YES

Problem:
 terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
 sqr(0()) -> 0()
 sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
 dbl(0()) -> 0()
 dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
 add(0(),X) -> X
 add(s(X),Y) -> s(n__add(activate(X),Y))
 first(0(),X) -> nil()
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
 terms(X) -> n__terms(X)
 add(X1,X2) -> n__add(X1,X2)
 s(X) -> n__s(X)
 dbl(X) -> n__dbl(X)
 first(X1,X2) -> n__first(X1,X2)
 activate(n__terms(X)) -> terms(X)
 activate(n__add(X1,X2)) -> add(X1,X2)
 activate(n__s(X)) -> s(X)
 activate(n__dbl(X)) -> dbl(X)
 activate(n__first(X1,X2)) -> first(X1,X2)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   terms#(N) -> s#(N)
   terms#(N) -> sqr#(N)
   sqr#(s(X)) -> dbl#(activate(X))
   sqr#(s(X)) -> activate#(X)
   sqr#(s(X)) -> sqr#(activate(X))
   sqr#(s(X)) -> s#(n__add(sqr(activate(X)),dbl(activate(X))))
   dbl#(s(X)) -> activate#(X)
   dbl#(s(X)) -> s#(n__s(n__dbl(activate(X))))
   add#(s(X),Y) -> activate#(X)
   add#(s(X),Y) -> s#(n__add(activate(X),Y))
   first#(s(X),cons(Y,Z)) -> activate#(Z)
   first#(s(X),cons(Y,Z)) -> activate#(X)
   activate#(n__terms(X)) -> terms#(X)
   activate#(n__add(X1,X2)) -> add#(X1,X2)
   activate#(n__s(X)) -> s#(X)
   activate#(n__dbl(X)) -> dbl#(X)
   activate#(n__first(X1,X2)) -> first#(X1,X2)
  TRS:
   terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
   sqr(0()) -> 0()
   sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
   dbl(0()) -> 0()
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   add(0(),X) -> X
   add(s(X),Y) -> s(n__add(activate(X),Y))
   first(0(),X) -> nil()
   first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
   terms(X) -> n__terms(X)
   add(X1,X2) -> n__add(X1,X2)
   s(X) -> n__s(X)
   dbl(X) -> n__dbl(X)
   first(X1,X2) -> n__first(X1,X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1,X2)) -> add(X1,X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1,X2)) -> first(X1,X2)
   activate(X) -> X
  TDG Processor:
   DPs:
    terms#(N) -> s#(N)
    terms#(N) -> sqr#(N)
    sqr#(s(X)) -> dbl#(activate(X))
    sqr#(s(X)) -> activate#(X)
    sqr#(s(X)) -> sqr#(activate(X))
    sqr#(s(X)) -> s#(n__add(sqr(activate(X)),dbl(activate(X))))
    dbl#(s(X)) -> activate#(X)
    dbl#(s(X)) -> s#(n__s(n__dbl(activate(X))))
    add#(s(X),Y) -> activate#(X)
    add#(s(X),Y) -> s#(n__add(activate(X),Y))
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    first#(s(X),cons(Y,Z)) -> activate#(X)
    activate#(n__terms(X)) -> terms#(X)
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    activate#(n__s(X)) -> s#(X)
    activate#(n__dbl(X)) -> dbl#(X)
    activate#(n__first(X1,X2)) -> first#(X1,X2)
   TRS:
    terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
    sqr(0()) -> 0()
    sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
    dbl(0()) -> 0()
    dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
    add(0(),X) -> X
    add(s(X),Y) -> s(n__add(activate(X),Y))
    first(0(),X) -> nil()
    first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
    terms(X) -> n__terms(X)
    add(X1,X2) -> n__add(X1,X2)
    s(X) -> n__s(X)
    dbl(X) -> n__dbl(X)
    first(X1,X2) -> n__first(X1,X2)
    activate(n__terms(X)) -> terms(X)
    activate(n__add(X1,X2)) -> add(X1,X2)
    activate(n__s(X)) -> s(X)
    activate(n__dbl(X)) -> dbl(X)
    activate(n__first(X1,X2)) -> first(X1,X2)
    activate(X) -> X
   graph:
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__dbl(X)) -> dbl#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__s(X)) -> s#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__terms(X)) -> terms#(X)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__dbl(X)) -> dbl#(X)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(X)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(X)
    add#(s(X),Y) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    add#(s(X),Y) -> activate#(X) -> activate#(n__dbl(X)) -> dbl#(X)
    add#(s(X),Y) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    add#(s(X),Y) -> activate#(X) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    add#(s(X),Y) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(X)
