From c671743a83bbc7fff114e3e3694f628c0ec6685b Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sun, 1 Jun 2014 18:44:03 +0000
Subject: [PATCH] - advances on hereditarily free variables: now "frees" is
 primitive - some refactoring in etc

---
 .../etc/{cpy/cpy.etc => append/cpy2.etc}      |   0
 .../lambdadelta/basic_2/etc/append/cpys0.etc  |  24 +++
 .../etc/{cpy/cpys.etc => append/cpys2.etc}    |   0
 .../lambdadelta/basic_2/etc/append/lpys0.etc  |  29 ++++
 .../basic_2/etc/cofrees0/cofrees.etc          | 116 +++++++++++++
 .../basic_2/etc/cofrees0/cofrees_alt.etc      | 104 ++++++++++++
 .../basic_2/etc/cofrees0/cofrees_lift.etc     | 148 ++++++++++++++++
 .../basic_2/etc/cofrees0/cofreestar_3.etc     |  19 +++
 .../cofrees.ma => etc/cofrees1/cofrees.etc}   |   4 +
 .../cofrees1/cofrees_alt.etc}                 |   0
 .../cofrees1/cofrees_lift.etc}                |  36 ++++
 .../cofrees1/cofreestar_4.etc}                |   0
 .../basic_2/etc/{cpys => cpys0}/cpys.etc      |  95 ++++-------
 .../basic_2/etc/{cpys => cpys0}/cpys_cpys.etc |   8 +-
 .../basic_2/etc/{cpys => cpys0}/cpys_lift.etc |  78 ++++-----
 .../basic_2/etc/{cpys => cpys0}/lpys.etc      |  44 ++---
 .../etc/{cpys => cpys0}/lpys_ldrop.etc        |  32 ++--
 .../basic_2/etc/{cpys => cpys0}/lpys_lpys.etc |  16 --
 .../extlrsubeq_2.etc => cpys0/lrsubeq_2.etc}  |   4 +-
 .../basic_2/etc/{cpys => cpys0}/lsuby.etc     |  54 ++++--
 .../etc/{cpys => cpys0}/lsuby_lsuby.etc       |   2 +-
 .../psubstsnstar_3.etc}                       |   4 +-
 .../psubststar_4.etc}                         |   4 +-
 .../basic_2/etc/{cpy => cpys2}/cpx_cpys.etc   |   0
 .../basic_2/etc/{cpy => cpys2}/cpx_cpys2.etc  |   0
 .../basic_2/etc/{cpy => cpys2}/cpxs_cpys.etc  |   0
 .../basic_2/etc/{cpy => cpys2}/lpx_cpys.etc   |   0
 .../basic_2/etc/{cpy => cpys2}/lsuby.etc      |   0
 .../basic_2/notation/relations/freestar_4.ma  |  19 +++
 .../basic_2/reduction/lpx_frees.ma            |  82 +++++++++
 .../lambdadelta/basic_2/relocation/cpy.ma     |   1 -
 .../lambdadelta/basic_2/substitution/frees.ma | 160 ++++++++++++++++++
 .../basic_2/substitution/frees_lift.ma        |  73 ++++++++
 .../basic_2/substitution/lleq_alt.ma          |   4 +-
 .../basic_2/substitution/lleq_llor.ma         |  20 +--
 .../lambdadelta/basic_2/substitution/llor.ma  |   8 +-
 .../basic_2/substitution/llor_ldrop.ma        |  22 +++
 .../basic_2/substitution/llpx_sn_alt.ma       |  10 +-
 .../lambdadelta/basic_2/web/basic_2_src.tbl   |   6 +-
 39 files changed, 1012 insertions(+), 214 deletions(-)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpy/cpy.etc => append/cpy2.etc} (100%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/append/cpys0.etc
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpy/cpys.etc => append/cpys2.etc} (100%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/append/lpys0.etc
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofrees.etc
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofrees_alt.etc
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofrees_lift.etc
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofreestar_3.etc
 rename matita/matita/contribs/lambdadelta/basic_2/{substitution/cofrees.ma => etc/cofrees1/cofrees.etc} (96%)
 rename matita/matita/contribs/lambdadelta/basic_2/{substitution/cofrees_alt.ma => etc/cofrees1/cofrees_alt.etc} (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/{substitution/cofrees_lift.ma => etc/cofrees1/cofrees_lift.etc} (81%)
 rename matita/matita/contribs/lambdadelta/basic_2/{notation/relations/cofreestar_4.ma => etc/cofrees1/cofreestar_4.etc} (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpys => cpys0}/cpys.etc (63%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpys => cpys0}/cpys_cpys.etc (93%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpys => cpys0}/cpys_lift.etc (71%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpys => cpys0}/lpys.etc (55%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpys => cpys0}/lpys_ldrop.etc (71%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpys => cpys0}/lpys_lpys.etc (60%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpys/extlrsubeq_2.etc => cpys0/lrsubeq_2.etc} (92%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpys => cpys0}/lsuby.etc (55%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpys => cpys0}/lsuby_lsuby.etc (97%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpys/extpsubstsnstar_3.etc => cpys0/psubstsnstar_3.etc} (94%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpys/extpsubststar_4.etc => cpys0/psubststar_4.etc} (93%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpy => cpys2}/cpx_cpys.etc (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpy => cpys2}/cpx_cpys2.etc (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpy => cpys2}/cpxs_cpys.etc (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpy => cpys2}/lpx_cpys.etc (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/etc/{cpy => cpys2}/lsuby.etc (100%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/notation/relations/freestar_4.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_frees.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/substitution/frees.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/substitution/frees_lift.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/substitution/llor_ldrop.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/append/cpy2.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpy.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/append/cpy2.etc
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/append/cpys0.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/append/cpys0.etc
new file mode 100644
index 000000000..fa9bcf6de
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/append/cpys0.etc
@@ -0,0 +1,24 @@
+include "basic_2/grammar/cl_shift.ma".
+include "basic_2/relocation/ldrop_append.ma".
+
+lemma cpys_append: ∀G. l_appendable_sn … (cpys G).
+#G #K #T1 #T2 #H elim H -G -K -T1 -T2
+/2 width=3 by cpys_bind, cpys_flat/
+#I #G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
+lapply (ldrop_fwd_length_lt2 … HK0) #H
+@(cpys_delta … I … (L@@K0) V1 … HVW2) // 
+@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *)
+qed.
+
+lemma cpys_fwd_shift1: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*× T →
+                       ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
+#G #L1 @(lenv_ind_dx … L1) -L1 normalize
+[ #L #T1 #T #HT1 @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
+| #I #L1 #V1 #IH #L #T1 #X >shift_append_assoc normalize
+  #H elim (cpys_inv_bind1 … H) -H
+  #V0 #T0 #_ #HT10 #H destruct
+  elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
+  >append_length >HL12 -HL12
+  @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] /2 width=3 by trans_eq/ (**) (* explicit constructor *)
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/append/cpys2.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpys.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/append/cpys2.etc
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/append/lpys0.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/append/lpys0.etc
new file mode 100644
index 000000000..cb3dfd037
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/append/lpys0.etc
@@ -0,0 +1,29 @@
+lemma lpys_append: ∀G,K1,K2. ⦃G, K1⦄ ⊢ ▶*× K2 → ∀L1,L2. ⦃G, L1⦄ ⊢ ▶*× L2 →
+                   ⦃G, L1 @@ K1⦄ ⊢ ▶*× L2 @@ K2.
+/3 width=1 by lpx_sn_append, cpys_append/ qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lpys_fwd_append1: ∀G,K1,L1,L. ⦃G, K1 @@ L1⦄ ⊢ ▶*× L →
+                        ∃∃K2,L2. ⦃G, K1⦄ ⊢ ▶*× K2 & L = K2 @@ L2.
+/2 width=2 by lpx_sn_fwd_append1/ qed-.
+
+lemma lpys_fwd_append2: ∀G,L,K2,L2. ⦃G, L⦄ ⊢ ▶*× K2 @@ L2 →
+                        ∃∃K1,L1. ⦃G, K1⦄ ⊢ ▶*× K2 & L = K1 @@ L1.
+/2 width=2 by lpx_sn_fwd_append2/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma cpys_fwd_shift1_ext: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*× T →
+                           ∃∃L2,T2. ⦃G, L @@ L1⦄ ⊢ ▶*× L @@ L2 & ⦃G, L @@ L1⦄ ⊢ T1 ▶*× T2 &
