rewrite plus_assoc;f_equal;apply plus_comm.
rewrite <- !plus_assoc;f_equal;apply IHΓ1.
Qed.

(* Specifications: *)
Lemma sizes_comm_l : forall Γ1 Γ2 A Δ, sizes (A :: Δ) (Γ1 ++ Γ2) = sizes Δ (Γ1 ++ A::Γ2).
intros;rewrite sizes_comm;rewrite sizes_comm_r;apply sizes_comm.