+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/tdeq.ma".
+include "basic_2/relocation/lifts.ma".
+
+(* GENERIC RELOCATION FOR TERMS *********************************************)
+
+(* Properties with degree-based equivalence for terms ***********************)
+
+lemma tdeq_lifts: ∀h,o. liftable2 (tdeq h o).
+#h #o #T1 #T2 #H elim H -T1 -T2 [||| * ]
+[ #s1 #s2 #d #Hs1 #Hs2 #f #X #H >(lifts_inv_sort1 … H) -H
+ /3 width=5 by lifts_sort, tdeq_sort, ex2_intro/
+| #i #f #X #H elim (lifts_inv_lref1 … H) -H
+ /3 width=3 by lifts_lref, tdeq_lref, ex2_intro/
+| #l #f #X #H >(lifts_inv_gref1 … H) -H
+ /2 width=3 by lifts_gref, tdeq_gref, ex2_intro/
+| #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f #X #H elim (lifts_inv_bind1 … H) -H
+ #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV … HVW1) -V1 elim (IHT … HTU1) -T1
+ /3 width=5 by lifts_bind, tdeq_pair, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f #X #H elim (lifts_inv_flat1 … H) -H
+ #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV … HVW1) -V1 elim (IHT … HTU1) -T1
+ /3 width=5 by lifts_flat, tdeq_pair, ex2_intro/
+]
+qed-.
+
+(* Inversion lemmas with degree-based equivalence for terms *****************)
+
+lemma tdeq_inv_lifts: ∀h,o. deliftable2_sn (tdeq h o).
+#h #o #U1 #U2 #H elim H -U1 -U2 [||| * ]
+[ #s1 #s2 #d #Hs1 #Hs2 #f #X #H >(lifts_inv_sort2 … H) -H
+ /3 width=5 by lifts_sort, tdeq_sort, ex2_intro/
+| #i #f #X #H elim (lifts_inv_lref2 … H) -H
+ /3 width=3 by lifts_lref, tdeq_lref, ex2_intro/
+| #l #f #X #H >(lifts_inv_gref2 … H) -H
+ /2 width=3 by lifts_gref, tdeq_gref, ex2_intro/
+| #p #I #W1 #W2 #U1 #U2 #_ #_ #IHW #IHU #f #X #H elim (lifts_inv_bind2 … H) -H
+ #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW … HVW1) -W1 elim (IHU … HTU1) -U1
+ /3 width=5 by lifts_bind, tdeq_pair, ex2_intro/
+| #I #W1 #W2 #U1 #U2 #_ #_ #IHW #IHU #f #X #H elim (lifts_inv_flat2 … H) -H
+ #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW … HVW1) -W1 elim (IHU … HTU1) -U1
+ /3 width=5 by lifts_flat, tdeq_pair, ex2_intro/
+]
+qed-.