(**
  Accepts two monoid elements, x and y, and
  asserts  that y is x's left inverse.
*)
Definition op_is_inv_l := is_inv_l M {+} 0 op_id.(**
  Accepts two monoid elements, x and y, and
  asserts  that y is x's right inverse.
*)
Definition op_is_inv_r := is_inv_r M {+} 0 op_id.(**
  Accepts two monoid elements, x and y, and
  asserts  that y is x's inverse.
*)
Definition op_is_inv := is_inv M {+} 0 op_id.(**
  Accepts one argument, x, and asserts that
  x has a left inverse.
*)
Definition has_inv_l
  := fun x => exists y : M, op_is_inv_l x y.(**
  Accepts one argument, x, and asserts that
  x has a right inverse.
*)
Definition has_inv_r
  := fun x => exists y : M, op_is_inv_r x y.(**
  Accepts one argument, x, and asserts that
  x has an inverse.
*)
Definition has_inv
  := fun x => exists y : M, op_is_inv x y.

Theorem op_inv_sym : forall x y : M, op_is_inv x y <-> op_is_inv y x.