/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^1))
* Step 1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a() -> g(c())
            f(g(X),b()) -> f(a(),X)
            g(a()) -> b()
        - Signature:
            {a/0,f/2,g/1} / {b/0,c/0}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,f,g}
    + 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(a) = [9]                  
            p(b) = [0]                  
            p(c) = [0]                  
            p(f) = [1] x1 + [1] x2 + [0]
            p(g) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
             a() = [9]   
                 > [0]   
                 = g(c())
          
          g(a()) = [9]   
                 > [0]   
                 = b()   
          
          
          Following rules are (at-least) weakly oriented:
          f(g(X),b()) =  [1] X + [0]
                      >= [1] X + [9]
                      =  f(a(),X)   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(g(X),b()) -> f(a(),X)
        - Weak TRS:
            a() -> g(c())
            g(a()) -> b()
        - Signature:
            {a/0,f/2,g/1} / {b/0,c/0}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,f,g}
    + 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(a) = [6]                  
          p(b) = [5]                  
          p(c) = [0]                  
          p(f) = [1] x1 + [1] x2 + [3]
          p(g) = [1] x1 + [6]         
        
        Following rules are strictly oriented:
        f(g(X),b()) = [1] X + [14]
                    > [1] X + [9] 
                    = f(a(),X)    
        
        
        Following rules are (at-least) weakly oriented:
           a() =  [6]   
               >= [6]   
               =  g(c())
        
        g(a()) =  [12]  
               >= [5]   
               =  b()   
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a() -> g(c())
            f(g(X),b()) -> f(a(),X)
            g(a()) -> b()
        - Signature:
            {a/0,f/2,g/1} / {b/0,c/0}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))