* Step 1: DependencyPairs 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:
             runtime complexity wrt. defined symbols {activate,first,from,sel} and constructors {0,cons,n__first,n__from
            ,nil,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          activate#(X) -> c_1(X)
          activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          activate#(n__from(X)) -> c_3(from#(X))
          first#(X1,X2) -> c_4(X1,X2)
          first#(0(),Z) -> c_5()
          first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
          from#(X) -> c_7(X,X)
          from#(X) -> c_8(X)
          sel#(0(),cons(X,Z)) -> c_9(X)
          sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1(X)
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4(X1,X2)
            first#(0(),Z) -> c_5()
            first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
            from#(X) -> c_7(X,X)
            from#(X) -> c_8(X)
            sel#(0(),cons(X,Z)) -> c_9(X)
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
        - 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,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
            ,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          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)
          activate#(X) -> c_1(X)
          activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          activate#(n__from(X)) -> c_3(from#(X))
          first#(X1,X2) -> c_4(X1,X2)
          first#(0(),Z) -> c_5()
          first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
          from#(X) -> c_7(X,X)
          from#(X) -> c_8(X)
          sel#(0(),cons(X,Z)) -> c_9(X)
          sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1(X)
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4(X1,X2)
            first#(0(),Z) -> c_5()
            first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
            from#(X) -> c_7(X,X)
            from#(X) -> c_8(X)
            sel#(0(),cons(X,Z)) -> c_9(X)
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
        - 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)
        - Signature:
            {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
            ,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {5}
        by application of
          Pre({5}) = {1,2,4,6,7,8,9}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1(X)
          2: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
          3: activate#(n__from(X)) -> c_3(from#(X))
          4: first#(X1,X2) -> c_4(X1,X2)
          5: first#(0(),Z) -> c_5()
          6: first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
          7: from#(X) -> c_7(X,X)
          8: from#(X) -> c_8(X)
          9: sel#(0(),cons(X,Z)) -> c_9(X)
          10: sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
* Step 4: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1(X)
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4(X1,X2)
            first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
            from#(X) -> c_7(X,X)
            from#(X) -> c_8(X)
            sel#(0(),cons(X,Z)) -> c_9(X)
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
        - 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)
        - Weak DPs:
            first#(0(),Z) -> c_5()
        - Signature:
            {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
            ,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, 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(activate) = {1},
            uargs(cons) = {2},
            uargs(first) = {2},
            uargs(n__first) = {2},
            uargs(sel#) = {2},
            uargs(c_2) = {1},
            uargs(c_3) = {1},
            uargs(c_6) = {3},
            uargs(c_10) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]                  
             p(activate) = [1] x1 + [0]         
                 p(cons) = [1] x2 + [3]         
                p(first) = [1] x1 + [1] x2 + [0]
                 p(from) = [1] x1 + [0]         
             p(n__first) = [1] x1 + [1] x2 + [7]
              p(n__from) = [1] x1 + [0]         
                  p(nil) = [0]                  
                    p(s) = [1] x1 + [0]         
                  p(sel) = [0]                  
            p(activate#) = [0]                  
               p(first#) = [0]                  
                p(from#) = [0]                  
                 p(sel#) = [1] x2 + [0]         
                  p(c_1) = [0]                  
                  p(c_2) = [1] x1 + [0]         
                  p(c_3) = [1] x1 + [0]         
                  p(c_4) = [0]                  
                  p(c_5) = [0]                  
                  p(c_6) = [1] x3 + [0]         
                  p(c_7) = [0]                  
                  p(c_8) = [0]                  
                  p(c_9) = [0]                  
                 p(c_10) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
                sel#(0(),cons(X,Z)) = [1] Z + [3]              
                                    > [0]                      
                                    = c_9(X)                   
          
               sel#(s(X),cons(Y,Z)) = [1] Z + [3]              
                                    > [1] Z + [0]              
                                    = c_10(sel#(X,activate(Z)))
          
          activate(n__first(X1,X2)) = [1] X1 + [1] X2 + [7]    
                                    > [1] X1 + [1] X2 + [0]    
                                    = first(X1,X2)             
          
