Qed.

Lemma size_S : forall n Γ Δ, sizes Γ Δ = S n -> exists A B,
In (A→B) Γ \/ In (A→B) Δ \/ In (A∨B) Γ \/ In (A∨B) Δ \/ In (A∧B) Γ \/ In (A∧B) Δ.