/export/starexec/sandbox/solver/bin/starexec_run_tct_dci /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(X)
            activate(n__g(X)) -> g(activate(X))
            f(X) -> n__f(X)
            f(n__f(n__a())) -> f(n__g(n__f(n__a())))
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,activate,f,g,n__a,n__f,n__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) = [1]         
          p(activate) = [1] x1 + [2]
                 p(f) = [1] x1 + [2]
                 p(g) = [1] x1 + [0]
              p(n__a) = [0]         
              p(n__f) = [1] x1 + [1]
              p(n__g) = [1] x1 + [0]
        
        Following rules are strictly oriented:
                      a() = [1]        
                          > [0]        
                          = n__a()     
        
              activate(X) = [1] X + [2]
                          > [1] X + [0]
                          = X          
        
         activate(n__a()) = [2]        
                          > [1]        
                          = a()        
        
        activate(n__f(X)) = [1] X + [3]
                          > [1] X + [2]
                          = f(X)       
        
                     f(X) = [1] X + [2]
                          > [1] X + [1]
                          = n__f(X)    
        
        
        Following rules are (at-least) weakly oriented:
        activate(n__g(X)) =  [1] X + [2]          
                          >= [1] X + [2]          
                          =  g(activate(X))       
        
          f(n__f(n__a())) =  [3]                  
                          >= [3]                  
                          =  f(n__g(n__f(n__a())))
        
                     g(X) =  [1] X + [0]          
                          >= [1] X + [0]          
                          =  n__g(X)              
        
* Step 2: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            activate(n__g(X)) -> g(activate(X))
            f(n__f(n__a())) -> f(n__g(n__f(n__a())))
            g(X) -> n__g(X)
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            f(X) -> n__f(X)
        - Signature:
            {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,activate,f,g,n__a,n__f,n__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) = [5]         
            p(activate) = [1] x1 + [0]
                   p(f) = [1] x1 + [0]
                   p(g) = [1] x1 + [1]
                p(n__a) = [5]         
                p(n__f) = [1] x1 + [0]
                p(n__g) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          g(X) = [1] X + [1]
               > [1] X + [0]
               = n__g(X)    
          
          
          Following rules are (at-least) weakly oriented:
                        a() =  [5]                  
                            >= [5]                  
                            =  n__a()               
          
                activate(X) =  [1] X + [0]          
                            >= [1] X + [0]          
                            =  X                    
          
           activate(n__a()) =  [5]                  
                            >= [5]                  
                            =  a()                  
          
          activate(n__f(X)) =  [1] X + [0]          
                            >= [1] X + [0]          
                            =  f(X)                 
          
          activate(n__g(X)) =  [1] X + [0]          
                            >= [1] X + [1]          
                            =  g(activate(X))       
          
                       f(X) =  [1] X + [0]          
                            >= [1] X + [0]          
                            =  n__f(X)              
          
            f(n__f(n__a())) =  [5]                  
                            >= [5]                  
                            =  f(n__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__g(X)) -> g(activate(X))
            f(n__f(n__a())) -> f(n__g(n__f(n__a())))
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            f(X) -> n__f(X)
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,activate,f,g,n__a,n__f,n__g}
    + 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) = [3]           
                        [5]           
          p(activate) = [1 0] x1 + [3]
                        [0 1]      [7]
                 p(f) = [1 2] x1 + [3]
                        [0 1]      [1]
                 p(g) = [1 0] x1 + [0]
                        [0 0]      [3]
              p(n__a) = [0]           
                        [1]           
              p(n__f) = [1 2] x1 + [0]
                        [0 1]      [1]
              p(n__g) = [1 0] x1 + [0]
                        [0 0]      [0]
        
        Following rules are strictly oriented:
        f(n__f(n__a())) = [9]                  
                          [3]                  
                        > [5]                  
                          [1]                  
                        = f(n__g(n__f(n__a())))
        
        
        Following rules are (at-least) weakly oriented:
                      a() =  [3]           
                             [5]           
                          >= [0]           
                             [1]           
                          =  n__a()        
        
              activate(X) =  [1 0] X + [3] 
                             [0 1]     [7] 
                          >= [1 0] X + [0] 
                             [0 1]     [0] 
                          =  X             
        
         activate(n__a()) =  [3]           
                             [8]           
                          >= [3]           
                             [5]           
                          =  a()           
        
        activate(n__f(X)) =  [1 2] X + [3] 
                             [0 1]     [8] 
                          >= [1 2] X + [3] 
                             [0 1]     [1] 
                          =  f(X)          
        
        activate(n__g(X)) =  [1 0] X + [3] 
                             [0 0]     [7] 
                          >= [1 0] X + [3] 
                             [0 0]     [3] 
                          =  g(activate(X))
        
                     f(X) =  [1 2] X + [3] 
                             [0 1]     [1] 
                          >= [1 2] X + [0] 
                             [0 1]     [1] 
                          =  n__f(X)       
        
                     g(X) =  [1 0] X + [0] 
                             [0 0]     [3] 
                          >= [1 0] X + [0] 
                             [0 0]     [0] 
                          =  n__g(X)       
        
* Step 4: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            activate(n__g(X)) -> g(activate(X))
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            f(X) -> n__f(X)
            f(n__f(n__a())) -> f(n__g(n__f(n__a())))
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,activate,f,g,n__a,n__f,n__g}
    + 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) = [2]           
                        [0]           
          p(activate) = [1 2] x1 + [1]
                        [0 1]      [0]
                 p(f) = [1 0] x1 + [1]
                        [0 0]      [0]
                 p(g) = [1 0] x1 + [1]
                        [0 1]      [1]
              p(n__a) = [1]           
                        [0]           
              p(n__f) = [1 0] x1 + [0]
                        [0 0]      [0]
              p(n__g) = [1 0] x1 + [0]
                        [0 1]      [1]
        
        Following rules are strictly oriented:
        activate(n__g(X)) = [1 2] X + [3] 
                            [0 1]     [1] 
                          > [1 2] X + [2] 
                            [0 1]     [1] 
                          = g(activate(X))
        
        
        Following rules are (at-least) weakly oriented:
                      a() =  [2]                  
                             [0]                  
                          >= [1]                  
                             [0]                  
                          =  n__a()               
        
              activate(X) =  [1 2] X + [1]        
                             [0 1]     [0]        
                          >= [1 0] X + [0]        
                             [0 1]     [0]        
                          =  X                    
        
         activate(n__a()) =  [2]                  
                             [0]                  
                          >= [2]                  
                             [0]                  
                          =  a()                  
        
        activate(n__f(X)) =  [1 0] X + [1]        
                             [0 0]     [0]        
                          >= [1 0] X + [1]        
                             [0 0]     [0]        
                          =  f(X)                 
        
                     f(X) =  [1 0] X + [1]        
                             [0 0]     [0]        
                          >= [1 0] X + [0]        
                             [0 0]     [0]        
                          =  n__f(X)              
        
          f(n__f(n__a())) =  [2]                  
                             [0]                  
                          >= [2]                  
                             [0]                  
                          =  f(n__g(n__f(n__a())))
        
                     g(X) =  [1 0] X + [1]        
                             [0 1]     [1]        
                          >= [1 0] X + [0]        
                             [0 1]     [1]        
                          =  n__g(X)              
        
* Step 5: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            activate(n__g(X)) -> g(activate(X))
            f(X) -> n__f(X)
            f(n__f(n__a())) -> f(n__g(n__f(n__a())))
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,activate,f,g,n__a,n__f,n__g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))