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:
  a -> c
  a -> d
  b -> c
  b -> d
  f(x) -> x
  g(x, x) -> h(x, x)
  h(x, f(x)) -> x
  h(x, y) -> x
  h(x, y) -> y
  g(x, y) -> x
  g(x, y) -> y
  f(x) -> x

 DP Processor:
  DPs:
   g#(x,x) -> h#(x,x)
  TRS:
   a() -> c()
   a() -> d()
   b() -> c()
   b() -> d()
   f(x) -> x
   g(x,x) -> h(x,x)
   h(x,f(x)) -> x
   h(x,y) -> x
   h(x,y) -> y
   g(x,y) -> x
   g(x,y) -> y
  TDG Processor:
   DPs:
    g#(x,x) -> h#(x,x)
   TRS:
    a() -> c()
    a() -> d()
    b() -> c()
    b() -> d()
    f(x) -> x
    g(x,x) -> h(x,x)
    h(x,f(x)) -> x
    h(x,y) -> x
    h(x,y) -> y
    g(x,y) -> x
    g(x,y) -> y
   graph:
    
   Qed