YES

   1: f(x1) -> g(k(x1))
   3: g(x1) -> a()
   4: k(a()) -> k(k(a()))

@Development Closedness
--- R
   1: f(x1) -> g(k(x1))
   3: g(x1) -> a()
   4: k(a()) -> k(k(a()))

--- S
   1: f(x1) -> g(k(x1))
   3: g(x1) -> a()
   4: k(a()) -> k(k(a()))


NOTE: input TRS is reduced

original is
   1: f(x1) -> g(k(x1))
   2: f(x1) -> a()
   3: g(x1) -> a()
   4: k(a()) -> k(k(a()))

reduced to
   1: f(x1) -> g(k(x1))
   3: g(x1) -> a()
   4: k(a()) -> k(k(a()))