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
  Usable Rule 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:
    f8(x,y) -> x
    f8(x,y) -> y
   CDG 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:
     f8(x,y) -> x
     f8(x,y) -> y
    graph:
     f#(x,y) -> g2#(y,y,x) -> g2#(x,y,y) -> h#(x,y)
     f#(x,y) -> g2#(y,y,x) -> g2#(y,y,x) -> h#(x,y)
     f#(x,y) -> g2#(x,y,y) -> g2#(x,y,y) -> h#(x,y)
     f#(x,y) -> g2#(x,y,y) -> g2#(y,y,x) -> h#(x,y)
     f#(x,y) -> g1#(y,x,x) -> g1#(x,x,y) -> h#(x,y)
     f#(x,y) -> g1#(y,x,x) -> g1#(y,x,x) -> h#(x,y)
     f#(x,y) -> g1#(x,x,y) -> g1#(x,x,y) -> h#(x,y)
     f#(x,y) -> g1#(x,x,y) -> g1#(y,x,x) -> h#(x,y)
    Restore Modifier:
     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
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 8/64