(** The equality operator [=] is also a function that returns a [Prop].  The expression [n = m] is syntactic sugar for [eq n m], defined using Coq's [Notation] mechanism. Because [eq] can be used with elements of any type, it is also polymorphic: *)
Check @eq.
(* ===> forall A : Type, A -> A -> Prop *)
(** (Notice that we wrote [@eq] instead of [eq]: The type argument [A] to [eq] is declared as implicit, so we need to turn off implicit arguments to see the full type of [eq].) *)
(* ################################################################# *)
(** * Logical Connectives *)
(* ================================================================= *)
(** ** Conjunction *)
(** The _conjunction_ (or _logical and_) of propositions [A] and [B] is written [A /\ B], representing the claim that both [A] and [B] are true. *)
Example and_example : 3 + 4 = 7 /\ 2 * 2 = 4.
(** To prove a conjunction, use the [split] tactic.  It will generate two subgoals, one for each part of the statement: *)
(** For any propositions [A] and [B], if we assume that [A] is true and we assume that [B] is true, we can conclude that [A /\ B] is also true. *)

Lemma and_intro : forall A B : Prop, A -> B -> A /\ B.
Example and_example' : 3 + 4 = 7 /\ 2 * 2 = 4.