+                                    T = L2 @@ T2.
+#G #L1 @(lenv_ind_dx … L1) -L1
+[ #L #T1 #T #HT1 @ex3_2_intro
+  [3: // |4,5: // |1,2: skip ] (**) (* auto does not work *)
+| #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H <append_assoc
+  elim (cpys_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+  elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
+  lapply (lpys_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1 by lpys_append, lpys_pair/ #HL12
+  @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3 by trans_eq/ | skip ] <append_assoc // (**) (* explicit constructor *)
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofrees.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofrees.etc
new file mode 100644
index 000000000..aa0ab9ec0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofrees.etc
@@ -0,0 +1,116 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/cofreestar_3.ma".
+include "basic_2/relocation/lift_neg.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
+
+definition cofrees: relation3 lenv nat term ≝
+                    λL,i,U1. ∀U2. ⦃⋆, L⦄ ⊢ U1 ▶* U2 → ∃T2. ⇧[i, 1] T2 ≡ U2.
+
+interpretation
+   "context-sensitive exclusion from free variables (term)"
+   'CoFreeStar L i U = (cofrees L i U).
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cofrees_fwd_lift: ∀L,U,i. L ⊢ i ~ϵ 𝐅*⦃U⦄ → ∃T. ⇧[i, 1] T ≡ U.
+/2 width=1 by/ qed-.
+
+lemma cofrees_fwd_bind_sn: ∀a,I,L,W,U,i. L ⊢ i ~ϵ 𝐅*⦃ⓑ{a,I}W.U⦄ →
+                           L ⊢ i ~ϵ 𝐅*⦃W⦄.
+#a #I #L #W1 #U #i #H #W2 #HW12 elim (H (ⓑ{a,I}W2.U)) /2 width=1 by cpys_bind/ -W1
+#X #H elim (lift_inv_bind2 … H) -H /2 width=2 by ex_intro/
+qed-.
+
+lemma cofrees_fwd_bind_dx: ∀a,I,L,W,U,i. L ⊢ i ~ϵ 𝐅*⦃ⓑ{a,I}W.U⦄ →
+                           L.ⓑ{I}W ⊢ i+1 ~ϵ 𝐅*⦃U⦄.
+#a #I #L #W #U1 #i #H #U2 #HU12 elim (H (ⓑ{a,I}W.U2)) /2 width=1 by cpys_bind/ -U1
+#X #H elim (lift_inv_bind2 … H) -H /2 width=2 by ex_intro/
+qed-.
+
+lemma cofrees_fwd_flat_sn: ∀I,L,W,U,i. L ⊢ i ~ϵ 𝐅*⦃ⓕ{I}W.U⦄ →
+                           L ⊢ i ~ϵ 𝐅*⦃W⦄.
+#I #L #W1 #U #i #H #W2 #HW12 elim (H (ⓕ{I}W2.U)) /2 width=1 by cpys_flat/ -W1
+#X #H elim (lift_inv_flat2 … H) -H /2 width=2 by ex_intro/
+qed-.
+
+lemma cofrees_fwd_flat_dx: ∀I,L,W,U,i. L ⊢ i ~ϵ 𝐅*⦃ⓕ{I}W.U⦄ →
+                           L ⊢ i ~ϵ 𝐅*⦃U⦄.
+#I #L #W #U1 #i #H #U2 #HU12 elim (H (ⓕ{I}W.U2)) /2 width=1 by cpys_flat/ -U1
+#X #H elim (lift_inv_flat2 … H) -H /2 width=2 by ex_intro/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cofrees_inv_gen: ∀L,U,U0,i. ⦃⋆, L⦄ ⊢ U ▶* U0 → (∀T. ⇧[i, 1] T ≡ U0 → ⊥) →
+                       L ⊢ i ~ϵ 𝐅*⦃U⦄ → ⊥.
+#L #U #U0 #i #HU0 #HnU0 #HU elim (HU … HU0) -L -U /2 width=2 by/
+qed-.
+
+lemma cofrees_inv_lref_eq: ∀L,i. L ⊢ i ~ϵ 𝐅*⦃#i⦄ → ⊥.
+#L #i #H elim (H (#i)) -H //
+#X #H elim (lift_inv_lref2_be … H) -H //
+qed-. 
+
+lemma cofrees_inv_bind: ∀a,I,L,W,U,i. L ⊢ i ~ϵ 𝐅*⦃ⓑ{a,I}W.U⦄ →
+                        L ⊢ i ~ϵ 𝐅*⦃W⦄ ∧ L.ⓑ{I}W ⊢ i+1 ~ϵ 𝐅*⦃U⦄.
+/3 width=7 by cofrees_fwd_bind_sn, cofrees_fwd_bind_dx, conj/ qed-.
+
+lemma cofrees_inv_flat: ∀I,L,W,U,i. L ⊢ i ~ϵ 𝐅*⦃ⓕ{I}W.U⦄ →
+                        L ⊢ i ~ϵ 𝐅*⦃W⦄ ∧ L ⊢ i ~ϵ 𝐅*⦃U⦄.
+/3 width=6 by cofrees_fwd_flat_sn, cofrees_fwd_flat_dx, conj/ qed-.
+
+(* Basic Properties *********************************************************)
+
+lemma cofrees_sort: ∀L,i,k. L ⊢ i ~ϵ 𝐅*⦃⋆k⦄.
+#L #i #k #X #H >(cpys_inv_sort1 … H) -X /2 width=2 by ex_intro/
+qed.
+
+lemma cofrees_gref: ∀L,i,p. L ⊢ i ~ϵ 𝐅*⦃§p⦄.
+#L #i #p #X #H >(cpys_inv_gref1 … H) -X /2 width=2 by ex_intro/
+qed.
+
+lemma cofrees_bind: ∀L,V,i. L ⊢ i ~ϵ 𝐅*⦃V⦄ →
+                    ∀I,T. L.ⓑ{I}V ⊢ i+1 ~ϵ 𝐅*⦃T⦄ →
+                    ∀a. L ⊢ i ~ϵ 𝐅*⦃ⓑ{a,I}V.T⦄.
+#L #W1 #i #HW1 #I #U1 #HU1 #a #X #H elim (cpys_inv_bind1 … H) -H
+#W2 #U2 #HW12 #HU12 #H destruct
+elim (HW1 … HW12) elim (HU1 … HU12) -W1 -U1 /3 width=2 by lift_bind, ex_intro/
+qed.
+
+lemma cofrees_flat: ∀L,V,i. L ⊢ i ~ϵ 𝐅*⦃V⦄ → ∀T. L ⊢ i ~ϵ 𝐅*⦃T⦄ →
+                    ∀I. L ⊢ i ~ϵ 𝐅*⦃ⓕ{I}V.T⦄.
+#L #W1 #i #HW1 #U1 #HU1 #I #X #H elim (cpys_inv_flat1 … H) -H
+#W2 #U2 #HW12 #HU12 #H destruct
+elim (HW1 … HW12) elim (HU1 … HU12) -W1 -U1 /3 width=2 by lift_flat, ex_intro/
+qed.
+
+axiom cofrees_dec: ∀L,T,i. Decidable (L ⊢ i ~ϵ 𝐅*⦃T⦄).
+
+(* Basic negated inversion lemmas *******************************************)
+
+lemma frees_inv_bind: ∀a,I,L,V,T,i. (L ⊢ i ~ϵ 𝐅*⦃ⓑ{a,I}V.T⦄ → ⊥) →
+                      (L ⊢ i ~ϵ 𝐅*⦃V⦄ → ⊥) ∨ (L.ⓑ{I}V ⊢ i+1 ~ϵ 𝐅*⦃T⦄ → ⊥).
+#a #I #L #W #U #i #H elim (cofrees_dec L W i)
+/4 width=8 by cofrees_bind, or_intror, or_introl/
+qed-.
+
+lemma frees_inv_flat: ∀I,L,V,T,i. (L ⊢ i ~ϵ 𝐅*⦃ⓕ{I}V.T⦄ → ⊥) →
+                      (L ⊢ i ~ϵ 𝐅*⦃V⦄ → ⊥) ∨ (L ⊢ i ~ϵ 𝐅*⦃T⦄ → ⊥).
+#I #L #W #U #i #H elim (cofrees_dec L W i)
+/4 width=7 by cofrees_flat, or_intror, or_introl/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofrees_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofrees_alt.etc
new file mode 100644
index 000000000..8443f38e3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofrees_alt.etc
@@ -0,0 +1,104 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/lift_neg.ma".
+include "basic_2/relocation/lift_lift.ma".
+include "basic_2/substitution/cpys.ma".
+include "basic_2/substitution/cofrees_lift.ma".
+
+(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
+
+(* Alternative definition of frees_ge ***************************************)
+
+(*
+lemma cpys_fwd_nlift2: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶* U2 →
+                       ∀i. (∀T2. ⇧[i, 1] T2 ≡ U2 → ⊥) →
+                       (∀T1. ⇧[i, 1] T1 ≡ U1 → ⊥) ∨
+                       ∃∃I,K,W,j. j < i & ⇩[j]L ≡ K.ⓑ{I}W &
+                                  (∀V. ⇧[i-j-1, 1] V ≡ W → ⊥) & (∀T1. ⇧[j, 1] T1 ≡ U1 → ⊥).
+#G #L #U1 #U2 #H elim H -G -L -U1 -U2
+[ /3 width=2 by or_introl/
+| #I #G #L #K #V1 #V2 #W2 #j #HLK #_ #HVW2 #IHV12 #i #HnW2
+  elim (lt_or_ge j i) #Hij
+  [ @or_intror (**) @(ex4_4_intro … HLK) //
+    [ #X #HXV elim (lift_trans_le … HXV … HVW ?) -V //
+      #Y #HXY >minus_plus <plus_minus_m_m /2 width=2 by/
+    | -HnW2 /2 width=7 by lift_inv_lref2_be/
+    ]
+  | elim (lift_split … HVW2 i j) -HVW2 //
+    #X #_ #H elim HnW2 -HnW2 //
+  ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #_ #_ #IHW12 #IHU12 #i #H elim (nlift_inv_bind … H) -H
+  [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12
+    [ /4 width=9 by nlift_bind_sn, or_introl/
+    | * /5 width=9 by nlift_bind_sn, ex4_4_intro, or_intror/
+    ]
+  | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12
+    [ /4 width=9 by nlift_bind_dx, or_introl/
+    | * #J #K #W #j @(nat_ind_plus … j) -j
+      [ #_ #H lapply (ldrop_inv_pair1 … H) -H
+        #H destruct /4 width=9 by nlift_bind_sn, or_introl/
+      | #j #_ #Hji #HLK #HnW
+        lapply (ldrop_inv_drop1_lt … HLK ?) // -HLK #HLK #HnU1
+        <minus_le_minus_minus_comm in HnW;
+        /5 width=9 by nlift_bind_dx, monotonic_lt_pred, ex4_4_intro, or_intror/
+      ]
+    ]
+  ]
+| #I #G #L #W1 #W2 #U1 #U2 #_ #_ #IHW12 #IHU12 #i #H elim (nlift_inv_flat … H) -H
+  [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12
+    [ /4 width=9 by nlift_flat_sn, or_introl/
+    | * /5 width=9 by nlift_flat_sn, ex4_4_intro, or_intror/
+    ]
+  | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12
+    [ /4 width=9 by nlift_flat_dx, or_introl/
+    | * /5 width=9 by nlift_flat_dx, ex4_4_intro, or_intror/
+  ]
+]
+qed-.
+*)
+lemma nlift_frees: ∀L,U,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → (L ⊢ i ~ϵ 𝐅*⦃U⦄ → ⊥).
+#L #U #i #HnTU #H elim (cofrees_fwd_lift … H) -H /2 width=2 by/
+qed-.
+(*
+lemma frees_inv_ge: ∀L,U,d,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
+                    (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
+                    ∃∃I,K,W,j. d ≤ yinj j & j < i & ⇩[j]L ≡ K.ⓑ{I}W &
+                               (K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄ → ⊥) & (∀T. ⇧[j, 1] T ≡ U → ⊥).
+#L #U #d #i #Hdi #H @(frees_ind … H) -U /3 width=2 by or_introl/
+#U1 #U2 #HU12 #HU2 *
+[ #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /3 width=2 by or_introl/
+  * /5 width=9 by nlift_frees, ex5_4_intro, or_intror/
