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diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbg.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbg.ma
index 4f092111a..97846fda4 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbg.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbg.ma
@@ -12,11 +12,46 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/computation/lpxs_lpxs.ma".
+include "basic_2/computation/fpbs_alt.ma".
 include "basic_2/computation/fpbg.ma".
 
-(* "BIG TREE" ORDER FOR CLOSURES ********************************************)
+(* GENERAL "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **************)
 
-(* Main properties **********************************************************)
+(* Advanced forward lemmas **************************************************)
 
-theorem fpbg_trans: ∀h,g. tri_transitive … (fpbg h g).
-/3 width=5 by fpbg_fwd_fpbs, fpbg_fpbs_trans/ qed-.
+lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ →
+                     ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
+/3 width=5 by cpxs_fqus_lpxs_fpbs, cpxs_fqup_fpbs, fpbs_trans, lpxs_fpbs, cpxs_fpbs/
+qed-.
+
+lemma fpbg_fwd_fpbc_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ →
+                        ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
+[ /4 width=5 by fpbc_cpxs, lpxs_fpbs, ex2_3_intro/
+| #G2 #L #L2 #T #T2 #HT1 #HT2 #HL2 elim (eq_term_dec T1 T) #H destruct
+  [ -HT1 /3 width=5 by fpbc_fqup, lpxs_fpbs, ex2_3_intro/
+  | /5 width=9 by fpbc_cpxs, fpbsa_inv_fpbs, fqup_fqus, ex3_2_intro, ex2_3_intro/
+  ]
+| #G2 #L #L0 #L2 #T #T2 #HT1 #HT2 #HL0 #H0 #HL02 #H02
+  lapply (lpxs_trans … HL0 … HL02) #HL2
+  elim (eq_term_dec T1 T) #H destruct
+  [ -HT1 elim (fqus_inv_gen … HT2) -HT2
+    [ /3 width=5 by fpbc_fqup, lpxs_fpbs, ex2_3_intro/
+    | * #H1 #H2 #H3 destruct
+      /4 width=5 by fpbc_lpxs, lpxs_fpbs, ex2_3_intro/
+    ]
+  | /4 width=9 by fpbc_cpxs, fpbsa_inv_fpbs, ex3_2_intro, ex2_3_intro/
+  ]
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fqu_fpbs_fpbg: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G, L, T⦄ →
+                     ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H elim(fpbs_fpbsa … H) -H
+#L0 #T0 #HT0 #HT02 #HL02 elim (fqu_cpxs_trans … HT0 … H1) -T
+/3 width=7 by fpbg_fqup, fqus_strap2_fqu/
+qed.