YES

1 decompositions
 #0 -----------
   1: g(g(a())) -> f(h(g(f(c(),c()))),f(f(g(c()),a()),g(f(a(),a()))))
   2: f(x,y) -> f(y,x)
   3: c() -> f(a(),h(b()))
   4: f(h(f(f(a(),a()),h(a()))),g(f(x,g(b())))) -> c()

@Jouannaud and Kirchner's criterion
--- R
   1: g(g(a())) -> f(h(g(f(c(),c()))),f(f(g(c()),a()),g(f(a(),a()))))
   2: f(x,y) -> f(y,x)
   3: c() -> f(a(),h(b()))
   4: f(h(f(f(a(),a()),h(a()))),g(f(x,g(b())))) -> c()

--- S
   1: g(g(a())) -> f(h(g(f(c(),c()))),f(f(g(c()),a()),g(f(a(),a()))))
   2: f(x,y) -> f(y,x)
   3: c() -> f(a(),h(b()))
   4: f(h(f(f(a(),a()),h(a()))),g(f(x,g(b())))) -> c()