Section MainLemma.
Variable (Gamma : ctx) (s: exp X) (n: nat).
Hypothesis (Ts: Gamma ⊢(3) s : Arr (repeat (alpha → alpha) n) alpha).
Hypothesis (Vu: ~ u ∈ vars s) (Vv: ~ v ∈ vars s).

Lemma main_lemma:
exists I, I ⊆ nats n /\ forall W, |W| = n -> exists t,
AppR s (Enc W) ≡ AppL (select I (Enc W)) t /\ nostring t.