YES

(VAR x y z)
(RULES
  g(x) -> b(f(s(x)),f(x))
  u(b(x,0())) -> x
  u(b(x,s(y))) -> s(u(b(x,y)))
  p(x,y) -> u(b(x,y))
  n(b(x,y)) -> b(u(b(x,y)),x)
  f(s(s(z))) -> u(b(f(s(z)),f(z)))
  f(s(0())) -> s(0())
  f(0()) -> 0()
)

(COMMENT
Termination is shown by ELPO with interpretations on natural numbers

  p_A(x1,x2) = x1
  0_A = 41379
  s_A(x1) = x1
  f_A(x1) = 41380
  g_A(x1) = 41380
  b_A(x1,x2) = x1
  n_A(x1) = x1
  u_A(x1) = x1
  p#_A(x1,x2) = x2
  0#_A = 0
  s#_A(x1) = 0
  f#_A(x1) = x1
  g#_A(x1) = x1
  n#_A(x1) = x1
  u#_A(x1) = 0

and precedence:

n > g > f > p > b > s > u > 0
)