YES

Problem:
 a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
 a__sqr(0()) -> 0()
 a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
 a__dbl(0()) -> 0()
 a__dbl(s(X)) -> s(s(dbl(X)))
 a__add(0(),X) -> mark(X)
 a__add(s(X),Y) -> s(add(X,Y))
 a__first(0(),X) -> nil()
 a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
 mark(terms(X)) -> a__terms(mark(X))
 mark(sqr(X)) -> a__sqr(mark(X))
 mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
 mark(dbl(X)) -> a__dbl(mark(X))
 mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(recip(X)) -> recip(mark(X))
 mark(s(X)) -> s(X)
 mark(0()) -> 0()
 mark(nil()) -> nil()
 a__terms(X) -> terms(X)
 a__sqr(X) -> sqr(X)
 a__add(X1,X2) -> add(X1,X2)
 a__dbl(X) -> dbl(X)
 a__first(X1,X2) -> first(X1,X2)

Proof:
 DP Processor:
  DPs:
   a__terms#(N) -> mark#(N)
   a__terms#(N) -> a__sqr#(mark(N))
   a__add#(0(),X) -> mark#(X)
   a__first#(s(X),cons(Y,Z)) -> mark#(Y)
   mark#(terms(X)) -> mark#(X)
   mark#(terms(X)) -> a__terms#(mark(X))
   mark#(sqr(X)) -> mark#(X)
   mark#(sqr(X)) -> a__sqr#(mark(X))
   mark#(add(X1,X2)) -> mark#(X2)
   mark#(add(X1,X2)) -> mark#(X1)
   mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2))
   mark#(dbl(X)) -> mark#(X)
   mark#(dbl(X)) -> a__dbl#(mark(X))
   mark#(first(X1,X2)) -> mark#(X2)
   mark#(first(X1,X2)) -> mark#(X1)
   mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
   mark#(cons(X1,X2)) -> mark#(X1)
   mark#(recip(X)) -> mark#(X)
  TRS:
   a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
   a__sqr(0()) -> 0()
   a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
   a__dbl(0()) -> 0()
   a__dbl(s(X)) -> s(s(dbl(X)))
   a__add(0(),X) -> mark(X)
   a__add(s(X),Y) -> s(add(X,Y))
   a__first(0(),X) -> nil()
   a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
   mark(terms(X)) -> a__terms(mark(X))
   mark(sqr(X)) -> a__sqr(mark(X))
   mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
   mark(dbl(X)) -> a__dbl(mark(X))
   mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
   mark(cons(X1,X2)) -> cons(mark(X1),X2)
   mark(recip(X)) -> recip(mark(X))
   mark(s(X)) -> s(X)
   mark(0()) -> 0()
   mark(nil()) -> nil()
   a__terms(X) -> terms(X)
   a__sqr(X) -> sqr(X)
   a__add(X1,X2) -> add(X1,X2)
   a__dbl(X) -> dbl(X)
   a__first(X1,X2) -> first(X1,X2)
  Matrix Interpretation Processor: dim=1
   
   usable rules:
    a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
    a__sqr(0()) -> 0()
    a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
    a__dbl(0()) -> 0()
    a__dbl(s(X)) -> s(s(dbl(X)))
    a__add(0(),X) -> mark(X)
    a__add(s(X),Y) -> s(add(X,Y))
    a__first(0(),X) -> nil()
    a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
    mark(terms(X)) -> a__terms(mark(X))
    mark(sqr(X)) -> a__sqr(mark(X))
    mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
    mark(dbl(X)) -> a__dbl(mark(X))
    mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
    mark(cons(X1,X2)) -> cons(mark(X1),X2)
    mark(recip(X)) -> recip(mark(X))
    mark(s(X)) -> s(X)
    mark(0()) -> 0()
    mark(nil()) -> nil()
    a__terms(X) -> terms(X)
    a__sqr(X) -> sqr(X)
    a__add(X1,X2) -> add(X1,X2)
    a__dbl(X) -> dbl(X)
    a__first(X1,X2) -> first(X1,X2)
   interpretation:
    [a__first#](x0, x1) = 2x1 + 3,
    
    [a__add#](x0, x1) = x1 + 2,
    
    [a__dbl#](x0) = x0,
    
    [a__sqr#](x0) = 0,
    
    [mark#](x0) = x0,
    
    [a__terms#](x0) = x0 + 1,
    
    [first](x0, x1) = 4x0 + 2x1 + 4,
    
    [nil] = 2,
    
    [a__first](x0, x1) = 4x0 + 2x1 + 4,
    
    [a__add](x0, x1) = x0 + x1 + 6,
    
    [a__dbl](x0) = x0 + 1,
    
    [add](x0, x1) = x0 + x1 + 6,
    
    [dbl](x0) = x0 + 1,
    
    [sqr](x0) = x0 + 2,
    
    [0] = 4,
    
    [cons](x0, x1) = x0 + 1,
    
    [terms](x0) = 2x0 + 7,
    
    [s](x0) = 0,
    
    [recip](x0) = 2x0 + 2,
    
    [a__sqr](x0) = x0 + 2,
    
    [mark](x0) = x0,
    
    [a__terms](x0) = 2x0 + 7
   orientation:
    a__terms#(N) = N + 1 >= N = mark#(N)
    
