YES

Proof:
This system is quasi-decreasing.
By \cite{O02}, p. 214, Proposition 7.2.50.
This system is of type 3 or smaller.
This system is deterministic.
System R transformed to U(R).
Call external tool:
ttt2 - trs 30
Input:
  ?2(h(x), x, y) -> c(x, y)
  ?1(y, x) -> ?2(y, x, y)
  f(x) -> ?1(g(x), x)
  a -> b
  g(a) -> h(b)

 DP Processor:
  DPs:
   ?1#(y,x) -> ?2#(y,x,y)
   f#(x) -> g#(x)
   f#(x) -> ?1#(g(x),x)
  TRS:
   ?2(h(x),x,y) -> c(x,y)
   ?1(y,x) -> ?2(y,x,y)
   f(x) -> ?1(g(x),x)
   a() -> b()
   g(a()) -> h(b())
  TDG Processor:
   DPs:
    ?1#(y,x) -> ?2#(y,x,y)
    f#(x) -> g#(x)
    f#(x) -> ?1#(g(x),x)
   TRS:
    ?2(h(x),x,y) -> c(x,y)
    ?1(y,x) -> ?2(y,x,y)
    f(x) -> ?1(g(x),x)
    a() -> b()
    g(a()) -> h(b())
   graph:
    f#(x) -> ?1#(g(x),x) -> ?1#(y,x) -> ?2#(y,x,y)
   SCC Processor:
    #sccs: 0
    #rules: 0
    #arcs: 1/9