Proof
fun x y z H
=>  eq_refl (x + z)
|| x + z = a + z @a by <- H.
(** 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.

(** Proves that the left and right inverses of an element must be equal. *)
Theorem op_inv_l_r_eq
:  forall x y : M, op_is_inv_l x y -> forall z : M, op_is_inv_r x z -> y = z.