+lemma cpms_inv_delta_sn (h) (n) (G) (K) (V):
+ ∀T2. ❪G,K.ⓓV❫ ⊢ #0 ➡*[h,n] T2 →
+ ∨∨ ∧∧ T2 = #0 & n = 0
+ | ∃∃V2. ❪G,K❫ ⊢ V ➡*[h,n] V2 & ⇧[1] V2 ≘ T2.
+#h #n #G #K #V #T2 #H
+elim (cpms_inv_lref1_drops … H) -H *
+[ /3 width=1 by or_introl, conj/
+| #Y #X #V2 #H #HV2 #HVT2
+ lapply (drops_fwd_isid … H ?) -H [ // ] #H destruct
+ /3 width=3 by ex2_intro, or_intror/
+| #m #Y #X #V2 #H #HV2 #HVT2
+ lapply (drops_fwd_isid … H ?) -H [ // ] #H destruct
+]
+qed-.
+
+lemma cpms_inv_ell_sn (h) (n) (G) (K) (V):
+ ∀T2. ❪G,K.ⓛV❫ ⊢ #0 ➡*[h,n] T2 →
+ ∨∨ ∧∧ T2 = #0 & n = 0
+ | ∃∃m,V2. ❪G,K❫ ⊢ V ➡*[h,m] V2 & ⇧[1] V2 ≘ T2 & n = ↑m.
+#h #n #G #K #V #T2 #H
+elim (cpms_inv_lref1_drops … H) -H *
+[ /3 width=1 by or_introl, conj/
+| #Y #X #V2 #H #HV2 #HVT2
+ lapply (drops_fwd_isid … H ?) -H [ // ] #H destruct
+| #m #Y #X #V2 #H #HV2 #HVT2 #H0 destruct
+ lapply (drops_fwd_isid … H ?) -H [ // ] #H destruct
+ /3 width=5 by ex3_2_intro, or_intror/
+]
+qed-.
+
+lemma cpms_inv_lref_sn (h) (n) (G) (I) (K):
+ ∀U2,i. ❪G,K.ⓘ[I]❫ ⊢ #↑i ➡*[h,n] U2 →
+ ∨∨ ∧∧ U2 = #↑i & n = 0
+ | ∃∃T2. ❪G,K❫ ⊢ #i ➡*[h,n] T2 & ⇧[1] T2 ≘ U2.
+#h #n #G #I #K #U2 #i #H
+elim (cpms_inv_lref1_drops … H) -H *
+[ /3 width=1 by or_introl, conj/
+| #L #V #V2 #H #HV2 #HVU2
+ lapply (drops_inv_drop1 … H) -H #HLK
+ elim (lifts_split_trans … HVU2 (𝐔❨↑i❩) (𝐔❨1❩)) -HVU2
+ [| // ] #T2 #HVT2 #HTU2
+ /4 width=6 by cpms_delta_drops, ex2_intro, or_intror/
+| #m #L #V #V2 #H #HV2 #HVU2 #H0 destruct
+ lapply (drops_inv_drop1 … H) -H #HLK
+ elim (lifts_split_trans … HVU2 (𝐔❨↑i❩) (𝐔❨1❩)) -HVU2
+ [| // ] #T2 #HVT2 #HTU2
+ /4 width=6 by cpms_ell_drops, ex2_intro, or_intror/
+]
+qed-.
+
+fact cpms_inv_succ_sn (h) (n) (G) (L):
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[h,↑n] T2 →
+ ∃∃T. ❪G,L❫ ⊢ T1 ➡*[h,1] T & ❪G,L❫ ⊢ T ➡*[h,n] T2.
+#h #n #G #L #T1 #T2