--- /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 "static_2/syntax/toeq.ma".
+include "static_2/relocation/lifts_lifts.ma".
+
+(* GENERIC RELOCATION FOR TERMS *********************************************)
+
+(* Properties with sort-irrelevant outer equivalence for terms **************)
+
+lemma toeq_lifts_sn: liftable2_sn toeq.
+#T1 #T2 #H elim H -T1 -T2 [||| * ]
+[ #s1 #s2 #f #X #H
+ >(lifts_inv_sort1 … H) -H
+ /2 width=3 by toeq_sort, ex2_intro/
+| #i #f #X #H
+ elim (lifts_inv_lref1 … H) -H #j #Hj #H destruct
+ /3 width=3 by toeq_lref, lifts_lref, ex2_intro/
+| #l #f #X #H
+ >(lifts_inv_gref1 … H) -H
+ /2 width=3 by toeq_gref, ex2_intro/
+| #p #I #V1 #V2 #T1 #T2 #f #X #H
+ elim (lifts_inv_bind1 … H) -H #W1 #U1 #_ #_ #H destruct -V1 -T1
+ elim (lifts_total V2 f) #W2 #HVW2
+ elim (lifts_total T2 (⫯f)) #U2 #HTU2
+ /3 width=3 by toeq_pair, lifts_bind, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #f #X #H
+ elim (lifts_inv_flat1 … H) -H #W1 #U1 #_ #_ #H destruct -V1 -T1
+ elim (lifts_total V2 f) #W2 #HVW2
+ elim (lifts_total T2 f) #U2 #HTU2
+ /3 width=3 by toeq_pair, lifts_flat, ex2_intro/
+]
+qed-.
+
+lemma toeq_lifts_dx: liftable2_dx toeq.
+/3 width=3 by toeq_lifts_sn, liftable2_sn_dx, toeq_sym/ qed-.
+
+lemma toeq_lifts_bi: liftable2_bi toeq.
+/3 width=6 by toeq_lifts_sn, liftable2_sn_bi/ qed-.
+
+(* Inversion lemmas with sort-irrelevant outer equivalence for terms ********)
+
+lemma toeq_inv_lifts_bi: deliftable2_bi toeq.
+#U1 #U2 #H elim H -U1 -U2 [||| * ]
+[ #s1 #s2 #f #X1 #H1 #X2 #H2
+ >(lifts_inv_sort2 … H1) -H1 >(lifts_inv_sort2 … H2) -H2
+ /1 width=1 by toeq_sort/
+| #j #f #X1 #H1 #X2 #H2
+ elim (lifts_inv_lref2 … H1) -H1 #i1 #Hj1 #H destruct
+ elim (lifts_inv_lref2 … H2) -H2 #i2 #Hj2 #H destruct
+ <(at_inj … Hj2 … Hj1) -j -f -i1
+ /1 width=1 by toeq_lref/
+| #l #f #X1 #H1 #X2 #H2
+ >(lifts_inv_gref2 … H1) -H1 >(lifts_inv_gref2 … H2) -H2
+ /1 width=1 by toeq_gref/
+| #p #I #W1 #W2 #U1 #U2 #f #X1 #H1 #X2 #H2
+ elim (lifts_inv_bind2 … H1) -H1 #V1 #T1 #_ #_ #H destruct -W1 -U1
+ elim (lifts_inv_bind2 … H2) -H2 #V2 #T2 #_ #_ #H destruct -W2 -U2
+ /1 width=1 by toeq_pair/
+| #I #W1 #W2 #U1 #U2 #f #X1 #H1 #X2 #H2
+ elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #_ #_ #H destruct -W1 -U1
+ elim (lifts_inv_flat2 … H2) -H2 #V2 #T2 #_ #_ #H destruct -W2 -U2
+ /1 width=1 by toeq_pair/
+]
+qed-.