          
          Following rules are (at-least) weakly oriented:
                        activate#(X) =  [0]                            
                                     >= [0]                            
                                     =  c_1(X)                         
          
          activate#(n__first(X1,X2)) =  [0]                            
                                     >= [0]                            
                                     =  c_2(first#(X1,X2))             
          
               activate#(n__from(X)) =  [0]                            
                                     >= [0]                            
                                     =  c_3(from#(X))                  
          
                       first#(X1,X2) =  [0]                            
                                     >= [0]                            
                                     =  c_4(X1,X2)                     
          
                       first#(0(),Z) =  [0]                            
                                     >= [0]                            
                                     =  c_5()                          
          
              first#(s(X),cons(Y,Z)) =  [0]                            
                                     >= [0]                            
                                     =  c_6(Y,X,activate#(Z))          
          
                            from#(X) =  [0]                            
                                     >= [0]                            
                                     =  c_7(X,X)                       
          
                            from#(X) =  [0]                            
                                     >= [0]                            
                                     =  c_8(X)                         
          
                         activate(X) =  [1] X + [0]                    
                                     >= [1] X + [0]                    
                                     =  X                              
          
                activate(n__from(X)) =  [1] X + [0]                    
                                     >= [1] X + [0]                    
                                     =  from(X)                        
          
                        first(X1,X2) =  [1] X1 + [1] X2 + [0]          
                                     >= [1] X1 + [1] X2 + [7]          
                                     =  n__first(X1,X2)                
          
                        first(0(),Z) =  [1] Z + [0]                    
                                     >= [0]                            
                                     =  nil()                          
          
               first(s(X),cons(Y,Z)) =  [1] X + [1] Z + [3]            
                                     >= [1] X + [1] Z + [10]           
                                     =  cons(Y,n__first(X,activate(Z)))
          
                             from(X) =  [1] X + [0]                    
                                     >= [1] X + [3]                    
                                     =  cons(X,n__from(s(X)))          
          
                             from(X) =  [1] X + [0]                    
                                     >= [1] X + [0]                    
                                     =  n__from(X)                     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1(X)
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4(X1,X2)
            first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
            from#(X) -> c_7(X,X)
            from#(X) -> c_8(X)
        - Strict TRS:
            activate(X) -> X
            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)
        - Weak DPs:
            first#(0(),Z) -> c_5()
            sel#(0(),cons(X,Z)) -> c_9(X)
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
        - Weak TRS:
            activate(n__first(X1,X2)) -> first(X1,X2)
        - Signature:
            {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
            ,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, 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(activate) = {1},
            uargs(cons) = {2},
            uargs(first) = {2},
            uargs(n__first) = {2},
            uargs(sel#) = {2},
            uargs(c_2) = {1},
            uargs(c_3) = {1},
            uargs(c_6) = {3},
            uargs(c_10) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]                  
             p(activate) = [1] x1 + [0]         
                 p(cons) = [1] x2 + [0]         
                p(first) = [1] x1 + [1] x2 + [3]
                 p(from) = [1] x1 + [0]         
             p(n__first) = [1] x1 + [1] x2 + [3]
              p(n__from) = [1] x1 + [0]         
                  p(nil) = [0]                  
                    p(s) = [1] x1 + [0]         
                  p(sel) = [0]                  
            p(activate#) = [0]                  
               p(first#) = [0]                  
                p(from#) = [1]                  
                 p(sel#) = [1] x2 + [0]         
                  p(c_1) = [0]                  
                  p(c_2) = [1] x1 + [0]         
                  p(c_3) = [1] x1 + [0]         
                  p(c_4) = [0]                  
                  p(c_5) = [0]                  
                  p(c_6) = [1] x3 + [0]         
                  p(c_7) = [0]                  
                  p(c_8) = [0]                  
                  p(c_9) = [0]                  
                 p(c_10) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
              from#(X) = [1]        
                       > [0]        
                       = c_7(X,X)   
          
              from#(X) = [1]        
                       > [0]        
                       = c_8(X)     
          
          first(0(),Z) = [1] Z + [3]
                       > [0]        
                       = nil()      
          
