YES

Problem:
 from(X) -> cons(X,n__from(n__s(X)))
 first(0(),Z) -> nil()
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
 sel(0(),cons(X,Z)) -> X
 sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
 from(X) -> n__from(X)
 s(X) -> n__s(X)
 first(X1,X2) -> n__first(X1,X2)
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   first#(s(X),cons(Y,Z)) -> activate#(Z)
   sel#(s(X),cons(Y,Z)) -> activate#(Z)
   sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))
   activate#(n__from(X)) -> activate#(X)
   activate#(n__from(X)) -> from#(activate(X))
   activate#(n__s(X)) -> activate#(X)
   activate#(n__s(X)) -> s#(activate(X))
   activate#(n__first(X1,X2)) -> activate#(X2)
   activate#(n__first(X1,X2)) -> activate#(X1)
   activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
  TRS:
   from(X) -> cons(X,n__from(n__s(X)))
   first(0(),Z) -> nil()
   first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
   sel(0(),cons(X,Z)) -> X
   sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   first(X1,X2) -> n__first(X1,X2)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
   activate(X) -> X
  TDG Processor:
   DPs:
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    sel#(s(X),cons(Y,Z)) -> activate#(Z)
    sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))
    activate#(n__from(X)) -> activate#(X)
    activate#(n__from(X)) -> from#(activate(X))
    activate#(n__s(X)) -> activate#(X)
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__first(X1,X2)) -> activate#(X2)
    activate#(n__first(X1,X2)) -> activate#(X1)
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
   TRS:
    from(X) -> cons(X,n__from(n__s(X)))
    first(0(),Z) -> nil()
    first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
    sel(0(),cons(X,Z)) -> X
    sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
    from(X) -> n__from(X)
    s(X) -> n__s(X)
    first(X1,X2) -> n__first(X1,X2)
    activate(n__from(X)) -> from(activate(X))
    activate(n__s(X)) -> s(activate(X))
    activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
    activate(X) -> X
   graph:
    sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) ->
    sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))
    sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z)) ->
    sel#(s(X),cons(Y,Z)) -> activate#(Z)
    sel#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
    sel#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__first(X1,X2)) -> activate#(X1)
    sel#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__first(X1,X2)) -> activate#(X2)
    sel#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__s(X)) -> s#(activate(X))
    sel#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__s(X)) -> activate#(X)
    sel#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__from(X)) -> from#(activate(X))
    sel#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__from(X)) -> activate#(X)
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__first(X1,X2)) -> activate#(X1)
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__first(X1,X2)) -> activate#(X2)
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__s(X)) -> activate#(X)
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__from(X)) -> from#(activate(X))
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__from(X)) -> activate#(X)
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__first(X1,X2)) -> activate#(X1)
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__first(X1,X2)) -> activate#(X2)
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__s(X)) -> activate#(X)
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__from(X)) -> from#(activate(X))
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__from(X)) -> activate#(X)
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) ->
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    activate#(n__from(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
    activate#(n__from(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> activate#(X1)
    activate#(n__from(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> activate#(X2)
    activate#(n__from(X)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__from(X)) -> activate#(X) ->
    activate#(n__s(X)) -> activate#(X)
    activate#(n__from(X)) -> activate#(X) ->
    activate#(n__from(X)) -> from#(activate(X))
    activate#(n__from(X)) -> activate#(X) ->
    activate#(n__from(X)) -> activate#(X)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> activate#(X1)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> activate#(X2)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__s(X)) -> activate#(X)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__from(X)) -> from#(activate(X))
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__from(X)) -> activate#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__first(X1,X2)) -> activate#(X1)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__first(X1,X2)) -> activate#(X2)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__s(X)) -> s#(activate(X))
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__s(X)) -> activate#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__from(X)) -> from#(activate(X))
    first#(s(X),cons(Y,Z)) -> activate#(Z) -> activate#(n__from(X)) -> activate#(X)
   SCC Processor:
    #sccs: 2
    #rules: 7
    #arcs: 45/100
    DPs:
     sel#(s(X),cons(Y,Z)) -> sel#(X,activate(Z))
    TRS:
     from(X) -> cons(X,n__from(n__s(X)))
     first(0(),Z) -> nil()
     first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
     sel(0(),cons(X,Z)) -> X
     sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
     from(X) -> n__from(X)
     s(X) -> n__s(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__from(X)) -> from(activate(X))
     activate(n__s(X)) -> s(activate(X))
     activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
     activate(X) -> X
    Subterm Criterion Processor:
     simple projection:
      pi(sel#) = 0
     problem:
      DPs:
       