+| * #I2 #K2 #W2 #j2 #Hdj2 #Hj2i #HLK2 #HnW2 #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /4 width=9 by ex5_4_intro, or_intror/
+  * #I1 #K1 #W1 #j1 #Hdj1 #Hj12 #HLK1 #HnW1 #HnU1
+  lapply (ldrop_conf_ge … HLK1 … HLK2 ?) -HLK2 /2 width=1 by lt_to_le/
+  #HK12 lapply (ldrop_inv_drop1_lt … HK12 ?) /2 width=1 by lt_plus_to_minus_r/ -HK12
+  #HK12
+  @or_intror @(ex5_4_intro … HLK1 … HnU1) -HLK1 -HnU1 /2 width=3 by transitive_lt/
+  @(frees_be … HK12 … HnW1) /2 width=1 by arith_k_sn/ -HK12 -HnW1
+  >minus_plus in ⊢ (??(?(?%?)?)??→?); >minus_plus in ⊢ (??(?(??%)?)??→?); >arith_b1 /2 width=1 by/
+]
+qed-.
+
+lemma frees_ind_ge: ∀R:relation4 ynat nat lenv term.
+                    (∀d,i,L,U. d ≤ yinj i → (∀T. ⇧[i, 1] T ≡ U → ⊥) → R d i L U) →
+                    (∀d,i,j,I,L,K,W,U. d ≤ yinj j → j < i → ⇩[j]L ≡ K.ⓑ{I}W → (K ⊢ i-j-1 ~ϵ 𝐅*[0]⦃W⦄ → ⊥) → (∀T. ⇧[j, 1] T ≡ U → ⊥) → R 0 (i-j-1) K W → R d i L U) →
+                    ∀d,i,L,U. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R d i L U.
+#R #IH1 #IH2 #d #i #L #U
+generalize in match d; -d generalize in match i; -i
+@(f2_ind … rfw … L U) -L -U
+#n #IHn #L #U #Hn #i #d #Hdi #H elim (frees_inv_ge … H) -H /3 width=2 by/
+-IH1 * #I #K #W #j #Hdj #Hji #HLK #HnW #HnU destruct /4 width=12 by ldrop_fwd_rfw/
+qed-.
+*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofrees_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofrees_lift.etc
new file mode 100644
index 000000000..02f227a7a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofrees_lift.etc
@@ -0,0 +1,148 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/substitution/cpys_lift.ma".
+include "basic_2/substitution/cofrees.ma".
+
+(* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cofrees_inv_lref_lt: ∀L,i,j. L ⊢ i ~ϵ 𝐅*⦃#j⦄ → j < i →
+                           ∀I,K,W. ⇩[j]L ≡ K.ⓑ{I}W → K ⊢ i-j-1 ~ϵ 𝐅*⦃W⦄.
+#L #i #j #Hj #Hji #I #K #W1 #HLK #W2 #HW12 elim (lift_total W2 0 (j+1))
+#X2 #HWX2 elim (Hj X2) /2 width=7 by cpys_delta/ -I -L -K -W1
+#Z2 #HZX2 elim (lift_div_le … HWX2 (i-j-1) 1 Z2) -HWX2 /2 width=2 by ex_intro/
+>minus_plus <plus_minus_m_m //
+qed-.
+
+lemma cofrees_inv_lt: ∀L,U,i. L ⊢ i ~ϵ 𝐅*⦃U⦄ → ∀j. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
+                      ∀I,K,W. ⇩[j]L ≡ K.ⓑ{I}W → j < i → K ⊢ i-j-1 ~ϵ 𝐅*⦃W⦄.
+#L #U @(f2_ind … rfw … L U) -L -U
+#n #IH #L * *
+[ -IH #k #_ #i #_ #j #H elim (H (⋆k)) -H //
+| -IH #j #_ #i #Hi0 #j0 #H <(nlift_inv_lref_be_SO … H) -j0
+  /2 width=7 by cofrees_inv_lref_lt/
+| -IH #p #_ #i #_ #j #H elim (H (§p)) -H //
+| #a #J #W #U #Hn #i #H1 #j #H2 #I #K #V #HLK #Hji destruct
+  elim (cofrees_inv_bind … H1) -H1 #HW #HU
+  elim (nlift_inv_bind … H2) -H2 [ -HU /3 width=7 by/ ]
+  -HW #HnU lapply (IH … HU … HnU I K V ? ?)
+  /2 width=1 by ldrop_drop, lt_minus_to_plus/ -a -I -J -L -W -U
+  >minus_plus_plus_l //
+| #J #W #U #Hn #i #H1 #j #H2 #I #K #V #HLK #Hji destruct
+  elim (cofrees_inv_flat … H1) -H1 #HW #HU
+  elim (nlift_inv_flat … H2) -H2 [ /3 width=7 by/ ]
+  #HnU @(IH … HU … HnU … HLK) // (**) (* full auto fails *)
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma cofrees_lref_gt: ∀L,i,j. i < j → L ⊢ i ~ϵ 𝐅*⦃#j⦄.
+#L #i #j #Hij #X #H elim (cpys_inv_lref1 … H) -H
+[ #H destruct /3 width=2 by lift_lref_ge_minus, ex_intro/
+| * #I #K #V1 #V2 #_ #_ #H -I -L -K -V1
+  elim (lift_split … H i j) /2 width=2 by lt_to_le, ex_intro/
+]
+qed.
+
+lemma cofrees_lref_lt: ∀I,L,K,W,i,j. j < i → ⇩[j] L ≡ K.ⓑ{I}W →
+                       K ⊢ (i-j-1) ~ϵ 𝐅*⦃W⦄ → L ⊢ i ~ϵ 𝐅*⦃#j⦄.
+#I #L #K #W1 #i #j #Hji #HLK #HW1 #X #H elim (cpys_inv_lref1 … H) -H
+[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
+| * #I0 #K0 #W0 #W2 #HLK0 #HW12 #HW2 lapply (ldrop_mono … HLK0 … HLK) -L
+  #H destruct elim (HW1 … HW12) -I -K -W1
+  #V2 #HVW2 elim (lift_trans_le … HVW2 … HW2) -W2 //
+  >minus_plus <plus_minus_m_m /2 width=2 by ex_intro/
+]
+qed.
+
+lemma cofrees_lref_free: ∀L,i,j. |L| ≤ j → j < i → L ⊢ i ~ϵ 𝐅*⦃#j⦄.
+#L #i #j #Hj #Hji #X #H elim (cpys_inv_lref1 … H) -H
+[ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
+| * #I #K #W1 #W2 #HLK lapply (ldrop_fwd_length_lt2 … HLK) -I
+  #H elim (lt_refl_false j) -K -W1 -W2 -X -i /2 width=3 by lt_to_le_to_lt/
+]
+qed.
+
+(* Advanced negated inversion lemmas ****************************************)
+
+lemma frees_inv_lref_lt: ∀L,i,j. j < i → (L ⊢ i ~ϵ 𝐅*⦃#j⦄ → ⊥) →
+                         ∃∃I,K,W. ⇩[j] L ≡ K.ⓑ{I}W & (K ⊢ (i-j-1) ~ϵ 𝐅*⦃W⦄ → ⊥).
+#L #i #j #Hji #H elim (lt_or_ge j (|L|)) #Hj
+[ elim (ldrop_O1_lt (Ⓕ) … Hj) -Hj /4 width=9 by cofrees_lref_lt, ex2_3_intro/
+| elim H -H /2 width=5 by cofrees_lref_free/
+]
+qed-.
+
+lemma frees_inv_lref_free: ∀L,i,j. (L ⊢ i ~ϵ 𝐅*⦃#j⦄  → ⊥) → |L| ≤ j → j = i.
+#L #i #j #H #Hj elim (lt_or_eq_or_gt i j) //
+#Hij elim H -H /2 width=5 by cofrees_lref_gt, cofrees_lref_free/
+qed-.
+
+lemma frees_inv_gen: ∀L,U,i. (L ⊢ i ~ϵ 𝐅*⦃U⦄ → ⊥) →
+                     ∃∃U0.  ⦃⋆, L⦄ ⊢ U ▶* U0 & (∀T. ⇧[i, 1] T ≡ U0 → ⊥).
+#L #U @(f2_ind … rfw … L U) -L -U
+#n #IH #L * *
+[ -IH #k #_ #i #H elim H -H //
+| #j #Hn #i #H elim (lt_or_eq_or_gt i j)
+  [ -n #Hij elim H -H /2 width=4 by cofrees_lref_gt/
+  | -H -n #H destruct /3 width=7 by lift_inv_lref2_be, ex2_intro/
+  | #Hji elim (frees_inv_lref_lt … H) // -H
+    #I #K #W1 #HLK #H elim (IH … H) /2 width=3 by ldrop_fwd_rfw/ -H -n
+    #W2 #HW12 #HnW2 elim (lift_total W2 0 (j+1))
+    #U2 #HWU2 @(ex2_intro … U2) /2 width=7 by cpys_delta/ -I -L -K -W1
+    #T2 #HTU2 elim (lift_div_le … HWU2 (i-j-1) 1 T2) /2 width=2 by/ -W2
+    >minus_plus <plus_minus_m_m //
+  ]
+| -IH #p #_ #i #H elim H -H //
+| #a #I #W #U #Hn #i #H elim (frees_inv_bind … H) -H
+  #H elim (IH … H) // -H -n
+  #X #HX #HnX [ @(ex2_intro … (ⓑ{a,I}X.U)) | @(ex2_intro … (ⓑ{a,I}W.X)) ] (**) (* explicit constructor *)
+  /3 width=9 by cpys_bind, nlift_bind_dx, nlift_bind_sn/
+| #I #W #U #Hn #i #H elim (frees_inv_flat … H) -H
+  #H elim (IH … H) // -H -n
+  #X #HX #HnX [ @(ex2_intro … (ⓕ{I}X.U)) | @(ex2_intro … (ⓕ{I}W.X)) ] (**) (* explicit constructor *)
+  /3 width=8 by cpys_flat, nlift_flat_dx, nlift_flat_sn/
+]
+qed-.
+
+(* Advanced negated properties **********************************************)
+
+lemma frees_lt: ∀I,L,K,W,j. ⇩[j]L ≡ K.ⓑ{I}W →
+                ∀i. j < i → (K ⊢ i-j-1 ~ϵ 𝐅*⦃W⦄ → ⊥) →
+                ∀U. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
+                (L ⊢ i ~ϵ 𝐅*⦃U⦄ → ⊥).
+/4 width=9 by cofrees_inv_lt/ qed-.
+
+(* Relocation properties ****************************************************)
+
+lemma cofrees_lift_be: ∀d,e,i. d ≤ i → i ≤ d + e →
+                       ∀L,K,s. ⇩[s, d, e+1] L ≡ K → ∀T,U. ⇧[d, e+1] T ≡ U →
+                       L ⊢ i ~ϵ 𝐅*⦃U⦄.
+#d #e #i #Hdi #Hide #L #K #s #HLK #T1 #U1 #HTU1 #U2 #HU12
+elim (cpys_inv_lift1 … HU12 … HLK … HTU1) #T2 #HTU2 #_ -s -L -K -U1 -T1
+elim (lift_split … HTU2 i e) /2 width=2 by ex_intro/
+qed.
+
+lemma cofrees_lift_ge: ∀d,e,i. d + e ≤ i →
+                       ∀L,K,s. ⇩[s, d, e] L ≡ K → ∀T,U. ⇧[d, e] T ≡ U →
+                       K ⊢ i-e ~ϵ 𝐅*⦃T⦄ → L ⊢ i ~ϵ 𝐅*⦃U⦄.
+#d #e #i #Hdei #L #K #s #HLK #T1 #U1 #HTU1 #HT1 #U2 #HU12
+elim (le_inv_plus_l … Hdei) -Hdei #Hdie #Hei
+elim (cpys_inv_lift1 … HU12 … HLK … HTU1) -s -L #T2 #HTU2 #HT12
+elim (HT1 … HT12) -T1 #V2 #HVT2
+elim (lift_trans_le … HVT2 … HTU2 ?) // <plus_minus_m_m /2 width=2 by ex_intro/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofreestar_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofreestar_3.etc
new file mode 100644
index 000000000..f723e7390
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees0/cofreestar_3.etc
@@ -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( L ⊢ break term 46 i ~ ϵ 𝐅 * ⦃ break term 46 T ⦄ )"
+   non associative with precedence 45
+   for @{ 'CoFreeStar $L $i $T }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees1/cofrees.etc
similarity index 96%
rename from matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cofrees1/cofrees.etc
index 59b8baab8..e1b9b06b2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees1/cofrees.etc
@@ -76,6 +76,10 @@ lemma cofrees_inv_flat: ∀I,L,W,U,i,d. L ⊢ i ~ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
 
