/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:
            f(0()) -> s(0())
            f(s(x)) -> s(s(g(x)))
            g(0()) -> 0()
            g(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) = [3]          
            p(f) = [1] x1 + [0] 
            p(g) = [1] x1 + [13]
            p(s) = [1] x1 + [0] 
          
          Following rules are strictly oriented:
           g(0()) = [16]        
                  > [3]         
                  = 0()         
          
          g(s(x)) = [1] x + [13]
                  > [1] x + [0] 
                  = f(x)        
          
          
          Following rules are (at-least) weakly oriented:
           f(0()) =  [3]         
                  >= [3]         
                  =  s(0())      
          
          f(s(x)) =  [1] x + [0] 
                  >= [1] x + [13]
                  =  s(s(g(x)))  
          
        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:
            f(0()) -> s(0())
            f(s(x)) -> s(s(g(x)))
        - Weak TRS:
            g(0()) -> 0()
            g(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 = 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 + [4]
          p(g) = [1] x1 + [2]
          p(s) = [1] x1 + [2]
        
        Following rules are strictly oriented:
        f(0()) = [4]   
               > [2]   
               = s(0())
        
        
        Following rules are (at-least) weakly oriented:
        f(s(x)) =  [1] x + [6]
                >= [1] x + [6]
                =  s(s(g(x))) 
        
         g(0()) =  [2]        
                >= [0]        
                =  0()        
        
        g(s(x)) =  [1] x + [4]
                >= [1] x + [4]
                =  f(x)       
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(s(x)) -> s(s(g(x)))
        - Weak TRS:
            f(0()) -> s(0())
            g(0()) -> 0()
            g(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 = 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(0) = [9]           
                 [2]           
          p(f) = [1 2] x1 + [1]
                 [0 1]      [1]
          p(g) = [1 2] x1 + [2]
                 [0 1]      [0]
          p(s) = [1 0] x1 + [0]
                 [0 1]      [1]
        
        Following rules are strictly oriented:
        f(s(x)) = [1 2] x + [3]
                  [0 1]     [2]
                > [1 2] x + [2]
                  [0 1]     [2]
                = s(s(g(x)))   
        
        
        Following rules are (at-least) weakly oriented:
         f(0()) =  [14]         
                   [3]          
                >= [9]          
                   [3]          
                =  s(0())       
        
         g(0()) =  [15]         
                   [2]          
                >= [9]          
                   [2]          
                =  0()          
        
        g(s(x)) =  [1 2] x + [4]
                   [0 1]     [1]
                >= [1 2] x + [1]
                   [0 1]     [1]
                =  f(x)         
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(0()) -> s(0())
            f(s(x)) -> s(s(g(x)))
            g(0()) -> 0()
            g(s(x)) -> f(x)
        - 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^2))