End BasicOperations.
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(** Cofinite quantification *)
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(* In the next example, we will reexamine the definition of [lc] in the [abs] case.  The lemma [subst_lc] says that local closure is preserved by substitution.  Let's start working through this proof. *)

Lemma subst_lc_0 : forall (x : atom) u e,
lc e ->
lc u ->
lc ([x ~> u] e).