YES

Proof:
This system is quasi-decreasing.
By \cite{A14}, Theorem 11.5.9.
This system is of type 3 or smaller.
This system is deterministic.
System R transformed to V(R) + Emb.
Call external tool:
ttt2 - trs 30
Input:
  f(x, y) -> x
  f(x, y) -> g(g(x))
  f(x, y) -> g(x)
  f(x, y) -> y
  f(x, y) -> g(g(y))
  f(x, y) -> g(y)
  g(x) -> x
  f(x, y) -> x
  f(x, y) -> y

 DP Processor:
  DPs:
   f#(x,y) -> g#(x)
   f#(x,y) -> g#(g(x))
   f#(x,y) -> g#(y)
   f#(x,y) -> g#(g(y))
  TRS:
   f(x,y) -> x
   f(x,y) -> g(g(x))
   f(x,y) -> g(x)
   f(x,y) -> y
   f(x,y) -> g(g(y))
   f(x,y) -> g(y)
   g(x) -> x
  TDG Processor:
   DPs:
    f#(x,y) -> g#(x)
    f#(x,y) -> g#(g(x))
    f#(x,y) -> g#(y)
    f#(x,y) -> g#(g(y))
   TRS:
    f(x,y) -> x
    f(x,y) -> g(g(x))
    f(x,y) -> g(x)
    f(x,y) -> y
    f(x,y) -> g(g(y))
    f(x,y) -> g(y)
    g(x) -> x
   graph:
    
   Qed