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diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fpbs_alt.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_2A/notation/relations/btpredstaralt_8.ma".
+include "basic_2A/multiple/lleq_fqus.ma".
+include "basic_2A/computation/cpxs_lleq.ma".
+include "basic_2A/computation/lpxs_lleq.ma".
+include "basic_2A/computation/fpbs.ma".
+
+(* "QREST" PARALLEL COMPUTATION FOR CLOSURES ********************************)
+
+(* Note: alternative definition of fpbs *)
+definition fpbsa: ∀h. sd h → tri_relation genv lenv term ≝
+                  λh,g,G1,L1,T1,G2,L2,T2.
+                  ∃∃L0,L,T. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T &
+                         ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ &
+                         ⦃G2, L0⦄ ⊢ ➡*[h, g] L & L ≡[T2, 0] L2.
+
+interpretation "'big tree' parallel computation (closure) alternative"
+   'BTPRedStarAlt h g G1 L1 T1 G2 L2 T2 = (fpbsa h g G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpb_fpbsa_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
+                       ∀G2,L2,T2. ⦃G, L, T⦄ ≥≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #L1 #L #T1 #T * -G -L -T [ #G #L #T #HG1 | #T #HT1 | #L #HL1 | #L #HL1 ]
+#G2 #L2 #T2 * #L00 #L0 #T0 #HT0 #HG2 #HL00 #HL02
+[ elim (fquq_cpxs_trans … HT0 … HG1) -T
+  /3 width=7 by fqus_strap2, ex4_3_intro/
+| /3 width=7 by cpxs_strap2, ex4_3_intro/
+| lapply (lpx_cpxs_trans … HT0 … HL1) -HT0 #HT10
+  elim (lpx_fqus_trans … HG2 … HL1) -L
+  /3 width=7 by lpxs_strap2, cpxs_trans, ex4_3_intro/
+| lapply (lleq_cpxs_trans … HT0 … HL1) -HT0 #HT0
+  lapply (cpxs_lleq_conf_sn … HT0 … HL1) -HL1 #HL1
+  elim (lleq_fqus_trans … HG2 … HL1) -L #K00 #HG12 #HKL00
+  elim (lleq_lpxs_trans … HL00 … HKL00) -L00
+  /3 width=9 by lleq_trans, ex4_3_intro/
+]
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem fpbs_fpbsa: ∀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 @(fpbs_ind_dx … H) -G1 -L1 -T1
+/2 width=7 by fpb_fpbsa_trans, ex4_3_intro/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem fpbsa_inv_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 *
+/3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_lleq/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fpbs_intro_alt: ∀h,g,G1,G2,L1,L0,L,L2,T1,T,T2.
+                      ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ →
+                      ⦃G2, L0⦄ ⊢ ➡*[h, g] L → L ≡[T2, 0] L2 →  ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ .
+/3 width=7 by fpbsa_inv_fpbs, ex4_3_intro/ qed.
+
+(* Advanced inversion lemmas *************************************************)
+
+lemma fpbs_inv_alt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                    ∃∃L0,L,T. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T &
+                              ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ &
+                              ⦃G2, L0⦄ ⊢ ➡*[h, g] L & L ≡[T2, 0] L2.
+/2 width=1 by  fpbs_fpbsa/ qed-.