 (* Basic Properties *********************************************************)
 
+lemma cofrees_lsuby_conf: ∀L1,U,d,i. L1 ⊢ i ~ϵ 𝐅*[d]⦃U⦄ →
+                          ∀L2. L1 ⊆[d, ∞] L2 → L2 ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+/3 width=3 by lsuby_cpys_trans/ qed-.
+
 lemma cofrees_sort: ∀L,d,i,k. L ⊢ i ~ϵ 𝐅*[d]⦃⋆k⦄.
 #L #d #i #k #X #H >(cpys_inv_sort1 … H) -X /2 width=2 by ex_intro/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees1/cofrees_alt.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees_alt.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cofrees1/cofrees_alt.etc
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees1/cofrees_lift.etc
similarity index 81%
rename from matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees_lift.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cofrees1/cofrees_lift.etc
index 7dea71858..f93f72216 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees1/cofrees_lift.etc
@@ -142,3 +142,39 @@ lemma frees_be: ∀I,L,K,W,j. ⇩[j]L ≡ K.ⓑ{I}W →
                 ∀U. (∀T. ⇧[j, 1] T ≡ U → ⊥) →
                 ∀d. d ≤ yinj j → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥).
 /4 width=11 by cofrees_inv_be/ qed-.
+
+(* Relocation properties ****************************************************)
+
+lemma cofrees_lift_be: ∀d0,e0,i. d0 ≤ i → i ≤ d0 + e0 →
+                       ∀L,K,s. ⇩[s, d0, e0+1] L ≡ K → ∀T,U. ⇧[d0, e0+1] T ≡ U →
+                       ∀d. L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+#d0 #e0 #i #Hd0i #Hide0 #L #K #s #HLK #T1 #U1 #HTU1 #d #U2 #HU12
+elim (yle_split d0 d) #H1
+[ elim (yle_split d (d0+e0+1)) #H2
+  [ letin cpys_inv ≝ cpys_inv_lift1_ge_up
+  | letin cpys_inv ≝ cpys_inv_lift1_ge
+  ]
+| letin cpys_inv ≝ cpys_inv_lift1_be
+]
+elim (cpys_inv … HU12 … HLK … HTU1) // #T2 #_ #HTU2 -s -L -K -U1 -T1 -d
+elim (lift_split … HTU2 i e0) /2 width=2 by ex_intro/
+qed.
+
+lemma cofrees_lift_ge: ∀d0,e0,i. d0 + e0 ≤ i →
+                       ∀L,K,s. ⇩[s, d0, e0] L ≡ K → ∀T,U. ⇧[d0, e0] T ≡ U →
+                       ∀d. K ⊢ i-e0 ~ϵ 𝐅*[d-yinj e0]⦃T⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃U⦄.
+#d0 #e0 #i #Hde0i #L #K #s #HLK #T1 #U1 #HTU1 #d #HT1 #U2 #HU12
+elim (le_inv_plus_l … Hde0i) -Hde0i #Hd0ie0 #He0i
+elim (yle_split d0 d) #H1
+[ elim (yle_split d (d0+e0)) #H2
+  [ elim (cpys_inv_lift1_ge_up … HU12 … HLK … HTU1) // >yplus_inj >yminus_Y_inj #T2 #HT12
+    lapply (cpys_weak … HT12 (d-yinj e0) (∞) ? ?) /2 width=1 by yle_plus2_to_minus_inj2/ -HT12
+  | elim (cpys_inv_lift1_ge … HU12 … HLK … HTU1) // #T2
+  ]
+| elim (cpys_inv_lift1_be … HU12 … HLK … HTU1) // >yminus_Y_inj #T2 #HT12
+  lapply (cpys_weak … HT12 (d-yinj e0) (∞) ? ?) // -HT12
+]
+-s -L #HT12 #HTU2
+elim (HT1 … HT12) -T1 #V2 #HVT2
+elim (lift_trans_le … HVT2 … HTU2 ?) // <plus_minus_m_m /2 width=2 by ex_intro/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/cofreestar_4.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/cofrees1/cofreestar_4.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/notation/relations/cofreestar_4.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cofrees1/cofreestar_4.etc
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/cpys.etc
similarity index 63%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/cpys.etc
index a0603f13e..b3b5bc639 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/cpys.etc
@@ -12,11 +12,9 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/extpsubststar_4.ma".
+include "basic_2/notation/relations/psubststar_4.ma".
 include "basic_2/grammar/genv.ma".
-include "basic_2/grammar/cl_shift.ma".
-include "basic_2/relocation/ldrop_append.ma".
-include "basic_2/relocation/lsuby.ma".
+include "basic_2/substitution/lsuby.ma".
 
 (* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
 
@@ -24,7 +22,7 @@ include "basic_2/relocation/lsuby.ma".
 inductive cpys: relation4 genv lenv term term ≝
 | cpys_atom : ∀I,G,L. cpys G L (⓪{I}) (⓪{I})
 | cpys_delta: ∀I,G,L,K,V,V2,W2,i.
-              ⇩[0, i] L ≡ K.ⓑ{I}V → cpys G K V V2 →
+              ⇩[i] L ≡ K.ⓑ{I}V → cpys G K V V2 →
               ⇧[0, i + 1] V2 ≡ W2 → cpys G L (#i) W2
 | cpys_bind : ∀a,I,G,L,V1,V2,T1,T2.
               cpys G L V1 V2 → cpys G (L.ⓑ{I}V1) T1 T2 →
@@ -36,7 +34,7 @@ inductive cpys: relation4 genv lenv term term ≝
 
 interpretation
    "context-sensitive extended multiple substitution (term)"
-   'ExtPSubstStar G L T1 T2 = (cpys G L T1 T2).
+   'PSubstStar G L T1 T2 = (cpys G L T1 T2).
 
 (* Basic properties *********************************************************)
 
@@ -44,7 +42,7 @@ lemma lsuby_cpys_trans: ∀G. lsub_trans … (cpys G) lsuby.
 #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
 [ //
 | #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
-  elim (lsuby_fwd_ldrop2_pair … HL12 … HLK1) -HL12 -HLK1 *
+  elim (lsuby_ldrop_trans … HL12 … HLK1) -HL12 -HLK1 *
   /3 width=7 by cpys_delta/
 | /4 width=1 by lsuby_pair, cpys_bind/
 | /3 width=1 by cpys_flat/
@@ -52,22 +50,22 @@ lemma lsuby_cpys_trans: ∀G. lsub_trans … (cpys G) lsuby.
 qed-.
 
 (* Note: this is "∀L. reflexive … (cpys L)" *)
-lemma cpys_refl: ∀G,T,L. ⦃G, L⦄ ⊢ T ▶*× T.
+lemma cpys_refl: ∀G,T,L. ⦃G, L⦄ ⊢ T ▶* T.
 #G #T elim T -T // * /2 width=1 by cpys_bind, cpys_flat/
 qed.
 
-lemma cpys_pair_sn: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ▶*× V2 →
-                    ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ▶*× ②{I}V2.T.
+lemma cpys_pair_sn: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ▶* V2 →
+                    ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ▶* ②{I}V2.T.
 * /2 width=1 by cpys_bind, cpys_flat/
 qed.
 
-lemma cpys_bind_ext: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ▶*× V2 →
-                     ∀J,T1,T2. ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ▶*× T2 →
-                     ∀a,I. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*× ⓑ{a,I}V2.T2.
+lemma cpys_bind_ext: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ▶* V2 →
+                     ∀J,T1,T2. ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ▶* T2 →
+                     ∀a,I. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶* ⓑ{a,I}V2.T2.
 /4 width=4 by lsuby_cpys_trans, cpys_bind, lsuby_pair/ qed.
 