    a__terms#(N) = N + 1 >= 0 = a__sqr#(mark(N))
    
    a__add#(0(),X) = X + 2 >= X = mark#(X)
    
    a__first#(s(X),cons(Y,Z)) = 2Y + 5 >= Y = mark#(Y)
    
    mark#(terms(X)) = 2X + 7 >= X = mark#(X)
    
    mark#(terms(X)) = 2X + 7 >= X + 1 = a__terms#(mark(X))
    
    mark#(sqr(X)) = X + 2 >= X = mark#(X)
    
    mark#(sqr(X)) = X + 2 >= 0 = a__sqr#(mark(X))
    
    mark#(add(X1,X2)) = X1 + X2 + 6 >= X2 = mark#(X2)
    
    mark#(add(X1,X2)) = X1 + X2 + 6 >= X1 = mark#(X1)
    
    mark#(add(X1,X2)) = X1 + X2 + 6 >= X2 + 2 = a__add#(mark(X1),mark(X2))
    
    mark#(dbl(X)) = X + 1 >= X = mark#(X)
    
    mark#(dbl(X)) = X + 1 >= X = a__dbl#(mark(X))
    
    mark#(first(X1,X2)) = 4X1 + 2X2 + 4 >= X2 = mark#(X2)
    
    mark#(first(X1,X2)) = 4X1 + 2X2 + 4 >= X1 = mark#(X1)
    
    mark#(first(X1,X2)) = 4X1 + 2X2 + 4 >= 2X2 + 3 = a__first#(mark(X1),mark(X2))
    
    mark#(cons(X1,X2)) = X1 + 1 >= X1 = mark#(X1)
    
    mark#(recip(X)) = 2X + 2 >= X = mark#(X)
    
    a__terms(N) = 2N + 7 >= 2N + 7 = cons(recip(a__sqr(mark(N))),terms(s(N)))
    
    a__sqr(0()) = 6 >= 4 = 0()
    
    a__sqr(s(X)) = 2 >= 0 = s(add(sqr(X),dbl(X)))
    
    a__dbl(0()) = 5 >= 4 = 0()
    
    a__dbl(s(X)) = 1 >= 0 = s(s(dbl(X)))
    
    a__add(0(),X) = X + 10 >= X = mark(X)
    
    a__add(s(X),Y) = Y + 6 >= 0 = s(add(X,Y))
    
    a__first(0(),X) = 2X + 20 >= 2 = nil()
    
    a__first(s(X),cons(Y,Z)) = 2Y + 6 >= Y + 1 = cons(mark(Y),first(X,Z))
    
    mark(terms(X)) = 2X + 7 >= 2X + 7 = a__terms(mark(X))
    
    mark(sqr(X)) = X + 2 >= X + 2 = a__sqr(mark(X))
    
    mark(add(X1,X2)) = X1 + X2 + 6 >= X1 + X2 + 6 = a__add(mark(X1),mark(X2))
    
    mark(dbl(X)) = X + 1 >= X + 1 = a__dbl(mark(X))
    
    mark(first(X1,X2)) = 4X1 + 2X2 + 4 >= 4X1 + 2X2 + 4 = a__first(mark(X1),mark(X2))
    
    mark(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(mark(X1),X2)
    
    mark(recip(X)) = 2X + 2 >= 2X + 2 = recip(mark(X))
    
    mark(s(X)) = 0 >= 0 = s(X)
    
    mark(0()) = 4 >= 4 = 0()
    
    mark(nil()) = 2 >= 2 = nil()
    
    a__terms(X) = 2X + 7 >= 2X + 7 = terms(X)
    
    a__sqr(X) = X + 2 >= X + 2 = sqr(X)
    
    a__add(X1,X2) = X1 + X2 + 6 >= X1 + X2 + 6 = add(X1,X2)
    
    a__dbl(X) = X + 1 >= X + 1 = dbl(X)
    
    a__first(X1,X2) = 4X1 + 2X2 + 4 >= 4X1 + 2X2 + 4 = first(X1,X2)
   problem:
    DPs:
     
    TRS:
     a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
     a__sqr(0()) -> 0()
     a__sqr(s(X)) -> s(add(sqr(X),dbl(X)))
     a__dbl(0()) -> 0()
     a__dbl(s(X)) -> s(s(dbl(X)))
     a__add(0(),X) -> mark(X)
     a__add(s(X),Y) -> s(add(X,Y))
     a__first(0(),X) -> nil()
     a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
     mark(terms(X)) -> a__terms(mark(X))
     mark(sqr(X)) -> a__sqr(mark(X))
     mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
     mark(dbl(X)) -> a__dbl(mark(X))
     mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
     mark(cons(X1,X2)) -> cons(mark(X1),X2)
     mark(recip(X)) -> recip(mark(X))
     mark(s(X)) -> s(X)
     mark(0()) -> 0()
     mark(nil()) -> nil()
     a__terms(X) -> terms(X)
     a__sqr(X) -> sqr(X)
     a__add(X1,X2) -> add(X1,X2)
     a__dbl(X) -> dbl(X)
     a__first(X1,X2) -> first(X1,X2)
   Qed