YES

1 decompositions
 #1 -----------
   2: f(x1) -> g(x1,f(x1))

@Development Closedness
--- R
   2: f(x1) -> g(x1,f(x1))

--- S
   2: f(x1) -> g(x1,f(x1))


NOTE: input TRS is reduced

original is
   1: f(f(f(f(x1)))) -> f(f(f(g(x1,f(x1)))))
   2: f(x1) -> g(x1,f(x1))

reduced to
   2: f(x1) -> g(x1,f(x1))