/export/starexec/sandbox2/solver/bin/starexec_run_tct_dci /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(x) -> g(x)
            f(a()) -> f(b())
            g(b()) -> g(a())
        - Signature:
            {f/1,g/1} / {a/0,b/0}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,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) = [0]         
          p(b) = [0]         
          p(f) = [1] x1 + [9]
          p(g) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        f(x) = [1] x + [9]
             > [1] x + [1]
             = g(x)       
        
        
        Following rules are (at-least) weakly oriented:
        f(a()) =  [9]   
               >= [9]   
               =  f(b())
        
        g(b()) =  [1]   
               >= [1]   
               =  g(a())
        
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a()) -> f(b())
            g(b()) -> g(a())
        - Weak TRS:
            f(x) -> g(x)
        - Signature:
            {f/1,g/1} / {a/0,b/0}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,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) = [0]         
            p(b) = [9]         
            p(f) = [1] x1 + [0]
            p(g) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          g(b()) = [9]   
                 > [0]   
                 = g(a())
          
          
          Following rules are (at-least) weakly oriented:
            f(x) =  [1] x + [0]
                 >= [1] x + [0]
                 =  g(x)       
          
          f(a()) =  [0]        
                 >= [9]        
                 =  f(b())     
          
        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(a()) -> f(b())
        - Weak TRS:
            f(x) -> g(x)
            g(b()) -> g(a())
        - Signature:
            {f/1,g/1} / {a/0,b/0}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,f,g}
    + Applied Processor:
        NaturalMI {miDimension = 3, 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(a) = [0]             
                 [3]             
                 [14]            
          p(b) = [1]             
                 [4]             
                 [8]             
          p(f) = [1 4 1]      [0]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [2]
          p(g) = [1 2 0]      [0]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [1]
        
        Following rules are strictly oriented:
        f(a()) = [26]  
                 [0]   
                 [2]   
               > [25]  
                 [0]   
                 [2]   
               = f(b())
        
        
        Following rules are (at-least) weakly oriented:
          f(x) =  [1 4 1]     [0]
                  [0 0 0] x + [0]
                  [0 0 0]     [2]
               >= [1 2 0]     [0]
                  [0 0 0] x + [0]
                  [0 0 0]     [1]
               =  g(x)           
        
        g(b()) =  [9]            
                  [0]            
                  [1]            
               >= [6]            
                  [0]            
                  [1]            
               =  g(a())         
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x) -> g(x)
            f(a()) -> f(b())
            g(b()) -> g(a())
        - Signature:
            {f/1,g/1} / {a/0,b/0}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))