          
          Following rules are (at-least) weakly oriented:
                        activate#(X) =  [0]                            
                                     >= [0]                            
                                     =  c_1(X)                         
          
          activate#(n__first(X1,X2)) =  [0]                            
                                     >= [0]                            
                                     =  c_2(first#(X1,X2))             
          
               activate#(n__from(X)) =  [0]                            
                                     >= [1]                            
                                     =  c_3(from#(X))                  
          
                       first#(X1,X2) =  [0]                            
                                     >= [0]                            
                                     =  c_4(X1,X2)                     
          
                       first#(0(),Z) =  [0]                            
                                     >= [0]                            
                                     =  c_5()                          
          
              first#(s(X),cons(Y,Z)) =  [0]                            
                                     >= [0]                            
                                     =  c_6(Y,X,activate#(Z))          
          
                 sel#(0(),cons(X,Z)) =  [1] Z + [0]                    
                                     >= [0]                            
                                     =  c_9(X)                         
          
                sel#(s(X),cons(Y,Z)) =  [1] Z + [0]                    
                                     >= [1] Z + [0]                    
                                     =  c_10(sel#(X,activate(Z)))      
          
                         activate(X) =  [1] X + [0]                    
                                     >= [1] X + [0]                    
                                     =  X                              
          
           activate(n__first(X1,X2)) =  [1] X1 + [1] X2 + [3]          
                                     >= [1] X1 + [1] X2 + [3]          
                                     =  first(X1,X2)                   
          
                activate(n__from(X)) =  [1] X + [0]                    
                                     >= [1] X + [0]                    
                                     =  from(X)                        
          
                        first(X1,X2) =  [1] X1 + [1] X2 + [3]          
                                     >= [1] X1 + [1] X2 + [3]          
                                     =  n__first(X1,X2)                
          
               first(s(X),cons(Y,Z)) =  [1] X + [1] Z + [3]            
                                     >= [1] X + [1] Z + [3]            
                                     =  cons(Y,n__first(X,activate(Z)))
          
                             from(X) =  [1] X + [0]                    
                                     >= [1] X + [0]                    
                                     =  cons(X,n__from(s(X)))          
          
                             from(X) =  [1] X + [0]                    
                                     >= [1] X + [0]                    
                                     =  n__from(X)                     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1(X)
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4(X1,X2)
            first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
        - Strict TRS:
            activate(X) -> X
            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)
        - Weak DPs:
            first#(0(),Z) -> c_5()
            from#(X) -> c_7(X,X)
            from#(X) -> c_8(X)
            sel#(0(),cons(X,Z)) -> c_9(X)
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
        - Weak TRS:
            activate(n__first(X1,X2)) -> first(X1,X2)
            first(0(),Z) -> nil()
        - Signature:
            {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
            ,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, 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(activate) = {1},
            uargs(cons) = {2},
            uargs(first) = {2},
            uargs(n__first) = {2},
            uargs(sel#) = {2},
            uargs(c_2) = {1},
            uargs(c_3) = {1},
            uargs(c_6) = {3},
            uargs(c_10) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]                  
             p(activate) = [1] x1 + [0]         
                 p(cons) = [1] x2 + [0]         
                p(first) = [1] x1 + [1] x2 + [0]
                 p(from) = [0]                  
             p(n__first) = [1] x1 + [1] x2 + [0]
              p(n__from) = [0]                  
                  p(nil) = [0]                  
                    p(s) = [1] x1 + [1]         
                  p(sel) = [0]                  
            p(activate#) = [1] x1 + [0]         
               p(first#) = [1] x1 + [1] x2 + [0]
                p(from#) = [0]                  
                 p(sel#) = [1] x2 + [0]         
                  p(c_1) = [1] x1 + [0]         
                  p(c_2) = [1] x1 + [0]         
                  p(c_3) = [1] x1 + [0]         
                  p(c_4) = [1] x1 + [1] x2 + [0]
                  p(c_5) = [0]                  
                  p(c_6) = [1] x2 + [1] x3 + [0]
                  p(c_7) = [0]                  
                  p(c_8) = [0]                  
                  p(c_9) = [0]                  
                 p(c_10) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          first#(s(X),cons(Y,Z)) = [1] X + [1] Z + [1]            
                                 > [1] X + [1] Z + [0]            
                                 = c_6(Y,X,activate#(Z))          
          
           first(s(X),cons(Y,Z)) = [1] X + [1] Z + [1]            
                                 > [1] X + [1] Z + [0]            
                                 = cons(Y,n__first(X,activate(Z)))
          
