+(**************************************************************************)
+(* ___ *)
+(* ||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/rt_computation/fpbs_fpbc.ma".
+include "basic_2/rt_computation/fpbg_fpbs.ma".
+
+(* PROPER PARALLEL RST-COMPUTATION FOR CLOSURES *****************************)
+
+(* Properties with generic equivalence for closures *************************)
+
+(* Basic_2A1: uses: fpbg_fleq_trans *)
+lemma fpbg_feqg_trans (S) (G) (L) (T):
+ reflexive … S → symmetric … S →
+ ∀G1,L1,T1. ❪G1,L1,T1❫ > ❪G,L,T❫ →
+ ∀G2,L2,T2. ❪G,L,T❫ ≛[S] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
+/3 width=8 by fpbg_fpb_trans, feqg_fpb/ qed-.
+
+(* Basic_2A1: uses: fleq_fpbg_trans *)
+lemma feqg_fpbg_trans (S) (G) (L) (T):
+ reflexive … S → symmetric … S →
+ ∀G1,L1,T1. ❪G1,L1,T1❫ ≛[S] ❪G,L,T❫ →
+ ∀G2,L2,T2. ❪G,L,T❫ > ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
+/3 width=8 by fpb_fpbg_trans, feqg_fpb/ qed-.
+
+(* Properties with generic equivalence for terms ****************************)
+
+lemma fpbg_teqg_div (S):
+ reflexive … S → symmetric … S →
+ ∀G1,G2,L1,L2,T1,T. ❪G1,L1,T1❫ > ❪G2,L2,T❫ →
+ ∀T2. T2 ≛[S] T → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
+/4 width=8 by fpbg_feqg_trans, teqg_feqg, teqg_sym/ qed-.
+
+(* Advanced inversion lemmas of parallel rst-computation on closures ********)
+
+(* Basic_2A1: was: fpbs_fpbg *)
+lemma fpbs_inv_fpbg:
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ →
+ ∨∨ ❪G1,L1,T1❫ ≅ ❪G2,L2,T2❫
+ | ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
+#G1 #G2 #L1 #L2 #T1 #T2 #H
+elim (fpbs_inv_fpbc_sn … H) -H
+[ /2 width=1 by or_introl/
+| * #G #L #T #H1 #H2
+ /3 width=9 by fpbg_intro, or_intror/
+]
+qed-.
+
+(* Basic_2A1: this was the definition of fpbg *)
+lemma fpbg_inv_fpbc_fpbs (G1) (G2) (L1) (L2) (T1) (T2):
+ ❪G1,L1,T1❫ > ❪G2,L2,T2❫ →
+ ∃∃G,L,T. ❪G1,L1,T1❫ ≻ ❪G,L,T❫ & ❪G,L,T❫ ≥ ❪G2,L2,T2❫.
+#G1 #G2 #L1 #L2 #T1 #T2 #H
+elim (fpbg_inv_gen … H) -H #G3 #L3 #T3 #G4 #L4 #T4 #H13 #H34 #H42
+elim (fpbs_inv_fpbc_sn … H13) -H13
+[ /3 width=9 by feqg_fpbc_trans, ex2_3_intro/
+| * #G #L #T #H1 #H3
+ /4 width=13 by fpbg_fwd_fpbs,fpbg_intro, ex2_3_intro/
+]
+qed-.
+
+(* Advanced properties of parallel rst-computation on closures **************)
+
+lemma fpbs_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 #H elim (fpbs_inv_fpbg … H) -H
+[ #H12 #G2 #L2 #U2 #H22
+ lapply (feqg_fpbc_trans … H12 … H22) -F2 -K2 -T2
+ /3 width=5 by feqg_fpbs, ex2_3_intro/
+| #H12 #G2 #L2 #U2 #H22
+ elim (fpbg_inv_fpbc_fpbs … H12) -H12 #F #K #T #H1 #H2
+ /4 width=9 by fpbs_strap1, fpbc_fwd_fpb, ex2_3_intro/
+]
+qed-.