Section Theorems.
(** Represents an arbitrary commutative ring.  Note: we use Variable rather than Parameter to ensure that the following theorems are generalized w.r.t r. *)
Variable r : Commutative_Ring.
(** Represents the set of group elements.  Note: We use Let to define E as a local abbreviation. *)
Let E := E r.
(** Accepts one ring element, x, and asserts that x is the left identity element. *)
Definition sum_is_id_l := Monoid.is_id_l E {+}.
(** Accepts one ring element, x, and asserts that x is the right identity element. *)
Definition sum_is_id_r := Monoid.is_id_r E {+}.
(** Accepts one ring element, x, and asserts that x is the identity element. *)
Definition sum_is_id := Monoid.is_id E {+}.
(** Accepts one ring element, x, and asserts that x is the left identity element. *)
Definition prod_is_id_l := Monoid.is_id_l E {#}.
(** Accepts one ring element, x, and asserts that x is the right identity element. *)
Definition prod_is_id_r := Monoid.is_id_r E {#}.
(** Accepts one ring element, x, and asserts that x is the identity element. *)
Definition prod_is_id := Monoid.is_id E {#}.
Proof
fun x : E
=> eq_ind_r
(fun a => a = x)
(prod_id_l x)
(prod_is_comm x 1).
Proof
fun x y z : E
=>  prod_sum_distrib_l x y z
|| x # (y + z) = a + (x # z) @a by <- prod_is_comm x y
|| x # (y + z) = (y # x) + a @a by <- prod_is_comm x z
|| a = (y # x) + (z # x)     @a by <- prod_is_comm x (y + z).
(** Represents the non-commutative ring formed by addition and multiplication over E. *)
Definition ring := Ring.ring E 0 1 {+} {#} distinct_0_1 sum_is_assoc sum_is_comm sum_id_l sum_inv_l_ex prod_is_assoc prod_id_l prod_id_r prod_sum_distrib_l prod_sum_distrib_r.
(** Represents the abelian group formed by addition over E. *)
Definition sum_abelian_group := Ring.sum_abelian_group ring.
(** Represents the group formed by addition over E. *)
Definition sum_group := Ring.sum_group ring.
(** Represents the monoid formed by addition over E. *)
Definition sum_monoid := Ring.sum_monoid ring.
(** Represents the monoid formed by multiplication over E. *)
Definition prod_monoid := Ring.prod_monoid ring.

(** Proves that 1 <> 0. *)
Theorem distinct_1_0
:  E_1 (c := r) <> E_0 (c := r).