/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^1))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(f(x)) -> f(x)
            f(s(x)) -> f(x)
            g(s(0())) -> g(f(s(0())))
        - Signature:
            {f/1,g/1} / {0/0,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,f,g,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) = [0]         
          p(f) = [1] x1 + [0]
          p(g) = [1] x1 + [2]
          p(s) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        f(s(x)) = [1] x + [1]
                > [1] x + [0]
                = f(x)       
        
        
        Following rules are (at-least) weakly oriented:
          f(f(x)) =  [1] x + [0] 
                  >= [1] x + [0] 
                  =  f(x)        
        
        g(s(0())) =  [3]         
                  >= [3]         
                  =  g(f(s(0())))
        
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(f(x)) -> f(x)
            g(s(0())) -> g(f(s(0())))
        - Weak TRS:
            f(s(x)) -> f(x)
        - Signature:
            {f/1,g/1} / {0/0,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,f,g,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) = [1]         
            p(f) = [1] x1 + [1]
            p(g) = [1] x1 + [0]
            p(s) = [1] x1 + [1]
          
          Following rules are strictly oriented:
          f(f(x)) = [1] x + [2]
                  > [1] x + [1]
                  = f(x)       
          
          
          Following rules are (at-least) weakly oriented:
            f(s(x)) =  [1] x + [2] 
                    >= [1] x + [1] 
                    =  f(x)        
          
          g(s(0())) =  [2]         
                    >= [3]         
                    =  g(f(s(0())))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(s(0())) -> g(f(s(0())))
        - Weak TRS:
            f(f(x)) -> f(x)
            f(s(x)) -> f(x)
        - Signature:
            {f/1,g/1} / {0/0,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,f,g,s}
    + 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(0) = [4]           
                 [4]           
          p(f) = [1 1] x1 + [0]
                 [0 0]      [0]
          p(g) = [1 3] x1 + [0]
                 [0 0]      [0]
          p(s) = [1 2] x1 + [4]
                 [0 0]      [1]
        
        Following rules are strictly oriented:
        g(s(0())) = [19]        
                    [0]         
                  > [17]        
                    [0]         
                  = g(f(s(0())))
        
        
        Following rules are (at-least) weakly oriented:
        f(f(x)) =  [1 1] x + [0]
                   [0 0]     [0]
                >= [1 1] x + [0]
                   [0 0]     [0]
                =  f(x)         
        
        f(s(x)) =  [1 2] x + [5]
                   [0 0]     [0]
                >= [1 1] x + [0]
                   [0 0]     [0]
                =  f(x)         
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(f(x)) -> f(x)
            f(s(x)) -> f(x)
            g(s(0())) -> g(f(s(0())))
        - Signature:
            {f/1,g/1} / {0/0,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,f,g,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))