YES
Input TRS:
      1: g(c(x1)) -> g(f(c(x1)))
      2: g(f(c(x1))) -> g(f(f(c(x1))))
      3: g(g(x1)) -> g(f(g(x1)))
      4: f(f(g(x1))) -> g(f(x1))
Number of strict rules: 4
Direct POLO(bPol) ... failed.
Uncurrying ... failed.
Dependency Pairs:
     #1: #g(f(c(x1))) -> #g(f(f(c(x1))))
     #2: #g(f(c(x1))) -> #f(f(c(x1)))
     #3: #g(g(x1)) -> #g(f(g(x1)))
     #4: #g(g(x1)) -> #f(g(x1))
     #5: #g(c(x1)) -> #g(f(c(x1)))
     #6: #g(c(x1)) -> #f(c(x1))
     #7: #f(f(g(x1))) -> #g(f(x1))
     #8: #f(f(g(x1))) -> #f(x1)
Number of SCCs: 1, DPs: 1
  SCC { #8 }
POLO(Sum)... succeeded.
      c 	w: 0
      f 	w: x1
      #f 	w: x1
      #g 	w: 0
      g 	w: x1 + 1
    USABLE RULES: { }
    Removed DPs: #8
Number of SCCs: 0, DPs: 0