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

Lemma all_remove_map : forall f E xs,
(map f E) ∖ xs = map f (E ∖ xs).