+(**************************************************************************)
+(* ___ *)
+(* ||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/static/reqx_reqx.ma".
+include "static_2/static/feqx.ma".
+
+(* SORT-IRRELEVANT EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *************)
+
+(* Advanced properties ******************************************************)
+
+lemma feqx_sym: tri_symmetric … feqx.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1
+/3 width=1 by feqx_intro_dx, reqx_sym, teqx_sym/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem feqx_trans: tri_transitive … feqx.
+#G1 #G #L1 #L #T1 #T * -G -L -T
+#L #T #HL1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
+/4 width=5 by feqx_intro_sn, reqx_trans, teqx_reqx_div, teqx_trans/
+qed-.
+
+theorem feqx_canc_sn: ∀G,G1,L,L1,T,T1. ❪G,L,T❫ ≛ ❪G1,L1,T1❫→
+ ∀G2,L2,T2. ❪G,L,T❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫.
+/3 width=5 by feqx_trans, feqx_sym/ qed-.
+
+theorem feqx_canc_dx: ∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ≛ ❪G,L,T❫ →
+ ∀G2,L2,T2. ❪G2,L2,T2❫ ≛ ❪G,L,T❫ → ❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫.
+/3 width=5 by feqx_trans, feqx_sym/ qed-.
+
+(* Main inversion lemmas with degree-based equivalence on terms *************)
+
+theorem feqx_tneqx_repl_dx: ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫ →
+ ∀U1,U2. ❪G1,L1,U1❫ ≛ ❪G2,L2,U2❫ →
+ (T2 ≛ U2 → ⊥) → (T1 ≛ U1 → ⊥).
+#G1 #G2 #L1 #L2 #T1 #T2 #HT #U1 #U2 #HU #HnTU2 #HTU1
+elim (feqx_inv_gen_sn … HT) -HT #_ #_ #HT
+elim (feqx_inv_gen_sn … HU) -HU #_ #_ #HU
+/3 width=5 by teqx_repl/
+qed-.