Section Theorems.
(** Represents an arbitrary monoid.  Note: we use Variable rather than Parameter to ensure that the following theorems are generalized w.r.t m. *)
Variable m : Monoid.
(** Represents the set of monoid elements. *)
Let M := E m.
(** Accepts one monoid element, x, and asserts that x is the left identity element. *)
Definition op_is_id_l := is_id_l M {+}.
(** Accepts one monoid element, x, and asserts that x is the right identity element. *)
Definition op_is_id_r := is_id_r M {+}.
(** Accepts one monoid element, x, and asserts that x is the identity element. *)
Definition op_is_id := is_id M {+}.
Proof conj op_id_l op_id_r.
Proof
fun x H
=> H 0 || a = 0 @a by <- op_id_r x.
Proof
fun x H
=> H 0 || a = 0 @a by <- op_id_l x.
Proof
fun x
=> and_ind (fun H _ => op_id_l_uniq x H).

(** Proves the left introduction rule. *)
Theorem op_intro_l
:  forall x y z : M, x = y -> z + x = z + y.