| lift_sort : ∀k,d,e. lift d e (⋆k) (⋆k)
| lift_lref_lt: ∀i,d,e. i < d → lift d e (#i) (#i)
| lift_lref_ge: ∀i,d,e. d ≤ i → lift d e (#i) (#(i + e))
+| lift_gref : ∀p,d,e. lift d e (§p) (§p)
| lift_bind : ∀I,V1,V2,T1,T2,d,e.
lift d e V1 V2 → lift (d + 1) e T1 T2 →
lift d e (𝕓{I} V1. T1) (𝕓{I} V2. T2)
| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
<(arith_d1 i e2 e1) // /3/
+| /3/
| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
elim (IHT (d2+1) … ? ? He12) /3 width = 5/
[ #k #d #e #i #H destruct
| #j #d #e #Hj #i #Hi destruct /3/
| #j #d #e #Hj #i #Hi destruct /3/
+| #p #d #e #i #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
]
elim (lt_refl_false … Hdd)
qed.
+fact lift_inv_gref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
+#d #e #T1 #T2 * -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_gref1: ∀d,e,T2,p. ↑[d,e] §p ≡ T2 → T2 = §p.
+/2 width=5/ qed.
+
fact 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 &
[ #k #d #e #I #V1 #U1 #H destruct
| #i #d #e #_ #I #V1 #U1 #H destruct
| #i #d #e #_ #I #V1 #U1 #H destruct
+| #p #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
]
[ #k #d #e #I #V1 #U1 #H destruct
| #i #d #e #_ #I #V1 #U1 #H destruct
| #i #d #e #_ #I #V1 #U1 #H destruct
+| #p #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/
]
[ #k #d #e #i #H destruct
| #j #d #e #Hj #i #Hi destruct /3/
| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4/
+| #p #d #e #i #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
]
elim (plus_lt_false … Hdd)
qed.
+fact lift_inv_gref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
+#d #e #T1 #T2 * -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_gref2: ∀d,e,T1,p. ↑[d,e] T1 ≡ §p → T1 = §p.
+/2 width=5/ qed.
+
fact lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
∀I,V2,U2. T2 = 𝕓{I} V2.U2 →
∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
[ #k #d #e #I #V2 #U2 #H destruct
| #i #d #e #_ #I #V2 #U2 #H destruct
| #i #d #e #_ #I #V2 #U2 #H destruct
+| #p #d #e #I #V2 #U2 #H destruct
| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
]
[ #k #d #e #I #V2 #U2 #H destruct
| #i #d #e #_ #I #V2 #U2 #H destruct
| #i #d #e #_ #I #V2 #U2 #H destruct
+| #p #d #e #I #V2 #U2 #H destruct
| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width = 5/
]