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) = 0   
          p(activate) = 8*x1
              p(cons) = x2  
             p(first) = 8*x2
              p(from) = 2   
          p(n__first) = x2  
           p(n__from) = 2   
               p(nil) = 0   
                 p(s) = 0   
        
        Following rules are strictly oriented:
        activate(n__from(X)) = 16     
                             > 2      
                             = from(X)
        
        
        Following rules are (at-least) weakly oriented:
                      activate(X) =  8*X                            
                                  >= X                              
                                  =  X                              
        
        activate(n__first(X1,X2)) =  8*X2                           
                                  >= 8*X2                           
                                  =  first(X1,X2)                   
        
                     first(X1,X2) =  8*X2                           
                                  >= X2                             
                                  =  n__first(X1,X2)                
        
                     first(0(),X) =  8*X                            
                                  >= 0                              
                                  =  nil()                          
        
            first(s(X),cons(Y,Z)) =  8*Z                            
                                  >= 8*Z                            
                                  =  cons(Y,n__first(X,activate(Z)))
        
                          from(X) =  2                              
                                  >= 2                              
                                  =  cons(X,n__from(s(X)))          
        
                          from(X) =  2                              
                                  >= 2                              
                                  =  n__from(X)                     
        
* Step 3: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            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__from(X)) -> 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) = 0       
          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) = 1       
        
        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 4: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> 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)
            activate(n__from(X)) -> 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) = 8       
          p(activate) = 8*x1    
              p(cons) = x2      
             p(first) = 8 + 8*x2
              p(from) = 0       
          p(n__first) = 1 + x2  
           p(n__from) = 0       
               p(nil) = 8       
                 p(s) = 0       
        
        Following rules are strictly oriented:
                 first(X1,X2) = 8 + 8*X2                       
                              > 1 + X2                         
                              = n__first(X1,X2)                
        
        first(s(X),cons(Y,Z)) = 8 + 8*Z                        
                              > 1 + 8*Z                        
                              = cons(Y,n__first(X,activate(Z)))
        
        
        Following rules are (at-least) weakly oriented:
                      activate(X) =  8*X                  
                                  >= X                    
                                  =  X                    
        
        activate(n__first(X1,X2)) =  8 + 8*X2             
                                  >= 8 + 8*X2             
                                  =  first(X1,X2)         
        
             activate(n__from(X)) =  0                    
                                  >= 0                    
                                  =  from(X)              
        
                     first(0(),X) =  8 + 8*X              
                                  >= 8                    
                                  =  nil()                
        
                          from(X) =  0                    
                                  >= 0                    
                                  =  cons(X,n__from(s(X)))
        
                          from(X) =  0                    
                                  >= 0                    
                                  =  n__from(X)           
        
* Step 5: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            first(0(),X) -> nil()
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Weak TRS:
            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)))
        - 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) = 3               
          p(activate) = 14 + 8*x1       
              p(cons) = x2              
             p(first) = 10 + 3*x1 + 8*x2
              p(from) = 0               
          p(n__first) = 2 + x1 + x2     
           p(n__from) = 0               
               p(nil) = 0               
                 p(s) = 2 + x1          
        
        Following rules are strictly oriented:
         activate(X) = 14 + 8*X
                     > X       
                     = X       
        
        first(0(),X) = 19 + 8*X
                     > 0       
                     = nil()   
        
        
        Following rules are (at-least) weakly oriented:
        activate(n__first(X1,X2)) =  30 + 8*X1 + 8*X2               
                                  >= 10 + 3*X1 + 8*X2               
                                  =  first(X1,X2)                   
        
             activate(n__from(X)) =  14                             
                                  >= 0                              
                                  =  from(X)                        
        
                     first(X1,X2) =  10 + 3*X1 + 8*X2               
                                  >= 2 + X1 + X2                    
                                  =  n__first(X1,X2)                
        
            first(s(X),cons(Y,Z)) =  16 + 3*X + 8*Z                 
                                  >= 16 + X + 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 6: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - 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)))
        - 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) = 6        
          p(activate) = 10 + 8*x1
              p(cons) = x2       
             p(first) = x1 + 8*x2
              p(from) = 10       
          p(n__first) = x1 + x2  
           p(n__from) = 0        
               p(nil) = 6        
                 p(s) = 10 + x1  
        
        Following rules are strictly oriented:
        from(X) = 10                   
                > 0                    
                = cons(X,n__from(s(X)))
        
        from(X) = 10                   
                > 0                    
                = n__from(X)           
        
        
        Following rules are (at-least) weakly oriented:
                      activate(X) =  10 + 8*X                       
                                  >= X                              
                                  =  X                              
        
        activate(n__first(X1,X2)) =  10 + 8*X1 + 8*X2               
                                  >= X1 + 8*X2                      
                                  =  first(X1,X2)                   
        
             activate(n__from(X)) =  10                             
                                  >= 10                             
                                  =  from(X)                        
        
                     first(X1,X2) =  X1 + 8*X2                      
                                  >= X1 + X2                        
                                  =  n__first(X1,X2)                
        
                     first(0(),X) =  6 + 8*X                        
                                  >= 6                              
                                  =  nil()                          
        
            first(s(X),cons(Y,Z)) =  10 + X + 8*Z                   
                                  >= 10 + X + 8*Z                   
                                  =  cons(Y,n__first(X,activate(Z)))
        
* Step 7: 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))