YES

Problem:
 f(x,y) -> g1(x,x,y)
 f(x,y) -> g1(y,x,x)
 f(x,y) -> g2(x,y,y)
 f(x,y) -> g2(y,y,x)
 g1(x,x,y) -> h(x,y)
 g1(y,x,x) -> h(x,y)
 g2(x,y,y) -> h(x,y)
 g2(y,y,x) -> h(x,y)
 h(x,x) -> x

Proof:
 DP Processor:
  DPs:
   f#(x,y) -> g1#(x,x,y)
   f#(x,y) -> g1#(y,x,x)
   f#(x,y) -> g2#(x,y,y)
   f#(x,y) -> g2#(y,y,x)
   g1#(x,x,y) -> h#(x,y)
   g1#(y,x,x) -> h#(x,y)
   g2#(x,y,y) -> h#(x,y)
   g2#(y,y,x) -> h#(x,y)
  TRS:
   f(x,y) -> g1(x,x,y)
   f(x,y) -> g1(y,x,x)
   f(x,y) -> g2(x,y,y)
   f(x,y) -> g2(y,y,x)
   g1(x,x,y) -> h(x,y)
   g1(y,x,x) -> h(x,y)
   g2(x,y,y) -> h(x,y)
   g2(y,y,x) -> h(x,y)
   h(x,x) -> x
  TDG Processor:
   DPs:
    f#(x,y) -> g1#(x,x,y)
    f#(x,y) -> g1#(y,x,x)
    f#(x,y) -> g2#(x,y,y)
    f#(x,y) -> g2#(y,y,x)
    g1#(x,x,y) -> h#(x,y)
    g1#(y,x,x) -> h#(x,y)
    g2#(x,y,y) -> h#(x,y)
    g2#(y,y,x) -> h#(x,y)
   TRS:
    f(x,y) -> g1(x,x,y)
    f(x,y) -> g1(y,x,x)
    f(x,y) -> g2(x,y,y)
    f(x,y) -> g2(y,y,x)
    g1(x,x,y) -> h(x,y)
    g1(y,x,x) -> h(x,y)
    g2(x,y,y) -> h(x,y)
    g2(y,y,x) -> h(x,y)
    h(x,x) -> x
   graph:
    f#(x,y) -> g2#(y,y,x) -> g2#(y,y,x) -> h#(x,y)
    f#(x,y) -> g2#(y,y,x) -> g2#(x,y,y) -> h#(x,y)
    f#(x,y) -> g2#(x,y,y) -> g2#(y,y,x) -> h#(x,y)
    f#(x,y) -> g2#(x,y,y) -> g2#(x,y,y) -> h#(x,y)
    f#(x,y) -> g1#(y,x,x) -> g1#(y,x,x) -> h#(x,y)
    f#(x,y) -> g1#(y,x,x) -> g1#(x,x,y) -> h#(x,y)
    f#(x,y) -> g1#(x,x,y) -> g1#(y,x,x) -> h#(x,y)
    f#(x,y) -> g1#(x,x,y) -> g1#(x,x,y) -> h#(x,y)
   SCC Processor:
    #sccs: 0
    #rules: 0
    #arcs: 8/64