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. *)
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(** **** Properties of singulars *)
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(** **** Properties of get *)
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(** **** Properties of dom *)
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(** **** Properties of img *)
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(** **** Properties of map *)
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(** **** Properties of update_one *)
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(** **** Properties of update *)
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(** **** Properties of remove *)

Lemma remove_empty : forall x,
(@empty A) ∖ {x} = (@empty A).