From a76f56fdad6348b167376093920650379c9936d4 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Mon, 7 Apr 2014 19:53:54 +0000
Subject: [PATCH] some resuls on pointwise extensions (all of them are now in
 the "relocation" component

---
 .../lambdadelta/basic_2/computation/lprs.ma   |   2 +-
 .../basic_2/notation/relations/lazyor_4.ma    |  19 ++
 .../lambdadelta/basic_2/reduction/lpr.ma      |   2 +-
 .../basic_2/reduction/lpr_ldrop.ma            |   2 +-
 .../lambdadelta/basic_2/reduction/lpr_lpr.ma  |   2 +-
 .../lift_neg.etc => relocation/lift_neg.ma}   |   0
 .../lambdadelta/basic_2/relocation/llor.ma    |  55 ++++++
 .../llpx_sn_alt.ma}                           | 178 ++++++++----------
 .../basic_2/relocation/llpx_sn_lpx_sn.ma      |   2 +-
 .../basic_2/{grammar => relocation}/lpx_sn.ma |   0
 .../basic_2/relocation/lpx_sn_alt.ma          | 150 +++++++++++++++
 .../{ldrop_lpx_sn.ma => lpx_sn_ldrop.ma}      |   6 +-
 .../{grammar => relocation}/lpx_sn_lpx_sn.ma  |   2 +-
 .../{grammar => relocation}/lpx_sn_tc.ma      |   2 +-
 .../lleq_alt.etc => substitution/lleq_alt.ma} |  10 +-
 .../lambdadelta/basic_2/web/basic_2_src.tbl   |   8 +-
 16 files changed, 319 insertions(+), 121 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyor_4.ma
 rename matita/matita/contribs/lambdadelta/basic_2/{etc/llpx_sn/lift_neg.etc => relocation/lift_neg.ma} (100%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/relocation/llor.ma
 rename matita/matita/contribs/lambdadelta/basic_2/{etc/llpx_sn/llpx_sn_alt.etc => relocation/llpx_sn_alt.ma} (62%)
 rename matita/matita/contribs/lambdadelta/basic_2/{grammar => relocation}/lpx_sn.ma (100%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_alt.ma
 rename matita/matita/contribs/lambdadelta/basic_2/relocation/{ldrop_lpx_sn.ma => lpx_sn_ldrop.ma} (96%)
 rename matita/matita/contribs/lambdadelta/basic_2/{grammar => relocation}/lpx_sn_lpx_sn.ma (98%)
 rename matita/matita/contribs/lambdadelta/basic_2/{grammar => relocation}/lpx_sn_tc.ma (99%)
 rename matita/matita/contribs/lambdadelta/basic_2/{etc/llpx_sn/lleq_alt.etc => substitution/lleq_alt.ma} (89%)

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lprs.ma
index 6e8431959..c3d7203df 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lprs.ma
@@ -13,7 +13,7 @@
 (**************************************************************************)
 
 include "basic_2/notation/relations/predsnstar_3.ma".
-include "basic_2/grammar/lpx_sn_tc.ma".
+include "basic_2/relocation/lpx_sn_tc.ma".
 include "basic_2/reduction/lpr.ma".
 
 (* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyor_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyor_4.ma
new file mode 100644
index 000000000..cc97e6bbe
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyor_4.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⩖ break [ term 46 T ] break term 46 L2 ≡ break term 46 L )"
+   non associative with precedence 45
+   for @{ 'LazyOr $L1 $T $L2 $L }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr.ma
index 7bcf22df5..01966eefb 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr.ma
@@ -13,7 +13,7 @@
 (**************************************************************************)
 
 include "basic_2/notation/relations/predsn_3.ma".
-include "basic_2/grammar/lpx_sn.ma".
+include "basic_2/relocation/lpx_sn.ma".
 include "basic_2/reduction/cpr.ma".
 
