qed.
lemma delift_delifta: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼▼*[d, e] T1 ≡ T2.
-#L #T1 @(fw_ind … L T1) -L -T1 #L #T1 elim T1 -T1
-[ * #i #IH #T2 #d #e #H
+#L #T1 @(f2_ind … fw … L T1) -L -T1 #n #IH #L *
+[ * #i #Hn #T2 #d #e #H destruct
[ >(delift_inv_sort1 … H) -H //
| elim (delift_inv_lref1 … H) -H * /2 width=1/
#K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2
lapply (IH … HV12) // -H /2 width=6/
| >(delift_inv_gref1 … H) -H //
]
-| * [ #a ] #I #V1 #T1 #_ #_ #IH #X #d #e #H
+| * [ #a ] #I #V1 #T1 #Hn #X #d #e #H
[ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
lapply (delift_lsubs_trans … HT12 (L.ⓑ{I}V1) ?) -HT12 /2 width=1/ #HT12
lapply (IH … HV12) -HV12 // #HV12
(∀L,d,e,i. i < d → R d e L (#i) (#i)) →
(∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
⇩[O, i] L ≡ K.ⓓV1 → K ⊢ ▼*[O, d + e - i - 1] V1 ≡ V2 →
- ⇧[O, d] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L #i W2
+ ⇧[O, d] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L (#i) W2
) →
(∀L,d,e,i. d + e ≤ i → R d e L (#i) (#(i - e))) →
(∀L,d,e,p. R d e L (§p) (§p)) →