(* RELOCATION ***************************************************************)
inductive lift: term → nat → nat → term → Prop ≝
- | lift_sort : ∀k,d,e. lift (⋆k) d e (⋆k)
- | lift_lref_lt: ∀i,d,e. i < d → lift (#i) d e (#i)
- | lift_lref_ge: ∀i,d,e. d ≤ i → lift (#i) d e (#(i + e))
- | lift_con2 : ∀I,V1,V2,T1,T2,d,e.
- lift V1 d e V2 → lift T1 (d + 1) e T2 →
- lift (♭I V1. T1) d e (♭I V2. T2)
+| lift_sort : ∀k,d,e. lift (⋆k) d e (⋆k)
+| lift_lref_lt: ∀i,d,e. i < d → lift (#i) d e (#i)
+| lift_lref_ge: ∀i,d,e. d ≤ i → lift (#i) d e (#(i + e))
+| lift_bind : ∀I,V1,V2,T1,T2,d,e.
+ lift V1 d e V2 → lift T1 (d + 1) e T2 →
+ lift (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
+| lift_flat : ∀I,V1,V2,T1,T2,d,e.
+ lift V1 d e V2 → lift T1 d e T2 →
+ lift (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
.
interpretation "relocation" 'RLift T1 d e T2 = (lift T1 d e T2).
+
+(* The basic properties *****************************************************)
+
+lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d,e] #(i - e) ≡ #i.
+#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3/
+qed.
+
+(* The basic inversion lemmas ***********************************************)
+
+lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆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_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
+#d #e #T1 #k #H lapply (lift_inv_sort2_aux … H) /2/
+qed.
+
+lemma lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i →
+ (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(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 <minus_plus_m_m /4/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
+]
+qed.
+
+lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i →
+ (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
+#d #e #T1 #i #H lapply (lift_inv_lref2_aux … H) /2/
+qed.
+
+lemma 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 &
+ T1 = 𝕓{I} V1.U1.
+#d #e #T1 #T2 #H elim H -H d e T1 T2
+[ #k #d #e #I #V2 #U2 #H destruct
+| #i #d #e #_ #I #V2 #U2 #H destruct
+| #i #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
+]
+qed.
+
+lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕓{I} V2. U2 →
+ ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+ T1 = 𝕓{I} V1. U1.
+#d #e #T1 #I #V2 #U2 #H lapply (lift_inv_bind2_aux … H) /2/
+qed.
+
+lemma lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+ ∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
+ ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+ T1 = 𝕗{I} V1.U1.
+#d #e #T1 #T2 #H elim H -H d e T1 T2
+[ #k #d #e #I #V2 #U2 #H destruct
+| #i #d #e #_ #I #V2 #U2 #H destruct
+| #i #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/
+]
+qed.
+
+lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕗{I} V2. U2 →
+ ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+ T1 = 𝕗{I} V1. U1.
+#d #e #T1 #I #V2 #U2 #H lapply (lift_inv_flat2_aux … H) /2/
+qed.
+
+(* the main properies *******************************************************)
+
+theorem lift_trans_rev: ∀d1,e1,T1,T. ↑[d1,e1] T1 ≡ T →
+ ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T →
+ d1 ≤ d2 →
+ ∃∃T0. ↑[d1, e1] T0 ≡ T2 & ↑[d2, e2] T0 ≡ T1.
+#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
+[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+ lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
+ lapply (lift_inv_lref2 … Hi) -Hi * * #Hid2 #H destruct -T2
+ [ -Hid2 /4/
+ | elim (lt_false d1 ?)
+ @(le_to_lt_to_lt … Hd12) -Hd12 @(le_to_lt_to_lt … Hid1) -Hid1 /2/
+ ]
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
+ lapply (lift_inv_lref2 … Hi) -Hi * * #Hid2 #H destruct -T2
+ [ -Hd12; lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3/
+ | -Hid1; lapply (arith1 … Hid2) -Hid2 #Hid2
+ @(ex2_1_intro … #(i - e2))
+ [ >le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ]
+ | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /3/
+ ]
+ ]
+| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
+ lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
+ elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
+ >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /3 width = 5/
+| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
+ lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
+ elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
+ elim (IHU … HU2 ?) /3 width = 5/
+]
+qed.
+
+theorem lift_free: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
+ d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
+ ∃∃T. ↑[d1, e1] T1 ≡ T & ↑[d2, e2 - e1] T ≡ T2.
+#d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2
+[ /3/
+| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
+ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/
+| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
+ lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
+ <(plus_plus_minus_m_m e1 e2 i) /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/
+| #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 … ? ? He12) /3 width = 5/
+]
+qed.
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