+lemma lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
+#d #e #T1 #T2 #H elim H -H d e T1 T2 //
+[ #i #d #e #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #k #H destruct
+]
+qed.
+
+lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
+#d #e #T2 #k #H lapply (lift_inv_sort1_aux … H) /2/
+qed.
+
+lemma lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i →
+ (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
+#d #e #T1 #T2 #H elim H -H d e T1 T2
+[ #k #d #e #i #H destruct
+| #j #d #e #Hj #i #Hi destruct /3/
+| #j #d #e #Hj #i #Hi destruct /3/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
+]
+qed.
+
+lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 →
+ (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
+#d #e #T2 #i #H lapply (lift_inv_lref1_aux … H) /2/
+qed.
+
+lemma lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+ ∀I,V1,U1. T1 = 𝕓{I} V1.U1 →
+ ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+ T2 = 𝕓{I} V2. U2.
+#d #e #T1 #T2 #H elim H -H d e T1 T2
+[ #k #d #e #I #V1 #U1 #H destruct
+| #i #d #e #_ #I #V1 #U1 #H destruct
+| #i #d #e #_ #I #V1 #U1 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V1 #U1 #H destruct /2 width=5/
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V1 #U1 #H destruct
+]
+qed.
+
+lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+ T2 = 𝕓{I} V2. U2.
+#d #e #T2 #I #V1 #U1 #H lapply (lift_inv_bind1_aux … H) /2/
+qed.
+
+lemma lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+ ∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
+ ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+ T2 = 𝕗{I} V2. U2.
+#d #e #T1 #T2 #H elim H -H d e T1 T2
+[ #k #d #e #I #V1 #U1 #H destruct
+| #i #d #e #_ #I #V1 #U1 #H destruct
+| #i #d #e #_ #I #V1 #U1 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V1 #U1 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V1 #U1 #H destruct /2 width=5/
+]
+qed.
+
+lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+ T2 = 𝕗{I} V2. U2.
+#d #e #T2 #I #V1 #U1 #H lapply (lift_inv_flat1_aux … H) /2/
+qed.
+