YES

(VAR x1 x0 x2 x y z)
(RULES
  i2(f(x1,i1(i1(x0)))) -> i2(f(x1,x0))
  f(x0,i2(x1)) -> i2(f(x1,i1(x0)))
  i1(f(x1,x0)) -> f(i1(x0),i1(x1))
  i2(f(f(x0,i1(x1)),x1)) -> i2(x0)
  f(f(f(x0,i1(x2)),x2),x1) -> f(x0,x1)
  i2(f(i1(x0),x0)) -> i2(a1())
  i2(i1(i1(x0))) -> i2(x0)
  i1(i1(i1(x0))) -> i1(x0)
  f(f(i1(x0),x0),x1) -> x1
  f(f(x0,i1(i1(x1))),x2) -> f(f(x0,x1),x2)
  f(i2(x0),x1) -> f(i1(x0),x1)
  i2(i2(x0)) -> i2(i1(x0))
  f(i1(i1(x0)),x1) -> f(x0,x1)
  i1(i2(x0)) -> i1(i1(x0))
  f(x0,a1()) -> i1(i1(x0))
  f(f(x0,x1),i1(x1)) -> i1(i1(x0))
  i1(a1()) -> a1()
  a2() -> i2(a1())
  f(x,i1(x)) -> a1()
  f(a1(),y) -> y
  f(x,f(y,z)) -> f(f(x,y),z)
)

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

  f_A(x1,x2) = x1 + x2 + 2
  a1_A = 0
  a2_A = 6
  i1_A(x1) = x1
  i2_A(x1) = x1 + 1
  f#_A(x1,x2) = x2
  a1#_A = 0
  a2#_A = 0

weights

  w0 = 1
  w(f) = 1
  w(a1) = 1
  w(a2) = 2
  w(i1) = 0
  w(i2) = 1

and precedence:

i1 > f > a1 > a2 > i2
)