YES

(VAR x0 x1 x2 a b)
(RULES
  divide(inverse(x0),divide(x1,divide(x0,x2))) -> divide(inverse(x2),x1)
  divide(x2,divide(x0,divide(inverse(x2),x1))) -> divide(inverse(x1),x0)
  divide(divide(x0,x1),x2) -> divide(x0,divide(x2,inverse(x1)))
  divide(inverse(x0),divide(x1,x0)) -> inverse(x1)
  divide(x1,divide(x0,inverse(x1))) -> inverse(x0)
  inverse(divide(x1,x0)) -> divide(x0,x1)
  divide(x0,identity()) -> x0
  inverse(inverse(x0)) -> x0
  multiply(a,b) -> divide(a,inverse(b))
  inverse(identity()) -> identity()
  divide(a,a) -> identity()
  divide(identity(),a) -> inverse(a)
)

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

  divide_A(x1,x2) = x1 + x2 + 2
  identity_A = 1
  multiply_A(x1,x2) = x1 + x2 + 3
  inverse_A(x1) = x1
  identity#_A = 0

weights

  w0 = 1
  w(divide) = 0
  w(identity) = 1
  w(multiply) = 2
  w(inverse) = 0

and precedence:

inverse > multiply > divide > identity
)