    activate#(n__first(X1,X2)) -> first#(X1,X2) ->
    first#(s(X),cons(Y,Z)) -> activate#(X)
    activate#(n__first(X1,X2)) -> first#(X1,X2) ->
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    activate#(n__dbl(X)) -> dbl#(X) ->
    dbl#(s(X)) -> s#(n__s(n__dbl(activate(X))))
    activate#(n__dbl(X)) -> dbl#(X) ->
    dbl#(s(X)) -> activate#(X)
    activate#(n__add(X1,X2)) -> add#(X1,X2) ->
    add#(s(X),Y) -> s#(n__add(activate(X),Y))
    activate#(n__add(X1,X2)) -> add#(X1,X2) ->
    add#(s(X),Y) -> activate#(X)
    activate#(n__terms(X)) -> terms#(X) -> terms#(N) -> sqr#(N)
    activate#(n__terms(X)) -> terms#(X) -> terms#(N) -> s#(N)
    dbl#(s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    dbl#(s(X)) -> activate#(X) -> activate#(n__dbl(X)) -> dbl#(X)
    dbl#(s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    dbl#(s(X)) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(X1,X2)
    dbl#(s(X)) -> activate#(X) -> activate#(n__terms(X)) -> terms#(X)
    sqr#(s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    sqr#(s(X)) -> activate#(X) -> activate#(n__dbl(X)) -> dbl#(X)
    sqr#(s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    sqr#(s(X)) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(X1,X2)
    sqr#(s(X)) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(X)
    sqr#(s(X)) -> dbl#(activate(X)) ->
    dbl#(s(X)) -> s#(n__s(n__dbl(activate(X))))
    sqr#(s(X)) -> dbl#(activate(X)) -> dbl#(s(X)) -> activate#(X)
    sqr#(s(X)) -> sqr#(activate(X)) ->
    sqr#(s(X)) -> s#(n__add(sqr(activate(X)),dbl(activate(X))))
    sqr#(s(X)) -> sqr#(activate(X)) ->
    sqr#(s(X)) -> sqr#(activate(X))
    sqr#(s(X)) -> sqr#(activate(X)) -> sqr#(s(X)) -> activate#(X)
    sqr#(s(X)) -> sqr#(activate(X)) -> sqr#(s(X)) -> dbl#(activate(X))
    terms#(N) -> sqr#(N) ->
    sqr#(s(X)) -> s#(n__add(sqr(activate(X)),dbl(activate(X))))
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> sqr#(activate(X))
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> activate#(X)
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> dbl#(activate(X))
   SCC Processor:
    #sccs: 1
    #rules: 12
    #arcs: 43/289
    DPs:
     first#(s(X),cons(Y,Z)) -> activate#(Z)
     activate#(n__terms(X)) -> terms#(X)
     terms#(N) -> sqr#(N)
     sqr#(s(X)) -> dbl#(activate(X))
     dbl#(s(X)) -> activate#(X)
     activate#(n__add(X1,X2)) -> add#(X1,X2)
     add#(s(X),Y) -> activate#(X)
     activate#(n__dbl(X)) -> dbl#(X)
     activate#(n__first(X1,X2)) -> first#(X1,X2)
     first#(s(X),cons(Y,Z)) -> activate#(X)
     sqr#(s(X)) -> activate#(X)
     sqr#(s(X)) -> sqr#(activate(X))
    TRS:
     terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
     sqr(0()) -> 0()
     sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
     dbl(0()) -> 0()
     dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
     add(0(),X) -> X
     add(s(X),Y) -> s(n__add(activate(X),Y))
     first(0(),X) -> nil()
     first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
     terms(X) -> n__terms(X)
     add(X1,X2) -> n__add(X1,X2)
     s(X) -> n__s(X)
     dbl(X) -> n__dbl(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__terms(X)) -> terms(X)
     activate(n__add(X1,X2)) -> add(X1,X2)
     activate(n__s(X)) -> s(X)
     activate(n__dbl(X)) -> dbl(X)
     activate(n__first(X1,X2)) -> first(X1,X2)
     activate(X) -> X
    Arctic Interpretation Processor:
     dimension: 1
     usable rules:
      terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
      dbl(0()) -> 0()
      dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
      add(0(),X) -> X
      add(s(X),Y) -> s(n__add(activate(X),Y))
      first(0(),X) -> nil()
      first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
      terms(X) -> n__terms(X)
      add(X1,X2) -> n__add(X1,X2)
      s(X) -> n__s(X)
      dbl(X) -> n__dbl(X)
      first(X1,X2) -> n__first(X1,X2)
      activate(n__terms(X)) -> terms(X)
      activate(n__add(X1,X2)) -> add(X1,X2)
      activate(n__s(X)) -> s(X)
      activate(n__dbl(X)) -> dbl(X)
      activate(n__first(X1,X2)) -> first(X1,X2)
      activate(X) -> X
     interpretation:
      [sqr#](x0) = x0 + 0,
      
