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:
  a -> c
  a -> d
  ?2(z, x, y) -> z
  ?1(x, x, y) -> ?2(y, x, y)
  f(x, y) -> ?1(c, x, y)

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