YES

Problem:
 f(g(x)) -> g(f(f(x)))
 f(h(x)) -> h(g(x))
 f'(s(x),y,y) -> f'(y,x,s(x))

Proof:
 DP Processor:
  DPs:
   f#(g(x)) -> f#(x)
   f#(g(x)) -> f#(f(x))
   f'#(s(x),y,y) -> f'#(y,x,s(x))
  TRS:
   f(g(x)) -> g(f(f(x)))
   f(h(x)) -> h(g(x))
   f'(s(x),y,y) -> f'(y,x,s(x))
  TDG Processor:
   DPs:
    f#(g(x)) -> f#(x)
    f#(g(x)) -> f#(f(x))
    f'#(s(x),y,y) -> f'#(y,x,s(x))
   TRS:
    f(g(x)) -> g(f(f(x)))
    f(h(x)) -> h(g(x))
    f'(s(x),y,y) -> f'(y,x,s(x))
   graph:
    f'#(s(x),y,y) -> f'#(y,x,s(x)) -> f'#(s(x),y,y) -> f'#(y,x,s(x))
    f#(g(x)) -> f#(f(x)) -> f#(g(x)) -> f#(f(x))
    f#(g(x)) -> f#(f(x)) -> f#(g(x)) -> f#(x)
    f#(g(x)) -> f#(x) -> f#(g(x)) -> f#(f(x))
    f#(g(x)) -> f#(x) -> f#(g(x)) -> f#(x)
   SCC Processor:
    #sccs: 2
    #rules: 3
    #arcs: 5/9
    DPs:
     f#(g(x)) -> f#(f(x))
     f#(g(x)) -> f#(x)
    TRS:
     f(g(x)) -> g(f(f(x)))
     f(h(x)) -> h(g(x))
     f'(s(x),y,y) -> f'(y,x,s(x))
    KBO Processor:
     argument filtering:
      pi(g) = [0]
      pi(f) = 0
      pi(h) = []
      pi(s) = []
      pi(f') = []
      pi(f#) = 0
     weight function:
      w0 = 1
      w(f') = w(s) = w(h) = 1
      w(f#) = w(f) = w(g) = 0
     precedence:
      f# ~ f' ~ s ~ h ~ f ~ g
     problem:
      DPs:
       
      TRS:
       f(g(x)) -> g(f(f(x)))
       f(h(x)) -> h(g(x))
       f'(s(x),y,y) -> f'(y,x,s(x))
     Qed
    
    DPs:
     f'#(s(x),y,y) -> f'#(y,x,s(x))
    TRS:
     f(g(x)) -> g(f(f(x)))
     f(h(x)) -> h(g(x))
     f'(s(x),y,y) -> f'(y,x,s(x))
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [f'#](x0, x1, x2) = -3x0 + x1 + -8x2,
      
      [f'](x0, x1, x2) = -7x0 + 1x1 + -9x2 + -16,
      
      [s](x0) = 8x0,
      
      [h](x0) = -2x0 + 4,
      
      [f](x0) = x0 + -6,
      
      [g](x0) = x0 + 0
     orientation:
      f'#(s(x),y,y) = 5x + y >= x + -3y = f'#(y,x,s(x))
      
      f(g(x)) = x + 0 >= x + 0 = g(f(f(x)))
      
      f(h(x)) = -2x + 4 >= -2x + 4 = h(g(x))
      
      f'(s(x),y,y) = 1x + 1y + -16 >= 1x + -7y + -16 = f'(y,x,s(x))
     problem:
      DPs:
       
      TRS:
       f(g(x)) -> g(f(f(x)))
       f(h(x)) -> h(g(x))
       f'(s(x),y,y) -> f'(y,x,s(x))
     Qed