          
          Following rules are (at-least) weakly oriented:
                        activate#(X) =  [1] X + [0]              
                                     >= [1] X + [0]              
                                     =  c_1(X)                   
          
          activate#(n__first(X1,X2)) =  [1] X1 + [1] X2 + [0]    
                                     >= [1] X1 + [1] X2 + [0]    
                                     =  c_2(first#(X1,X2))       
          
               activate#(n__from(X)) =  [0]                      
                                     >= [0]                      
                                     =  c_3(from#(X))            
          
                       first#(X1,X2) =  [1] X1 + [1] X2 + [0]    
                                     >= [1] X1 + [1] X2 + [0]    
                                     =  c_4(X1,X2)               
          
                       first#(0(),Z) =  [1] Z + [0]              
                                     >= [0]                      
                                     =  c_5()                    
          
                            from#(X) =  [0]                      
                                     >= [0]                      
                                     =  c_7(X,X)                 
          
                            from#(X) =  [0]                      
                                     >= [0]                      
                                     =  c_8(X)                   
          
                 sel#(0(),cons(X,Z)) =  [1] Z + [0]              
                                     >= [0]                      
                                     =  c_9(X)                   
          
                sel#(s(X),cons(Y,Z)) =  [1] Z + [0]              
                                     >= [1] Z + [0]              
                                     =  c_10(sel#(X,activate(Z)))
          
                         activate(X) =  [1] X + [0]              
                                     >= [1] X + [0]              
                                     =  X                        
          
           activate(n__first(X1,X2)) =  [1] X1 + [1] X2 + [0]    
                                     >= [1] X1 + [1] X2 + [0]    
                                     =  first(X1,X2)             
          
                activate(n__from(X)) =  [0]                      
                                     >= [0]                      
                                     =  from(X)                  
          
                        first(X1,X2) =  [1] X1 + [1] X2 + [0]    
                                     >= [1] X1 + [1] X2 + [0]    
                                     =  n__first(X1,X2)          
          
                        first(0(),Z) =  [1] Z + [0]              
                                     >= [0]                      
                                     =  nil()                    
          
                             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 7: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1(X)
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4(X1,X2)
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            first(X1,X2) -> n__first(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Weak DPs:
            first#(0(),Z) -> c_5()
            first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
            from#(X) -> c_7(X,X)
            from#(X) -> c_8(X)
            sel#(0(),cons(X,Z)) -> c_9(X)
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
        - Weak TRS:
            activate(n__first(X1,X2)) -> first(X1,X2)
            first(0(),Z) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
        - Signature:
            {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
            ,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, 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(activate) = {1},
            uargs(cons) = {2},
            uargs(first) = {2},
            uargs(n__first) = {2},
            uargs(sel#) = {2},
            uargs(c_2) = {1},
            uargs(c_3) = {1},
            uargs(c_6) = {3},
            uargs(c_10) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]                  
             p(activate) = [1] x1 + [1]         
                 p(cons) = [1] x2 + [3]         
                p(first) = [1] x2 + [2]         
                 p(from) = [1] x1 + [0]         
             p(n__first) = [1] x2 + [1]         
              p(n__from) = [1] x1 + [7]         
                  p(nil) = [1]                  
                    p(s) = [1] x1 + [1]         
                  p(sel) = [2] x1 + [2]         
            p(activate#) = [1]                  
               p(first#) = [5]                  
                p(from#) = [0]                  
                 p(sel#) = [2] x1 + [1] x2 + [6]
                  p(c_1) = [0]                  
                  p(c_2) = [1] x1 + [1]         
                  p(c_3) = [1] x1 + [0]         
                  p(c_4) = [0]                  
                  p(c_5) = [5]                  
                  p(c_6) = [1] x3 + [4]         
                  p(c_7) = [0]                  
                  p(c_8) = [0]                  
                  p(c_9) = [0]                  
                 p(c_10) = [1] x1 + [4]         
          
