YES

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

Proof:
 DP Processor:
  DPs:
   f#(x,y,w,w,a()) -> g1#(x,x,y,w)
   f#(x,y,w,a(),a()) -> g1#(y,x,x,w)
   f#(x,y,a(),a(),w) -> g2#(x,y,y,w)
   f#(x,y,a(),w,w) -> g2#(y,y,x,w)
   g1#(x,x,y,a()) -> h#(x,y)
   g1#(y,x,x,a()) -> h#(x,y)
   g2#(x,y,y,a()) -> h#(x,y)
   g2#(y,y,x,a()) -> h#(x,y)
  TRS:
   f(x,y,w,w,a()) -> g1(x,x,y,w)
   f(x,y,w,a(),a()) -> g1(y,x,x,w)
   f(x,y,a(),a(),w) -> g2(x,y,y,w)
   f(x,y,a(),w,w) -> g2(y,y,x,w)
   g1(x,x,y,a()) -> h(x,y)
   g1(y,x,x,a()) -> h(x,y)
   g2(x,y,y,a()) -> h(x,y)
   g2(y,y,x,a()) -> h(x,y)
   h(x,x) -> x
  Usable Rule Processor:
   DPs:
    f#(x,y,w,w,a()) -> g1#(x,x,y,w)
    f#(x,y,w,a(),a()) -> g1#(y,x,x,w)
    f#(x,y,a(),a(),w) -> g2#(x,y,y,w)
    f#(x,y,a(),w,w) -> g2#(y,y,x,w)
    g1#(x,x,y,a()) -> h#(x,y)
    g1#(y,x,x,a()) -> h#(x,y)
    g2#(x,y,y,a()) -> h#(x,y)
    g2#(y,y,x,a()) -> h#(x,y)
   TRS:
    
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     
    interpretation:
     [h#](x0, x1) = x0 + 0,
     
     [g2#](x0, x1, x2, x3) = 5x0 + 2x1 + 1x2 + 2,
     
     [g1#](x0, x1, x2, x3) = 8x0 + 7x1 + 7x2 + x3,
     
     [f#](x0, x1, x2, x3, x4) = 15x0 + 9x1 + 6x2 + x3 + 1,
     
     [a] = 1
    orientation:
     f#(x,y,w,w,a()) = 6w + 15x + 9y + 1 >= w + 8x + 7y = g1#(x,x,y,w)
     
     f#(x,y,w,a(),a()) = 6w + 15x + 9y + 1 >= w + 7x + 8y = g1#(y,x,x,w)
     
     f#(x,y,a(),a(),w) = 15x + 9y + 7 >= 5x + 2y + 2 = g2#(x,y,y,w)
     
     f#(x,y,a(),w,w) = w + 15x + 9y + 7 >= 1x + 5y + 2 = g2#(y,y,x,w)
     
     g1#(x,x,y,a()) = 8x + 7y + 1 >= x + 0 = h#(x,y)
     
     g1#(y,x,x,a()) = 7x + 8y + 1 >= x + 0 = h#(x,y)
     
     g2#(x,y,y,a()) = 5x + 2y + 2 >= x + 0 = h#(x,y)
     
     g2#(y,y,x,a()) = 1x + 5y + 2 >= x + 0 = h#(x,y)
    problem:
     DPs:
      
     TRS:
      
    Qed