      [n__terms](x0) = x0 + 0,
      
      [n__first](x0, x1) = 7x0 + x1 + 7,
      
      [dbl](x0) = x0 + 5,
      
      [sqr](x0) = x0 + 4,
      
      [s](x0) = x0 + 0,
      
      [terms#](x0) = x0 + 0,
      
      [add#](x0, x1) = x0 + x1 + 0,
      
      [dbl#](x0) = x0,
      
      [first#](x0, x1) = 7x0 + x1,
      
      [cons](x0, x1) = x1 + 0,
      
      [n__dbl](x0) = x0 + 5,
      
      [terms](x0) = x0 + 0,
      
      [activate](x0) = x0 + 0,
      
      [activate#](x0) = x0,
      
      [nil] = 0,
      
      [add](x0, x1) = x0 + x1 + 0,
      
      [n__s](x0) = x0,
      
      [n__add](x0, x1) = x0 + x1 + 0,
      
      [first](x0, x1) = 7x0 + x1 + 7,
      
      [0] = 0,
      
      [recip](x0) = x0 + 0
     orientation:
      first#(s(X),cons(Y,Z)) = 7X + Z + 7 >= Z = activate#(Z)
      
      activate#(n__terms(X)) = X + 0 >= X + 0 = terms#(X)
      
      terms#(N) = N + 0 >= N + 0 = sqr#(N)
      
      sqr#(s(X)) = X + 0 >= X + 0 = dbl#(activate(X))
      
      dbl#(s(X)) = X + 0 >= X = activate#(X)
      
      activate#(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = add#(X1,X2)
      
      add#(s(X),Y) = X + Y + 0 >= X = activate#(X)
      
      activate#(n__dbl(X)) = X + 5 >= X = dbl#(X)
      
      activate#(n__first(X1,X2)) = 7X1 + X2 + 7 >= 7X1 + X2 = first#(X1,X2)
      
      first#(s(X),cons(Y,Z)) = 7X + Z + 7 >= X = activate#(X)
      
      sqr#(s(X)) = X + 0 >= X = activate#(X)
      
      sqr#(s(X)) = X + 0 >= X + 0 = sqr#(activate(X))
      
      terms(N) = N + 0 >= N + 0 = cons(recip(sqr(N)),n__terms(s(N)))
      
      sqr(0()) = 4 >= 0 = 0()
      
      sqr(s(X)) = X + 4 >= X + 5 = s(n__add(sqr(activate(X)),dbl(activate(X))))
      
      dbl(0()) = 5 >= 0 = 0()
      
      dbl(s(X)) = X + 5 >= X + 5 = s(n__s(n__dbl(activate(X))))
      
      add(0(),X) = X + 0 >= X = X
      
      add(s(X),Y) = X + Y + 0 >= X + Y + 0 = s(n__add(activate(X),Y))
      
      first(0(),X) = X + 7 >= 0 = nil()
      
      first(s(X),cons(Y,Z)) = 7X + Z + 7 >= 7X + Z + 7 = cons(Y,n__first(activate(X),activate(Z)))
      
      terms(X) = X + 0 >= X + 0 = n__terms(X)
      
      add(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = n__add(X1,X2)
      
      s(X) = X + 0 >= X = n__s(X)
      
      dbl(X) = X + 5 >= X + 5 = n__dbl(X)
      
      first(X1,X2) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = n__first(X1,X2)
      
      activate(n__terms(X)) = X + 0 >= X + 0 = terms(X)
      
      activate(n__add(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = add(X1,X2)
      
      activate(n__s(X)) = X + 0 >= X + 0 = s(X)
      
