+
+lemma delift_inv_lref1: ∀L,U2,i,d,e. L ⊢ #i [d, e] ≡ U2 →
+ ∨∨ (i < d ∧ U2 = #i)
+ | (∃∃K,V1,V2. d ≤ i & i < d + e &
+ ⇩[0, i] L ≡ K. ⓓV1 &
+ K ⊢ V1 [0, d + e - i - 1] ≡ V2 &
+ ⇧[0, d] V2 ≡ U2
+ )
+ | (d + e ≤ i ∧ U2 = #(i - e)).
+#L #U2 #i #d #e #H
+elim (lt_or_ge i d) #Hdi
+[ elim (delift_inv_lref1_lt … H Hdi) -H /3 width=1/
+| elim (lt_or_ge i (d+e)) #Hide
+ [ elim (delift_inv_lref1_be … H Hdi Hide) -H /3 width=6/
+ | elim (delift_inv_lref1_ge … H Hide) -H /3 width=1/
+ ]
+]
+qed-.
+
+(* Relocation properties ****************************************************)
+
+lemma delift_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≡ T2 →
+ ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d - et, e] T2 ≡ U2 →
+ L ⊢ U1 [dt, et] ≡ U2.
+#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdetd #HLK #HTU1 #U2 #HTU2
+elim (lift_total T d e) #U #HTU
+lapply (tpss_lift_le … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
+elim (lift_trans_ge … HT2 … HTU ?) -T // -Hdetd #T #HT2 #HTU
+>(lift_mono … HTU2 … HT2) -T2 /2 width=3/
+qed.
+
+lemma delift_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≡ T2 →
+ ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
+ ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
+ L ⊢ U1 [dt, et + e] ≡ T2.
+#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1
+elim (lift_total T d e) #U #HTU
+lapply (tpss_lift_be … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
+lapply (lift_trans_be … HT2 … HTU ? ?) -T // -Hdtd -Hddet /2 width=3/
+qed.
+
+lemma delift_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≡ T2 →
+ ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ L ⊢ U1 [dt + e, et] ≡ U2.
+#K #T1 #T2 #dt #et * #T #HT1 #HT2 #L #U1 #d #e #Hddt #HLK #HTU1 #U2 #HTU2
+elim (lift_total T d e) #U #HTU
+lapply (tpss_lift_ge … HT1 … HLK HTU1 … HTU) -T1 -HLK // #HU1
+elim (lift_trans_le … HT2 … HTU ?) -T // -Hddt #T #HT2 #HTU
+>(lift_mono … HTU2 … HT2) -T2 /2 width=3/
+qed.