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(),Z) -> 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)
            sel(0(),cons(X,Z)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,first/2,from/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,first,from,sel} 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(),Z) -> 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)
            sel(0(),cons(X,Z)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,first/2,from/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,first,from,sel} 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},
          uargs(sel) = {2}
        
        Following symbols are considered usable:
          {activate,first,from,sel}
        TcT has computed the following interpretation:
                 p(0) = 1      
          p(activate) = x1     
              p(cons) = x1 + x2
             p(first) = 8 + x2 
              p(from) = 4 + x1 
          p(n__first) = 8 + x2 
           p(n__from) = 4 + x1 
               p(nil) = 6      
                 p(s) = 0      
               p(sel) = 1 + x2 
        
        Following rules are strictly oriented:
              first(0(),Z) = 8 + Z    
                           > 6        
                           = nil()    
        
        sel(0(),cons(X,Z)) = 1 + X + Z
                           > X        
                           = X        
        
        
        Following rules are (at-least) weakly oriented:
                      activate(X) =  X                              
                                  >= X                              
                                  =  X                              
        
        activate(n__first(X1,X2)) =  8 + X2                         
                                  >= 8 + X2                         
                                  =  first(X1,X2)                   
        
             activate(n__from(X)) =  4 + X                          
                                  >= 4 + X                          
                                  =  from(X)                        
        
                     first(X1,X2) =  8 + X2                         
                                  >= 8 + X2                         
                                  =  n__first(X1,X2)                
        
            first(s(X),cons(Y,Z)) =  8 + Y + Z                      
                                  >= 8 + Y + Z                      
                                  =  cons(Y,n__first(X,activate(Z)))
        
                          from(X) =  4 + X                          
                                  >= 4 + X                          
                                  =  cons(X,n__from(s(X)))          
        
                          from(X) =  4 + X                          
                                  >= 4 + X                          
                                  =  n__from(X)                     
        
              sel(s(X),cons(Y,Z)) =  1 + Y + Z                      
                                  >= 1 + Z                          
                                  =  sel(X,activate(Z))             
        
* Step 3: 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(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)
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Weak TRS:
            first(0(),Z) -> nil()
            sel(0(),cons(X,Z)) -> X
        - Signature:
            {activate/1,first/2,from/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,first,from,sel} 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},
          uargs(sel) = {2}
        
        Following symbols are considered usable:
          {activate,first,from,sel}
        TcT has computed the following interpretation:
                 p(0) = 8          
          p(activate) = 1 + x1     
              p(cons) = 1 + x1 + x2
             p(first) = 2 + x2     
              p(from) = 1 + x1     
          p(n__first) = 1 + x2     
           p(n__from) = x1         
               p(nil) = 1          
                 p(s) = 0          
               p(sel) = 2*x2       
        
        Following rules are strictly oriented:
         activate(X) = 1 + X          
                     > X              
                     = X              
        
        first(X1,X2) = 2 + X2         
                     > 1 + X2         
                     = n__first(X1,X2)
        
             from(X) = 1 + X          
                     > X              
                     = n__from(X)     
        
        
        Following rules are (at-least) weakly oriented:
        activate(n__first(X1,X2)) =  2 + X2                         
                                  >= 2 + X2                         
                                  =  first(X1,X2)                   
        
             activate(n__from(X)) =  1 + X                          
                                  >= 1 + X                          
                                  =  from(X)                        
        
                     first(0(),Z) =  2 + Z                          
                                  >= 1                              
                                  =  nil()                          
        
            first(s(X),cons(Y,Z)) =  3 + Y + Z                      
                                  >= 3 + Y + Z                      
                                  =  cons(Y,n__first(X,activate(Z)))
        
                          from(X) =  1 + X                          
                                  >= 1 + X                          
                                  =  cons(X,n__from(s(X)))          
        
               sel(0(),cons(X,Z)) =  2 + 2*X + 2*Z                  
                                  >= X                              
                                  =  X                              
        
              sel(s(X),cons(Y,Z)) =  2 + 2*Y + 2*Z                  
                                  >= 2 + 2*Z                        
                                  =  sel(X,activate(Z))             
        
* Step 4: Ara WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(n__from(X)) -> from(X)
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
            from(X) -> cons(X,n__from(s(X)))
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Weak TRS:
            activate(X) -> X
            first(X1,X2) -> n__first(X1,X2)
            first(0(),Z) -> nil()
            from(X) -> n__from(X)
            sel(0(),cons(X,Z)) -> X
        - Signature:
            {activate/1,first/2,from/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,first,from,sel} and constructors {0,cons
            ,n__first,n__from,nil,s}
    + Applied Processor:
        Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 1, araTimeout = 5, araRuleShifting = Nothing}
    + Details:
        Signatures used:
        ----------------
          0 :: [] -(0)-> "A"(8)
          0 :: [] -(0)-> "A"(10)
          activate :: ["A"(8)] -(5)-> "A"(8)
          cons :: ["A"(0) x "A"(8)] -(0)-> "A"(8)
          first :: ["A"(8) x "A"(8)] -(3)-> "A"(8)
          from :: ["A"(0)] -(4)-> "A"(8)
          n__first :: ["A"(8) x "A"(8)] -(0)-> "A"(8)
          n__from :: ["A"(0)] -(0)-> "A"(8)
          n__from :: ["A"(0)] -(0)-> "A"(14)
          nil :: [] -(0)-> "A"(14)
          s :: ["A"(8)] -(8)-> "A"(8)
          s :: ["A"(10)] -(10)-> "A"(10)
          s :: ["A"(0)] -(0)-> "A"(0)
          sel :: ["A"(10) x "A"(8)] -(12)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
          "0_A" :: [] -(0)-> "A"(1)
          "cons_A" :: ["A"(0) x "A"(1)] -(0)-> "A"(1)
          "n__first_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1)
          "n__from_A" :: ["A"(0)] -(0)-> "A"(1)
          "nil_A" :: [] -(0)-> "A"(1)
          "s_A" :: ["A"(1)] -(1)-> "A"(1)
        

WORST_CASE(?,O(n^1))