      activate(n__dbl(X)) = X + 5 >= X + 5 = dbl(X)
      
      activate(n__first(X1,X2)) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = first(X1,X2)
      
      activate(X) = X + 0 >= X = X
     problem:
      DPs:
       first#(s(X),cons(Y,Z)) -> activate#(Z)
       activate#(n__terms(X)) -> terms#(X)
       terms#(N) -> sqr#(N)
       sqr#(s(X)) -> dbl#(activate(X))
       dbl#(s(X)) -> activate#(X)
       activate#(n__add(X1,X2)) -> add#(X1,X2)
       add#(s(X),Y) -> activate#(X)
       activate#(n__dbl(X)) -> dbl#(X)
       activate#(n__first(X1,X2)) -> first#(X1,X2)
       sqr#(s(X)) -> activate#(X)
       sqr#(s(X)) -> sqr#(activate(X))
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
       sqr(0()) -> 0()
       sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
       dbl(0()) -> 0()
       dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
       add(0(),X) -> X
       add(s(X),Y) -> s(n__add(activate(X),Y))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
       terms(X) -> n__terms(X)
       add(X1,X2) -> n__add(X1,X2)
       s(X) -> n__s(X)
       dbl(X) -> n__dbl(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(X)
       activate(n__add(X1,X2)) -> add(X1,X2)
       activate(n__s(X)) -> s(X)
       activate(n__dbl(X)) -> dbl(X)
       activate(n__first(X1,X2)) -> first(X1,X2)
       activate(X) -> X
     Restore Modifier:
      DPs:
       first#(s(X),cons(Y,Z)) -> activate#(Z)
       activate#(n__terms(X)) -> terms#(X)
       terms#(N) -> sqr#(N)
       sqr#(s(X)) -> dbl#(activate(X))
       dbl#(s(X)) -> activate#(X)
       activate#(n__add(X1,X2)) -> add#(X1,X2)
       add#(s(X),Y) -> activate#(X)
       activate#(n__dbl(X)) -> dbl#(X)
       activate#(n__first(X1,X2)) -> first#(X1,X2)
       sqr#(s(X)) -> activate#(X)
       sqr#(s(X)) -> sqr#(activate(X))
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
       sqr(0()) -> 0()
       sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
       dbl(0()) -> 0()
       dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
       add(0(),X) -> X
       add(s(X),Y) -> s(n__add(activate(X),Y))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
       terms(X) -> n__terms(X)
       add(X1,X2) -> n__add(X1,X2)
       s(X) -> n__s(X)
       dbl(X) -> n__dbl(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(X)
       activate(n__add(X1,X2)) -> add(X1,X2)
       activate(n__s(X)) -> s(X)
       activate(n__dbl(X)) -> dbl(X)
       activate(n__first(X1,X2)) -> first(X1,X2)
       activate(X) -> X
      Matrix Interpretation Processor: dim=1
       
       usable rules:
        terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
        dbl(0()) -> 0()
        dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
        add(0(),X) -> X
        add(s(X),Y) -> s(n__add(activate(X),Y))
        first(0(),X) -> nil()
        first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
        terms(X) -> n__terms(X)
        add(X1,X2) -> n__add(X1,X2)
        s(X) -> n__s(X)
        dbl(X) -> n__dbl(X)
        first(X1,X2) -> n__first(X1,X2)
        activate(n__terms(X)) -> terms(X)
        activate(n__add(X1,X2)) -> add(X1,X2)
        activate(n__s(X)) -> s(X)
        activate(n__dbl(X)) -> dbl(X)
        activate(n__first(X1,X2)) -> first(X1,X2)
        activate(X) -> X
       interpretation:
        [sqr#](x0) = x0 + 2,
        
        [n__terms](x0) = 2x0 + 2,
        
        [n__first](x0, x1) = 5x0 + x1,
        
        [dbl](x0) = 2x0 + 4,
        
        [sqr](x0) = 1,
        
        [s](x0) = 2x0,
        
        [terms#](x0) = x0 + 2,
        
        [add#](x0, x1) = 2x0 + 2x1,
        
        [dbl#](x0) = x0 + 2,
        
        [first#](x0, x1) = x1,
        
        [cons](x0, x1) = x1,
        
        [n__dbl](x0) = x0 + 2,
        
        [terms](x0) = 4x0 + 2,
        
        [activate](x0) = 2x0,
        
        [activate#](x0) = x0,
        
        [nil] = 0,
        
        [add](x0, x1) = 4x0 + 4x1,
        
        [n__s](x0) = x0,
        
        [n__add](x0, x1) = 2x0 + 2x1,
        
        [first](x0, x1) = 5x0 + 2x1,
        
        [0] = 2,
        
        [recip](x0) = 2
       orientation:
        first#(s(X),cons(Y,Z)) = Z >= Z = activate#(Z)
        
