--- /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/s_transition/fqu_teqx.ma".
+include "static_2/static/feqx.ma".
+include "basic_2/rt_transition/fpb_reqx.ma".
+
+(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
+
+(* Properties with degree-based equivalence for closures ********************)
+
+(* Basic_2A1: uses: fleq_fpb_trans *)
+lemma feqx_fpb_trans:
+ ∀F1,F2,K1,K2,T1,T2. ❪F1,K1,T1❫ ≛ ❪F2,K2,T2❫ →
+ ∀G2,L2,U2. ❪F2,K2,T2❫ ≻ ❪G2,L2,U2❫ →
+ ∃∃G1,L1,U1. ❪F1,K1,T1❫ ≻ ❪G1,L1,U1❫ & ❪G1,L1,U1❫ ≛ ❪G2,L2,U2❫.
+#F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
+#K2 #T2 #HK12 #HT12 #G2 #L2 #U2 #H12
+elim (teqx_fpb_trans … HT12 … H12) -T2 #K0 #T0 #H #HT0 #HK0
+elim (reqx_fpb_trans … HK12 … H) -K2 #L0 #U0 #H #HUT0 #HLK0
+@(ex2_3_intro … H) -H (**) (* full auto too slow *)
+/4 width=3 by feqx_intro_dx, reqx_trans, teqx_reqx_conf_sn, teqx_trans/
+qed-.
+
+(* Inversion lemmas with degree-based equivalence for closures **************)
+
+(* Basic_2A1: uses: fpb_inv_fleq *)
+lemma fpb_inv_feqx:
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≻ ❪G2,L2,T2❫ →
+ ❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫ → ⊥.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #H elim (feqx_inv_gen_sn … H) -H
+ /3 width=11 by reqx_fwd_length, fqu_inv_teqx/
+| #T2 #_ #HnT #H elim (feqx_inv_gen_sn … H) -H /2 width=1 by/
+| #L2 #_ #HnL #H elim (feqx_inv_gen_sn … H) -H /2 width=1 by/
+]
+qed-.