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(x, x', x'', x) -> g(x)
  ?1(x, x', x'') -> ?2(x'', x', x'', x)
  f(x', x'') -> ?1(x', x', x'')
  ?4(y, y', y'', y) -> g(y)
  ?3(y, y', y'') -> ?4(y'', y', y'', y)
  f(y', h(y'')) -> ?3(y', y', y'')

 DP Processor:
  DPs:
   ?1#(x,x',x'') -> ?2#(x'',x',x'',x)
   f#(x',x'') -> ?1#(x',x',x'')
   ?3#(y,y',y'') -> ?4#(y'',y',y'',y)
   f#(y',h(y'')) -> ?3#(y',y',y'')
  TRS:
   ?2(x,x',x'',x) -> g(x)
   ?1(x,x',x'') -> ?2(x'',x',x'',x)
   f(x',x'') -> ?1(x',x',x'')
   ?4(y,y',y'',y) -> g(y)
   ?3(y,y',y'') -> ?4(y'',y',y'',y)
   f(y',h(y'')) -> ?3(y',y',y'')
  TDG Processor:
   DPs:
    ?1#(x,x',x'') -> ?2#(x'',x',x'',x)
    f#(x',x'') -> ?1#(x',x',x'')
    ?3#(y,y',y'') -> ?4#(y'',y',y'',y)
    f#(y',h(y'')) -> ?3#(y',y',y'')
   TRS:
    ?2(x,x',x'',x) -> g(x)
    ?1(x,x',x'') -> ?2(x'',x',x'',x)
    f(x',x'') -> ?1(x',x',x'')
    ?4(y,y',y'',y) -> g(y)
    ?3(y,y',y'') -> ?4(y'',y',y'',y)
    f(y',h(y'')) -> ?3(y',y',y'')
   graph:
    f#(x',x'') -> ?1#(x',x',x'') ->
    ?1#(x,x',x'') -> ?2#(x'',x',x'',x)
    f#(y',h(y'')) -> ?3#(y',y',y'') -> ?3#(y,y',y'') -> ?4#(y'',y',y'',y)
   SCC Processor:
    #sccs: 0
    #rules: 0
    #arcs: 2/16