 (* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_ldrop.ma
index 4156df6ea..c8443e08b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_ldrop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_ldrop.ma
@@ -12,8 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/relocation/lpx_sn_ldrop.ma".
 include "basic_2/relocation/fquq_alt.ma".
-include "basic_2/relocation/ldrop_lpx_sn.ma".
 include "basic_2/reduction/cpr_lift.ma".
 include "basic_2/reduction/lpr.ma".
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma
index 7b65286ab..2f653c61b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/grammar/lpx_sn_lpx_sn.ma".
+include "basic_2/relocation/lpx_sn_lpx_sn.ma".
 include "basic_2/substitution/fqup.ma".
 include "basic_2/reduction/lpr_ldrop.ma".
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/lift_neg.etc b/matita/matita/contribs/lambdadelta/basic_2/relocation/lift_neg.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/lift_neg.etc
rename to matita/matita/contribs/lambdadelta/basic_2/relocation/lift_neg.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/llor.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/llor.ma
new file mode 100644
index 000000000..8c76ecc02
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/llor.ma
@@ -0,0 +1,55 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/notation/relations/lazyor_4.ma".
+include "basic_2/relocation/lpx_sn_alt.ma".
+
+(* LAZY UNION FOR LOCAL ENVIRONMENTS ****************************************)
+
+inductive clor (T) (L2) (K1) (V1): predicate term ≝
+| clor_sn: ∀U. |K1| < |L2| → ⇧[|L2|-|K1|-1, 1] U ≡ T → clor T L2 K1 V1 V1
+| clor_dx: ∀I,K2,V2. |K1| < |L2| → (∀U. ⇧[|L2|-|K1|-1, 1] U ≡ T → ⊥) →
+           ⇩[|L2|-|K1|-1] L2 ≡ K2.ⓑ{I}V2 → clor T L2 K1 V1 V2
+.
+
+definition llor: relation4 term lenv lenv lenv ≝
+                 λT,L2. lpx_sn (clor T L2).
+
+interpretation
+   "lazy union (local environment)"
+   'LazyOr L1 T L2 L = (llor T L2 L1 L).
+
+(* Basic properties *********************************************************)
+
+lemma llor_pair_sn: ∀I,L1,L2,L,V,T,U. L1 ⩖[T] L2 ≡ L →
+                    |L1| < |L2| → ⇧[|L2|-|L1|-1, 1] U ≡ T →
+                    L1.ⓑ{I}V ⩖[T] L2 ≡ L.ⓑ{I}V.
+/3 width=2 by clor_sn, lpx_sn_pair/ qed.
+
+lemma llor_pair_dx: ∀I,J,L1,L2,L,K2,V1,V2,T. L1 ⩖[T] L2 ≡ L →
+                    |L1| < |L2| → (∀U. ⇧[|L2|-|L1|-1, 1] U ≡ T → ⊥) →
+                    ⇩[|L2|-|L1|-1] L2 ≡ K2.ⓑ{J}V2 →
+                    L1.ⓑ{I}V1 ⩖[T] L2 ≡ L.ⓑ{I}V2.
+/4 width=3 by clor_dx, lpx_sn_pair/ qed.
+
+lemma llor_total: ∀T,L2,L1. |L1| ≤ |L2| → ∃L. L1 ⩖[T] L2 ≡ L.
+#T #L2 #L1 elim L1 -L1 /2 width=2 by ex_intro/
+#L1 #I1 #V1 #IHL1 normalize
+#H elim IHL1 -IHL1 /2 width=3 by transitive_le/
+#L #HT elim (is_lift_dec T (|L2|-|L1|-1) 1)
+[ * /3 width=2 by llor_pair_sn, ex_intro/
+| elim (ldrop_O1_lt L2 (|L2|-|L1|-1))
+  /5 width=4 by llor_pair_dx, monotonic_lt_minus_l, ex_intro/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt.ma
similarity index 62%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_alt.etc
rename to matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt.ma
index 4356553ff..1da11d013 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_alt.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt.ma
@@ -22,16 +22,16 @@ include "basic_2/relocation/llpx_sn.ma".
 inductive llpx_sn_alt (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝
 | llpx_sn_alt_intro: ∀L1,L2,T,d.
                      (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                       ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2
+                        ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2
                      ) →
                      (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                       ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt R 0 V1 K1 K2
+                        ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt R 0 V1 K1 K2
                      ) → |L1| = |L2| → llpx_sn_alt R d T L1 L2
 .
 
-(* Basic forward lemmas ******************************************************)
+(* Basic inversion lemmas ****************************************************)
 
-lemma llpx_sn_alt_fwd_gen: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 →
+lemma llpx_sn_alt_inv_gen: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 →
                            |L1| = |L2| ∧
                            ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
                            ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
@@ -42,26 +42,35 @@ lemma llpx_sn_alt_fwd_gen: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 →
 elim (IH1 … HnT HLK1 HLK2) -IH1 /4 width=8 by and3_intro/
 qed-.
 
