+fact lsubc_inv_atom1_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → L1 = ⋆ → L2 = ⋆.
+#RP #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+]
+qed.
+
+(* Basic_1: was: csubc_gen_sort_r *)
+lemma lsubc_inv_atom1: ∀RP,L2. ⋆ [RP] ⊑ L2 → L2 = ⋆.
+/2 width=4/ qed-.
+
+fact lsubc_inv_pair1_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
+ (∃∃K2. K1 [RP] ⊑ K2 & L2 = K2. ⓑ{I} V) ∨
+ ∃∃K2,W,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
+ K1 [RP] ⊑ K2 &
+ L2 = K2. ⓛW & I = Abbr.
+#RP #L1 #L2 * -L1 -L2
+[ #I #K1 #V #H destruct
+| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
+| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/
+]
+qed.
+
+(* Basic_1: was: csubc_gen_head_r *)
+lemma lsubc_inv_pair1: ∀RP,I,K1,L2,V. K1. ⓑ{I} V [RP] ⊑ L2 →
+ (∃∃K2. K1 [RP] ⊑ K2 & L2 = K2. ⓑ{I} V) ∨
+ ∃∃K2,W,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
+ K1 [RP] ⊑ K2 &
+ L2 = K2. ⓛW & I = Abbr.
+/2 width=3/ qed-.
+
+fact lsubc_inv_atom2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → L2 = ⋆ → L1 = ⋆.
+#RP #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+]
+qed.
+
+(* Basic_1: was: csubc_gen_sort_l *)
+lemma lsubc_inv_atom2: ∀RP,L1. L1 [RP] ⊑ ⋆ → L1 = ⋆.
+/2 width=4/ qed-.
+
+fact lsubc_inv_pair2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
+ (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. ⓑ{I} W) ∨