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(y, x, x', y) -> g(y, y)
  ?1(y, x, x') -> ?2(x', x, x', y)
  f(x, x') -> ?1(x, x, x')
  ?5(y, x, x', x'', y) -> c
  ?4(y, x, x', x'', y) -> ?5(x'', x, x', x'', y)
  ?3(y, x, x', x'') -> ?4(x', x, x', x'', y)
  h(x, x', x'') -> ?3(x, x, x', x'')

 DP Processor:
  DPs:
   ?1#(y,x,x') -> ?2#(x',x,x',y)
   f#(x,x') -> ?1#(x,x,x')
   ?4#(y,x,x',x'',y) -> ?5#(x'',x,x',x'',y)
   ?3#(y,x,x',x'') -> ?4#(x',x,x',x'',y)
   h#(x,x',x'') -> ?3#(x,x,x',x'')
  TRS:
   ?2(y,x,x',y) -> g(y,y)
   ?1(y,x,x') -> ?2(x',x,x',y)
   f(x,x') -> ?1(x,x,x')
   ?5(y,x,x',x'',y) -> c()
   ?4(y,x,x',x'',y) -> ?5(x'',x,x',x'',y)
   ?3(y,x,x',x'') -> ?4(x',x,x',x'',y)
   h(x,x',x'') -> ?3(x,x,x',x'')
  TDG Processor:
   DPs:
    ?1#(y,x,x') -> ?2#(x',x,x',y)
    f#(x,x') -> ?1#(x,x,x')
    ?4#(y,x,x',x'',y) -> ?5#(x'',x,x',x'',y)
    ?3#(y,x,x',x'') -> ?4#(x',x,x',x'',y)
    h#(x,x',x'') -> ?3#(x,x,x',x'')
   TRS:
    ?2(y,x,x',y) -> g(y,y)
    ?1(y,x,x') -> ?2(x',x,x',y)
    f(x,x') -> ?1(x,x,x')
    ?5(y,x,x',x'',y) -> c()
    ?4(y,x,x',x'',y) -> ?5(x'',x,x',x'',y)
    ?3(y,x,x',x'') -> ?4(x',x,x',x'',y)
    h(x,x',x'') -> ?3(x,x,x',x'')
   graph:
    h#(x,x',x'') -> ?3#(x,x,x',x'') ->
    ?3#(y,x,x',x'') -> ?4#(x',x,x',x'',y)
    ?3#(y,x,x',x'') -> ?4#(x',x,x',x'',y) ->
    ?4#(y,x,x',x'',y) -> ?5#(x'',x,x',x'',y)
    f#(x,x') -> ?1#(x,x,x') -> ?1#(y,x,x') -> ?2#(x',x,x',y)
   SCC Processor:
    #sccs: 0
    #rules: 0
    #arcs: 3/25