-lemma llpx_sn_alt_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #T #d * -L1 -L2 -T -d //
+lemma llpx_sn_alt_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓕ{I}V.T) L1 L2 →
+                            llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R d T L1 L2.
+#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_inv_gen … H) -H
+#HL12 #IH @conj @llpx_sn_alt_intro // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
+/3 width=8 by nlift_flat_sn, nlift_flat_dx, conj/
 qed-.
 
-fact llpx_sn_alt_fwd_lref_aux: ∀R,L1,L2,X,d. llpx_sn_alt R d X L1 L2 → ∀i. X = #i →
-                               ∨∨ |L1| ≤ i ∧ |L2| ≤ i
-                                | yinj i < d
-                                | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
-                                                   ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
-                                                   llpx_sn_alt R (yinj 0) V1 K1 K2 &
-                                                   R K1 V1 V2 & d ≤ yinj i.
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-#L1 #L2 #X #d #H1X #H2X #HL12 #i #H destruct
-elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
-elim (ylt_split i d) /3 width=1 by or3_intro1/
-#Hdi #HL1 elim (ldrop_O1_lt … HL1) #I1 #K1 #V1 #HLK1
-elim (ldrop_O1_lt L2 i) // #I2 #K2 #V2 #HLK2
-elim (H1X … HLK1 HLK2) -H1X /2 width=3 by nlift_lref_be_SO/ #H #HV12 destruct
-lapply (H2X … HLK1 HLK2) -H2X /2 width=3 by nlift_lref_be_SO/
-/3 width=9 by or3_intro2, ex5_5_intro/
+lemma llpx_sn_alt_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2 →
+                            llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_inv_gen … H) -H
+#HL12 #IH @conj @llpx_sn_alt_intro [3,6: normalize // ] -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+[1,2: elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
+  /3 width=9 by nlift_bind_sn, conj/
+|3,4: lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
+  lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
+  lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
+  elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
+  [1,3,4,6: /2 width=1 by conj/ ]
+  @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
+]
+qed-.
+
+(* Basic forward lemmas ******************************************************)
+
+lemma llpx_sn_alt_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #T #d * -L1 -L2 -T -d //
 qed-.
 
 lemma llpx_sn_alt_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt R d (#i) L1 L2 →
@@ -71,44 +80,15 @@ lemma llpx_sn_alt_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt R d (#i) L1 L2 →
                                                 ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
                                                 llpx_sn_alt R (yinj 0) V1 K1 K2 &
                                                 R K1 V1 V2 & d ≤ yinj i.
-/2 width=3 by llpx_sn_alt_fwd_lref_aux/ qed-.
-
-(* Basic inversion lemmas ****************************************************)
-
-fact llpx_sn_alt_inv_flat_aux: ∀R,L1,L2,X,d. llpx_sn_alt R d X L1 L2 →
-                               ∀I,V,T. X = ⓕ{I}V.T →
-                               llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R d T L1 L2.
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-#L1 #L2 #X #d #H1X #H2X #HL12
-#I #V #T #H destruct
-@conj @llpx_sn_alt_intro // -HL12
-/4 width=8 by nlift_flat_sn, nlift_flat_dx/
-qed-.
-
-lemma llpx_sn_alt_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓕ{I}V.T) L1 L2 →
-                            llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R d T L1 L2.
-/2 width=4 by llpx_sn_alt_inv_flat_aux/ qed-.
-
-fact llpx_sn_alt_inv_bind_aux: ∀R,L1,L2,X,d. llpx_sn_alt R d X L1 L2 →
-                               ∀a,I,V,T. X = ⓑ{a,I}V.T →
-                               llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-#L1 #L2 #X #d #H1X #H2X #HL12
-#a #I #V #T #H destruct
-@conj @llpx_sn_alt_intro [3,6: normalize /2 width=1 by eq_f2/ ] -HL12
-#I1 #I2 #K1 #K2 #W1 #W2 #i #Hdi #H #HLK1 #HLK2
-[1,2: /4 width=9 by nlift_bind_sn/ ]
-lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
-lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
-lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
-[ @(H1X … HLK1 HLK2) | @(H2X … HLK1 HLK2) ] // -I1 -I2 -L1 -L2 -K1 -K2 -W1 -W2
-@nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
+#R #L1 #L2 #d #i #H elim (llpx_sn_alt_inv_gen … H) -H
+#HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
+elim (ylt_split i d) /3 width=1 by or3_intro1/
+#Hdi #HL1 elim (ldrop_O1_lt … HL1)
+#I1 #K1 #V1 #HLK1 elim (ldrop_O1_lt L2 i) //
+#I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
+/3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
 qed-.
 