      TRS:
       from(X) -> cons(X,n__from(n__s(X)))
       first(0(),Z) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
       sel(0(),cons(X,Z)) -> X
       sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
       from(X) -> n__from(X)
       s(X) -> n__s(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__from(X)) -> from(activate(X))
       activate(n__s(X)) -> s(activate(X))
       activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
       activate(X) -> X
     Qed
    
    DPs:
     activate#(n__from(X)) -> activate#(X)
     activate#(n__s(X)) -> activate#(X)
     activate#(n__first(X1,X2)) -> activate#(X2)
     activate#(n__first(X1,X2)) -> activate#(X1)
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     first#(s(X),cons(Y,Z)) -> activate#(Z)
    TRS:
     from(X) -> cons(X,n__from(n__s(X)))
     first(0(),Z) -> nil()
     first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
     sel(0(),cons(X,Z)) -> X
     sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
     from(X) -> n__from(X)
     s(X) -> n__s(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__from(X)) -> from(activate(X))
     activate(n__s(X)) -> s(activate(X))
     activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
     activate(X) -> X
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [activate#](x0) = x0 + 0,
      
      [first#](x0, x1) = x1 + 2,
      
      [sel](x0, x1) = x0 + 1x1 + 14,
      
      [n__first](x0, x1) = 1x0 + x1 + 4,
      
      [activate](x0) = x0,
      
      [s](x0) = x0,
      
      [nil] = 4,
      
      [first](x0, x1) = 1x0 + x1 + 4,
      
      [0] = 6,
      
      [cons](x0, x1) = x0 + x1,
      
      [n__from](x0) = 1x0,
      
      [n__s](x0) = x0,
      
      [from](x0) = 1x0
     orientation:
      activate#(n__from(X)) = 1X + 0 >= X + 0 = activate#(X)
      
      activate#(n__s(X)) = X + 0 >= X + 0 = activate#(X)
      
      activate#(n__first(X1,X2)) = 1X1 + X2 + 4 >= X2 + 0 = activate#(X2)
      
      activate#(n__first(X1,X2)) = 1X1 + X2 + 4 >= X1 + 0 = activate#(X1)
      
      activate#(n__first(X1,X2)) = 1X1 + X2 + 4 >= X2 + 2 = first#(activate(X1),activate(X2))
      
      first#(s(X),cons(Y,Z)) = Y + Z + 2 >= Z + 0 = activate#(Z)
      
      from(X) = 1X >= 1X = cons(X,n__from(n__s(X)))
      
      first(0(),Z) = Z + 7 >= 4 = nil()
      
      first(s(X),cons(Y,Z)) = 1X + Y + Z + 4 >= 1X + Y + Z + 4 = cons(Y,n__first(X,activate(Z)))
      
      sel(0(),cons(X,Z)) = 1X + 1Z + 14 >= X = X
      
      sel(s(X),cons(Y,Z)) = X + 1Y + 1Z + 14 >= X + 1Z + 14 = sel(X,activate(Z))
      
      from(X) = 1X >= 1X = n__from(X)
      
      s(X) = X >= X = n__s(X)
      
      first(X1,X2) = 1X1 + X2 + 4 >= 1X1 + X2 + 4 = n__first(X1,X2)
      
      activate(n__from(X)) = 1X >= 1X = from(activate(X))
      
      activate(n__s(X)) = X >= X = s(activate(X))
      