          Following rules are strictly oriented:
                   activate#(X) = [1]            
                                > [0]            
                                = c_1(X)         
          
          activate#(n__from(X)) = [1]            
                                > [0]            
                                = c_3(from#(X))  
          
                  first#(X1,X2) = [5]            
                                > [0]            
                                = c_4(X1,X2)     
          
                    activate(X) = [1] X + [1]    
                                > [1] X + [0]    
                                = X              
          
           activate(n__from(X)) = [1] X + [8]    
                                > [1] X + [0]    
                                = from(X)        
          
                   first(X1,X2) = [1] X2 + [2]   
                                > [1] X2 + [1]   
                                = n__first(X1,X2)
          
          
          Following rules are (at-least) weakly oriented:
          activate#(n__first(X1,X2)) =  [1]                            
                                     >= [6]                            
                                     =  c_2(first#(X1,X2))             
          
                       first#(0(),Z) =  [5]                            
                                     >= [5]                            
                                     =  c_5()                          
          
              first#(s(X),cons(Y,Z)) =  [5]                            
                                     >= [5]                            
                                     =  c_6(Y,X,activate#(Z))          
          
                            from#(X) =  [0]                            
                                     >= [0]                            
                                     =  c_7(X,X)                       
          
                            from#(X) =  [0]                            
                                     >= [0]                            
                                     =  c_8(X)                         
          
                 sel#(0(),cons(X,Z)) =  [1] Z + [9]                    
                                     >= [0]                            
                                     =  c_9(X)                         
          
                sel#(s(X),cons(Y,Z)) =  [2] X + [1] Z + [11]           
                                     >= [2] X + [1] Z + [11]           
                                     =  c_10(sel#(X,activate(Z)))      
          
           activate(n__first(X1,X2)) =  [1] X2 + [2]                   
                                     >= [1] X2 + [2]                   
                                     =  first(X1,X2)                   
          
                        first(0(),Z) =  [1] Z + [2]                    
                                     >= [1]                            
                                     =  nil()                          
          
               first(s(X),cons(Y,Z)) =  [1] Z + [5]                    
                                     >= [1] Z + [5]                    
                                     =  cons(Y,n__first(X,activate(Z)))
          
                             from(X) =  [1] X + [0]                    
                                     >= [1] X + [11]                   
                                     =  cons(X,n__from(s(X)))          
          
                             from(X) =  [1] X + [0]                    
                                     >= [1] X + [7]                    
                                     =  n__from(X)                     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 8: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
        - Strict TRS:
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Weak DPs:
            activate#(X) -> c_1(X)
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4(X1,X2)
            first#(0(),Z) -> c_5()
            first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
            from#(X) -> c_7(X,X)
            from#(X) -> c_8(X)
            sel#(0(),cons(X,Z)) -> c_9(X)
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
        - 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(),Z) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
        - Signature:
            {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
            ,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, 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(activate) = {1},
            uargs(cons) = {2},
            uargs(first) = {2},
            uargs(n__first) = {2},
            uargs(sel#) = {2},
            uargs(c_2) = {1},
            uargs(c_3) = {1},
            uargs(c_6) = {3},
            uargs(c_10) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]         
             p(activate) = [1] x1 + [0]
                 p(cons) = [1] x2 + [4]
                p(first) = [1] x2 + [5]
                 p(from) = [0]         
             p(n__first) = [1] x2 + [5]
              p(n__from) = [0]         
                  p(nil) = [5]         
                    p(s) = [1] x1 + [0]
                  p(sel) = [0]         
            p(activate#) = [1] x1 + [2]
               p(first#) = [1] x2 + [0]
                p(from#) = [0]         
                 p(sel#) = [1] x2 + [0]
                  p(c_1) = [1] x1 + [2]
                  p(c_2) = [1] x1 + [0]
                  p(c_3) = [1] x1 + [2]
                  p(c_4) = [1] x2 + [0]
                  p(c_5) = [0]         
                  p(c_6) = [1] x3 + [2]
                  p(c_7) = [0]         
                  p(c_8) = [0]         
                  p(c_9) = [4]         
                 p(c_10) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          activate#(n__first(X1,X2)) = [1] X2 + [7]      
                                     > [1] X2 + [0]      
                                     = c_2(first#(X1,X2))
          