-lemma llpx_sn_alt_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2 →
-                            llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-/2 width=4 by llpx_sn_alt_inv_bind_aux/ qed-.
-
 (* Basic properties **********************************************************)
 
 lemma llpx_sn_alt_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
@@ -117,29 +97,29 @@ lemma llpx_sn_alt_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
                                 ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2
                              ) → llpx_sn_alt R d T L1 L2.
 #R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_intro // -HL12
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
 qed.
 
 lemma llpx_sn_alt_sort: ∀R,L1,L2,d,k. |L1| = |L2| → llpx_sn_alt R d (⋆k) L1 L2.
-#R #L1 #L2 #d #k #HL12 @llpx_sn_alt_intro // -HL12
+#R #L1 #L2 #d #k #HL12 @llpx_sn_alt_intro_alt // -HL12
 #I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
 qed.
 
 lemma llpx_sn_alt_gref: ∀R,L1,L2,d,p. |L1| = |L2| → llpx_sn_alt R d (§p) L1 L2.
-#R #L1 #L2 #d #p #HL12 @llpx_sn_alt_intro // -HL12
+#R #L1 #L2 #d #p #HL12 @llpx_sn_alt_intro_alt // -HL12
 #I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
 qed.
 
 lemma llpx_sn_alt_skip: ∀R,L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn_alt R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_intro // -HL12
+#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_intro_alt // -HL12
 #I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
 /4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
 qed.
 
 lemma llpx_sn_alt_free: ∀R,L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
                         llpx_sn_alt R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_intro // -HL12
+#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_intro_alt // -HL12
 #I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
 lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
 /3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
@@ -149,53 +129,47 @@ lemma llpx_sn_alt_lref: ∀R,I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
                         ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
                         llpx_sn_alt R 0 V1 K1 K2 → R K1 V1 V2 →
                         llpx_sn_alt R d (#i) L1 L2.
-#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_intro
-[1,2: #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
+#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_intro_alt
+[ lapply (llpx_sn_alt_fwd_length … HK12) -HK12 #HK12
+  @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize //
+| #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
   elim (lt_or_eq_or_gt i j) #Hij destruct
-  [1,4: elim (H (#i)) -H /2 width=1 by lift_lref_lt/
-  |2,5: lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
-    lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by conj/
-  |3,6: elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
+  [ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
+  | lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
+    lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
+  | elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
   ]
-| lapply (llpx_sn_alt_fwd_length … HK12) -HK12 #HK12
-  @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize /2 width=1 by eq_f2/
 ]
 qed.
 
-fact llpx_sn_alt_flat_aux: ∀R,I,L1,L2,V,d. llpx_sn_alt R d V L1 L2 →
-                           ∀Y1,Y2,T,m. llpx_sn_alt R m T Y1 Y2 →
-                           Y1 = L1 → Y2 = L2 → m = d →
-                           llpx_sn_alt R d (ⓕ{I}V.T) L1 L2.
-#R #I #L1 #L2 #V #d * -L1 -L2 -V -d #L1 #L2 #V #d #H1V #H2V #HL12
-#Y1 #Y2 #T #m * -Y1 -Y2 -T -m #Y1 #Y2 #T #m #H1T #H2T #_
-#HT1 #HY2 #Hm destruct
-@llpx_sn_alt_intro // -HL12
-#J1 #J2 #K1 #K2 #W1 #W2 #i #Hdi #HnVT #HLK1 #HLK2
-elim (nlift_inv_flat … HnVT) -HnVT /3 width=8 by/
-qed-.
-
 lemma llpx_sn_alt_flat: ∀R,I,L1,L2,V,T,d.
                         llpx_sn_alt R d V L1 L2 → llpx_sn_alt R d T L1 L2 →
                         llpx_sn_alt R d (ⓕ{I}V.T) L1 L2.
-/2 width=7 by llpx_sn_alt_flat_aux/ qed.
-
-fact llpx_sn_alt_bind_aux: ∀R,a,I,L1,L2,V,d. llpx_sn_alt R d V L1 L2 →
-                           ∀Y1,Y2,T,m. llpx_sn_alt R m T Y1 Y2 →
-                           Y1 = L1.ⓑ{I}V → Y2 = L2.ⓑ{I}V → m = ⫯d →
-                           llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2.
-#R #a #I #L1 #L2 #V #d * -L1 -L2 -V -d #L1 #L2 #V #d #H1V #H2V #HL12
-#Y1 #Y2 #T #m * -Y1 -Y2 -T -m #Y1 #Y2 #T #m #H1T #H2T #_
-#HT1 #HY2 #Hm destruct
-@llpx_sn_alt_intro // -HL12
-#J1 #J2 #K1 #K2 #W1 #W2 #i #Hdi #HnVT #HLK1 #HLK2
-elim (nlift_inv_bind … HnVT) -HnVT /3 width=8 by ldrop_drop, yle_succ/
-qed-.
+#R #I #L1 #L2 #V #T #d #HV #HT
+elim (llpx_sn_alt_inv_gen … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_inv_gen … HT) -HT #_ #IHT
+@llpx_sn_alt_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+elim (nlift_inv_flat … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
+]
+qed.
 