        activate#(n__terms(X)) = 2X + 2 >= X + 2 = terms#(X)
        
        terms#(N) = N + 2 >= N + 2 = sqr#(N)
        
        sqr#(s(X)) = 2X + 2 >= 2X + 2 = dbl#(activate(X))
        
        dbl#(s(X)) = 2X + 2 >= X = activate#(X)
        
        activate#(n__add(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = add#(X1,X2)
        
        add#(s(X),Y) = 4X + 2Y >= X = activate#(X)
        
        activate#(n__dbl(X)) = X + 2 >= X + 2 = dbl#(X)
        
        activate#(n__first(X1,X2)) = 5X1 + X2 >= X2 = first#(X1,X2)
        
        sqr#(s(X)) = 2X + 2 >= X = activate#(X)
        
        sqr#(s(X)) = 2X + 2 >= 2X + 2 = sqr#(activate(X))
        
        terms(N) = 4N + 2 >= 4N + 2 = cons(recip(sqr(N)),n__terms(s(N)))
        
        sqr(0()) = 1 >= 2 = 0()
        
        sqr(s(X)) = 1 >= 16X + 20 = s(n__add(sqr(activate(X)),dbl(activate(X))))
        
        dbl(0()) = 8 >= 2 = 0()
        
        dbl(s(X)) = 4X + 4 >= 4X + 4 = s(n__s(n__dbl(activate(X))))
        
        add(0(),X) = 4X + 8 >= X = X
        
        add(s(X),Y) = 8X + 4Y >= 8X + 4Y = s(n__add(activate(X),Y))
        
        first(0(),X) = 2X + 10 >= 0 = nil()
        
        first(s(X),cons(Y,Z)) = 10X + 2Z >= 10X + 2Z = cons(Y,n__first(activate(X),activate(Z)))
        
        terms(X) = 4X + 2 >= 2X + 2 = n__terms(X)
        
        add(X1,X2) = 4X1 + 4X2 >= 2X1 + 2X2 = n__add(X1,X2)
        
        s(X) = 2X >= X = n__s(X)
        
        dbl(X) = 2X + 4 >= X + 2 = n__dbl(X)
        
        first(X1,X2) = 5X1 + 2X2 >= 5X1 + X2 = n__first(X1,X2)
        
        activate(n__terms(X)) = 4X + 4 >= 4X + 2 = terms(X)
        
        activate(n__add(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = add(X1,X2)
        
        activate(n__s(X)) = 2X >= 2X = s(X)
        
        activate(n__dbl(X)) = 2X + 4 >= 2X + 4 = dbl(X)
        
        activate(n__first(X1,X2)) = 10X1 + 2X2 >= 5X1 + 2X2 = first(X1,X2)
        
