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(z, x, y, z) -> x
  ?1(z, x, y) -> ?2(g(y), x, y, z)
  f(x, y) -> ?1(g(x), x, y)
  ?3(c, x) -> c
  g(x) -> ?3(d, x)

 DP Processor:
  DPs:
   ?1#(z,x,y) -> g#(y)
   ?1#(z,x,y) -> ?2#(g(y),x,y,z)
   f#(x,y) -> g#(x)
   f#(x,y) -> ?1#(g(x),x,y)
   g#(x) -> ?3#(d(),x)
  TRS:
   ?2(z,x,y,z) -> x
   ?1(z,x,y) -> ?2(g(y),x,y,z)
   f(x,y) -> ?1(g(x),x,y)
   ?3(c(),x) -> c()
   g(x) -> ?3(d(),x)
  TDG Processor:
   DPs:
    ?1#(z,x,y) -> g#(y)
    ?1#(z,x,y) -> ?2#(g(y),x,y,z)
    f#(x,y) -> g#(x)
    f#(x,y) -> ?1#(g(x),x,y)
    g#(x) -> ?3#(d(),x)
   TRS:
    ?2(z,x,y,z) -> x
    ?1(z,x,y) -> ?2(g(y),x,y,z)
    f(x,y) -> ?1(g(x),x,y)
    ?3(c(),x) -> c()
    g(x) -> ?3(d(),x)
   graph:
    f#(x,y) -> g#(x) -> g#(x) -> ?3#(d(),x)
    f#(x,y) -> ?1#(g(x),x,y) -> ?1#(z,x,y) -> ?2#(g(y),x,y,z)
    f#(x,y) -> ?1#(g(x),x,y) -> ?1#(z,x,y) -> g#(y)
    ?1#(z,x,y) -> g#(y) -> g#(x) -> ?3#(d(),x)
   SCC Processor:
    #sccs: 0
    #rules: 0
    #arcs: 4/25