-lemma cpys_delift: ∀I,G,K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓑ{I}V) →
-                   ∃∃T2,T.  ⦃G, L⦄ ⊢ T1 ▶*× T2 & ⇧[d, 1] T ≡ T2.
+lemma cpys_delift: ∀I,G,K,V,T1,L,d. ⇩[d] L ≡ (K.ⓑ{I}V) →
+                   ∃∃T2,T.  ⦃G, L⦄ ⊢ T1 ▶* T2 & ⇧[d, 1] T ≡ T2.
 #I #G #K #V #T1 elim T1 -T1
 [ * /2 width=4 by cpys_atom, lift_sort, lift_gref, ex2_2_intro/
   #i #L #d elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4 by cpys_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
@@ -76,26 +74,17 @@ lemma cpys_delift: ∀I,G,K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓑ{I}V) →
   elim (lift_split … HVW i i) /3 width=7 by cpys_delta, ex2_2_intro/
 | * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
   elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
-  [ elim (IHU1 (L. ⓑ{I}W1) (d+1)) -IHU1 /3 width=9 by cpys_bind, ldrop_ldrop, lift_bind, ex2_2_intro/
+  [ elim (IHU1 (L. ⓑ{I}W1) (d+1)) -IHU1 /3 width=9 by cpys_bind, ldrop_drop, lift_bind, ex2_2_intro/
   | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpys_flat, lift_flat, ex2_2_intro/
   ]
 ]
 qed-.
 
-lemma cpys_append: ∀G. l_appendable_sn … (cpys G).
-#G #K #T1 #T2 #H elim H -G -K -T1 -T2
-/2 width=3 by cpys_bind, cpys_flat/
-#I #G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
-lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpys_delta … I … (L@@K0) V1 … HVW2) // 
-@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *)
-qed.
-
 (* Basic inversion lemmas ***************************************************)
 
-fact cpys_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*× T2 → ∀J. T1 = ⓪{J} →
+fact cpys_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶* T2 → ∀J. T1 = ⓪{J} →
                          T2 = ⓪{J} ∨
-                         ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ▶*× V2 &
+                         ∃∃I,K,V,V2,i. ⇩[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ▶* V2 &
                                        ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
 #G #L #T1 #T2 * -L -T1 -T2
 [ #I #G #L #J #H destruct /2 width=1 by or_introl/
@@ -105,39 +94,39 @@ fact cpys_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*× T2 → ∀J. T1
 ]
 qed-.
 
-lemma cpys_inv_atom1: ∀J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ▶*× T2 →
+lemma cpys_inv_atom1: ∀J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ▶* T2 →
                       T2 = ⓪{J} ∨
-                      ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ▶*× V2 &
+                      ∃∃I,K,V,V2,i. ⇩[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ▶* V2 &
                                     ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
 /2 width=3 by cpys_inv_atom1_aux/ qed-.
 
-lemma cpys_inv_sort1: ∀G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ▶*× T2 → T2 = ⋆k.
+lemma cpys_inv_sort1: ∀G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ▶* T2 → T2 = ⋆k.
 #G #L #T2 #k #H elim (cpys_inv_atom1 … H) -H // *
 #I #K #V #V2 #i #_ #_ #_ #H destruct
 qed-.
 
-lemma cpys_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ▶*× T2 →
+lemma cpys_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ▶* T2 →
                       T2 = #i ∨
-                      ∃∃I,K,V,V2. ⇩[O, i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ▶*× V2 &
+                      ∃∃I,K,V,V2. ⇩[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ▶* V2 &
                                   ⇧[O, i + 1] V2 ≡ T2.
 #G #L #T2 #i #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/ *
 #I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7 by ex3_4_intro, or_intror/
 qed-.
 
-lemma cpys_inv_lref1_ge: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ▶*× T2 → |L| ≤ i → T2 = #i.
+lemma cpys_inv_lref1_ge: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ▶* T2 → |L| ≤ i → T2 = #i.
 #G #L #T2 #i #H elim (cpys_inv_lref1 … H) -H // *
 #I #K #V1 #V2 #HLK #_ #_ #HL -V2 lapply (ldrop_fwd_length_lt2 … HLK) -K -I -V1
 #H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
 qed-.
 
-lemma cpys_inv_gref1: ∀G,L,T2,p.  ⦃G, L⦄ ⊢ §p ▶*× T2 → T2 = §p.
+lemma cpys_inv_gref1: ∀G,L,T2,p.  ⦃G, L⦄ ⊢ §p ▶* T2 → T2 = §p.
 #G #L #T2 #p #H elim (cpys_inv_atom1 … H) -H // *
 #I #K #V #V2 #i #_ #_ #_ #H destruct
 qed-.
 
-fact cpys_inv_bind1_aux: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶*× U2 →
+fact cpys_inv_bind1_aux: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶* U2 →
                          ∀a,J,V1,T1. U1 = ⓑ{a,J}V1.T1 →
-                         ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*× V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ▶*× T2 &
+                         ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ▶* T2 &
                                   U2 = ⓑ{a,J}V2.T2.
 #G #L #U1 #U2 * -L -U1 -U2
 [ #I #G #L #b #J #W #U1 #H destruct
@@ -147,22 +136,22 @@ fact cpys_inv_bind1_aux: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ▶*× U2 →
 ]
 qed-.
 
-lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*× U2 →
-                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*× V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*× T2 &
+lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶* U2 →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶* T2 &
                                U2 = ⓑ{a,I}V2.T2.
 /2 width=3 by cpys_inv_bind1_aux/ qed-.
 
-lemma cpys_inv_bind1_ext: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*× U2 → ∀J.
-                          ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*× V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ▶*× T2 &
+lemma cpys_inv_bind1_ext: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶* U2 → ∀J.
+                          ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ▶* T2 &
                                    U2 = ⓑ{a,I}V2.T2.
 #a #I #G #L #V1 #T1 #U2 #H #J elim (cpys_inv_bind1 … H) -H
 #V2 #T2 #HV12 #HT12 #H destruct
 /4 width=5 by lsuby_cpys_trans, lsuby_pair, ex3_2_intro/
 qed-.
 
-fact cpys_inv_flat1_aux: ∀G,L,U,U2. ⦃G, L⦄ ⊢ U ▶*× U2 →
+fact cpys_inv_flat1_aux: ∀G,L,U,U2. ⦃G, L⦄ ⊢ U ▶* U2 →
                          ∀J,V1,U1. U = ⓕ{J}V1.U1 →
-                         ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*× V2 & ⦃G, L⦄ ⊢ U1 ▶*× T2 &
+                         ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L⦄ ⊢ U1 ▶* T2 &
                                   U2 = ⓕ{J}V2.T2.
 #G #L #U #U2 * -L -U -U2
 [ #I #G #L #J #W #U1 #H destruct
@@ -172,29 +161,17 @@ fact cpys_inv_flat1_aux: ∀G,L,U,U2. ⦃G, L⦄ ⊢ U ▶*× U2 →
 ]
 qed-.
 
-lemma cpys_inv_flat1: ∀I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ▶*× U2 →
-                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*× V2 & ⦃G, L⦄ ⊢ U1 ▶*× T2 &
+(* Note: lemma 1250 *)
+lemma cpys_inv_flat1: ∀I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ▶* U2 →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶* V2 & ⦃G, L⦄ ⊢ U1 ▶* T2 &
                                U2 = ⓕ{I}V2.T2.
 /2 width=3 by cpys_inv_flat1_aux/ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpys_fwd_bind1: ∀a,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*× T → ∀b,J.
-                      ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,J}V1.T1 ▶*× ⓑ{b,J}V2.T2 &
+lemma cpys_fwd_bind1: ∀a,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶* T → ∀b,J.
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,J}V1.T1 ▶* ⓑ{b,J}V2.T2 &
                                T = ⓑ{a,I}V2.T2.
 #a #I #G #L #V1 #T1 #T #H #b #J elim (cpys_inv_bind1_ext … H J) -H
 #V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpys_bind, ex2_2_intro/
 qed-.
-
-lemma cpys_fwd_shift1: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*× T →
-                       ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#G #L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #HT1 @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X >shift_append_assoc normalize
-  #H elim (cpys_inv_bind1 … H) -H
-  #V0 #T0 #_ #HT10 #H destruct
-  elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
-  >append_length >HL12 -HL12
-  @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] /2 width=3 by trans_eq/ (**) (* explicit constructor *)
-]
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/cpys_cpys.etc
similarity index 93%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_cpys.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/cpys_cpys.etc
index e0abed64d..7f423f69f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_cpys.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/cpys_cpys.etc
@@ -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/substitution/lpys_ldrop.ma".
 
@@ -20,9 +20,9 @@ include "basic_2/substitution/lpys_ldrop.ma".
 
 (* Main properties **********************************************************)
 
-theorem cpys_antisym: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*× T2 → ⦃G, L⦄ ⊢ T2 ▶*× T1 → T1 = T2.
+theorem cpys_antisym: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶* T2 → ⦃G, L⦄ ⊢ T2 ▶* T1 → T1 = T2.
 #G #L #T1 #T2 #H elim H -G -L -T1 -T2 //
-[ #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #_ #HW2 lapply (ldrop_fwd_ldrop2 … HLK) -I -V1
+[ #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #_ #HW2 lapply (ldrop_fwd_drop2 … HLK) -I -V1
   #HLK elim (cpys_inv_lift1 … HW2 … HLK … HVW2) -L -HVW2
   #X #H #_ elim (lift_inv_lref2_be … H) -G -K -V2 -W2 -X //
 | #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #H elim (cpys_inv_bind1 … H) -H
@@ -48,7 +48,7 @@ theorem cpys_trans_lpys: ∀G. lpx_sn_transitive (cpys G) (cpys G).
   | * #I1 #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
     elim (lpys_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
     elim (lpys_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
-    lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+    lapply (ldrop_fwd_drop2 … HLK2) -W2 #HLK2
     elim (cpys_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
     lapply (fqup_lref … G1 … HLK1) /3 width=10 by cpys_delta/
   ]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/cpys_lift.etc
similarity index 71%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_lift.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/cpys_lift.etc
index d798c8a62..f435a862f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_lift.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/cpys_lift.etc
@@ -22,20 +22,20 @@ include "basic_2/substitution/cpys.ma".
 