        activate(X) = 2X >= X = X
       problem:
        DPs:
         first#(s(X),cons(Y,Z)) -> activate#(Z)
         activate#(n__terms(X)) -> terms#(X)
         terms#(N) -> sqr#(N)
         sqr#(s(X)) -> dbl#(activate(X))
         activate#(n__add(X1,X2)) -> add#(X1,X2)
         add#(s(X),Y) -> activate#(X)
         activate#(n__dbl(X)) -> dbl#(X)
         activate#(n__first(X1,X2)) -> first#(X1,X2)
         sqr#(s(X)) -> sqr#(activate(X))
        TRS:
         terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
         sqr(0()) -> 0()
         sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
         dbl(0()) -> 0()
         dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
         add(0(),X) -> X
         add(s(X),Y) -> s(n__add(activate(X),Y))
         first(0(),X) -> nil()
         first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
         terms(X) -> n__terms(X)
         add(X1,X2) -> n__add(X1,X2)
         s(X) -> n__s(X)
         dbl(X) -> n__dbl(X)
         first(X1,X2) -> n__first(X1,X2)
         activate(n__terms(X)) -> terms(X)
         activate(n__add(X1,X2)) -> add(X1,X2)
         activate(n__s(X)) -> s(X)
         activate(n__dbl(X)) -> dbl(X)
         activate(n__first(X1,X2)) -> first(X1,X2)
         activate(X) -> X
       Restore Modifier:
        DPs:
         first#(s(X),cons(Y,Z)) -> activate#(Z)
         activate#(n__terms(X)) -> terms#(X)
         terms#(N) -> sqr#(N)
         sqr#(s(X)) -> dbl#(activate(X))
         activate#(n__add(X1,X2)) -> add#(X1,X2)
         add#(s(X),Y) -> activate#(X)
         activate#(n__dbl(X)) -> dbl#(X)
         activate#(n__first(X1,X2)) -> first#(X1,X2)
         sqr#(s(X)) -> sqr#(activate(X))
        TRS:
         terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
         sqr(0()) -> 0()
         sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
         dbl(0()) -> 0()
         dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
         add(0(),X) -> X
         add(s(X),Y) -> s(n__add(activate(X),Y))
         first(0(),X) -> nil()
         first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
         terms(X) -> n__terms(X)
         add(X1,X2) -> n__add(X1,X2)
         s(X) -> n__s(X)
         dbl(X) -> n__dbl(X)
         first(X1,X2) -> n__first(X1,X2)
         activate(n__terms(X)) -> terms(X)
         activate(n__add(X1,X2)) -> add(X1,X2)
         activate(n__s(X)) -> s(X)
         activate(n__dbl(X)) -> dbl(X)
         activate(n__first(X1,X2)) -> first(X1,X2)
         activate(X) -> X
        SCC Processor:
         #sccs: 2
         #rules: 5
         #arcs: 32/81
         DPs:
          first#(s(X),cons(Y,Z)) -> activate#(Z)
          activate#(n__add(X1,X2)) -> add#(X1,X2)
          add#(s(X),Y) -> activate#(X)
          activate#(n__first(X1,X2)) -> first#(X1,X2)
         TRS:
          terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
          sqr(0()) -> 0()
          sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
          dbl(0()) -> 0()
          dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
          add(0(),X) -> X
          add(s(X),Y) -> s(n__add(activate(X),Y))
          first(0(),X) -> nil()
          first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
          terms(X) -> n__terms(X)
          add(X1,X2) -> n__add(X1,X2)
          s(X) -> n__s(X)
          dbl(X) -> n__dbl(X)
          first(X1,X2) -> n__first(X1,X2)
          activate(n__terms(X)) -> terms(X)
          activate(n__add(X1,X2)) -> add(X1,X2)
          activate(n__s(X)) -> s(X)
          activate(n__dbl(X)) -> dbl(X)
          activate(n__first(X1,X2)) -> first(X1,X2)
          activate(X) -> X
         Subterm Criterion Processor:
          simple projection:
           pi(activate#) = 0
           pi(add#) = 0
           pi(first#) = 1
          problem:
           DPs:
            
           TRS:
            terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
            add(0(),X) -> X
            add(s(X),Y) -> s(n__add(activate(X),Y))
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
            terms(X) -> n__terms(X)
            add(X1,X2) -> n__add(X1,X2)
            s(X) -> n__s(X)
            dbl(X) -> n__dbl(X)
            first(X1,X2) -> n__first(X1,X2)
            activate(n__terms(X)) -> terms(X)
            activate(n__add(X1,X2)) -> add(X1,X2)
            activate(n__s(X)) -> s(X)
            activate(n__dbl(X)) -> dbl(X)
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(X) -> X
          Qed
         
         DPs:
          sqr#(s(X)) -> sqr#(activate(X))
         TRS:
          terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
          sqr(0()) -> 0()
          sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
          dbl(0()) -> 0()
          dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
          add(0(),X) -> X
          add(s(X),Y) -> s(n__add(activate(X),Y))
          first(0(),X) -> nil()
          first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
          terms(X) -> n__terms(X)
          add(X1,X2) -> n__add(X1,X2)
          s(X) -> n__s(X)
          dbl(X) -> n__dbl(X)
          first(X1,X2) -> n__first(X1,X2)
          activate(n__terms(X)) -> terms(X)
          activate(n__add(X1,X2)) -> add(X1,X2)
          activate(n__s(X)) -> s(X)
          activate(n__dbl(X)) -> dbl(X)
          activate(n__first(X1,X2)) -> first(X1,X2)
          activate(X) -> X
         Matrix Interpretation Processor: dim=1
          
