YES

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

(COMMENT
Termination is shown by EKBO with interpretations on natural numbers

  f_A(x1,x2) = x2
  a1_A = 1
  a2_A = 1
  i1_A(x1) = 1
  i2_A(x1) = 2
  a1#_A = 0
  a2#_A = 0

weights

  w0 = 3
  w(f) = 0
  w(a1) = 5
  w(a2) = 3
  w(i1) = 0
  w(i2) = 4

and precedence:

i1 > a1 > f > i2 > a2
)