bits(s(x)) =  [1] x + [1]        
                     >= [1] x + [15]       
                     =  s(bits(half(s(x))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            bits(s(x)) -> s(bits(half(s(x))))
        - Weak TRS:
            bits(0()) -> 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:
             derivational complexity wrt. signature {0,bits,half,s}
    + Applied Processor:
        MI {miKind = Automaton (Just 2), miDimension = 3, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind Automaton (Just 2):
        
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
             p(0) = [0]              
                    [0]              
                    [0]              
          p(bits) = [1 0 4]       [4]
                    [0 0 1] x_1 + [3]
                    [0 0 1]       [4]
          p(half) = [1 0 0]       [0]
                    [0 1 0] x_1 + [0]
                    [0 1 0]       [0]
             p(s) = [1 0 0]       [1]
                    [0 1 0] x_1 + [1]
                    [0 1 0]       [2]
        
        Following rules are strictly oriented:
        bits(s(x)) = [1 4 0]     [13]   
                     [0 1 0] x + [5]    
                     [0 1 0]     [6]    
                   > [1 4 0]     [10]   
                     [0 1 0] x + [5]    
                     [0 1 0]     [6]    
                   = s(bits(half(s(x))))
        
        
        Following rules are (at-least) weakly oriented:
            bits(0()) =  [4]            
                         [3]            
                         [4]            
                      >= [0]            
                         [0]            
                         [0]            
                      =  0()            
        
            half(0()) =  [0]            
                         [0]            
                         [0]            
                      >= [0]            
                         [0]            
                         [0]            
                      =  0()            
        
         half(s(0())) =  [1]            
                         [1]            
                         [1]            
                      >= [0]            
                         [0]            
                         [0]            
                      =  0()            
        
        half(s(s(x))) =  [1 0 0]     [2]
                         [0 1 0] x + [2]
                         [0 1 0]     [2]
                      >= [1 0 0]     [1]
                         [0 1 0] x + [1]
                         [0 1 0]     [2]
                      =  s(half(x))     
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak 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:
             derivational complexity wrt. signature {0,bits,half,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))