/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^2))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(activate(X))
            f(X) -> n__f(X)
            f(f(a())) -> f(g(n__f(n__a())))
        - Signature:
            {a/0,activate/1,f/1} / {g/1,n__a/0,n__f/1}
        - Obligation:
             derivational complexity wrt. signature {a,activate,f,g,n__a,n__f}
    + 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) = [8]          
          p(activate) = [1] x1 + [12]
                 p(f) = [1] x1 + [9] 
                 p(g) = [1] x1 + [5] 
              p(n__a) = [2]          
              p(n__f) = [1] x1 + [9] 
        
        Following rules are strictly oriented:
                     a() = [8]               
                         > [2]               
                         = n__a()            
        
             activate(X) = [1] X + [12]      
                         > [1] X + [0]       
                         = X                 
        
        activate(n__a()) = [14]              
                         > [8]               
                         = a()               
        
               f(f(a())) = [26]              
                         > [25]              
                         = f(g(n__f(n__a())))
        
        
        Following rules are (at-least) weakly oriented:
        activate(n__f(X)) =  [1] X + [21]  
                          >= [1] X + [21]  
                          =  f(activate(X))
        
                     f(X) =  [1] X + [9]   
                          >= [1] X + [9]   
                          =  n__f(X)       
        
* Step 2: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            activate(n__f(X)) -> f(activate(X))
            f(X) -> n__f(X)
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            f(f(a())) -> f(g(n__f(n__a())))
        - Signature:
            {a/0,activate/1,f/1} / {g/1,n__a/0,n__f/1}
        - Obligation:
             derivational complexity wrt. signature {a,activate,f,g,n__a,n__f}
    + 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) = [7]         
            p(activate) = [1] x1 + [0]
                   p(f) = [1] x1 + [5]
                   p(g) = [1] x1 + [5]
                p(n__a) = [7]         
                p(n__f) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          f(X) = [1] X + [5]
               > [1] X + [0]
               = n__f(X)    
          
          
          Following rules are (at-least) weakly oriented:
                        a() =  [7]               
                            >= [7]               
                            =  n__a()            
          
                activate(X) =  [1] X + [0]       
                            >= [1] X + [0]       
                            =  X                 
          
           activate(n__a()) =  [7]               
                            >= [7]               
                            =  a()               
          
          activate(n__f(X)) =  [1] X + [0]       
                            >= [1] X + [5]       
                            =  f(activate(X))    
          
                  f(f(a())) =  [17]              
                            >= [17]              
                            =  f(g(n__f(n__a())))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            activate(n__f(X)) -> f(activate(X))
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            f(X) -> n__f(X)
            f(f(a())) -> f(g(n__f(n__a())))
        - Signature:
            {a/0,activate/1,f/1} / {g/1,n__a/0,n__f/1}
        - Obligation:
             derivational complexity wrt. signature {a,activate,f,g,n__a,n__f}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, 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]           
                        [4]           
          p(activate) = [1 2] x1 + [1]
                        [0 1]      [4]
                 p(f) = [1 0] x1 + [2]
                        [0 1]      [8]
                 p(g) = [1 0] x1 + [1]
                        [0 1]      [0]
              p(n__a) = [0]           
                        [4]           
              p(n__f) = [1 0] x1 + [1]
                        [0 1]      [8]
        
        Following rules are strictly oriented:
        activate(n__f(X)) = [1 2] X + [18]
                            [0 1]     [12]
                          > [1 2] X + [3] 
                            [0 1]     [12]
                          = f(activate(X))
        
        
        Following rules are (at-least) weakly oriented:
                     a() =  [0]               
                            [4]               
                         >= [0]               
                            [4]               
                         =  n__a()            
        
             activate(X) =  [1 2] X + [1]     
                            [0 1]     [4]     
                         >= [1 0] X + [0]     
                            [0 1]     [0]     
                         =  X                 
        
        activate(n__a()) =  [9]               
                            [8]               
                         >= [0]               
                            [4]               
                         =  a()               
        
                    f(X) =  [1 0] X + [2]     
                            [0 1]     [8]     
                         >= [1 0] X + [1]     
                            [0 1]     [8]     
                         =  n__f(X)           
        
               f(f(a())) =  [4]               
                            [20]              
                         >= [4]               
                            [20]              
                         =  f(g(n__f(n__a())))
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(activate(X))
            f(X) -> n__f(X)
            f(f(a())) -> f(g(n__f(n__a())))
        - Signature:
            {a/0,activate/1,f/1} / {g/1,n__a/0,n__f/1}
        - Obligation:
             derivational complexity wrt. signature {a,activate,f,g,n__a,n__f}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))