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

 DP Processor:
  DPs:
   f#(x) -> g#(x)
   f#(x) -> c#(x,g(x))
   g#(a()) -> h#(b())
  TRS:
   f(x) -> c(x,g(x))
   f(x) -> g(x)
   a() -> b()
   g(a()) -> h(b())
   h(x) -> x
   c(x,y) -> x
   c(x,y) -> y
   g(x) -> x
   f(x) -> x
  TDG Processor:
   DPs:
    f#(x) -> g#(x)
    f#(x) -> c#(x,g(x))
    g#(a()) -> h#(b())
   TRS:
    f(x) -> c(x,g(x))
    f(x) -> g(x)
    a() -> b()
    g(a()) -> h(b())
    h(x) -> x
    c(x,y) -> x
    c(x,y) -> y
    g(x) -> x
    f(x) -> x
   graph:
    f#(x) -> g#(x) -> g#(a()) -> h#(b())
   SCC Processor:
    #sccs: 0
    #rules: 0
    #arcs: 1/9