--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L ⊢ break ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'TSubst $L $T1 $d $e $T2 }.
+
+include "basic_2/unfold/tpss.ma".
+
+(* INVERSE BASIC TERM RELOCATION *******************************************)
+
+definition delift: nat → nat → lenv → relation term ≝
+ λd,e,L,T1,T2. ∃∃T. L ⊢ T1 ▶* [d, e] T & ⇧[d, e] T2 ≡ T.
+
+interpretation "inverse basic relocation (term)"
+ 'TSubst L T1 d e T2 = (delift d e L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lift_delift: ∀T1,T2,d,e. ⇧[d, e] T1 ≡ T2 →
+ ∀L. L ⊢ ▼*[d, e] T2 ≡ T1.
+/2 width=3/ qed.
+
+lemma delift_refl_O2: ∀L,T,d. L ⊢ ▼*[d, 0] T ≡ T.
+/2 width=3/ qed.
+
+lemma delift_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼*[d, e] T1 ≡ T2 →
+ ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ ▼*[d, e] T1 ≡ T2.
+#L1 #T1 #T2 #d #e * /3 width=3/
+qed.
+
+lemma delift_sort: ∀L,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ ⋆k.
+/2 width=3/ qed.
+
+lemma delift_lref_lt: ∀L,d,e,i. i < d → L ⊢ ▼*[d, e] #i ≡ #i.
+/3 width=3/ qed.
+
+lemma delift_lref_ge: ∀L,d,e,i. d + e ≤ i → L ⊢ ▼*[d, e] #i ≡ #(i - e).
+/3 width=3/ qed.
+
+lemma delift_gref: ∀L,d,e,p. L ⊢ ▼*[d, e] §p ≡ §p.
+/2 width=3/ qed.
+
+lemma delift_bind: ∀a,I,L,V1,V2,T1,T2,d,e.
+ L ⊢ ▼*[d, e] V1 ≡ V2 → L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 →
+ L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ ⓑ{a,I} V2. T2.
+#a #I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2
+lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ /3 width=5/
+qed.
+
+lemma delift_flat: ∀I,L,V1,V2,T1,T2,d,e.
+ L ⊢ ▼*[d, e] V1 ≡ V2 → L ⊢ ▼*[d, e] T1 ≡ T2 →
+ L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
+#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * /3 width=5/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma delift_inv_sort1: ∀L,U2,d,e,k. L ⊢ ▼*[d, e] ⋆k ≡ U2 → U2 = ⋆k.
+#L #U2 #d #e #k * #U #HU
+>(tpss_inv_sort1 … HU) -HU #HU2
+>(lift_inv_sort2 … HU2) -HU2 //
+qed-.
+
+lemma delift_inv_gref1: ∀L,U2,d,e,p. L ⊢ ▼*[d, e] §p ≡ U2 → U2 = §p.
+#L #U #d #e #p * #U #HU
+>(tpss_inv_gref1 … HU) -HU #HU2
+>(lift_inv_gref2 … HU2) -HU2 //
+qed-.
+
+lemma delift_inv_bind1: ∀a,I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓑ{a,I} V1. T1 ≡ U2 →
+ ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 &
+ L. ⓑ{I} V2 ⊢ ▼*[d+1, e] T1 ≡ T2 &
+ U2 = ⓑ{a,I} V2. T2.
+#a #I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
+elim (tpss_inv_bind1 … HU) -HU #V #T #HV1 #HT1 #X destruct
+elim (lift_inv_bind2 … HU2) -HU2 #V2 #T2 #HV2 #HT2
+lapply (tpss_lsubr_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
+qed-.
+
+lemma delift_inv_flat1: ∀I,L,V1,T1,U2,d,e. L ⊢ ▼*[d, e] ⓕ{I} V1. T1 ≡ U2 →
+ ∃∃V2,T2. L ⊢ ▼*[d, e] V1 ≡ V2 &
+ L ⊢ ▼*[d, e] T1 ≡ T2 &
+ U2 = ⓕ{I} V2. T2.
+#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
+elim (tpss_inv_flat1 … HU) -HU #V #T #HV1 #HT1 #X destruct
+elim (lift_inv_flat2 … HU2) -HU2 /3 width=5/
+qed-.
+
+lemma delift_inv_refl_O2: ∀L,T1,T2,d. L ⊢ ▼*[d, 0] T1 ≡ T2 → T1 = T2.
+#L #T1 #T2 #d * #T #HT1
+>(tpss_inv_refl_O2 … HT1) -HT1 #HT2
+>(lift_inv_refl_O2 … HT2) -HT2 //
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma delift_fwd_tw: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → ♯{T1} ≤ ♯{T2}.
+#L #T1 #T2 #d #e * #T #HT1 #HT2
+>(lift_fwd_tw … HT2) -T2 /2 width=4 by tpss_fwd_tw/
+qed-.