/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__from(X)) -> from(X)
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil
            ,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__from(X)) -> from(X)
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil
            ,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__from(X)) -> from(X)
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil
            ,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          activate(v){v -> n__first(s(x),cons(z,v))} =
            activate(n__first(s(x),cons(z,v))) ->^+ cons(z,n__first(x,activate(v)))
              = C[activate(v) = activate(v){}]

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__from(X)) -> from(X)
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil
            ,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(cons) = {2},
          uargs(n__first) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                 p(0) = [0]         
          p(activate) = [4] x1 + [9]
              p(cons) = [1] x2 + [2]
             p(first) = [4] x2 + [6]
              p(from) = [13]        
          p(n__first) = [1] x2 + [3]
           p(n__from) = [1]         
               p(nil) = [1]         
                 p(s) = [2]         
        
        Following rules are strictly oriented:
                      activate(X) = [4] X + [9]          
                                  > [1] X + [0]          
                                  = X                    
        
        activate(n__first(X1,X2)) = [4] X2 + [21]        
                                  > [4] X2 + [6]         
                                  = first(X1,X2)         
        
                     first(X1,X2) = [4] X2 + [6]         
                                  > [1] X2 + [3]         
                                  = n__first(X1,X2)      
        
                     first(0(),X) = [4] X + [6]          
                                  > [1]                  
                                  = nil()                
        
                          from(X) = [13]                 
                                  > [3]                  
                                  = cons(X,n__from(s(X)))
        
                          from(X) = [13]                 
                                  > [1]                  
                                  = n__from(X)           
        
        
        Following rules are (at-least) weakly oriented:
         activate(n__from(X)) =  [13]                           
                              >= [13]                           
                              =  from(X)                        
        
        first(s(X),cons(Y,Z)) =  [4] Z + [14]                   
                              >= [4] Z + [14]                   
                              =  cons(Y,n__first(X,activate(Z)))
        
** Step 1.b:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(n__from(X)) -> from(X)
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
        - Weak TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil
            ,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(cons) = {2},
          uargs(n__first) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                 p(0) = [0]                  
          p(activate) = [8] x1 + [0]         
              p(cons) = [1] x2 + [0]         
             p(first) = [1] x1 + [8] x2 + [0]
              p(from) = [1] x1 + [3]         
          p(n__first) = [1] x1 + [1] x2 + [0]
           p(n__from) = [1] x1 + [3]         
               p(nil) = [0]                  
                 p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        activate(n__from(X)) = [8] X + [24]
                             > [1] X + [3] 
                             = from(X)     
        
        
        Following rules are (at-least) weakly oriented:
                      activate(X) =  [8] X + [0]                    
                                  >= [1] X + [0]                    
                                  =  X                              
        
        activate(n__first(X1,X2)) =  [8] X1 + [8] X2 + [0]          
                                  >= [1] X1 + [8] X2 + [0]          
                                  =  first(X1,X2)                   
        
                     first(X1,X2) =  [1] X1 + [8] X2 + [0]          
                                  >= [1] X1 + [1] X2 + [0]          
                                  =  n__first(X1,X2)                
        
                     first(0(),X) =  [8] X + [0]                    
                                  >= [0]                            
                                  =  nil()                          
        
            first(s(X),cons(Y,Z)) =  [1] X + [8] Z + [0]            
                                  >= [1] X + [8] Z + [0]            
                                  =  cons(Y,n__first(X,activate(Z)))
        
                          from(X) =  [1] X + [3]                    
                                  >= [1] X + [3]                    
                                  =  cons(X,n__from(s(X)))          
        
                          from(X) =  [1] X + [3]                    
                                  >= [1] X + [3]                    
                                  =  n__from(X)                     
        
** Step 1.b:3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
        - Weak TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__from(X)) -> from(X)
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil
            ,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(cons) = {2},
          uargs(n__first) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                 p(0) = [5]                  
          p(activate) = [2] x1 + [0]         
              p(cons) = [1] x2 + [0]         
             p(first) = [2] x1 + [2] x2 + [4]
              p(from) = [0]                  
          p(n__first) = [1] x1 + [1] x2 + [2]
           p(n__from) = [0]                  
               p(nil) = [1]                  
                 p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        first(s(X),cons(Y,Z)) = [2] X + [2] Z + [4]            
                              > [1] X + [2] Z + [2]            
                              = cons(Y,n__first(X,activate(Z)))
        
        
        Following rules are (at-least) weakly oriented:
                      activate(X) =  [2] X + [0]          
                                  >= [1] X + [0]          
                                  =  X                    
        
        activate(n__first(X1,X2)) =  [2] X1 + [2] X2 + [4]
                                  >= [2] X1 + [2] X2 + [4]
                                  =  first(X1,X2)         
        
             activate(n__from(X)) =  [0]                  
                                  >= [0]                  
                                  =  from(X)              
        
                     first(X1,X2) =  [2] X1 + [2] X2 + [4]
                                  >= [1] X1 + [1] X2 + [2]
                                  =  n__first(X1,X2)      
        
                     first(0(),X) =  [2] X + [14]         
                                  >= [1]                  
                                  =  nil()                
        
                          from(X) =  [0]                  
                                  >= [0]                  
                                  =  cons(X,n__from(s(X)))
        
                          from(X) =  [0]                  
                                  >= [0]                  
                                  =  n__from(X)           
        
** Step 1.b:4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__from(X)) -> from(X)
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first,n__from,nil
            ,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))