      activate(n__first(X1,X2)) = 1X1 + X2 + 4 >= 1X1 + X2 + 4 = first(activate(X1),activate(X2))
      
      activate(X) = X >= X = X
     problem:
      DPs:
       activate#(n__from(X)) -> activate#(X)
       activate#(n__s(X)) -> activate#(X)
       activate#(n__first(X1,X2)) -> activate#(X2)
       activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
       first#(s(X),cons(Y,Z)) -> activate#(Z)
      TRS:
       from(X) -> cons(X,n__from(n__s(X)))
       first(0(),Z) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
       sel(0(),cons(X,Z)) -> X
       sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
       from(X) -> n__from(X)
       s(X) -> n__s(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__from(X)) -> from(activate(X))
       activate(n__s(X)) -> s(activate(X))
       activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
       activate(X) -> X
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [activate#](x0) = x0 + 0,
       
       [first#](x0, x1) = x1 + 0,
       
       [sel](x0, x1) = 3x0 + 7x1 + 0,
       
       [n__first](x0, x1) = x0 + x1,
       
       [activate](x0) = x0,
       
       [s](x0) = x0 + 0,
       
       [nil] = 0,
       
       [first](x0, x1) = x0 + x1,
       
       [0] = 3,
       
       [cons](x0, x1) = x0 + x1 + 0,
       
       [n__from](x0) = 1x0 + 1,
       
       [n__s](x0) = x0 + 0,
       
       [from](x0) = 1x0 + 1
      orientation:
       activate#(n__from(X)) = 1X + 1 >= X + 0 = activate#(X)
       
       activate#(n__s(X)) = X + 0 >= X + 0 = activate#(X)
       
       activate#(n__first(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = activate#(X2)
       
       activate#(n__first(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = first#(activate(X1),activate(X2))
       
       first#(s(X),cons(Y,Z)) = Y + Z + 0 >= Z + 0 = activate#(Z)
       
       from(X) = 1X + 1 >= 1X + 1 = cons(X,n__from(n__s(X)))
       
       first(0(),Z) = Z + 3 >= 0 = nil()
       
       first(s(X),cons(Y,Z)) = X + Y + Z + 0 >= X + Y + Z + 0 = cons(Y,n__first(X,activate(Z)))
       
       sel(0(),cons(X,Z)) = 7X + 7Z + 7 >= X = X
       
       sel(s(X),cons(Y,Z)) = 3X + 7Y + 7Z + 7 >= 3X + 7Z + 0 = sel(X,activate(Z))
       
       from(X) = 1X + 1 >= 1X + 1 = n__from(X)
       
       s(X) = X + 0 >= X + 0 = n__s(X)
       
       first(X1,X2) = X1 + X2 >= X1 + X2 = n__first(X1,X2)
       
       activate(n__from(X)) = 1X + 1 >= 1X + 1 = from(activate(X))
       
       activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X))
       
       activate(n__first(X1,X2)) = X1 + X2 >= X1 + X2 = first(activate(X1),activate(X2))
       
       activate(X) = X >= X = X
      problem:
       DPs:
        activate#(n__s(X)) -> activate#(X)
        activate#(n__first(X1,X2)) -> activate#(X2)
        activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
        first#(s(X),cons(Y,Z)) -> activate#(Z)
       TRS:
        from(X) -> cons(X,n__from(n__s(X)))
        first(0(),Z) -> nil()
        first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
        sel(0(),cons(X,Z)) -> X
        sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        from(X) -> n__from(X)
        s(X) -> n__s(X)
        first(X1,X2) -> n__first(X1,X2)
        activate(n__from(X)) -> from(activate(X))
        activate(n__s(X)) -> s(activate(X))
        activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
        activate(X) -> X
      Arctic Interpretation Processor:
       dimension: 1
       interpretation:
        [activate#](x0) = 3x0,
        