 lemma cpys_lift: ∀G. l_liftable (cpys G).
 #G #K #T1 #T2 #H elim H -G -K -T1 -T2
-[ #I #G #K #L #d #e #_ #U1 #H1 #U2 #H2
+[ #I #G #K #L #s #d #e #_ #U1 #H1 #U2 #H2
   >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #d #e #HLK #U1 #H #U2 #HWU2
+| #I #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #d #e #HLK #U1 #H #U2 #HWU2
   elim (lift_inv_lref1 … H) * #Hid #H destruct
   [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
     elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
-    elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #HKV #HVY #H destruct /3 width=9 by cpys_delta/
+    elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #HKV #HVY #H destruct /3 width=10 by cpys_delta/
   | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
-    lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K /3 width=7 by cpys_delta/
+    lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K /3 width=7 by cpys_delta, ldrop_inv_gen/
   ]
-| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
+| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #d #e #HLK #U1 #H1 #U2 #H2
   elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
-  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=5 by cpys_bind, ldrop_skip/
-| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
+  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=6 by cpys_bind, ldrop_skip/
+| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #d #e #HLK #U1 #H1 #U2 #H2
   elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
   elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6 by cpys_flat/
 ]
@@ -43,12 +43,12 @@ qed.
 
 lemma cpys_inv_lift1: ∀G. l_deliftable_sn (cpys G).
 #G #L #U1 #U2 #H elim H -G -L -U1 -U2
-[ * #G #L #i #K #d #e #_ #T1 #H
+[ * #G #L #i #K #s #d #e #_ #T1 #H
   [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpys_atom, lift_sort, ex2_intro/
   | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by cpys_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
   | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpys_atom, lift_gref, ex2_intro/
   ]
-| #I #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #d #e #HLK #T1 #H
+| #I #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #d #e #HLK #T1 #H
   elim (lift_inv_lref2 … H) -H * #Hid #H destruct
   [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
     elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
@@ -58,11 +58,11 @@ lemma cpys_inv_lift1: ∀G. l_deliftable_sn (cpys G).
     elim (lift_split … HVW2 d (i - e + 1)) -HVW2 /3 width=1 by le_S, le_S_S/ -Hid -Hdie
     #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O /3 width=9 by cpys_delta, ex2_intro/
   ]
-| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
+| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
   elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
   elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
-  elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=5 by cpys_bind, ldrop_skip, lift_bind, ex2_intro/
-| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
+  elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=6 by cpys_bind, ldrop_skip, lift_bind, ex2_intro/
+| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
   elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
   elim (IHV12 … HLK … HWV1) -V1
   elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpys_flat, lift_flat, ex2_intro/
@@ -71,32 +71,32 @@ qed-.
 
 (* Properties on supclosure *************************************************)
 
-lemma fqu_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
-                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶*× U2 →
-                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶*× U1 & ⦃G1, L1, U1⦄ ⊃ ⦃G2, L2, U2⦄.
+lemma fqu_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 
 /3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpys_pair_sn, cpys_bind, cpys_flat, ex2_intro/
 [ #I #G #L #V2 #U2 #HVU2
   elim (lift_total U2 0 1)
-  /4 width=7 by fqu_drop, cpys_delta, ldrop_pair, ldrop_ldrop, ex2_intro/
+  /4 width=7 by fqu_drop, cpys_delta, ldrop_pair, ldrop_drop, ex2_intro/
 | #G #L #K #T1 #U1 #e #HLK1 #HTU1 #T2 #HTU2
   elim (lift_total T2 0 (e+1))
   /3 width=11 by cpys_lift, fqu_drop, ex2_intro/
 ]
 qed-.
 
-lemma fquq_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
-                       ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶*× U2 →
-                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶*× U1 & ⦃G1, L1, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
+lemma fquq_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                       ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 →
+                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
 #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
 [ #HT12 elim (fqu_cpys_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
 ]
 qed-.
 
-lemma fqup_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
-                       ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶*× U2 →
-                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶*× U1 & ⦃G1, L1, U1⦄ ⊃+ ⦃G2, L2, U2⦄.
+lemma fqup_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                       ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 →
+                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
 #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
 [ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpys_trans … H12 … HTU2) -T2
   /3 width=3 by fqu_fqup, ex2_intro/
@@ -106,22 +106,22 @@ lemma fqup_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2
 ]
 qed-.
 
-lemma fqus_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
-                       ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶*× U2 →
-                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶*× U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
+lemma fqus_cpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                       ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 →
+                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
 #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fqus_inv_gen … H) -H
 [ #HT12 elim (fqup_cpys_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
 ]
 qed-.
 
-lemma fqu_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
-                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶*× U2 → (T2 = U2 → ⊥) →
-                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶*× U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊃ ⦃G2, L2, U2⦄.
+lemma fqu_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 → (T2 = U2 → ⊥) →
+                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
 [ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
   #U2 #HVU2 @(ex3_intro … U2)
-  [1,3: /3 width=7 by fqu_drop, cpys_delta, ldrop_pair, ldrop_ldrop/
+  [1,3: /3 width=7 by fqu_drop, cpys_delta, ldrop_pair, ldrop_drop/
   | #H destruct /2 width=7 by lift_inv_lref2_be/
   ]
 | #I #G #L #V1 #T #V2 #HV12 #H @(ex3_intro … (②{I}V2.T))
@@ -138,14 +138,14 @@ lemma fqu_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2,
   ]
 | #G #L #K #T1 #U1 #e #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (e+1))
   #U2 #HTU2 @(ex3_intro … U2)
-  [1,3: /2 width=9 by cpys_lift, fqu_drop/
+  [1,3: /2 width=10 by cpys_lift, fqu_drop/
   | #H0 destruct /3 width=5 by lift_inj/
 ]
 qed-.
 
-lemma fquq_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
-                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶*× U2 → (T2 = U2 → ⊥) →
-                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶*× U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
+lemma fquq_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
 #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
 [ #H12 elim (fqu_cpys_trans_neq … H12 … HTU2 H) -T2
   /3 width=4 by fqu_fquq, ex3_intro/
@@ -153,9 +153,9 @@ lemma fquq_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2,
 ]
 qed-.
 
-lemma fqup_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
-                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶*× U2 → (T2 = U2 → ⊥) →
-                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶*× U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊃+ ⦃G2, L2, U2⦄.
+lemma fqup_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
 #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
 [ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpys_trans_neq … H12 … HTU2 H) -T2
   /3 width=4 by fqu_fqup, ex3_intro/
@@ -165,9 +165,9 @@ lemma fqup_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2
 ]
 qed-.
 
-lemma fqus_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
-                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶*× U2 → (T2 = U2 → ⊥) →
-                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶*× U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
+lemma fqus_cpys_trans_neq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ▶* U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ▶* U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
 #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
 [ #H12 elim (fqup_cpys_trans_neq … H12 … HTU2 H) -T2
   /3 width=4 by fqup_fqus, ex3_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lpys.etc
similarity index 55%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lpys.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lpys.etc
index 083fbbe46..0a9563a6a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lpys.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lpys.etc
@@ -12,8 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/extpsubstsnstar_3.ma".
-include "basic_2/grammar/lpx_sn.ma".
+include "basic_2/notation/relations/psubstsnstar_3.ma".
+include "basic_2/relocation/lpx_sn.ma".
 include "basic_2/substitution/cpys.ma".
 
 (* SN EXTENDED MULTIPLE SUBSTITUTION FOR LOCAL ENVIRONMENTS *****************)
@@ -22,54 +22,40 @@ definition lpys: relation3 genv lenv lenv ≝ λG. lpx_sn (cpys G).
 
 interpretation
    "extended multiple substitution (local environment, sn variant)"
-   'ExtPSubstSnStar G L1 L2 = (lpys G L1 L2).
+   'PSubstSnStar G L1 L2 = (lpys G L1 L2).
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma lpys_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ▶*× L2 → L2 = ⋆.
+lemma lpys_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ▶* L2 → L2 = ⋆.
 /2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
 
-lemma lpys_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ▶*× L2 →
-                      ∃∃K2,V2. ⦃G, K1⦄ ⊢ ▶*× K2 & ⦃G, K1⦄ ⊢ V1 ▶*× V2 &
+lemma lpys_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ▶* L2 →
+                      ∃∃K2,V2. ⦃G, K1⦄ ⊢ ▶* K2 & ⦃G, K1⦄ ⊢ V1 ▶* V2 &
                                L2 = K2. ⓑ{I} V2.
 /2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
 
-lemma lpys_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ▶*× ⋆ → L1 = ⋆.
+lemma lpys_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ▶* ⋆ → L1 = ⋆.
 /2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
 
-lemma lpys_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ▶*× K2.ⓑ{I}V2 →
-                      ∃∃K1,V1. ⦃G, K1⦄ ⊢ ▶*× K2 & ⦃G, K1⦄ ⊢ V1 ▶*× V2 &
+lemma lpys_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ▶* K2.ⓑ{I}V2 →
+                      ∃∃K1,V1. ⦃G, K1⦄ ⊢ ▶* K2 & ⦃G, K1⦄ ⊢ V1 ▶* V2 &
                                L1 = K1. ⓑ{I} V1.
 /2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
 
-lemma lpys_inv_pair: ∀I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ▶*× L2.ⓑ{I2}V2 →
-                     ∧∧ ⦃G, L1⦄ ⊢ ▶*× L2 & ⦃G, L1⦄ ⊢ V1 ▶*× V2 & I1 = I2.
+lemma lpys_inv_pair: ∀I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ▶* L2.ⓑ{I2}V2 →
+                     ∧∧ ⦃G, L1⦄ ⊢ ▶* L2 & ⦃G, L1⦄ ⊢ V1 ▶* V2 & I1 = I2.
 /2 width=1 by lpx_sn_inv_pair/ qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lpys_refl: ∀G,L. ⦃G, L⦄ ⊢ ▶*× L.
+lemma lpys_refl: ∀G,L. ⦃G, L⦄ ⊢ ▶* L.
 /2 width=1 by lpx_sn_refl/ qed.
 
-lemma lpys_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ▶*× K2 → ⦃G, K1⦄ ⊢ V1 ▶*× V2 →
-                 ⦃G, K1.ⓑ{I}V1⦄ ⊢ ▶*× K2.ⓑ{I}V2.
+lemma lpys_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ▶* K2 → ⦃G, K1⦄ ⊢ V1 ▶* V2 →
+                 ⦃G, K1.ⓑ{I}V1⦄ ⊢ ▶* K2.ⓑ{I}V2.
 /2 width=1 by lpx_sn_pair/ qed.
 
