From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Mon, 24 Feb 2014 18:56:46 +0000 (+0000)
Subject: - we bypassed another false conjecture :) ...
X-Git-Tag: make_still_working~970
X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=commitdiff_plain;h=f7994db705d6c1200cc3e9f1827b7d9f6d0ad001

- we bypassed another false conjecture :) ...
- minor corrections
---

diff --git a/matita/matita/contribs/lambdadelta/apps_2/web/apps_2.ldw.xml b/matita/matita/contribs/lambdadelta/apps_2/web/apps_2.ldw.xml
index dac657eb3..f13d943f8 100644
--- a/matita/matita/contribs/lambdadelta/apps_2/web/apps_2.ldw.xml
+++ b/matita/matita/contribs/lambdadelta/apps_2/web/apps_2.ldw.xml
@@ -1,8 +1,8 @@
 <?xml version="1.0" encoding="UTF-8"?>
 
 <page xmlns="http://lambdadelta.info/"
-      description = "applications of lambdadelta version 2"
-      title = "applications of lambdadelta version 2"
+      description = "applications of \lambda\delta version 2"
+      title = "applications of \lambda\delta version 2"
       head = "cic:/matita/lambdadelta/apps_2/ (applications of λδ version 2)"
 >
    <section>Contents of the Specification</section>
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs.ma
index 29eee60e4..7ad3bb47a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs.ma
@@ -93,6 +93,10 @@ lemma fpbs_fqus_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g
 /3 width=5 by fpbs_strap1, fpb_fquq/
 qed-.
 
+lemma fpbs_fqup_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+                       ⦃G, L, T⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_fqus_trans, fqup_fqus/ qed-.
+
 lemma fpbs_cpxs_trans: ∀h,g,G1,G,L1,L,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
                        ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
 #h #g #G1 #G #L1 #L #T1 #T #T2 #H1 #H @(cpxs_ind … H) -T2
@@ -105,6 +109,22 @@ lemma fpbs_lpxs_trans: ∀h,g,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G
 /3 width=5 by fpbs_strap1, fpb_lpx/
 qed-.
 
+lemma cpxs_fqus_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
+                      ⦃G1, L1, T⦄ ⊃* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.
+
+lemma cpxs_fqup_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
+                      ⦃G1, L1, T⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed-.
+
+lemma fqus_lpxs_fpbs: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L, T2⦄ →
+                      ⦃G2, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
+
+lemma cpxs_fqus_lpxs_fpbs: ∀h,g,G1,G2,L1,L,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
+                           ⦃G1, L1, T⦄ ⊃* ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
+
 lemma fqus_fpbs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
                        ⦃G1, L1, T1⦄ ⊃* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind_dx … H) -G1 -L1 -T1
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_ext.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_ext.ma
new file mode 100644
index 000000000..515753860
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_ext.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/computation/fpbs_alt.ma".
+
+(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
+
+(* Properties on extended context-sensitive parallel computation for terms **)
+
+lemma fpbs_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 #HnTU2 elim (fpbs_fpbsa … H) -H
+#L0 #T0 #HT10 #H10 #HL02 elim (eq_term_dec T1 T0) [ -HT10 | -HnTU2 ]
+[ #H destruct lapply (lpxs_cpxs_trans … HTU2 … HL02) -HTU2
+  #HTU2 elim (fqus_cpxs_trans_neq … H10 … HTU2 HnTU2) -T2
+  /3 width=6 by fqus_lpxs_fpbs, ex3_intro/
+| /4 width=6 by fpbs_cpxs_trans, fqus_lpxs_fpbs, ex3_intro/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_fpbs.ma
index 6a868cfa9..86cbe3ee3 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_fpbs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_fpbs.ma
@@ -28,15 +28,3 @@ lemma cpr_lpr_ssta_fpbs: ∀h,g,G,L1,L2,T1,T2,U2,l2.
                          ⦃G, L2⦄ ⊢ T2 ▪[h, g] l2+1 → ⦃G, L2⦄ ⊢ T2 •[h, g] U2 →
                          ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, U2⦄.
 /3 width=5 by fpbs_trans, cpr_lpr_fpbs, ssta_fpbs/ qed.
-
-lemma cpxs_fqup_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
-                      ⦃G1, L1, T⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_trans, cpxs_fpbs, fqup_fpbs/ qed.
-
-lemma cpxs_fqus_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
-                      ⦃G1, L1, T⦄ ⊃* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_trans, cpxs_fpbs, fqus_fpbs/ qed.
-
-lemma cpxs_fqus_lpxs_fpbs: ∀h,g,G1,G2,L1,L,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
-                           ⦃G1, L1, T⦄ ⊃* ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_trans, cpxs_fqus_fpbs, lpxs_fpbs/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_csx.ma
index d40f30a85..f983e8d06 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_csx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_csx.ma
@@ -12,35 +12,42 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/computation/fpbs_ext.ma".
+include "basic_2/computation/csx_fpbs.ma".
 include "basic_2/computation/lsx_csx.ma".
 include "basic_2/computation/fsb_alt.ma".
 