 lemma llpx_sn_alt_bind: ∀R,a,I,L1,L2,V,T,d.
                         llpx_sn_alt R d V L1 L2 →
                         llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
                         llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2.
-/2 width=7 by llpx_sn_alt_bind_aux/ qed.
+#R #a #I #L1 #L2 #V #T #d #HV #HT
+elim (llpx_sn_alt_inv_gen … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_inv_gen … HT) -HT #_ #IHT
+@llpx_sn_alt_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+elim (nlift_inv_bind … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim IHT -IHT /2 width=12 by ldrop_drop, yle_succ, and3_intro/
+]
+qed.
 
 (* Main properties **********************************************************)
 
@@ -231,20 +205,20 @@ lemma llpx_sn_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
                          ) → llpx_sn R d T L1 L2.
 #R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_inv_lpx_sn
 @llpx_sn_alt_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt, and3_intro/
 qed.
 
-(* Advanced forward lemmas lemmas *******************************************)
+(* Advanced inversion lemmas ************************************************)
 
-lemma llpx_sn_fwd_alt: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
+lemma llpx_sn_inv_gen: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
                        |L1| = |L2| ∧
                        ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
                        ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
                        ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
 #R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt … H) -H
-#H elim (llpx_sn_alt_fwd_gen … H) -H
+#H elim (llpx_sn_alt_inv_gen … H) -H
 #HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_inv_lpx_sn, and3_intro/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_lpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_lpx_sn.ma
index 84c81f3b6..defd02731 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_lpx_sn.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_lpx_sn.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/relocation/ldrop_lpx_sn.ma".
+include "basic_2/relocation/lpx_sn_ldrop.ma".
 include "basic_2/relocation/llpx_sn.ma".
 
