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(** * Size based reasoning                                   *)
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(** Some properties about nominal terms require calling the induction hypothesis not on a direct subterm, but on one that has first had a swapping applied to it.  However, swapping names does not change the size of terms, so that means we can prove such properties by induction on that size.  *)
Fixpoint size (t : n_exp) : nat :=
match t with
| n_var x => 1
| n_abs x t => 1 + size t
| n_app t1 t2 => 1 + size t1 + size t2
end.

Lemma swap_size_eq : forall x y t,
size (swap x y t) = size t.