+(* Advanced inversion lemmas ************************************************)
+
+lemma lsubr_inv_abst1: ∀K1,L2,W. K1.ⓛW ⫃ L2 →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW
+ | ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓤ{I2}.
+#K1 #L2 #W #H elim (lsubr_inv_bind1 … H) -H *
+/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
+#K2 #V2 #W2 #_ #_ #H destruct
+qed-.
+
+lemma lsubr_inv_unit1: ∀I,K1,L2. K1.ⓤ{I} ⫃ L2 →
+ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓤ{I}.
+#I #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
+[ #K2 #HK12 #H destruct /2 width=3 by ex2_intro/
+| #K2 #V #W #_ #_ #H destruct
+| #I1 #I2 #K2 #V #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_pair2: ∀I,L1,K2,W. L1 ⫃ K2.ⓑ{I}W →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓑ{I}W
+ | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = Abst.
+#I #L1 #K2 #W #H elim (lsubr_inv_bind2 … H) -H *
+[ /3 width=3 by ex2_intro, or_introl/
+| #K2 #X #V #HK12 #H1 #H2 destruct /3 width=4 by ex3_2_intro, or_intror/
+| #I1 #I1 #K2 #V #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_abbr2: ∀L1,K2,V. L1 ⫃ K2.ⓓV →
+ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓV.
+#L1 #K2 #V #H elim (lsubr_inv_pair2 … H) -H *
+[ /2 width=3 by ex2_intro/
+| #K1 #X #_ #_ #H destruct
+]
+qed-.