* 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:
             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) = [1] x1 + [4]
                p(cons) = [1] x2 + [0]
               p(first) = [1] x2 + [0]
                p(from) = [0]         
            p(n__first) = [1] x2 + [0]
             p(n__from) = [0]         
                 p(nil) = [0]         
                   p(s) = [1] x1 + [0]
          
          Following rules are strictly oriented:
                        activate(X) = [1] X + [4] 
                                    > [1] X + [0] 
                                    = X           
          
          activate(n__first(X1,X2)) = [1] X2 + [4]
                                    > [1] X2 + [0]
                                    = first(X1,X2)
          
               activate(n__from(X)) = [4]         
                                    > [0]         
                                    = from(X)     
          
          
          Following rules are (at-least) weakly oriented:
                   first(X1,X2) =  [1] X2 + [0]                   
                                >= [1] X2 + [0]                   
                                =  n__first(X1,X2)                
          
                   first(0(),X) =  [1] X + [0]                    
                                >= [0]                            
                                =  nil()                          
          
          first(s(X),cons(Y,Z)) =  [1] Z + [0]                    
                                >= [1] Z + [4]                    
                                =  cons(Y,n__first(X,activate(Z)))
          
                        from(X) =  [0]                            
                                >= [0]                            
                                =  cons(X,n__from(s(X)))          
          
                        from(X) =  [0]                            
                                >= [0]                            
                                =  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:
            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(X) -> X
            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:
             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) = [10] x1 + [8]         
                p(cons) = [1] x2 + [0]          
               p(first) = [8] x1 + [10] x2 + [0]
                p(from) = [2]                   
            p(n__first) = [1] x1 + [1] x2 + [1] 
             p(n__from) = [0]                   
                 p(nil) = [0]                   
                   p(s) = [1] x1 + [2]          
          
          Following rules are strictly oriented:
          first(s(X),cons(Y,Z)) = [8] X + [10] Z + [16]          
                                > [1] X + [10] Z + [9]           
                                = cons(Y,n__first(X,activate(Z)))
          
                        from(X) = [2]                            
                                > [0]                            
                                = cons(X,n__from(s(X)))          
          
                        from(X) = [2]                            
                                > [0]                            
                                = n__from(X)                     
          
          
          Following rules are (at-least) weakly oriented:
                        activate(X) =  [10] X + [8]            
                                    >= [1] X + [0]             
                                    =  X                       
          
          activate(n__first(X1,X2)) =  [10] X1 + [10] X2 + [18]
                                    >= [8] X1 + [10] X2 + [0]  
                                    =  first(X1,X2)            
          
               activate(n__from(X)) =  [8]                     
                                    >= [2]                     
                                    =  from(X)                 
          
                       first(X1,X2) =  [8] X1 + [10] X2 + [0]  
                                    >= [1] X1 + [1] X2 + [1]   
                                    =  n__first(X1,X2)         
          
                       first(0(),X) =  [10] X + [0]            
                                    >= [0]                     
                                    =  nil()                   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            first(X1,X2) -> n__first(X1,X2)
            first(0(),X) -> nil()
        - Weak TRS:
            activate(X) -> X
            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)))
            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:
        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) = [2]         
            p(activate) = [2] x1 + [8]
                p(cons) = [1] x2 + [1]
               p(first) = [2] x2 + [7]
                p(from) = [4]         
            p(n__first) = [1] x2 + [0]
             p(n__from) = [0]         
                 p(nil) = [1]         
                   p(s) = [1] x1 + [8]
          
          Following rules are strictly oriented:
          first(X1,X2) = [2] X2 + [7]   
                       > [1] X2 + [0]   
                       = n__first(X1,X2)
          
          first(0(),X) = [2] X + [7]    
                       > [1]            
                       = nil()          
          
          
          Following rules are (at-least) weakly oriented:
                        activate(X) =  [2] X + [8]                    
                                    >= [1] X + [0]                    
                                    =  X                              
          
          activate(n__first(X1,X2)) =  [2] X2 + [8]                   
                                    >= [2] X2 + [7]                   
                                    =  first(X1,X2)                   
          
               activate(n__from(X)) =  [8]                            
                                    >= [4]                            
                                    =  from(X)                        
          
              first(s(X),cons(Y,Z)) =  [2] Z + [9]                    
                                    >= [2] Z + [9]                    
                                    =  cons(Y,n__first(X,activate(Z)))
          
                            from(X) =  [4]                            
                                    >= [1]                            
                                    =  cons(X,n__from(s(X)))          
          
                            from(X) =  [4]                            
                                    >= [0]                            
                                    =  n__from(X)                     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* 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:
             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))