WORST_CASE(?,O(n^1))
* Step 1: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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) = [2] x1 + [0] 
                p(cons) = [1] x2 + [4] 
               p(first) = [2] x2 + [13]
                p(from) = [2] x1 + [0] 
            p(n__first) = [1] x2 + [3] 
             p(n__from) = [1] x1 + [1] 
                 p(nil) = [0]          
                   p(s) = [1] x1 + [13]
          
          Following rules are strictly oriented:
           activate(n__from(X)) = [2] X + [2]                    
                                > [2] X + [0]                    
                                = from(X)                        
          
                   first(X1,X2) = [2] X2 + [13]                  
                                > [1] X2 + [3]                   
                                = n__first(X1,X2)                
          
                   first(0(),X) = [2] X + [13]                   
                                > [0]                            
                                = nil()                          
          
          first(s(X),cons(Y,Z)) = [2] Z + [21]                   
                                > [2] Z + [7]                    
                                = 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] X2 + [6]         
                                    >= [2] X2 + [13]        
                                    =  first(X1,X2)         
          
                            from(X) =  [2] X + [0]          
                                    >= [1] X + [18]         
                                    =  cons(X,n__from(s(X)))
          
                            from(X) =  [2] X + [0]          
                                    >= [1] X + [1]          
                                    =  n__from(X)           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Weak TRS:
            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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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) = [2] x1 + [0]
                p(cons) = [1] x2 + [0]
               p(first) = [2] x2 + [0]
                p(from) = [8]         
            p(n__first) = [1] x2 + [0]
             p(n__from) = [4]         
                 p(nil) = [0]         
                   p(s) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          from(X) = [8]                  
                  > [4]                  
                  = cons(X,n__from(s(X)))
          
          from(X) = [8]                  
                  > [4]                  
                  = n__from(X)           
          
          
          Following rules are (at-least) weakly oriented:
                        activate(X) =  [2] X + [0]                    
                                    >= [1] X + [0]                    
                                    =  X                              
          
          activate(n__first(X1,X2)) =  [2] X2 + [0]                   
                                    >= [2] X2 + [0]                   
                                    =  first(X1,X2)                   
          
               activate(n__from(X)) =  [8]                            
                                    >= [8]                            
                                    =  from(X)                        
          
                       first(X1,X2) =  [2] X2 + [0]                   
                                    >= [1] X2 + [0]                   
                                    =  n__first(X1,X2)                
          
                       first(0(),X) =  [2] X + [0]                    
                                    >= [0]                            
                                    =  nil()                          
          
              first(s(X),cons(Y,Z)) =  [2] Z + [0]                    
                                    >= [2] Z + [0]                    
                                    =  cons(Y,n__first(X,activate(Z)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__first(X1,X2)) -> first(X1,X2)
        - Weak TRS:
            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:
        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) = [2] x1 + [1]         
              p(cons) = [1] x2 + [0]         
             p(first) = [2] x1 + [2] x2 + [9]
              p(from) = [6]                  
          p(n__first) = [1] x1 + [1] x2 + [8]
           p(n__from) = [6]                  
               p(nil) = [13]                 
                 p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
                      activate(X) = [2] X + [1]           
                                  > [1] X + [0]           
                                  = X                     
        
        activate(n__first(X1,X2)) = [2] X1 + [2] X2 + [17]
                                  > [2] X1 + [2] X2 + [9] 
                                  = first(X1,X2)          
        
        
        Following rules are (at-least) weakly oriented:
         activate(n__from(X)) =  [13]                           
                              >= [6]                            
                              =  from(X)                        
        
                 first(X1,X2) =  [2] X1 + [2] X2 + [9]          
                              >= [1] X1 + [1] X2 + [8]          
                              =  n__first(X1,X2)                
        
                 first(0(),X) =  [2] X + [13]                   
                              >= [13]                           
                              =  nil()                          
        
        first(s(X),cons(Y,Z)) =  [2] X + [2] Z + [9]            
                              >= [1] X + [2] Z + [9]            
                              =  cons(Y,n__first(X,activate(Z)))
        
                      from(X) =  [6]                            
                              >= [6]                            
                              =  cons(X,n__from(s(X)))          
        
                      from(X) =  [6]                            
                              >= [6]                            
                              =  n__from(X)                     
        
* 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))