          
          Following rules are (at-least) weakly oriented:
                       activate#(X) =  [1] X + [2]                    
                                    >= [1] X + [2]                    
                                    =  c_1(X)                         
          
              activate#(n__from(X)) =  [2]                            
                                    >= [2]                            
                                    =  c_3(from#(X))                  
          
                      first#(X1,X2) =  [1] X2 + [0]                   
                                    >= [1] X2 + [0]                   
                                    =  c_4(X1,X2)                     
          
                      first#(0(),Z) =  [1] Z + [0]                    
                                    >= [0]                            
                                    =  c_5()                          
          
             first#(s(X),cons(Y,Z)) =  [1] Z + [4]                    
                                    >= [1] Z + [4]                    
                                    =  c_6(Y,X,activate#(Z))          
          
                           from#(X) =  [0]                            
                                    >= [0]                            
                                    =  c_7(X,X)                       
          
                           from#(X) =  [0]                            
                                    >= [0]                            
                                    =  c_8(X)                         
          
                sel#(0(),cons(X,Z)) =  [1] Z + [4]                    
                                    >= [4]                            
                                    =  c_9(X)                         
          
               sel#(s(X),cons(Y,Z)) =  [1] Z + [4]                    
                                    >= [1] Z + [0]                    
                                    =  c_10(sel#(X,activate(Z)))      
          
                        activate(X) =  [1] X + [0]                    
                                    >= [1] X + [0]                    
                                    =  X                              
          
          activate(n__first(X1,X2)) =  [1] X2 + [5]                   
                                    >= [1] X2 + [5]                   
                                    =  first(X1,X2)                   
          
               activate(n__from(X)) =  [0]                            
                                    >= [0]                            
                                    =  from(X)                        
          
                       first(X1,X2) =  [1] X2 + [5]                   
                                    >= [1] X2 + [5]                   
                                    =  n__first(X1,X2)                
          
                       first(0(),Z) =  [1] Z + [5]                    
                                    >= [5]                            
                                    =  nil()                          
          
              first(s(X),cons(Y,Z)) =  [1] Z + [9]                    
                                    >= [1] Z + [9]                    
                                    =  cons(Y,n__first(X,activate(Z)))
          
                            from(X) =  [0]                            
                                    >= [4]                            
                                    =  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 9: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Weak DPs:
            activate#(X) -> c_1(X)
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4(X1,X2)
            first#(0(),Z) -> c_5()
            first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
            from#(X) -> c_7(X,X)
            from#(X) -> c_8(X)
            sel#(0(),cons(X,Z)) -> c_9(X)
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
        - 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(),Z) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
        - Signature:
            {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
            ,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, 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(activate) = {1},
            uargs(cons) = {2},
            uargs(first) = {2},
            uargs(n__first) = {2},
            uargs(sel#) = {2},
            uargs(c_2) = {1},
            uargs(c_3) = {1},
            uargs(c_6) = {3},
            uargs(c_10) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]                  
             p(activate) = [1] x1 + [1]         
                 p(cons) = [1] x1 + [1] x2 + [1]
                p(first) = [1] x2 + [5]         
                 p(from) = [1] x1 + [1]         
             p(n__first) = [1] x2 + [4]         
              p(n__from) = [1] x1 + [0]         
                  p(nil) = [0]                  
                    p(s) = [6]                  
                  p(sel) = [1] x2 + [0]         
            p(activate#) = [5]                  
               p(first#) = [5]                  
                p(from#) = [1]                  
                 p(sel#) = [1] x2 + [5]         
                  p(c_1) = [0]                  
                  p(c_2) = [1] x1 + [0]         
                  p(c_3) = [1] x1 + [0]         
                  p(c_4) = [0]                  
                  p(c_5) = [0]                  
                  p(c_6) = [1] x3 + [0]         
                  p(c_7) = [1]                  
                  p(c_8) = [1]                  
                  p(c_9) = [6]                  
                 p(c_10) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          from(X) = [1] X + [1]
                  > [1] X + [0]
                  = n__from(X) 
          
