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:
    
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     
    interpretation:
     [h#](x0, x1) = 0,
     
     [g2#](x0, x1, x2) = -12x0 + 8,
     
     [g1#](x0, x1, x2) = -16x0 + -3x1 + -1x2 + 2,
     
     [f#](x0, x1) = 8x0 + x1 + 11
    orientation:
     f#(x,y) = 8x + y + 11 >= -3x + -1y + 2 = g1#(x,x,y)
     
     f#(x,y) = 8x + y + 11 >= -1x + -16y + 2 = g1#(y,x,x)
     
     f#(x,y) = 8x + y + 11 >= -12x + 8 = g2#(x,y,y)
     
     f#(x,y) = 8x + y + 11 >= -12y + 8 = g2#(y,y,x)
     
     g1#(x,x,y) = -3x + -1y + 2 >= 0 = h#(x,y)
     
     g1#(y,x,x) = -1x + -16y + 2 >= 0 = h#(x,y)
     
     g2#(x,y,y) = -12x + 8 >= 0 = h#(x,y)
     
     g2#(y,y,x) = -12y + 8 >= 0 = h#(x,y)
    problem:
     DPs:
      
     TRS:
      
    Qed