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 -> b
  f(x) -> A
  f(x) -> x
  g(x, x) -> h(x)
  h(x) -> i(x)
  h(x) -> x
  g(x, y) -> x
  g(x, y) -> y
  f(x) -> x
  i(x) -> x

 DP Processor:
  DPs:
   g#(x,x) -> h#(x)
   h#(x) -> i#(x)
  TRS:
   a() -> b()
   f(x) -> A()
   f(x) -> x
   g(x,x) -> h(x)
   h(x) -> i(x)
   h(x) -> x
   g(x,y) -> x
   g(x,y) -> y
   i(x) -> x
  TDG Processor:
   DPs:
    g#(x,x) -> h#(x)
    h#(x) -> i#(x)
   TRS:
    a() -> b()
    f(x) -> A()
    f(x) -> x
    g(x,x) -> h(x)
    h(x) -> i(x)
    h(x) -> x
    g(x,y) -> x
    g(x,y) -> y
    i(x) -> x
   graph:
    g#(x,x) -> h#(x) -> h#(x) -> i#(x)
   SCC Processor:
    #sccs: 0
    #rules: 0
    #arcs: 1/4