Section Theorems.
(** Represents an arbitrary abelian group.  Note: we use Variable rather than Parameter to ensure that the following theorems are generalized w.r.t ag. *)
Variable g : Abelian_Group.
(** Represents the set of group elements.  Note: We use Let to define E as a local abbreviation. *)
Let E := E g.
(** Accepts one group element, x, and asserts that x is the left identity element. *)
Definition op_is_id_l := Monoid.is_id_l E {+}.
(** Accepts one group element, x, and asserts that x is the right identity element. *)
Definition op_is_id_r := Monoid.is_id_r E {+}.
(** Accepts one group element, x, and asserts that x is the identity element. *)
Definition op_is_id := Monoid.is_id E {+}.
Proof
fun x H y
=> H y
|| a = y @a by op_is_comm y x.

(** Proves that every left identity is an identity. *)
Theorem op_is_id_lid
: forall x : E, op_is_id_l x -> op_is_id x.