YES

(VAR x1 x0 x y z)
(RULES
  i2(f(x1,x0)) -> f(i2(x0),i2(x1))
  i1(x0) -> f(i2(x0),a1())
  i2(i2(i2(x0))) -> i2(x0)
  f(i2(i2(x0)),x1) -> f(x0,x1)
  f(i2(x0),f(x0,x1)) -> x1
  f(x0,i2(a1())) -> i2(i2(x0))
  f(x,i2(x)) -> i2(a1())
  f(x1,f(i2(x1),x0)) -> x0
  f(i2(a1()),y) -> y
  i2(i2(a1())) -> i2(a1())
  a2() -> i2(a1())
  f(a1(),y) -> y
  f(f(x,y),z) -> f(x,f(y,z))
)

(COMMENT
Termination is shown by KBO with weight

  w0 = 1
  w(f) = 0
  w(a1) = 1
  w(a2) = 3
  w(i1) = 4
  w(i2) = 0

and precedence:

  i2 > i1 > f > a1 > a2
)