/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:
            __(X,nil()) -> X
            __(__(X,Y),Z) -> __(X,__(Y,Z))
            __(nil(),X) -> X
            activate(X) -> X
            and(tt(),X) -> activate(X)
            isNePal(__(I,__(P,I))) -> tt()
        - Signature:
            {__/2,activate/1,and/2,isNePal/1} / {nil/0,tt/0}
        - Obligation:
             derivational complexity wrt. signature {__,activate,and,isNePal,nil,tt}
    + 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(__) = [1] x1 + [1] x2 + [0]
          p(activate) = [1] x1 + [0]         
               p(and) = [1] x1 + [1] x2 + [0]
           p(isNePal) = [1] x1 + [0]         
               p(nil) = [2]                  
                p(tt) = [0]                  
        
        Following rules are strictly oriented:
        __(X,nil()) = [1] X + [2]
                    > [1] X + [0]
                    = X          
        
        __(nil(),X) = [1] X + [2]
                    > [1] X + [0]
                    = X          
        
        
        Following rules are (at-least) weakly oriented:
                 __(__(X,Y),Z) =  [1] X + [1] Y + [1] Z + [0]
                               >= [1] X + [1] Y + [1] Z + [0]
                               =  __(X,__(Y,Z))              
        
                   activate(X) =  [1] X + [0]                
                               >= [1] X + [0]                
                               =  X                          
        
                   and(tt(),X) =  [1] X + [0]                
                               >= [1] X + [0]                
                               =  activate(X)                
        
        isNePal(__(I,__(P,I))) =  [2] I + [1] P + [0]        
                               >= [0]                        
                               =  tt()                       
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            __(__(X,Y),Z) -> __(X,__(Y,Z))
            activate(X) -> X
            and(tt(),X) -> activate(X)
            isNePal(__(I,__(P,I))) -> tt()
        - Weak TRS:
            __(X,nil()) -> X
            __(nil(),X) -> X
        - Signature:
            {__/2,activate/1,and/2,isNePal/1} / {nil/0,tt/0}
        - Obligation:
             derivational complexity wrt. signature {__,activate,and,isNePal,nil,tt}
    + 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(__) = [1] x1 + [1] x2 + [2] 
          p(activate) = [1] x1 + [8]          
               p(and) = [1] x1 + [1] x2 + [10]
           p(isNePal) = [1] x1 + [8]          
               p(nil) = [3]                   
                p(tt) = [6]                   
        
        Following rules are strictly oriented:
                   activate(X) = [1] X + [8]         
                               > [1] X + [0]         
                               = X                   
        
                   and(tt(),X) = [1] X + [16]        
                               > [1] X + [8]         
                               = activate(X)         
        
        isNePal(__(I,__(P,I))) = [2] I + [1] P + [12]
                               > [6]                 
                               = tt()                
        
        
        Following rules are (at-least) weakly oriented:
          __(X,nil()) =  [1] X + [5]                
                      >= [1] X + [0]                
                      =  X                          
        
        __(__(X,Y),Z) =  [1] X + [1] Y + [1] Z + [4]
                      >= [1] X + [1] Y + [1] Z + [4]
                      =  __(X,__(Y,Z))              
        
          __(nil(),X) =  [1] X + [5]                
                      >= [1] X + [0]                
                      =  X                          
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            __(__(X,Y),Z) -> __(X,__(Y,Z))
        - Weak TRS:
            __(X,nil()) -> X
            __(nil(),X) -> X
            activate(X) -> X
            and(tt(),X) -> activate(X)
            isNePal(__(I,__(P,I))) -> tt()
        - Signature:
            {__/2,activate/1,and/2,isNePal/1} / {nil/0,tt/0}
        - Obligation:
             derivational complexity wrt. signature {__,activate,and,isNePal,nil,tt}
    + 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(__) = [1 2] x1 + [1 0] x2 + [0]
                        [0 1]      [0 1]      [4]
          p(activate) = [1 2] x1 + [0]           
                        [0 1]      [0]           
               p(and) = [1 3] x1 + [1 4] x2 + [2]
                        [0 0]      [0 1]      [0]
           p(isNePal) = [1 0] x1 + [2]           
                        [0 1]      [2]           
               p(nil) = [4]                      
                        [5]                      
                p(tt) = [1]                      
                        [4]                      
        
        Following rules are strictly oriented:
        __(__(X,Y),Z) = [1 4] X + [1 2] Y + [1 0] Z + [8]
                        [0 1]     [0 1]     [0 1]     [8]
                      > [1 2] X + [1 2] Y + [1 0] Z + [0]
                        [0 1]     [0 1]     [0 1]     [8]
                      = __(X,__(Y,Z))                    
        
        
        Following rules are (at-least) weakly oriented:
                   __(X,nil()) =  [1 2] X + [4]           
                                  [0 1]     [9]           
                               >= [1 0] X + [0]           
                                  [0 1]     [0]           
                               =  X                       
        
                   __(nil(),X) =  [1 0] X + [14]          
                                  [0 1]     [9]           
                               >= [1 0] X + [0]           
                                  [0 1]     [0]           
                               =  X                       
        
                   activate(X) =  [1 2] X + [0]           
                                  [0 1]     [0]           
                               >= [1 0] X + [0]           
                                  [0 1]     [0]           
                               =  X                       
        
                   and(tt(),X) =  [1 4] X + [15]          
                                  [0 1]     [0]           
                               >= [1 2] X + [0]           
                                  [0 1]     [0]           
                               =  activate(X)             
        
        isNePal(__(I,__(P,I))) =  [2 2] I + [1 2] P + [2] 
                                  [0 2]     [0 1]     [10]
                               >= [1]                     
                                  [4]                     
                               =  tt()                    
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            __(X,nil()) -> X
            __(__(X,Y),Z) -> __(X,__(Y,Z))
            __(nil(),X) -> X
            activate(X) -> X
            and(tt(),X) -> activate(X)
            isNePal(__(I,__(P,I))) -> tt()
        - Signature:
            {__/2,activate/1,and/2,isNePal/1} / {nil/0,tt/0}
        - Obligation:
             derivational complexity wrt. signature {__,activate,and,isNePal,nil,tt}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))