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(** * Regularity                                                         *)
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(** To be fully confident about the exist-fresh rules, we can show their "regularity" --- that these relations only include locally closed terms. (For the exists-fresh definition of local closure.) *)

(** Importantly: to prove regularity, we need to know subst_lc, and that we most directly prove by using the connection between lc and lc_c. *)
Lemma subst_lc : forall (x : atom) u e,
lc e ->
lc u ->
lc ([x ~> u] e).