Section CoreDefinitions.
Variable A B : Type.
Implicit Types x y : TVar.
Implicit Types xs ys : list TVar.
Implicit Types v w : A.
Implicit Types vs ws : list A.
Implicit Types E F G : gen_env A.
Implicit Types f g : A -> B.
Definition eq_keys_dec := VarType.eq_dec.
(** Preliminary properties of eq_keys_dec. *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of singulars *)
(* ---------------------------------------------------------------------- *)
(** **** Properties of get *)

Lemma get_concat_inv : forall x y v w E,
get x (E & (y ∶ v)) = Some w ->
(x = y /\ v = w) \/ (x <> y /\ get x E = Some w).