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
  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:
    [first#](x0, x1) = x0 + x1,
    
    [add#](x0, x1) = x0,
    
    [activate#](x0) = x0,
    
    [dbl#](x0) = x0,
    
    [sqr#](x0) = 2x0,
    
    [s#](x0) = 0,
    
    [terms#](x0) = 2x0 + 1,
    
    [n__first](x0, x1) = 2x0 + x1 + 4,
    
    [nil] = 0,
    
    [first](x0, x1) = 4x0 + 2x1 + 6,
    
    [add](x0, x1) = 2x0 + 2x1 + 2,
    
    [n__s](x0) = x0 + 1,
    
    [n__dbl](x0) = 2x0 + 4,
    
    [n__add](x0, x1) = x0 + x1 + 1,
    
    [dbl](x0) = 4x0 + 7,
    
    [activate](x0) = 2x0,
    
    [0] = 5,
    
    [cons](x0, x1) = 2x0 + x1,
    
    [n__terms](x0) = 2x0 + 2,
    
    [s](x0) = 2x0 + 1,
    
    [recip](x0) = 0,
    
    [sqr](x0) = x0,
    
    [terms](x0) = 4x0 + 4
   orientation:
    terms#(N) = 2N + 1 >= 0 = s#(N)
    
    terms#(N) = 2N + 1 >= 2N = sqr#(N)
    
    sqr#(s(X)) = 4X + 2 >= 2X = dbl#(activate(X))
    
    sqr#(s(X)) = 4X + 2 >= X = activate#(X)
    
    sqr#(s(X)) = 4X + 2 >= 4X = sqr#(activate(X))
    
    sqr#(s(X)) = 4X + 2 >= 0 = s#(n__add(sqr(activate(X)),dbl(activate(X))))
    
    dbl#(s(X)) = 2X + 1 >= X = activate#(X)
    
    dbl#(s(X)) = 2X + 1 >= 0 = s#(n__s(n__dbl(activate(X))))
    
    add#(s(X),Y) = 2X + 1 >= X = activate#(X)
    
    add#(s(X),Y) = 2X + 1 >= 0 = s#(n__add(activate(X),Y))
    
    first#(s(X),cons(Y,Z)) = 2X + 2Y + Z + 1 >= Z = activate#(Z)
    
    first#(s(X),cons(Y,Z)) = 2X + 2Y + Z + 1 >= X = activate#(X)
    
    activate#(n__terms(X)) = 2X + 2 >= 2X + 1 = terms#(X)
    
    activate#(n__add(X1,X2)) = X1 + X2 + 1 >= X1 = add#(X1,X2)
    
    activate#(n__s(X)) = X + 1 >= 0 = s#(X)
    
    activate#(n__dbl(X)) = 2X + 4 >= X = dbl#(X)
    
    activate#(n__first(X1,X2)) = 2X1 + X2 + 4 >= X1 + X2 = first#(X1,X2)
    
    terms(N) = 4N + 4 >= 4N + 4 = cons(recip(sqr(N)),n__terms(s(N)))
    
    sqr(0()) = 5 >= 5 = 0()
    
    sqr(s(X)) = 2X + 1 >= 20X + 17 = s(n__add(sqr(activate(X)),dbl(activate(X))))
    
    dbl(0()) = 27 >= 5 = 0()
    
    dbl(s(X)) = 8X + 11 >= 8X + 11 = s(n__s(n__dbl(activate(X))))
    
    add(0(),X) = 2X + 12 >= X = X
    
    add(s(X),Y) = 4X + 2Y + 4 >= 4X + 2Y + 3 = s(n__add(activate(X),Y))
    
    first(0(),X) = 2X + 26 >= 0 = nil()
    
    first(s(X),cons(Y,Z)) = 8X + 4Y + 2Z + 10 >= 4X + 2Y + 2Z + 4 = cons(Y,n__first(activate(X),activate(Z)))
    
    terms(X) = 4X + 4 >= 2X + 2 = n__terms(X)
    
    add(X1,X2) = 2X1 + 2X2 + 2 >= X1 + X2 + 1 = n__add(X1,X2)
    
    s(X) = 2X + 1 >= X + 1 = n__s(X)
    
    dbl(X) = 4X + 7 >= 2X + 4 = n__dbl(X)
    
    first(X1,X2) = 4X1 + 2X2 + 6 >= 2X1 + X2 + 4 = n__first(X1,X2)
    
    activate(n__terms(X)) = 4X + 4 >= 4X + 4 = terms(X)
    
    activate(n__add(X1,X2)) = 2X1 + 2X2 + 2 >= 2X1 + 2X2 + 2 = add(X1,X2)
    
    activate(n__s(X)) = 2X + 2 >= 2X + 1 = s(X)
    
    activate(n__dbl(X)) = 4X + 8 >= 4X + 7 = dbl(X)
    
    activate(n__first(X1,X2)) = 4X1 + 2X2 + 8 >= 4X1 + 2X2 + 6 = 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