YES
Time: 0.297

Problem:
 Equations:
  
 TRS:
  1() -> s(0())
  2() -> s(1())
  3() -> s(2())
  4() -> s(3())
  5() -> s(4())
  6() -> s(5())
  7() -> s(6())
  8() -> s(7())
  9() -> s(8())
  max'C(0(),x) -> x
  max'C(s(x),s(y)) -> s(max'C(x,y))
  app(empty(),X) -> X
  max(singl(x)) -> x
  max(app(singl(x),Y)) -> max2(x,Y)
  max2(x,empty()) -> x
  max2(x,singl(y)) -> max'C(x,y)
  max2(x,app(singl(y),Z)) -> max2(max'C(x,y),Z)

Proof:
 AC-KBO Processor:
  precedence:
   8 ~ 7 ~ 6 > max > 9 > 3 > max2 ~ singl ~ empty ~ 5 ~ 4 ~ 2 ~ s ~ 0 ~ 1 ~ max'C ~ app
   weight function:
    [max2](x0, x1) = x0 + 3x1 + 4,
    
    [max](x0) = 6x0 + 10,
    
    [singl](x0) = 4x0 + 13,
    
    [empty] = 1,
    
    [9] = 14,
    
    [8] = 13,
    
    [7] = 12,
    
    [6] = 11,
    
    [5] = 10,
    
    [4] = 8,
    
    [3] = 6,
    
    [2] = 5,
    
    [s](x0) = x0 + 1,
    
    [0] = 1,
    
    [1] = 3,
    
    [max'C](x0, x1) = x0 + 10x1 + 8,
    
    [app](x0, x1) = 13x0 + 2x1
   problem:
    Equations:
     
    TRS:
     
   Qed