-axiom fqup_lpxs_trans_nlleq: ∀h,g,G1,G2,K1,K2,T1,T2. ⦃G1, K1, T1⦄ ⊃+ ⦃G2, K2, T2⦄ →
-                             ∀L2. ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 → (K2 ⋕[T2, 0] L2 →⊥) →
-                             ∃∃L1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 &
-                                   K1 ⋕[T1, 0] L1 → ⊥ & ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄.
-
 (* "BIG TREE" STRONGLY NORMALIZING TERMS ************************************)
 
 (* Advanced propreties on context-senstive extended bormalizing terms *******)
 
-lemma csx_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
+lemma csx_fsb_fpbs: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
+                    ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⦥[h, g] T2.
 #h #g #G1 #L1 #T1 #H @(csx_ind_alt … H) -T1
-#T1 #HT1 @(csx_ind_lsx … (yinj 0) … HT1) -L1
-#L1 @(fqup_wf_ind … G1 L1 T1) -G1 -L1 -T1
-#G1 #L1 #T1 #IHu #H1 #IHl #IHc @fsb_intro
-#G2 #L2 #T2 * -G2 -L2 -T2
-[ #G2 #L2 #T2 #H12 @IHu -IHu // /2 width=5 by csx_fqup_conf/ -H1 [| -IHl ]
-  [ #L0 #HL20 #HnL20 #_ elim (fqup_lpxs_trans_nlleq … H12 … HL20 HnL20) -L2
-    /6 width=5 by fsb_fpbs_trans, lpxs_fpbs, fqup_fpbs, lpxs_cpxs_trans/
-  | #T0 #HT20 #HnT20 elim (fqup_cpxs_trans_neq … H12 … HT20 HnT20) -T2
-    /4 width=5 by fsb_fpbs_trans, fqup_fpbs/
+#T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind … G2 L2 T2) -G2 -L2 -T2
+#G0 #L0 #T0 #IHu #H10 lapply (csx_fpbs_conf … H10) // -HT1
+#HT0 generalize in match IHu; -IHu generalize in match H10; -H10
+@(lsx_ind … (csx_lsx … HT0 0)) -L0
+#L0 #_ #IHl #H10 #IHu @fsb_intro
+#G2 #L2 #T2 * -G2 -L2 -T2 [ -IHl -IHc | -IHu -IHl |  ]
+[ /3 width=5 by fpbs_fqup_trans/
+| #T2 #HT02 #HnT02 elim (fpbs_cpxs_trans_neq … H10 … HT02 HnT02) -T0
+  /3 width=4 by/
+| #L2 #HL02 #HnL02 @(IHl … HL02 HnL02) -IHl -HnL02 [ -IHu -IHc | ]
+  [ /2 width=3 by fpbs_lpxs_trans/
+  | #G3 #L3 #T3 #H03 #_ elim (lpxs_fqup_trans … H03 … HL02) -L2
+    #L4 #T4 elim (eq_term_dec T0 T4) [ -IHc | -IHu ]
+    [ #H destruct /4 width=5 by fsb_fpbs_trans, lpxs_fpbs, fpbs_fqup_trans/
+    | #HnT04 #HT04 #H04 elim (fpbs_cpxs_trans_neq … H10 … HT04 HnT04) -T0
+      /4 width=8 by fpbs_fqup_trans, fpbs_lpxs_trans/
+    ]
   ]
-| -H1 -IHu -IHl /3 width=1 by/
-| -H1 -IHu /5 width=5 by fsb_fpbs_trans, lpxs_fpbs, lpxs_cpxs_trans/
 ]
 qed.
 
+lemma csx_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
+/2 width=5 by csx_fsb_fpbs/ qed.
+
 (* Advanced eliminators *****************************************************)
 
 lemma csx_ind_fpbu: ∀h,g. ∀R:relation3 genv lenv term.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
index b22ee7c89..2f21a8d70 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
@@ -89,6 +89,29 @@ lemma lpxs_pair2: ∀h,g,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
 
 (* Properties on supclosure *************************************************)
 
+lemma lpx_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
+                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
+                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (lpx_fqu_trans … H12 … HKL1) -L1
+  /3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
+| #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+  #L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fqu_trans … H2 … HL0) -L
+  #L #T3 #HT3 #HT32 #HL2 elim (fqup_cpx_trans … HT0 … HT3) -T
+  /3 width=7 by cpxs_strap1, fqup_strap1, ex3_2_intro/
+]
+qed-.
+
+lemma lpx_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
+                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
+                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/
+#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+#L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fquq_trans … H2 … HL0) -L
+#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans … HT0 … HT3) -T
+/3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/
+qed-.
+
 lemma lpxs_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
                        ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
                        ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
@@ -101,6 +124,18 @@ lemma lpxs_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2,
 ]
 qed-.
 
