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 *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of map *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of update_one *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of update *)

(** Notin commutes with update. *)
Axiom notin_update : forall A x (E F : gen_env A),
x ∉ E ->
x ∉ (E ::= F).