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 *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of remove *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of all_remove *)

(** All_remove and concatenation. *)
Axiom all_remove_belongs_concat_r : forall A xs (E F : gen_env A),
List.NoDup xs ->
xs ⊂ F ->
(E & F) ∖ xs = E & (F ∖ xs).