/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: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(a(f(x,y))) -> f(a(b(a(b(a(x))))),a(b(a(b(a(y))))))
            f(a(x),a(y)) -> a(f(x,y))
            f(b(x),b(y)) -> b(f(x,y))
        - Signature:
            {a/1,f/2} / {b/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,f}
    + 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(a) = [1] x1 + [1]         
            p(b) = [1] x1 + [0]         
            p(f) = [1] x1 + [1] x2 + [1]
          
          Following rules are strictly oriented:
          f(a(x),a(y)) = [1] x + [1] y + [3]
                       > [1] x + [1] y + [2]
                       = a(f(x,y))          
          
          
          Following rules are (at-least) weakly oriented:
          a(a(f(x,y))) =  [1] x + [1] y + [3]                 
                       >= [1] x + [1] y + [7]                 
                       =  f(a(b(a(b(a(x))))),a(b(a(b(a(y))))))
          
          f(b(x),b(y)) =  [1] x + [1] y + [1]                 
                       >= [1] x + [1] y + [1]                 
                       =  b(f(x,y))                           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(a(f(x,y))) -> f(a(b(a(b(a(x))))),a(b(a(b(a(y))))))
            f(b(x),b(y)) -> b(f(x,y))
        - Weak TRS:
            f(a(x),a(y)) -> a(f(x,y))
        - Signature:
            {a/1,f/2} / {b/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,f}
    + 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(a) = [1] x1 + [0]          
            p(b) = [1] x1 + [4]          
            p(f) = [1] x1 + [1] x2 + [13]
          
          Following rules are strictly oriented:
          f(b(x),b(y)) = [1] x + [1] y + [21]
                       > [1] x + [1] y + [17]
                       = b(f(x,y))           
          
          
          Following rules are (at-least) weakly oriented:
          a(a(f(x,y))) =  [1] x + [1] y + [13]                
                       >= [1] x + [1] y + [29]                
                       =  f(a(b(a(b(a(x))))),a(b(a(b(a(y))))))
          
          f(a(x),a(y)) =  [1] x + [1] y + [13]                
                       >= [1] x + [1] y + [13]                
                       =  a(f(x,y))                           
          
        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:
            a(a(f(x,y))) -> f(a(b(a(b(a(x))))),a(b(a(b(a(y))))))
        - Weak TRS:
            f(a(x),a(y)) -> a(f(x,y))
            f(b(x),b(y)) -> b(f(x,y))
        - Signature:
            {a/1,f/2} / {b/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,f}
    + 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(a) = [1 1] x1 + [0]           
                 [0 1]      [1]           
          p(b) = [1 0] x1 + [0]           
                 [0 0]      [0]           
          p(f) = [1 0] x1 + [1 0] x2 + [0]
                 [0 1]      [0 1]      [0]
        
        Following rules are strictly oriented:
        a(a(f(x,y))) = [1 2] x + [1 2] y + [1]             
                       [0 1]     [0 1]     [2]             
                     > [1 1] x + [1 1] y + [0]             
                       [0 0]     [0 0]     [2]             
                     = f(a(b(a(b(a(x))))),a(b(a(b(a(y))))))
        
        
        Following rules are (at-least) weakly oriented:
        f(a(x),a(y)) =  [1 1] x + [1 1] y + [0]
                        [0 1]     [0 1]     [2]
                     >= [1 1] x + [1 1] y + [0]
                        [0 1]     [0 1]     [1]
                     =  a(f(x,y))              
        
        f(b(x),b(y)) =  [1 0] x + [1 0] y + [0]
                        [0 0]     [0 0]     [0]
                     >= [1 0] x + [1 0] y + [0]
                        [0 0]     [0 0]     [0]
                     =  b(f(x,y))              
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(a(f(x,y))) -> f(a(b(a(b(a(x))))),a(b(a(b(a(y))))))
            f(a(x),a(y)) -> a(f(x,y))
            f(b(x),b(y)) -> b(f(x,y))
        - Signature:
            {a/1,f/2} / {b/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,f}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))