WORST_CASE(?,O(n^1))
* Step 1: Sum.  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:
            innermost 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 2: NaturalPI.  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:
            innermost runtime complexity wrt. defined symbols {activate,first,from} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(cons) = {2},
          uargs(n__first) = {2}
        
        Following symbols are considered usable:
          {activate,first,from}
        TcT has computed the following interpretation:
                 p(0) = 4       
          p(activate) = 8*x1    
              p(cons) = x2      
             p(first) = 2 + 8*x2
              p(from) = 0       
          p(n__first) = 2 + x2  
           p(n__from) = 0       
               p(nil) = 2       
                 p(s) = 2       
        
        Following rules are strictly oriented:
        activate(n__first(X1,X2)) = 16 + 8*X2   
                                  > 2 + 8*X2    
                                  = first(X1,X2)
        
        
        Following rules are (at-least) weakly oriented:
                  activate(X) =  8*X                            
                              >= X                              
                              =  X                              
        
         activate(n__from(X)) =  0                              
                              >= 0                              
                              =  from(X)                        
        
                 first(X1,X2) =  2 + 8*X2                       
                              >= 2 + X2                         
                              =  n__first(X1,X2)                
        
                 first(0(),X) =  2 + 8*X                        
                              >= 2                              
                              =  nil()                          
        
        first(s(X),cons(Y,Z)) =  2 + 8*Z                        
                              >= 2 + 8*Z                        
                              =  cons(Y,n__first(X,activate(Z)))
        
                      from(X) =  0                              
                              >= 0                              
                              =  cons(X,n__from(s(X)))          
        
                      from(X) =  0                              
                              >= 0                              
                              =  n__from(X)                     
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            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)
        - Weak TRS:
            activate(n__first(X1,X2)) -> first(X1,X2)
        - Signature:
            {activate/1,first/2,from/1} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
            innermost 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:
          {activate,first,from}
        TcT has computed the following interpretation:
                 p(0) = [2]                  
          p(activate) = [1] x1 + [6]         
              p(cons) = [1] x2 + [0]         
             p(first) = [1] x1 + [1] x2 + [5]
              p(from) = [6]                  
          p(n__first) = [1] x1 + [1] x2 + [0]
           p(n__from) = [1]                  
               p(nil) = [0]                  
                 p(s) = [1] x1 + [2]         
        
        Following rules are strictly oriented:
                  activate(X) = [1] X + [6]                    
                              > [1] X + [0]                    
                              = X                              
        
         activate(n__from(X)) = [7]                            
                              > [6]                            
                              = from(X)                        
        
                 first(X1,X2) = [1] X1 + [1] X2 + [5]          
                              > [1] X1 + [1] X2 + [0]          
                              = n__first(X1,X2)                
        
                 first(0(),X) = [1] X + [7]                    
                              > [0]                            
                              = nil()                          
        
        first(s(X),cons(Y,Z)) = [1] X + [1] Z + [7]            
                              > [1] X + [1] Z + [6]            
                              = cons(Y,n__first(X,activate(Z)))
        
                      from(X) = [6]                            
                              > [1]                            
                              = cons(X,n__from(s(X)))          
        
                      from(X) = [6]                            
                              > [1]                            
                              = n__from(X)                     
        
        
        Following rules are (at-least) weakly oriented:
        activate(n__first(X1,X2)) =  [1] X1 + [1] X2 + [6]
                                  >= [1] X1 + [1] X2 + [5]
                                  =  first(X1,X2)         
        
* Step 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:
            innermost 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(?,O(n^1))