+(* Basic properties *********************************************************)
+
+lemma lleq_sort: ∀L1,L2,d,k. |L1| = |L2| → L1 ⋕[⋆k, d] L2.
+/2 width=1 by llpx_sn_sort/ qed.
+
+lemma lleq_skip: ∀L1,L2,d,i. yinj i < d → |L1| = |L2| → L1 ⋕[#i, d] L2.
+/2 width=1 by llpx_sn_skip/ qed.
+
+lemma lleq_lref: ∀I,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
+ ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
+ K1 ⋕[V, 0] K2 → L1 ⋕[#i, d] L2.
+/2 width=9 by llpx_sn_lref/ qed.
+
+lemma lleq_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ⋕[#i, d] L2.
+/2 width=1 by llpx_sn_free/ qed.
+
+lemma lleq_gref: ∀L1,L2,d,p. |L1| = |L2| → L1 ⋕[§p, d] L2.
+/2 width=1 by llpx_sn_gref/ qed.
+
+lemma lleq_bind: ∀a,I,L1,L2,V,T,d.
+ L1 ⋕[V, d] L2 → L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V →
+ L1 ⋕[ⓑ{a,I}V.T, d] L2.
+/2 width=1 by llpx_sn_bind/ qed.
+
+lemma lleq_flat: ∀I,L1,L2,V,T,d.
+ L1 ⋕[V, d] L2 → L1 ⋕[T, d] L2 → L1 ⋕[ⓕ{I}V.T, d] L2.
+/2 width=1 by llpx_sn_flat/ qed.
+
+lemma lleq_refl: ∀d,T. reflexive … (lleq d T).
+/2 width=1 by llpx_sn_refl/ qed.
+
+lemma lleq_Y: ∀L1,L2,T. |L1| = |L2| → L1 ⋕[T, ∞] L2.
+/2 width=1 by llpx_sn_Y/ qed.
+
+lemma lleq_sym: ∀d,T. symmetric … (lleq d T).
+#d #T #L1 #L2 #H @(lleq_ind … H) -d -T -L1 -L2
+/2 width=7 by lleq_sort, lleq_skip, lleq_lref, lleq_free, lleq_gref, lleq_bind, lleq_flat/