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
  Arctic Interpretation Processor:
   dimension: 1
   interpretation:
    [h#](x0, x1) = x0 + x1 + 1,
    
    [g2#](x0, x1, x2) = 2x0 + 2x1 + 2x2 + 5,
    
    [g1#](x0, x1, x2) = 2x0 + 2x1 + 2x2 + 2,
    
    [f#](x0, x1) = 3x0 + 3x1 + 7,
    
    [h](x0, x1) = 1x1,
    
    [g2](x0, x1, x2) = 4x1 + x2,
    
    [g1](x0, x1, x2) = 2x0 + 6x1 + 5x2,
    
    [f](x0, x1) = 7x0 + 8x1 + 0
   orientation:
    f#(x,y) = 3x + 3y + 7 >= 2x + 2y + 2 = g1#(x,x,y)
    
    f#(x,y) = 3x + 3y + 7 >= 2x + 2y + 2 = g1#(y,x,x)
    
    f#(x,y) = 3x + 3y + 7 >= 2x + 2y + 5 = g2#(x,y,y)
    
    f#(x,y) = 3x + 3y + 7 >= 2x + 2y + 5 = g2#(y,y,x)
    
    g1#(x,x,y) = 2x + 2y + 2 >= x + y + 1 = h#(x,y)
    
    g1#(y,x,x) = 2x + 2y + 2 >= x + y + 1 = h#(x,y)
    
    g2#(x,y,y) = 2x + 2y + 5 >= x + y + 1 = h#(x,y)
    
    g2#(y,y,x) = 2x + 2y + 5 >= x + y + 1 = h#(x,y)
    
    f(x,y) = 7x + 8y + 0 >= 6x + 5y = g1(x,x,y)
    
    f(x,y) = 7x + 8y + 0 >= 6x + 2y = g1(y,x,x)
    
    f(x,y) = 7x + 8y + 0 >= 4y = g2(x,y,y)
    
    f(x,y) = 7x + 8y + 0 >= x + 4y = g2(y,y,x)
    
    g1(x,x,y) = 6x + 5y >= 1y = h(x,y)
    
    g1(y,x,x) = 6x + 2y >= 1y = h(x,y)
    
    g2(x,y,y) = 4y >= 1y = h(x,y)
    
    g2(y,y,x) = x + 4y >= 1y = h(x,y)
    
    h(x,x) = 1x >= x = x
   problem:
    DPs:
     
    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
   Qed