Section Properties.
Implicit Types x y : TVar.
Implicit Types xs ys : list TVar.
(* ---------------------------------------------------------------------- *)
(** *** Primary properties *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of singulars *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of concatenation *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of get *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of dom *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of img *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of belongs *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of all_belongs *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of notin *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of all_notin *)

Axiom all_notin_notin : forall A x xs (E : gen_env A),
(x :: xs) ⊄ E ->
x ∉ E ∧ xs ⊄ E.