+lemma lpxs_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
+                       ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
+                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(lpxs_ind_dx … H) -K1
+[ /2 width=5 by ex3_2_intro/
+| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
+  lapply (lpx_cpxs_trans … HT1 … HK1) -HT1
+  elim (lpx_fqup_trans … HT2 … HK1) -K
+  /3 width=7 by lpxs_strap2, cpxs_trans, ex3_2_intro/
+]
+qed-.
+
 lemma lpxs_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
                        ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
                        ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
@@ -110,13 +145,3 @@ lemma lpxs_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2
 #L #T3 #HT3 #HT32 #HL2 elim (fqus_cpxs_trans … HT3 … HT0) -T
 /3 width=7 by cpxs_trans, fqus_strap1, ex3_2_intro/
 qed-.
-
-lemma lpx_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
-                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
-                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/
-#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
-#L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fquq_trans … H2 … HL0) -L
-#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans … HT0 … HT3) -T
-/3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.ma
index 13e753a6c..61dca7f92 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.ma
@@ -58,25 +58,3 @@ theorem csx_lsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀d. G ⊢ ⋕
   elim (csx_fwd_flat … H) -H /3 width=1 by lsx_flat/
 ]
 qed.
-
-(* Advanced eliminators *****************************************************)
-
-fact csx_ind_lsx_aux: ∀h,g,G,T,d. ∀R:predicate lenv.
-                      (∀L1. ⦃G, L1⦄ ⊢ ⬊*[h, g] T →
-                            (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ⋕[T, d] L2 → ⊥) → R L2) →
-                            R L1
-                      ) →
-                      ∀L. G ⊢ ⋕⬊*[h, g, T, d] L → ⦃G, L⦄ ⊢ ⬊*[h, g] T → R L.
-#h #g #G #T #d #R #IH #L #H @(lsx_ind … H) -L
-#L1 #_ #IHL1 #HL1 @IH -IH //
-#L2 #HL12 #HnT @IHL1 -IHL1 /2 width=3 by csx_lpxs_conf/
-qed-.
-
-lemma csx_ind_lsx: ∀h,g,G,T,d. ∀R:predicate lenv.
-                   (∀L1. ⦃G, L1⦄ ⊢ ⬊*[h, g] T →
-                         (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ⋕[T, d] L2 → ⊥) → R L2) →
-                         R L1
-                   ) →
-                   ∀L. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R L.
-#h #g #G #T #d #R #IH #L #HL @(csx_ind_lsx_aux … d … HL) /4 width=1 by csx_lsx/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml
index 06a65ff53..cc0a402b0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml
+++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml
@@ -1,8 +1,8 @@
 <?xml version="1.0" encoding="UTF-8"?>
 
 <page xmlns="http://lambdadelta.info/"
-      description = "lambdadelta version 2"
-      title = "lambdadelta version 2"
+      description = "\lambda\delta version 2"
+      title = "\lambda\delta version 2"
       head = "cic:/matita/lambdadelta/basic_2/ (λδ version 2)"
 >
    <section>System's Syntax and Behavior</section>
diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
index b4eb847a0..9a292eeff 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
+++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
@@ -94,7 +94,7 @@ table {
              [ "fpbg ( ⦃?,?,?⦄ >⋕[?,?] ⦃?,?,?⦄ )" "fpbg_lift" + "fpbg_fpns" + "fpbg_fpbg" * ]
              [ "fpbc ( ⦃?,?,?⦄ ≻⋕[?,?] ⦃?,?,?⦄ )" "fpbc_fpns" + "fpbc_fpbs" * ]
              [ "fpbu ( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" "fpbu_lift" + "fpbu_fpns" * ]
-             [ "fpbs ( ⦃?,?,?⦄ ≥[?,?] ⦃?,?,?⦄ )" "fpbs_alt ( ⦃?,?,?⦄ ≥≥[?,?] ⦃?,?,?⦄ )" "fpbs_lift" + "fpbs_aaa" + "fpbs_fpns" + "fpbs_fpbs" * ]
+             [ "fpbs ( ⦃?,?,?⦄ ≥[?,?] ⦃?,?,?⦄ )" "fpbs_alt ( ⦃?,?,?⦄ ≥≥[?,?] ⦃?,?,?⦄ )" "fpbs_lift" + "fpbs_aaa" + "fpbs_fpns" + "fpbs_fpbs" + "fpbs_ext" * ]
           }
         ]
         [ { "parallel computation for \"big tree\" normal forms" * } {
diff --git a/matita/matita/contribs/lambdadelta/ground_2/web/ground_2.ldw.xml b/matita/matita/contribs/lambdadelta/ground_2/web/ground_2.ldw.xml
index f1ea686a1..3123e6807 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/web/ground_2.ldw.xml
+++ b/matita/matita/contribs/lambdadelta/ground_2/web/ground_2.ldw.xml
@@ -1,8 +1,8 @@
 <?xml version="1.0" encoding="UTF-8"?>
 
 <page xmlns="http://lambdadelta.info/"
-      description = "lambdadelta version 2"
-      title = "lambdadelta version 2"
+      description = "background for \lambda\delta version 2"
+      title = "background for \lambda\delta version 2"
       head = "cic:/matita/lambdadelta/ground_2/ (background for λδ version 2)"
 >
    <section>Summary of the Specification</section>