 (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/lpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/grammar/lpx_sn.ma
rename to matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_alt.ma
new file mode 100644
index 000000000..b5aa43cf8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_alt.ma
@@ -0,0 +1,150 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/relocation/ldrop.ma".
+include "basic_2/relocation/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* alternative definition of lpx_sn_alt *)
+inductive lpx_sn_alt (R:relation3 lenv term term): relation lenv ≝
+| lpx_sn_alt_intro: ∀L1,L2.
+                    (∀I1,I2,K1,K2,V1,V2,i.
+                       ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2
+                    ) →
+                    (∀I1,I2,K1,K2,V1,V2,i.
+                       ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → lpx_sn_alt R K1 K2
+                    ) → |L1| = |L2| → lpx_sn_alt R L1 L2
+.
+
+(* Basic forward lemmas ******************************************************)
+
+lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 * -L1 -L2 //
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpx_sn_alt_inv_gen: ∀R,L1,L2. lpx_sn_alt R L1 L2 →
+                          |L1| = |L2| ∧
+                          ∀I1,I2,K1,K2,V1,V2,i.
+                          ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                          ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn_alt R K1 K2.
+#R #L1 #L2 * -L1 -L2
+#L1 #L2 #IH1 #IH2 #HL12 @conj //
+#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #HLK1 #HLK2
+elim (IH1 … HLK1 HLK2) -IH1 /3 width=7 by and3_intro/
+qed-.
+
+lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆.
+#R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H
+normalize /2 width=1 by length_inv_zero_sn/
+qed-.
+
+lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 →
+                            ∃∃K2,V2. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
+#R #I1 #L2 #K1 #V1 #H elim (lpx_sn_alt_inv_gen … H) -H
+#H #IH elim (length_inv_pos_sn … H) -H
+#I2 #K2 #V2 #HK12 #H destruct
+elim (IH I1 I2 K1 K2 V1 V2 0) -IH /2 width=5 by ex3_2_intro/
+qed-.
+
+lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆.
+#R #L1 #H lapply (lpx_sn_alt_fwd_length … H) -H
+normalize /2 width=1 by length_inv_zero_dx/
+qed-.
+
+lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) →
+                            ∃∃K1,V1. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
+#R #I2 #L1 #K2 #V2 #H elim (lpx_sn_alt_inv_gen … H) -H
+#H #IH elim (length_inv_pos_dx … H) -H
+#I1 #K1 #V1 #HK12 #H destruct
+elim (IH I1 I2 K1 K2 V1 V2 0) -IH /2 width=5 by ex3_2_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_sn_alt_intro_alt: ∀R,L1,L2. |L1| = |L2| →
+                            (∀I1,I2,K1,K2,V1,V2,i.
+                               ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                               ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn_alt R K1 K2
+                            ) → lpx_sn_alt R L1 L2.
+#R #L1 #L2 #HL12 #IH @lpx_sn_alt_intro // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
+elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by conj/
+qed.
+
+lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
+#R @lpx_sn_alt_intro_alt //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (ldrop_inv_atom1 … HLK1) -HLK1
+#H destruct
+qed.
+
+lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2.
+                       lpx_sn_alt R L1 L2 → R L1 V1 V2 →
+                       lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2).
+#R #I #L1 #L2 #V1 #V2 #H #HV12 elim (lpx_sn_alt_inv_gen … H) -H
+#HL12 #IH @lpx_sn_alt_intro_alt normalize //
+#I1 #I2 #K1 #K2 #W1 #W2 #i @(nat_ind_plus … i) -i
+[ #HLK1 #HLK2
+  lapply (ldrop_inv_O2 … HLK1) -HLK1 #H destruct
+  lapply (ldrop_inv_O2 … HLK2) -HLK2 #H destruct
+  /4 width=3 by lpx_sn_alt_intro_alt, and3_intro/
+| -HL12 -HV12 /3 width=5 by ldrop_inv_drop1/
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem lpx_sn_lpx_sn_alt: ∀R,L1,L2. lpx_sn R L1 L2 → lpx_sn_alt R L1 L2.
+#R #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by lpx_sn_alt_atom, lpx_sn_alt_pair/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2.
+#R #L1 elim L1 -L1
+[ #L2 #H lapply (lpx_sn_alt_inv_atom1 … H) -H //
+| #L1 #I #V1 #IH #X #H elim (lpx_sn_alt_inv_pair1 … H) -H
+  #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by lpx_sn_pair/
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
+                        (∀I1,I2,K1,K2,V1,V2,i.
+                           ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                           ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn R K1 K2
+                        ) → lpx_sn R L1 L2.
+#R #L1 #L2 #HL12 #IH @lpx_sn_alt_inv_lpx_sn
+@lpx_sn_alt_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
+elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /3 width=1 by lpx_sn_lpx_sn_alt, and3_intro/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 →
+                      |L1| = |L2| ∧
+                      ∀I1,I2,K1,K2,V1,V2,i.
+                      ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                      ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn R K1 K2.
+#R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
+#H elim (lpx_sn_alt_inv_gen … H) -H
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
+elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /3 width=1 by lpx_sn_alt_inv_lpx_sn, and3_intro/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_lpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_ldrop.ma
similarity index 96%
rename from matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_lpx_sn.ma
rename to matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_ldrop.ma
index c392bf54c..1edda7640 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_lpx_sn.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_ldrop.ma
@@ -12,12 +12,12 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/grammar/lpx_sn.ma".
 include "basic_2/relocation/ldrop_leq.ma".
+include "basic_2/relocation/lpx_sn.ma".
 
-(* DROPPING *****************************************************************)
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
 
-(* Properties on sn pointwise extension *************************************)
+(* Properies on dropping ****************************************************)
 
 lemma lpx_sn_ldrop_conf: ∀R,L1,L2. lpx_sn R L1 L2 →
                          ∀I,K1,V1,i. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/lpx_sn_lpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_lpx_sn.ma
similarity index 98%
rename from matita/matita/contribs/lambdadelta/basic_2/grammar/lpx_sn_lpx_sn.ma
rename to matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_lpx_sn.ma
index 69c010a19..3cd97e96d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/lpx_sn_lpx_sn.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_lpx_sn.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/grammar/lpx_sn.ma".
+include "basic_2/relocation/lpx_sn.ma".
 
