+lemma nta_inv_ldef_sn (h) (a) (G) (K) (V):
+ ∀X2. ❪G,K.ⓓV❫ ⊢ #0 :[h,a] X2 →
+ ∃∃W,U. ❪G,K❫ ⊢ V :[h,a] W & ⇧[1] W ≘ U & ❪G,K.ⓓV❫ ⊢ U ⬌*[h] X2 & ❪G,K.ⓓV❫ ⊢ X2 ![h,a].
+#h #a #G #Y #X #X2 #H
+elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
+elim (cnv_inv_zero … H1) -H1 #Z #K #V #HV #H destruct
+elim (cpms_inv_delta_sn … H2) -H2 *
+[ #_ #H destruct
+| #W #HVW #HWX1
+ /3 width=5 by cnv_cpms_nta, cpcs_cprs_sn, ex4_2_intro/
+]
+qed-.
+
+lemma nta_inv_lref_sn (h) (a) (G) (L):
+ ∀X2,i. ❪G,L❫ ⊢ #↑i :[h,a] X2 →
+ ∃∃I,K,T2,U2. ❪G,K❫ ⊢ #i :[h,a] T2 & ⇧[1] T2 ≘ U2 & ❪G,K.ⓘ[I]❫ ⊢ U2 ⬌*[h] X2 & ❪G,K.ⓘ[I]❫ ⊢ X2 ![h,a] & L = K.ⓘ[I].
+#h #a #G #L #X2 #i #H
+elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
+elim (cnv_inv_lref … H1) -H1 #I #K #Hi #H destruct
+elim (cpms_inv_lref_sn … H2) -H2 *
+[ #_ #H destruct
+| #X #HX #HX1
+ /3 width=9 by cnv_cpms_nta, cpcs_cprs_sn, ex5_4_intro/
+]
+qed-.
+
+lemma nta_inv_lref_sn_drops_cnv (h) (a) (G) (L):
+ ∀X2,i. ❪G,L❫ ⊢ #i :[h,a] X2 →
+ ∨∨ ∃∃K,V,W,U. ⇩[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V :[h,a] W & ⇧[↑i] W ≘ U & ❪G,L❫ ⊢ U ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,a]
+ | ∃∃K,W,U. ⇩[i] L ≘ K. ⓛW & ❪G,K❫ ⊢ W ![h,a] & ⇧[↑i] W ≘ U & ❪G,L❫ ⊢ U ⬌*[h] X2 & ❪G,L❫ ⊢ X2 ![h,a].
+#h #a #G #L #X2 #i #H
+elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
+elim (cnv_inv_lref_drops … H1) -H1 #I #K #V #HLK #HV
+elim (cpms_inv_lref1_drops … H2) -H2 *
+[ #_ #H destruct
+| #Y #X #W #H #HVW #HUX1
+ lapply (drops_mono … H … HLK) -H #H destruct
+ /4 width=8 by cnv_cpms_nta, cpcs_cprs_sn, ex5_4_intro, or_introl/
+| #n #Y #X #U #H #HVU #HUX1 #H0 destruct
+ lapply (drops_mono … H … HLK) -H #H destruct
+ elim (lifts_total V (𝐔❨↑i❩)) #W #HVW
+ lapply (cpms_lifts_bi … HVU (Ⓣ) … L … HVW … HUX1) -U
+ [ /2 width=2 by drops_isuni_fwd_drop2/ ] #HWX1
+ /4 width=9 by cprs_div, ex5_3_intro, or_intror/
+]
+qed-.
+
+lemma nta_inv_bind_sn_cnv (h) (a) (p) (I) (G) (K) (X2):
+ ∀V,T. ❪G,K❫ ⊢ ⓑ[p,I]V.T :[h,a] X2 →
+ ∃∃U. ❪G,K❫ ⊢ V ![h,a] & ❪G,K.ⓑ[I]V❫ ⊢ T :[h,a] U & ❪G,K❫ ⊢ ⓑ[p,I]V.U ⬌*[h] X2 & ❪G,K❫ ⊢ X2 ![h,a].
+#h #a #p * #G #K #X2 #V #T #H