/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: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(0()) -> s(0())
            f(s(0())) -> s(0())
            f(s(s(x))) -> f(f(s(x)))
        - Signature:
            {f/1} / {0/0,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,f,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) = [2]         
            p(f) = [1] x1 + [0]
            p(s) = [1] x1 + [8]
          
          Following rules are strictly oriented:
          f(s(s(x))) = [1] x + [16]
                     > [1] x + [8] 
                     = f(f(s(x)))  
          
          
          Following rules are (at-least) weakly oriented:
             f(0()) =  [2]   
                    >= [10]  
                    =  s(0())
          
          f(s(0())) =  [10]  
                    >= [10]  
                    =  s(0())
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(0()) -> s(0())
            f(s(0())) -> s(0())
        - Weak TRS:
            f(s(s(x))) -> f(f(s(x)))
        - Signature:
            {f/1} / {0/0,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,f,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) = [7]         
            p(f) = [1] x1 + [4]
            p(s) = [1] x1 + [5]
          
          Following rules are strictly oriented:
          f(s(0())) = [16]  
                    > [12]  
                    = s(0())
          
          
          Following rules are (at-least) weakly oriented:
              f(0()) =  [11]        
                     >= [12]        
                     =  s(0())      
          
          f(s(s(x))) =  [1] x + [14]
                     >= [1] x + [13]
                     =  f(f(s(x)))  
          
        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:
            f(0()) -> s(0())
        - Weak TRS:
            f(s(0())) -> s(0())
            f(s(s(x))) -> f(f(s(x)))
        - Signature:
            {f/1} / {0/0,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,f,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) = [7]           
                 [10]          
          p(f) = [1 2] x1 + [0]
                 [0 0]      [4]
          p(s) = [1 0] x1 + [8]
                 [0 0]      [0]
        
        Following rules are strictly oriented:
        f(0()) = [27]  
                 [4]   
               > [15]  
                 [0]   
               = s(0())
        
        
        Following rules are (at-least) weakly oriented:
         f(s(0())) =  [15]          
                      [4]           
                   >= [15]          
                      [0]           
                   =  s(0())        
        
        f(s(s(x))) =  [1 0] x + [16]
                      [0 0]     [4] 
                   >= [1 0] x + [16]
                      [0 0]     [4] 
                   =  f(f(s(x)))    
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(0()) -> s(0())
            f(s(0())) -> s(0())
            f(s(s(x))) -> f(f(s(x)))
        - Signature:
            {f/1} / {0/0,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,f,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))