YES

(VAR x0 x1 x y z)
(RULES
  f(i(a()),f(i(a()),f(i(a()),f(i(a()),x0)))) -> f(a(),f(a(),f(a(),f(a(),x0))))
  f(a(),f(a(),f(a(),f(a(),f(a(),x0))))) -> f(i(a()),f(i(a()),f(i(a()),x0)))
  f(i(a()),f(i(a()),f(i(a()),i(a())))) -> f(a(),f(a(),f(a(),a())))
  f(b(),f(a(),x0)) -> f(i(a()),f(b(),x0))
  f(a(),f(a(),f(a(),f(a(),a())))) -> f(i(a()),f(i(a()),i(a())))
  f(b(),a()) -> f(i(a()),b())
  i(f(x1,x0)) -> f(i(x0),i(x1))
  f(i(x0),f(x0,x1)) -> x1
  f(x1,f(i(x1),x0)) -> x0
  f(i(x0),x0) -> e()
  i(i(x0)) -> x0
  i(e()) -> e()
  i(b()) -> b()
  f(e(),x0) -> x0
  f(b(),f(b(),x0)) -> x0
  f(b(),f(i(a()),x0)) -> f(a(),f(b(),x0))
  f(b(),i(a())) -> f(a(),b())
  f(b(),b()) -> e()
  f(x,e()) -> x
  f(x,i(x)) -> e()
  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) = x1 + x2 + 1
  i_A(x1) = x1
  e_A = 1
  a_A = 1
  b_A = 7
  e#_A = 0
  a#_A = 0
  b#_A = 0

weights

  w0 = 1
  w(f) = 0
  w(i) = 0
  w(e) = 1
  w(a) = 2
  w(b) = 15

and precedence:

i > a > b > f > e
)