-lemma lpys_append: ∀G,K1,K2. ⦃G, K1⦄ ⊢ ▶*× K2 → ∀L1,L2. ⦃G, L1⦄ ⊢ ▶*× L2 →
-                   ⦃G, L1 @@ K1⦄ ⊢ ▶*× L2 @@ K2.
-/3 width=1 by lpx_sn_append, cpys_append/ qed.
-
 (* Basic forward lemmas *****************************************************)
 
-lemma lpys_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ▶*× L2 → |L1| = |L2|.
+lemma lpys_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ▶* L2 → |L1| = |L2|.
 /2 width=2 by lpx_sn_fwd_length/ qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lpys_fwd_append1: ∀G,K1,L1,L. ⦃G, K1 @@ L1⦄ ⊢ ▶*× L →
-                        ∃∃K2,L2. ⦃G, K1⦄ ⊢ ▶*× K2 & L = K2 @@ L2.
-/2 width=2 by lpx_sn_fwd_append1/ qed-.
-
-lemma lpys_fwd_append2: ∀G,L,K2,L2. ⦃G, L⦄ ⊢ ▶*× K2 @@ L2 →
-                        ∃∃K1,L1. ⦃G, K1⦄ ⊢ ▶*× K2 & L = K1 @@ L1.
-/2 width=2 by lpx_sn_fwd_append2/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lpys_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lpys_ldrop.etc
similarity index 71%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lpys_ldrop.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lpys_ldrop.etc
index df77274de..db5037c38 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lpys_ldrop.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lpys_ldrop.etc
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/relocation/ldrop_lpx_sn.ma".
+include "basic_2/relocation/lpx_sn_ldrop.ma".
 include "basic_2/substitution/cpys_lift.ma".
 include "basic_2/substitution/lpys.ma".
 
@@ -21,19 +21,19 @@ include "basic_2/substitution/lpys.ma".
 (* Properies on local environment slicing ***********************************)
 
 lemma lpys_ldrop_conf: ∀G. dropable_sn (lpys G).
-/3 width=5 by lpx_sn_deliftable_dropable, cpys_inv_lift1/ qed-.
+/3 width=6 by lpx_sn_deliftable_dropable, cpys_inv_lift1/ qed-.
 
 lemma ldrop_lpys_trans: ∀G. dedropable_sn (lpys G).
-/3 width=9 by lpx_sn_liftable_dedropable, cpys_lift/ qed-.
+/3 width=10 by lpx_sn_liftable_dedropable, cpys_lift/ qed-.
 
 lemma lpys_ldrop_trans_O1: ∀G. dropable_dx (lpys G).
 /2 width=3 by lpx_sn_dropable/ qed-.
 
 (* Properties on supclosure *************************************************)
 
-lemma fqu_lpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
-                      ∀K2. ⦃G2, L2⦄ ⊢ ▶*× K2 →
-                      ∃∃K1,T. ⦃G1, L1⦄ ⊢ ▶*× K1 & ⦃G1, L1⦄ ⊢ T1 ▶*× T & ⦃G1, K1, T⦄ ⊃ ⦃G2, K2, T2⦄.
+lemma fqu_lpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                      ∀K2. ⦃G2, L2⦄ ⊢ ▶* K2 →
+                      ∃∃K1,T. ⦃G1, L1⦄ ⊢ ▶* K1 & ⦃G1, L1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄.
 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
 /3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpys_pair, ex3_2_intro/
 [ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpys_inv_pair1 … H) -H
@@ -45,32 +45,32 @@ lemma fqu_lpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2
 ]
 qed-.
 
-lemma fquq_lpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
-                       ∀K2. ⦃G2, L2⦄ ⊢ ▶*× K2 →
-                       ∃∃K1,T. ⦃G1, L1⦄ ⊢ ▶*× K1 & ⦃G1, L1⦄ ⊢ T1 ▶*× T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄.
+lemma fquq_lpys_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                       ∀K2. ⦃G2, L2⦄ ⊢ ▶* K2 →
+                       ∃∃K1,T. ⦃G1, L1⦄ ⊢ ▶* K1 & ⦃G1, L1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
 #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
 [ #HT12 elim (fqu_lpys_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
 ]
 qed-.
 
-lemma lpys_fqu_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
-                      ∀K1. ⦃G1, K1⦄ ⊢ ▶*× L1 →
-                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ▶*× T & ⦃G1, K1, T⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ▶*× L2.
+lemma lpys_fqu_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                      ∀K1. ⦃G1, K1⦄ ⊢ ▶* L1 →
+                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ▶* L2.
 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
 /3 width=7 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpys_pair, ex3_2_intro/
 [ #I #G1 #L1 #V1 #X #H elim (lpys_inv_pair2 … H) -H
   #K1 #W1 #HKL1 #HWV1 #H destruct elim (lift_total V1 0 1)
-  /4 width=7 by cpys_delta, fqu_drop, ldrop_ldrop, ex3_2_intro/
+  /4 width=7 by cpys_delta, fqu_drop, ldrop_drop, ex3_2_intro/
 | #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #L0 #HL01
   elim (lpys_ldrop_trans_O1 … HL01 … HLK1) -L1
   /3 width=5 by fqu_drop, ex3_2_intro/
 ]
 qed-.
 
-lemma lpys_fquq_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
-                       ∀K1. ⦃G1, K1⦄ ⊢ ▶*× L1 →
-                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ▶*× T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ▶*× L2.
+lemma lpys_fquq_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                       ∀K1. ⦃G1, K1⦄ ⊢ ▶* L1 →
+                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ▶* T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ▶* L2.
 #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H
 [ #HT12 elim (lpys_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/
 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lpys_lpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lpys_lpys.etc
similarity index 60%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lpys_lpys.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lpys_lpys.etc
index b9bf06694..f5524bfe2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lpys_lpys.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lpys_lpys.etc
@@ -21,19 +21,3 @@ include "basic_2/substitution/cpys_cpys.ma".
 theorem lpys_trans: ∀G. Transitive … (lpys G).
 /3 width=5 by lpx_sn_trans, cpys_trans_lpys/
 qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma cpys_fwd_shift1_ext: ∀G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*× T →
-                           ∃∃L2,T2. ⦃G, L @@ L1⦄ ⊢ ▶*× L @@ L2 & ⦃G, L @@ L1⦄ ⊢ T1 ▶*× T2 &
-                                    T = L2 @@ T2.
-#G #L1 @(lenv_ind_dx … L1) -L1
-[ #L #T1 #T #HT1 @ex3_2_intro
-  [3: // |4,5: // |1,2: skip ] (**) (* auto does not work *)
-| #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H <append_assoc
-  elim (cpys_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
-  elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
-  lapply (lpys_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1 by lpys_append, lpys_pair/ #HL12
-  @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3 by trans_eq/ | skip ] <append_assoc // (**) (* explicit constructor *)
-]
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/extlrsubeq_2.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lrsubeq_2.etc
similarity index 92%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpys/extlrsubeq_2.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lrsubeq_2.etc
index 97bdb1828..da283222e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/extlrsubeq_2.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lrsubeq_2.etc
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( L1 break ⊑ × break term 46 L2 )"
+notation "hvbox( L1 break ⊆ break term 46 L2 )"
    non associative with precedence 45
-   for @{ 'ExtLRSubEq $L1 $L2 }.
+   for @{ 'LRSubEq $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lsuby.etc
similarity index 55%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lsuby.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lsuby.etc
index 6d5c0072c..c06429d42 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lsuby.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lsuby.etc
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/extlrsubeq_2.ma".
+include "basic_2/notation/relations/lrsubeq_2.ma".
 include "basic_2/relocation/ldrop.ma".
 
 (* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
@@ -24,52 +24,72 @@ inductive lsuby: relation lenv ≝
 
 interpretation
   "local environment refinement (extended substitution)"
-  'ExtLRSubEq L1 L2 = (lsuby L1 L2).
+  'LRSubEq L1 L2 = (lsuby L1 L2).
 
 (* Basic properties *********************************************************)
 
-lemma lsuby_refl: ∀L. L ⊑× L.
-#L elim L -L // /2 width=1/
+lemma lsuby_refl: ∀L. L ⊆ L.
+#L elim L -L /2 width=1 by lsuby_pair/
 qed.
 
+lemma lsuby_sym: ∀L1,L2. L1 ⊆ L2 → |L1| = |L2| → L2 ⊆ L1.
+#L1 #L2 #H elim H -L1 -L2
+[ #L1 #H >(length_inv_zero_dx … H) -L1 //
+| #I1 #I2 #L1 #L2 #V #_ #IHL12 #H lapply (injective_plus_l … H) -H
+  /3 width=1 by lsuby_pair/
+]
+qed-.
+
 (* Basic inversion lemmas ***************************************************)
 
-fact lsuby_inv_atom1_aux: ∀L1,L2. L1 ⊑× L2 → L1 = ⋆ → L2 = ⋆.
+fact lsuby_inv_atom1_aux: ∀L1,L2. L1 ⊆ L2 → L1 = ⋆ → L2 = ⋆.
 #L1 #L2 * -L1 -L2 //
 #I1 #I2 #L1 #L2 #V #_ #H destruct
 qed-.
 
-lemma lsuby_inv_atom1: ∀L2. ⋆ ⊑× L2 → L2 = ⋆.
+lemma lsuby_inv_atom1: ∀L2. ⋆ ⊆ L2 → L2 = ⋆.
 /2 width=3 by lsuby_inv_atom1_aux/ qed-.
 
-fact lsuby_inv_pair1_aux: ∀L1,L2. L1 ⊑× L2 → ∀J1,K1,W. L1 = K1.ⓑ{J1}W →
-                          L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊑× K2 & L2 = K2.ⓑ{I2}W.
+fact lsuby_inv_pair1_aux: ∀L1,L2. L1 ⊆ L2 → ∀J1,K1,W. L1 = K1.ⓑ{J1}W →
+                          L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊆ K2 & L2 = K2.ⓑ{I2}W.
 #L1 #L2 * -L1 -L2
 [ #L #J1 #K1 #W #H destruct /2 width=1 by or_introl/
 | #I1 #I2 #L1 #L2 #V #HL12 #J1 #K1 #W #H destruct /3 width=4 by ex2_2_intro, or_intror/
 ]
 qed-.
 
-lemma lsuby_inv_pair1: ∀I1,K1,L2,W. K1.ⓑ{I1}W ⊑× L2 →
-                       L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊑× K2 & L2 = K2.ⓑ{I2}W.
+lemma lsuby_inv_pair1: ∀I1,K1,L2,W. K1.ⓑ{I1}W ⊆ L2 →
+                       L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊆ K2 & L2 = K2.ⓑ{I2}W.
 /2 width=4 by lsuby_inv_pair1_aux/ qed-.
 
