(* notation for lookups with default *)
Notation plook' i x := (odflt null (plook i x)).

(* Specifications: *)
Lemma interp_perm i : forall t1 t2, perm t1 t2 -> interp i t1 = interp i t2.
Lemma interp_subctx j k t: valid j t -> subctx j k -> interp j t = interp k t.