YES

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

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

  double_divide_A(x1,x2) = x1 + x2
  identity_A = 0
  multiply_A(x1,x2) = x1 + x2
  inverse_A(x1) = x1
  identity#_A = 0

weights

  w0 = 1
  w(double_divide) = 0
  w(identity) = 1
  w(multiply) = 0
  w(inverse) = 0

and precedence:

inverse > multiply > double_divide > identity
)