          
          Following rules are (at-least) weakly oriented:
                        activate#(X) =  [5]                            
                                     >= [0]                            
                                     =  c_1(X)                         
          
          activate#(n__first(X1,X2)) =  [5]                            
                                     >= [5]                            
                                     =  c_2(first#(X1,X2))             
          
               activate#(n__from(X)) =  [5]                            
                                     >= [1]                            
                                     =  c_3(from#(X))                  
          
                       first#(X1,X2) =  [5]                            
                                     >= [0]                            
                                     =  c_4(X1,X2)                     
          
                       first#(0(),Z) =  [5]                            
                                     >= [0]                            
                                     =  c_5()                          
          
              first#(s(X),cons(Y,Z)) =  [5]                            
                                     >= [5]                            
                                     =  c_6(Y,X,activate#(Z))          
          
                            from#(X) =  [1]                            
                                     >= [1]                            
                                     =  c_7(X,X)                       
          
                            from#(X) =  [1]                            
                                     >= [1]                            
                                     =  c_8(X)                         
          
                 sel#(0(),cons(X,Z)) =  [1] X + [1] Z + [6]            
                                     >= [6]                            
                                     =  c_9(X)                         
          
                sel#(s(X),cons(Y,Z)) =  [1] Y + [1] Z + [6]            
                                     >= [1] Z + [6]                    
                                     =  c_10(sel#(X,activate(Z)))      
          
                         activate(X) =  [1] X + [1]                    
                                     >= [1] X + [0]                    
                                     =  X                              
          
           activate(n__first(X1,X2)) =  [1] X2 + [5]                   
                                     >= [1] X2 + [5]                   
                                     =  first(X1,X2)                   
          
                activate(n__from(X)) =  [1] X + [1]                    
                                     >= [1] X + [1]                    
                                     =  from(X)                        
          
                        first(X1,X2) =  [1] X2 + [5]                   
                                     >= [1] X2 + [4]                   
                                     =  n__first(X1,X2)                
          
                        first(0(),Z) =  [1] Z + [5]                    
                                     >= [0]                            
                                     =  nil()                          
          
               first(s(X),cons(Y,Z)) =  [1] Y + [1] Z + [6]            
                                     >= [1] Y + [1] Z + [6]            
                                     =  cons(Y,n__first(X,activate(Z)))
          
                             from(X) =  [1] X + [1]                    
                                     >= [1] X + [7]                    
                                     =  cons(X,n__from(s(X)))          
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 10: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            from(X) -> cons(X,n__from(s(X)))
        - Weak DPs:
            activate#(X) -> c_1(X)
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4(X1,X2)
            first#(0(),Z) -> c_5()
            first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
            from#(X) -> c_7(X,X)
            from#(X) -> c_8(X)
            sel#(0(),cons(X,Z)) -> c_9(X)
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
        - 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(),Z) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
            ,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} and constructors {0,cons,n__first
            ,n__from,nil,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, 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(activate) = {1},
            uargs(cons) = {2},
            uargs(first) = {2},
            uargs(n__first) = {2},
            uargs(sel#) = {2},
            uargs(c_2) = {1},
            uargs(c_3) = {1},
            uargs(c_6) = {3},
            uargs(c_10) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]                  
             p(activate) = [1] x1 + [3]         
                 p(cons) = [1] x2 + [2]         
                p(first) = [1] x1 + [1] x2 + [4]
                 p(from) = [3]                  
             p(n__first) = [1] x1 + [1] x2 + [4]
              p(n__from) = [0]                  
                  p(nil) = [0]                  
                    p(s) = [1] x1 + [4]         
                  p(sel) = [0]                  
            p(activate#) = [0]                  
               p(first#) = [0]                  
                p(from#) = [0]                  
                 p(sel#) = [3] x1 + [1] x2 + [0]
                  p(c_1) = [0]                  
                  p(c_2) = [1] x1 + [0]         
                  p(c_3) = [1] x1 + [0]         
                  p(c_4) = [0]                  
                  p(c_5) = [0]                  
                  p(c_6) = [1] x3 + [0]         
                  p(c_7) = [0]                  
                  p(c_8) = [0]                  
                  p(c_9) = [0]                  
                 p(c_10) = [1] x1 + [2]         
          
