YES

(VAR x0 x1 x2 x3)
(RULES
  multiply(x0,multiply(x1,x2)) -> multiply(multiply(x0,x1),x2)
  inverse(multiply(x1,x0)) -> multiply(inverse(x0),inverse(x1))
  multiply(multiply(x2,inverse(x3)),x3) -> x2
  multiply(multiply(x0,x1),inverse(x1)) -> x0
  double_divide(x1,x0) -> multiply(inverse(x1),inverse(x0))
  multiply(x0,identity()) -> x0
  multiply(x1,inverse(x1)) -> identity()
  inverse(inverse(x0)) -> x0
  inverse(identity()) -> identity()
  multiply(identity(),x0) -> x0
  multiply(inverse(x0),x0) -> identity()
)

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

  double_divide_A(x1,x2) = x1 + x2 + 1
  identity_A = 1
  multiply_A(x1,x2) = x1 + x2 + 1
  inverse_A(x1) = x1
  identity#_A = 0
  multiply#_A(x1,x2) = x2

weights

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

and precedence:

inverse > multiply > double_divide > identity
)