/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(0()) -> 0()
            g(0()) -> 0()
            g(s(x)) -> f(g(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 + [8]
          p(g) = [1] x1 + [0]
          p(s) = [1] x1 + [8]
        
        Following rules are strictly oriented:
        f(0()) = [8]
               > [0]
               = 0()
        
        
        Following rules are (at-least) weakly oriented:
         g(0()) =  [0]        
                >= [0]        
                =  0()        
        
        g(s(x)) =  [1] x + [8]
                >= [1] x + [8]
                =  f(g(x))    
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(0()) -> 0()
            g(s(x)) -> f(g(x))
        - Weak TRS:
            f(0()) -> 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) = [5]          
          p(f) = [1] x1 + [0] 
          p(g) = [1] x1 + [13]
          p(s) = [1] x1 + [0] 
        
        Following rules are strictly oriented:
        g(0()) = [18]
               > [5] 
               = 0() 
        
        
        Following rules are (at-least) weakly oriented:
         f(0()) =  [5]         
                >= [5]         
                =  0()         
        
        g(s(x)) =  [1] x + [13]
                >= [1] x + [13]
                =  f(g(x))     
        
* Step 3: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(s(x)) -> f(g(x))
        - Weak TRS:
            f(0()) -> 0()
            g(0()) -> 0()
        - 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) = [5]         
            p(f) = [1] x1 + [0]
            p(g) = [1] x1 + [0]
            p(s) = [1] x1 + [1]
          
          Following rules are strictly oriented:
          g(s(x)) = [1] x + [1]
                  > [1] x + [0]
                  = f(g(x))    
          
          
          Following rules are (at-least) weakly oriented:
          f(0()) =  [5]
                 >= [5]
                 =  0()
          
          g(0()) =  [5]
                 >= [5]
                 =  0()
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(0()) -> 0()
            g(0()) -> 0()
            g(s(x)) -> f(g(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^1))