--- /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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/unfold/tpss_tpss.ma".
+include "basic_2/unfold/delift.ma".
+
+(* INVERSE TERM RELOCATION *************************************************)
+
+(* Properties on partial unfold on terms ************************************)
+
+lemma delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶* U2 →
+ ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+ ∃∃T2. K ⊢ T1 [d, e - ee] ▶* T2 &
+ L ⊢ U2 [dd, ee] ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2
+elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
+elim (tpss_inv_lift1_be … HXU1 … HLK … HTX1 ? ?) -X1 -HLK // /3 width=5/
+qed.
+
+lemma delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶ U2 →
+ ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
+ ∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+ ∃∃T2. K ⊢ T1 [d, e - ee] ▶* T2 &
+ L ⊢ U2 [dd, ee] ≡ T2.
+/3 width=3/ qed.
+
+lemma delift_tpss_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶* U2 →
+ ∀T. L ⊢ U1 [d, e] ≡ T → L ⊢ U2 [d, e] ≡ T.
+#L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
+elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
+lapply (tpss_inv_lift1_eq … HXU1 … HTX1) -HXU1 #H destruct /2 width=3/
+qed.
+
+lemma delift_tps_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶ U2 →
+ ∀T. L ⊢ U1 [d, e] ≡ T → L ⊢ U2 [d, e] ≡ T.
+/3 width=3/ qed.
+
+lemma tpss_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶* U2 →
+ ∀T. L ⊢ U2 [d, e] ≡ T → L ⊢ U1 [d, e] ≡ T.
+#L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
+lapply (tpss_trans_eq … HU12 … HUX1) -U2 /2 width=3/
+qed.
+
+lemma tps_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶ U2 →
+ ∀T. L ⊢ U2 [d, e] ≡ T → L ⊢ U1 [d, e] ≡ T.
+/3 width=3/ qed.