YES

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

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

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

and precedence:

i2 > i1 > f > a2 > a1
)