/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:
            b() -> a()
            f(a()) -> g(a())
            g(b()) -> f(b())
        - Signature:
            {b/0,f/1,g/1} / {a/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) = [6]         
          p(b) = [7]         
          p(f) = [1] x1 + [2]
          p(g) = [1] x1 + [2]
        
        Following rules are strictly oriented:
        b() = [7]
            > [6]
            = a()
        
        
        Following rules are (at-least) weakly oriented:
        f(a()) =  [8]   
               >= [8]   
               =  g(a())
        
        g(b()) =  [9]   
               >= [9]   
               =  f(b())
        
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a()) -> g(a())
            g(b()) -> f(b())
        - Weak TRS:
            b() -> a()
        - Signature:
            {b/0,f/1,g/1} / {a/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) = [10]        
            p(b) = [10]        
            p(f) = [1] x1 + [0]
            p(g) = [1] x1 + [1]
          
          Following rules are strictly oriented:
          g(b()) = [11]  
                 > [10]  
                 = f(b())
          
          
          Following rules are (at-least) weakly oriented:
             b() =  [10]  
                 >= [10]  
                 =  a()   
          
          f(a()) =  [10]  
                 >= [11]  
                 =  g(a())
          
        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()) -> g(a())
        - Weak TRS:
            b() -> a()
            g(b()) -> f(b())
        - Signature:
            {b/0,f/1,g/1} / {a/0}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,f,g}
    + 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(a) = [2]            
                 [0]            
          p(b) = [3]            
                 [2]            
          p(f) = [1 1] x1 + [2] 
                 [0 0]      [10]
          p(g) = [1 8] x1 + [0] 
                 [0 0]      [10]
        
        Following rules are strictly oriented:
        f(a()) = [4]   
                 [10]  
               > [2]   
                 [10]  
               = g(a())
        
        
        Following rules are (at-least) weakly oriented:
           b() =  [3]   
                  [2]   
               >= [2]   
                  [0]   
               =  a()   
        
        g(b()) =  [19]  
                  [10]  
               >= [7]   
                  [10]  
               =  f(b())
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            b() -> a()
            f(a()) -> g(a())
            g(b()) -> f(b())
        - Signature:
            {b/0,f/1,g/1} / {a/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))