 (* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/lpx_sn_tc.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_tc.ma
similarity index 99%
rename from matita/matita/contribs/lambdadelta/basic_2/grammar/lpx_sn_tc.ma
rename to matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_tc.ma
index 6dc76e174..e994bf076 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/lpx_sn_tc.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_tc.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/grammar/lpx_sn.ma".
+include "basic_2/relocation/lpx_sn.ma".
 
 (* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/lleq_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma
similarity index 89%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/lleq_alt.etc
rename to matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma
index ccf131d5a..470fa107f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/lleq_alt.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma
@@ -13,7 +13,7 @@
 (**************************************************************************)
 
 include "basic_2/relocation/llpx_sn_alt.ma".
-include "basic_2/relocation/lleq.ma".
+include "basic_2/substitution/lleq.ma".
 
 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
 
@@ -25,17 +25,17 @@ theorem lleq_intro_alt: ∀L1,L2,T,d. |L1| = |L2| →
                            ∧∧ I1 = I2 & V1 = V2 & K1 ⋕[V1, 0] K2
                         ) → L1 ⋕[T, d] L2.
 #L1 #L2 #T #d #HL12 #IH @llpx_sn_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
 qed.
 
-theorem lleq_fwd_alt: ∀L1,L2,T,d. L1 ⋕[T, d] L2 →
+theorem lleq_inv_gen: ∀L1,L2,T,d. L1 ⋕[T, d] L2 →
                       |L1| = |L2| ∧
                       ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
                       ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
                       ∧∧ I1 = I2 & V1 = V2 & K1 ⋕[V1, 0] K2.
-#L1 #L2 #T #d #H elim (llpx_sn_fwd_alt … H) -H
+#L1 #L2 #T #d #H elim (llpx_sn_inv_gen … H) -H
 #HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
 qed-.
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 a5f14f196..f65ec6eee 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
@@ -213,7 +213,7 @@ table {
    [ { "substitution" * } {
         [ { "lazy equivalence" * } {
              [ "fleq ( ⦃?,?,?⦄ ⋕[?] ⦃?,?,?⦄ )" "fleq_fleq" * ]
-             [ "lleq ( ? ⋕[?,?] ? )" "lleq_leq" + "lleq_ldrop" + "lleq_fqus" + "lleq_lleq" * ]
+             [ "lleq ( ? ⋕[?,?] ? )" "lleq_alt" + "lleq_leq" + "lleq_ldrop" + "lleq_fqus" + "lleq_lleq" * ]
           }
         ]
         [ { "iterated structural successor for closures" * } {
@@ -248,7 +248,7 @@ table {
           }
         ]
         [ { "lazy pointwise extension of a relation" * } {
-             [ "llpx_sn" "llpx_sn_tc" + "llpx_sn_leq" + "llpx_sn_ldrop" + "llpx_sn_lpx_sn" * ]
+             [ "llpx_sn" "llpx_sn_alt" + "llpx_sn_tc" + "llpx_sn_leq" + "llpx_sn_ldrop" + "llpx_sn_lpx_sn" * ]
           }
         ]
         [ { "basic local env. slicing" * } {
@@ -257,7 +257,7 @@ table {
         ]
         [ { "basic term relocation" * } {
              [ "lift_vector ( ⇧[?,?] ? ≡ ? )" "lift_lift_vector" * ]
-             [ "lift ( ⇧[?,?] ? ≡ ? )" "lift_lift" * ]
+             [ "lift ( ⇧[?,?] ? ≡ ? )" "lift_neq" + "lift_lift" * ]
           }
         ]
      }
@@ -269,7 +269,7 @@ table {
           }
         ]
         [ { "pointwise extension of a relation" * } {
-             [ "lpx_sn" "lpx_sn_tc" + "lpx_sn_lpx_sn" * ]
+             [ "lpx_sn" "lpx_sn_alt" + "lpx_sn_tc" + "lpx_sn_lpx_sn" * ]
           }
         ]
         [ { "same top term constructor" * } {
-- 
2.39.2