/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:
            rev(++(x,x)) -> rev(x)
            rev(++(x,y)) -> ++(rev(y),rev(x))
            rev(a()) -> a()
            rev(b()) -> b()
        - Signature:
            {rev/1} / {++/2,a/0,b/0}
        - Obligation:
             derivational complexity wrt. signature {++,a,b,rev}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
           p(++) = [1 0] x1 + [1 0] x2 + [4]
                   [0 0]      [0 0]      [0]
            p(a) = [4]                      
                   [0]                      
            p(b) = [1]                      
                   [0]                      
          p(rev) = [1 0] x1 + [0]           
                   [0 0]      [0]           
        
        Following rules are strictly oriented:
        rev(++(x,x)) = [2 0] x + [4]
                       [0 0]     [0]
                     > [1 0] x + [0]
                       [0 0]     [0]
                     = rev(x)       
        
        
        Following rules are (at-least) weakly oriented:
        rev(++(x,y)) =  [1 0] x + [1 0] y + [4]
                        [0 0]     [0 0]     [0]
                     >= [1 0] x + [1 0] y + [4]
                        [0 0]     [0 0]     [0]
                     =  ++(rev(y),rev(x))      
        
            rev(a()) =  [4]                    
                        [0]                    
                     >= [4]                    
                        [0]                    
                     =  a()                    
        
            rev(b()) =  [1]                    
                        [0]                    
                     >= [1]                    
                        [0]                    
                     =  b()                    
        
* Step 2: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            rev(++(x,y)) -> ++(rev(y),rev(x))
            rev(a()) -> a()
            rev(b()) -> b()
        - Weak TRS:
            rev(++(x,x)) -> rev(x)
        - Signature:
            {rev/1} / {++/2,a/0,b/0}
        - Obligation:
             derivational complexity wrt. signature {++,a,b,rev}
    + 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(++) = [1] x1 + [1] x2 + [0]
              p(a) = [0]                  
              p(b) = [0]                  
            p(rev) = [1] x1 + [15]        
          
          Following rules are strictly oriented:
          rev(a()) = [15]
                   > [0] 
                   = a() 
          
          rev(b()) = [15]
                   > [0] 
                   = b() 
          
          
          Following rules are (at-least) weakly oriented:
          rev(++(x,x)) =  [2] x + [15]        
                       >= [1] x + [15]        
                       =  rev(x)              
          
          rev(++(x,y)) =  [1] x + [1] y + [15]
                       >= [1] x + [1] y + [30]
                       =  ++(rev(y),rev(x))   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            rev(++(x,y)) -> ++(rev(y),rev(x))
        - Weak TRS:
            rev(++(x,x)) -> rev(x)
            rev(a()) -> a()
            rev(b()) -> b()
        - Signature:
            {rev/1} / {++/2,a/0,b/0}
        - Obligation:
             derivational complexity wrt. signature {++,a,b,rev}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, 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(++) = [1 0] x1 + [1 0] x2 + [3]
                   [0 1]      [0 1]      [1]
            p(a) = [3]                      
                   [3]                      
            p(b) = [0]                      
                   [1]                      
          p(rev) = [1 3] x1 + [2]           
                   [0 1]      [0]           
        
        Following rules are strictly oriented:
        rev(++(x,y)) = [1 3] x + [1 3] y + [8]
                       [0 1]     [0 1]     [1]
                     > [1 3] x + [1 3] y + [7]
                       [0 1]     [0 1]     [1]
                     = ++(rev(y),rev(x))      
        
        
        Following rules are (at-least) weakly oriented:
        rev(++(x,x)) =  [2 6] x + [8]
                        [0 2]     [1]
                     >= [1 3] x + [2]
                        [0 1]     [0]
                     =  rev(x)       
        
            rev(a()) =  [14]         
                        [3]          
                     >= [3]          
                        [3]          
                     =  a()          
        
            rev(b()) =  [5]          
                        [1]          
                     >= [0]          
                        [1]          
                     =  b()          
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            rev(++(x,x)) -> rev(x)
            rev(++(x,y)) -> ++(rev(y),rev(x))
            rev(a()) -> a()
            rev(b()) -> b()
        - Signature:
            {rev/1} / {++/2,a/0,b/0}
        - Obligation:
             derivational complexity wrt. signature {++,a,b,rev}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))