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diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbc_fpns.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbc_fpns.ma
index 672d2d6e4..dd15570ef 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbc_fpns.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbc_fpns.ma
@@ -12,29 +12,23 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/relocation/lleq_lleq.ma".
-include "basic_2/computation/fpns.ma".
-include "basic_2/computation/fpbs_alt.ma".
+include "basic_2/computation/fpns_fpns.ma".
+include "basic_2/computation/fpbu_fpns.ma".
 include "basic_2/computation/fpbc.ma".
 
-(* ATOMIC "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************)
+(* SINGLE-STEP "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **********)
 
-(* Forward lemmas on parallel computation for "big tree" normal forms *******)
+(* Properties on parallel computation for "big tree" normal forms ***********)
 
-lemma fpbs_fwd_fpbc_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, 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(fpbs_fpbsa … H) -H
-#L #T #HT1 #HT2 #HL2 elim (eq_term_dec T1 T) #H destruct
-[ -HT1 elim (fqus_inv_gen … HT2) -HT2
-  [ #HT2 @or_intror
-    /5 width=9 by fpbsa_inv_fpbs, fpbc_fqup, ex3_2_intro, ex2_3_intro, or_intror/
-  | * #HG #HL #HT destruct elim (lleq_dec T2 L L2 0) #H
-    [ /3 width=1 by fpns_intro, or_introl/
-    | /5 width=5 by fpbc_lpxs, ex2_3_intro, or_intror/
-    ]
-  ]
-| elim (cpxs_neq_inv_step_sn … HT1 H) -HT1 -H
-  /5 width=9 by fpbsa_inv_fpbs, fpbc_cpxs, cpx_cpxs, ex3_2_intro, ex2_3_intro, or_intror/
-]
+lemma fpbc_fpns_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≻⋕[h, g] ⦃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 *
+/3 width=9 by fpns_trans, ex2_3_intro/
+qed-.
+
+lemma fpns_fpbc_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃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 *
+#G0 #L0 #T0 #H0 #H02 elim (fpns_fpbu_trans … H1 … H0) -G -L -T
+/3 width=9 by fpns_trans, ex2_3_intro/
 qed-.