        [first#](x0, x1) = 5x0 + 4x1 + 0,
        
        [sel](x0, x1) = 4x0 + x1 + 0,
        
        [n__first](x0, x1) = 2x0 + 2x1 + 2,
        
        [activate](x0) = x0 + 0,
        
        [s](x0) = x0,
        
        [nil] = 0,
        
        [first](x0, x1) = 2x0 + 2x1 + 2,
        
        [0] = 2,
        
        [cons](x0, x1) = x0 + x1,
        
        [n__from](x0) = 2x0 + 2,
        
        [n__s](x0) = x0,
        
        [from](x0) = 2x0 + 2
       orientation:
        activate#(n__s(X)) = 3X >= 3X = activate#(X)
        
        activate#(n__first(X1,X2)) = 5X1 + 5X2 + 5 >= 3X2 = activate#(X2)
        
        activate#(n__first(X1,X2)) = 5X1 + 5X2 + 5 >= 5X1 + 4X2 + 5 = first#(activate(X1),activate(X2))
        
        first#(s(X),cons(Y,Z)) = 5X + 4Y + 4Z + 0 >= 3Z = activate#(Z)
        
        from(X) = 2X + 2 >= 2X + 2 = cons(X,n__from(n__s(X)))
        
        first(0(),Z) = 2Z + 4 >= 0 = nil()
        
        first(s(X),cons(Y,Z)) = 2X + 2Y + 2Z + 2 >= 2X + Y + 2Z + 2 = cons(Y,n__first(X,activate(Z)))
        
        sel(0(),cons(X,Z)) = X + Z + 6 >= X = X
        
        sel(s(X),cons(Y,Z)) = 4X + Y + Z + 0 >= 4X + Z + 0 = sel(X,activate(Z))
        
        from(X) = 2X + 2 >= 2X + 2 = n__from(X)
        
        s(X) = X >= X = n__s(X)
        
        first(X1,X2) = 2X1 + 2X2 + 2 >= 2X1 + 2X2 + 2 = n__first(X1,X2)
        
        activate(n__from(X)) = 2X + 2 >= 2X + 2 = from(activate(X))
        
        activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X))
        
        activate(n__first(X1,X2)) = 2X1 + 2X2 + 2 >= 2X1 + 2X2 + 2 = first(activate(X1),activate(X2))
        
        activate(X) = X + 0 >= X = X
       problem:
        DPs:
         activate#(n__s(X)) -> activate#(X)
         activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
        TRS:
         from(X) -> cons(X,n__from(n__s(X)))
         first(0(),Z) -> nil()
         first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
         sel(0(),cons(X,Z)) -> X
         sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
         from(X) -> n__from(X)
         s(X) -> n__s(X)
         first(X1,X2) -> n__first(X1,X2)
         activate(n__from(X)) -> from(activate(X))
         activate(n__s(X)) -> s(activate(X))
         activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
         activate(X) -> X
       SCC Processor:
        #sccs: 1
        #rules: 1
        #arcs: 26/4
        DPs:
         activate#(n__s(X)) -> activate#(X)
        TRS:
         from(X) -> cons(X,n__from(n__s(X)))
         first(0(),Z) -> nil()
         first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
         sel(0(),cons(X,Z)) -> X
         sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
         from(X) -> n__from(X)
         s(X) -> n__s(X)
         first(X1,X2) -> n__first(X1,X2)
         activate(n__from(X)) -> from(activate(X))
         activate(n__s(X)) -> s(activate(X))
         activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
         activate(X) -> X
        Subterm Criterion Processor:
         simple projection:
          pi(activate#) = 0
         problem:
          DPs:
           
          TRS:
           from(X) -> cons(X,n__from(n__s(X)))
           first(0(),Z) -> nil()
           first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
           sel(0(),cons(X,Z)) -> X
           sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
           from(X) -> n__from(X)
           s(X) -> n__s(X)
           first(X1,X2) -> n__first(X1,X2)
           activate(n__from(X)) -> from(activate(X))
           activate(n__s(X)) -> s(activate(X))
           activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
           activate(X) -> X
         Qed