          usable rules:
           terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
           dbl(0()) -> 0()
           dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
           add(0(),X) -> X
           add(s(X),Y) -> s(n__add(activate(X),Y))
           first(0(),X) -> nil()
           first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
           terms(X) -> n__terms(X)
           add(X1,X2) -> n__add(X1,X2)
           s(X) -> n__s(X)
           dbl(X) -> n__dbl(X)
           first(X1,X2) -> n__first(X1,X2)
           activate(n__terms(X)) -> terms(X)
           activate(n__add(X1,X2)) -> add(X1,X2)
           activate(n__s(X)) -> s(X)
           activate(n__dbl(X)) -> dbl(X)
           activate(n__first(X1,X2)) -> first(X1,X2)
           activate(X) -> X
          interpretation:
           [sqr#](x0) = 2x0 + 1,
           
           [n__terms](x0) = 0,
           
           [n__first](x0, x1) = x0 + 4x1,
           
           [dbl](x0) = 4x0 + 6,
           
           [sqr](x0) = 2x0 + 2,
           
           [s](x0) = 2x0 + 2,
           
           [cons](x0, x1) = 2x0,
           
           [n__dbl](x0) = 2x0 + 5,
           
           [terms](x0) = 0,
           
           [activate](x0) = 2x0,
           
           [nil] = 0,
           
           [add](x0, x1) = 2x0 + 2x1,
           
           [n__s](x0) = x0 + 1,
           
           [n__add](x0, x1) = x0 + x1,
           
           [first](x0, x1) = x0 + 7x1,
           
           [0] = 6,
           
           [recip](x0) = 0
          orientation:
           sqr#(s(X)) = 4X + 5 >= 4X + 1 = sqr#(activate(X))
           
           terms(N) = 0 >= 0 = cons(recip(sqr(N)),n__terms(s(N)))
           
           sqr(0()) = 14 >= 6 = 0()
           
           sqr(s(X)) = 4X + 6 >= 24X + 18 = s(n__add(sqr(activate(X)),dbl(activate(X))))
           
           dbl(0()) = 30 >= 6 = 0()
           
           dbl(s(X)) = 8X + 14 >= 8X + 14 = s(n__s(n__dbl(activate(X))))
           
           add(0(),X) = 2X + 12 >= X = X
           
           add(s(X),Y) = 4X + 2Y + 4 >= 4X + 2Y + 2 = s(n__add(activate(X),Y))
           
           first(0(),X) = 7X + 6 >= 0 = nil()
           
           first(s(X),cons(Y,Z)) = 2X + 14Y + 2 >= 2Y = cons(Y,n__first(activate(X),activate(Z)))
           
           terms(X) = 0 >= 0 = n__terms(X)
           
           add(X1,X2) = 2X1 + 2X2 >= X1 + X2 = n__add(X1,X2)
           
           s(X) = 2X + 2 >= X + 1 = n__s(X)
           
           dbl(X) = 4X + 6 >= 2X + 5 = n__dbl(X)
           
           first(X1,X2) = X1 + 7X2 >= X1 + 4X2 = n__first(X1,X2)
           
           activate(n__terms(X)) = 0 >= 0 = terms(X)
           
           activate(n__add(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = add(X1,X2)
           
           activate(n__s(X)) = 2X + 2 >= 2X + 2 = s(X)
           
           activate(n__dbl(X)) = 4X + 10 >= 4X + 6 = dbl(X)
           
           activate(n__first(X1,X2)) = 2X1 + 8X2 >= X1 + 7X2 = first(X1,X2)
           
           activate(X) = 2X >= X = X
          problem:
           DPs:
            
           TRS:
            terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
            add(0(),X) -> X
            add(s(X),Y) -> s(n__add(activate(X),Y))
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
            terms(X) -> n__terms(X)
            add(X1,X2) -> n__add(X1,X2)
            s(X) -> n__s(X)
            dbl(X) -> n__dbl(X)
            first(X1,X2) -> n__first(X1,X2)
            activate(n__terms(X)) -> terms(X)
            activate(n__add(X1,X2)) -> add(X1,X2)
            activate(n__s(X)) -> s(X)
            activate(n__dbl(X)) -> dbl(X)
            activate(n__first(X1,X2)) -> first(X1,X2)
            activate(X) -> X
          Qed