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


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WORST_CASE(?,O(n^2))
* Step 1: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            g(f(x),y) -> f(h(x,y))
            h(x,y) -> g(x,f(y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
             derivational complexity wrt. signature {f,g,h}
    + 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(f) = [1] x1 + [0]         
            p(g) = [1] x1 + [1] x2 + [0]
            p(h) = [1] x1 + [1] x2 + [5]
          
          Following rules are strictly oriented:
          h(x,y) = [1] x + [1] y + [5]
                 > [1] x + [1] y + [0]
                 = g(x,f(y))          
          
          
          Following rules are (at-least) weakly oriented:
          g(f(x),y) =  [1] x + [1] y + [0]
                    >= [1] x + [1] y + [5]
                    =  f(h(x,y))          
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            g(f(x),y) -> f(h(x,y))
        - Weak TRS:
            h(x,y) -> g(x,f(y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
             derivational complexity wrt. signature {f,g,h}
    + 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(f) = [1 0] x1 + [0]            
                 [0 1]      [1]            
          p(g) = [1 9] x1 + [1 6] x2 + [4] 
                 [0 1]      [0 0]      [0] 
          p(h) = [1 9] x1 + [1 6] x2 + [10]
                 [0 1]      [0 0]      [0] 
        
        Following rules are strictly oriented:
        g(f(x),y) = [1 9] x + [1 6] y + [13]
                    [0 1]     [0 0]     [1] 
                  > [1 9] x + [1 6] y + [10]
                    [0 1]     [0 0]     [1] 
                  = f(h(x,y))               
        
        
        Following rules are (at-least) weakly oriented:
        h(x,y) =  [1 9] x + [1 6] y + [10]
                  [0 1]     [0 0]     [0] 
               >= [1 9] x + [1 6] y + [10]
                  [0 1]     [0 0]     [0] 
               =  g(x,f(y))               
        
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            g(f(x),y) -> f(h(x,y))
            h(x,y) -> g(x,f(y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
             derivational complexity wrt. signature {f,g,h}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))