          Following rules are strictly oriented:
          from(X) = [3]                  
                  > [2]                  
                  = cons(X,n__from(s(X)))
          
          
          Following rules are (at-least) weakly oriented:
                        activate#(X) =  [0]                            
                                     >= [0]                            
                                     =  c_1(X)                         
          
          activate#(n__first(X1,X2)) =  [0]                            
                                     >= [0]                            
                                     =  c_2(first#(X1,X2))             
          
               activate#(n__from(X)) =  [0]                            
                                     >= [0]                            
                                     =  c_3(from#(X))                  
          
                       first#(X1,X2) =  [0]                            
                                     >= [0]                            
                                     =  c_4(X1,X2)                     
          
                       first#(0(),Z) =  [0]                            
                                     >= [0]                            
                                     =  c_5()                          
          
              first#(s(X),cons(Y,Z)) =  [0]                            
                                     >= [0]                            
                                     =  c_6(Y,X,activate#(Z))          
          
                            from#(X) =  [0]                            
                                     >= [0]                            
                                     =  c_7(X,X)                       
          
                            from#(X) =  [0]                            
                                     >= [0]                            
                                     =  c_8(X)                         
          
                 sel#(0(),cons(X,Z)) =  [1] Z + [2]                    
                                     >= [0]                            
                                     =  c_9(X)                         
          
                sel#(s(X),cons(Y,Z)) =  [3] X + [1] Z + [14]           
                                     >= [3] X + [1] Z + [5]            
                                     =  c_10(sel#(X,activate(Z)))      
          
                         activate(X) =  [1] X + [3]                    
                                     >= [1] X + [0]                    
                                     =  X                              
          
           activate(n__first(X1,X2)) =  [1] X1 + [1] X2 + [7]          
                                     >= [1] X1 + [1] X2 + [4]          
                                     =  first(X1,X2)                   
          
                activate(n__from(X)) =  [3]                            
                                     >= [3]                            
                                     =  from(X)                        
          
                        first(X1,X2) =  [1] X1 + [1] X2 + [4]          
                                     >= [1] X1 + [1] X2 + [4]          
                                     =  n__first(X1,X2)                
          
                        first(0(),Z) =  [1] Z + [4]                    
                                     >= [0]                            
                                     =  nil()                          
          
               first(s(X),cons(Y,Z)) =  [1] X + [1] Z + [10]           
                                     >= [1] X + [1] Z + [9]            
                                     =  cons(Y,n__first(X,activate(Z)))
          
                             from(X) =  [3]                            
                                     >= [0]                            
                                     =  n__from(X)                     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 11: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(X) -> c_1(X)
            activate#(n__first(X1,X2)) -> c_2(first#(X1,X2))
            activate#(n__from(X)) -> c_3(from#(X))
            first#(X1,X2) -> c_4(X1,X2)
            first#(0(),Z) -> c_5()
            first#(s(X),cons(Y,Z)) -> c_6(Y,X,activate#(Z))
            from#(X) -> c_7(X,X)
            from#(X) -> c_8(X)
            sel#(0(),cons(X,Z)) -> c_9(X)
            sel#(s(X),cons(Y,Z)) -> c_10(sel#(X,activate(Z)))
        - 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(),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)
        - Signature:
            {activate/1,first/2,from/1,sel/2,activate#/1,first#/2,from#/1,sel#/2} / {0/0,cons/2,n__first/2,n__from/1
            ,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/3,c_7/2,c_8/1,c_9/1,c_10/1}
        - Obligation:
             runtime complexity wrt. defined symbols {activate#,first#,from#,sel#} 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))