+fact lsuby_inv_pair2_aux: ∀L1,L2. L1 ⊆ L2 → ∀J2,K2,W. L2 = K2.ⓑ{J2}W →
+                          ∃∃I1,K1. K1 ⊆ K2 & L1 = K1.ⓑ{I1}W.
+#L1 #L2 * -L1 -L2
+[ #L #J2 #K2 #W #H destruct
+| #I1 #I2 #L1 #L2 #V #HL12 #J2 #K2 #W #H destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lsuby_inv_pair2: ∀I2,L1,K2,W. L1 ⊆ K2.ⓑ{I2}W  →
+                       ∃∃I1,K1. K1 ⊆ K2 & L1 = K1.ⓑ{I1}W.
+/2 width=4 by lsuby_inv_pair2_aux/ qed-.
+
 (* Basic forward lemmas *****************************************************)
 
-lemma lsuby_fwd_length: ∀L1,L2. L1 ⊑× L2 → |L2| ≤ |L1|.
+lemma lsuby_fwd_length: ∀L1,L2. L1 ⊆ L2 → |L2| ≤ |L1|.
 #L1 #L2 #H elim H -L1 -L2 /2 width=1 by monotonic_le_plus_l/
 qed-.
 
-lemma lsuby_fwd_ldrop2_pair: ∀L1,L2. L1 ⊑× L2 →
-                             ∀I2,K2,W,i. ⇩[0, i] L2 ≡ K2.ⓑ{I2}W →
-                             ∃∃I1,K1. K1 ⊑× K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I1}W.
+lemma lsuby_ldrop_trans: ∀L1,L2. L1 ⊆ L2 →
+                         ∀I2,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
+                         ∃∃I1,K1. K1 ⊆ K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
 #L1 #L2 #H elim H -L1 -L2
-[ #L #J2 #K2 #W #i #H
+[ #L #J2 #K2 #W #s #i #H
   elim (ldrop_inv_atom1 … H) -H #H destruct
-| #I1 #I2 #L1 #L2 #V #HL12 #IHL12 #J2 #K2 #W #i #H
+| #I1 #I2 #L1 #L2 #V #HL12 #IHL12 #J2 #K2 #W #s #i #H
   elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
   [ /3 width=4 by ldrop_pair, ex2_2_intro/
-  | elim (IHL12 … HLK2) -IHL12 -HLK2 * /3 width=4 by ldrop_ldrop_lt, ex2_2_intro/
+  | elim (IHL12 … HLK2) -IHL12 -HLK2 * /3 width=4 by ldrop_drop_lt, ex2_2_intro/
   ]
 ]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lsuby_lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lsuby_lsuby.etc
similarity index 97%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lsuby_lsuby.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lsuby_lsuby.etc
index 590e7d317..254d62928 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lsuby_lsuby.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/lsuby_lsuby.etc
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/relocation/lsuby.ma".
+include "basic_2/substitution/lsuby.ma".
 
 (* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/extpsubstsnstar_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/psubstsnstar_3.etc
similarity index 94%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpys/extpsubstsnstar_3.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/psubstsnstar_3.etc
index 64acfff4c..0d6dffee6 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/extpsubstsnstar_3.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/psubstsnstar_3.etc
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ▶ * × break term 46 L2 )"
+notation "hvbox( ⦃ term 46 G , break term 46 L1 ⦄ ⊢ ▶ * break term 46 L2 )"
    non associative with precedence 45
-   for @{ 'ExtPSubstSnStar $G $L1 $L2 }.
+   for @{ 'PSubstSnStar $G $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/extpsubststar_4.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/psubststar_4.etc
similarity index 93%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpys/extpsubststar_4.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/psubststar_4.etc
index 363dce487..0e76c6f71 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/extpsubststar_4.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys0/psubststar_4.etc
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * × break term 46 T2 )"
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * break term 46 T2 )"
    non associative with precedence 45
-   for @{ 'ExtPSubstStar $G $L $T1 $T2 }.
+   for @{ 'PSubstStar $G $L $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpx_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys2/cpx_cpys.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpx_cpys.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys2/cpx_cpys.etc
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpx_cpys2.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys2/cpx_cpys2.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpx_cpys2.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys2/cpx_cpys2.etc
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpxs_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys2/cpxs_cpys.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpy/cpxs_cpys.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys2/cpxs_cpys.etc
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/lpx_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys2/lpx_cpys.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpy/lpx_cpys.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys2/lpx_cpys.etc
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys2/lsuby.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/etc/cpy/lsuby.etc
rename to matita/matita/contribs/lambdadelta/basic_2/etc/cpys2/lsuby.etc
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/freestar_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/freestar_4.ma
new file mode 100644
index 000000000..4ce6d2dc7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/freestar_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( L ⊢ break term 46 i ϵ 𝐅 * [ break term 46 d ] ⦃ break term 46 T ⦄ )"
+   non associative with precedence 45
+   for @{ 'FreeStar $L $i $d $T }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_frees.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_frees.ma
new file mode 100644
index 000000000..e303b8386
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_frees.ma
@@ -0,0 +1,82 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/substitution/fqup.ma".
+include "basic_2/substitution/frees_lift.ma".
+include "basic_2/reduction/lpx_ldrop.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+(*
+lemma yle_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. x ≤ y + z → x - z ≤ y.
+/2 width=1 by monotonic_yle_minus_dx/ qed-.
+
+lemma yle_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. x ≤ z + y → x - z ≤ y.
+/2 width=1 by yle_plus2_to_minus_inj2/ qed-.
+
+lemma cofrees_lsuby_conf: ∀L1,U,i. L1 ⊢ i ~ϵ 𝐅*⦃U⦄ →
+                          ∀L2. lsuby L1 L2 → L2 ⊢ i ~ϵ 𝐅*⦃U⦄.
+/3 width=3 by lsuby_cpys_trans/ qed-.
+
+lemma lpx_cpx_cofrees_conf: ∀h,g,G,L1,U1,U2. ⦃G, L1⦄ ⊢ U1 ➡[h, g] U2 →
+                            ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+                            ∀i. L1 ⊢ i ~ϵ 𝐅*⦃U1⦄ → L2 ⊢ i ~ϵ 𝐅*⦃U2⦄.
+#h #g #G #L1 #U1 @(fqup_wf_ind_eq … G L1 U1) -G -L1 -U1
+#G0 #L0 #U0 #IH #G #L1 * *
+[ -IH #k #HG #HL #HU #U2 #H elim (cpx_inv_sort1 … H) -H
+  [| * #l #_ ] #H destruct //
+| #j #HG #HL #HU #U2 #H #L2 #HL12 #i #Hi elim (cpx_inv_lref1 … H) -H
+  [ #H destruct elim (lt_or_eq_or_gt i j) #Hji
+    [ -IH -HL12 /2 width=4 by cofrees_lref_gt/
+    | -IH -HL12 destruct elim (cofrees_inv_lref_eq … Hi)
+    | elim (lt_or_ge j (|L2|)) /2 width=5 by cofrees_lref_free/ #Hj
+      elim (ldrop_O1_lt (Ⓕ) L1 j) [2: >(lpx_fwd_length … HL12) // ] #I #K1 #W1 #HLK1
+      elim (lpx_ldrop_conf … HLK1 … HL12) -HL12 #Y #H #HLK2
+      elim (lpx_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
+      lapply (cofrees_inv_lref_lt … Hi … HLK1) -Hi // #HW1
+      lapply (IH … HW12 … HK12 … HW1) /2 width=2 by fqup_lref/ -L1 -K1 -W1 #HW2
+      /2 width=9 by cofrees_lref_lt/ (**) (* full auto too slow *)
+    ]
+  | * #I #K1 #W1 #W0 #HLK1 #HW10 #HW0U2 elim (lt_or_ge j i) #Hji
+    [ lapply (ldrop_fwd_drop2 … HLK1) #H0
+      elim (lpx_ldrop_conf … H0 … HL12) #K2 #HK12 #HLK2
+      @(cofrees_lift_ge … HLK2 … HW0U2) //
+      @(IH … HW10) /2 width=2 by fqup_lref/ -L2 -K2 -W0 -U2 -IH
+      <minus_plus /2 width=7 by cofrees_inv_lref_lt/
+    | -IH lapply (ldrop_fwd_drop2 … HLK1) -HLK1 #HLK1
+      elim (lpx_ldrop_conf … HLK1 … HL12) -I -L1 -W1
+      /2 width=12 by cofrees_lift_be/
+    ]
+  ]
+| -IH #p #HG #HL #HU #U2 #H lapply (cpx_inv_gref1 … H) -H
+  #H destruct //
+| #a #I #W1 #U1 #HG #HL #HU #X #HX #L2 #HL12 #i #Hi destruct
+  elim (cofrees_inv_bind … Hi) -Hi #HW1 #HU1
+  elim (cpx_inv_bind1 … HX) -HX
+  [ * #W2 #U2 #HW12 #HU12 #H destruct /4 width=8 by cofrees_bind, lpx_pair/
+  |
+  ]
+| #I #W1 #X1 #HG #HL #HU #X2 #HX2 #L2 #HL12 #i #Hi destruct
+  elim (cofrees_inv_flat … Hi) -Hi #HW1 #HX1
+  elim (cpx_inv_flat1 … HX2) -HX2 *
+  [ #W2 #U2 #HW12 #HU12 #H destruct /3 width=7 by cofrees_flat/
+  | #HU12 #H destruct /2 width=7 by/
+  | #HW12 #H destruct /2 width=7 by/
+  | #b #W2 #V1 #V2 #U1 #U2 #HW12 #HV12 #HU12 #H1 #H2 #H3 destruct
+    elim (cofrees_inv_bind … HX1) -HX1 #HV1 #HU1
+    @cofrees_bind [ /3 width=7 by cofrees_flat/ ]
+    @(cofrees_lsuby_conf (L2.ⓛV2)) /3 width=7 by lpx_pair, lsuby_succ/
+  | 
+    
+   
+*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
index cf29f1cc2..a1e538ae0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
@@ -83,7 +83,6 @@ lemma cpy_weak: ∀G,L,T1,T2,d1,e1. ⦃G, L⦄ ⊢ T1 ▶[d1, e1] T2 →
 ]
 qed-.
 
-(* Note: lemma 1250 *)
 lemma cpy_weak_top: ∀G,L,T1,T2,d,e.
                     ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶[d, |L| - d] T2.
 #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e //
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/frees.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/frees.ma
new file mode 100644
index 000000000..1b817427d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/frees.ma
@@ -0,0 +1,160 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "ground_2/ynat/ynat_plus.ma".
+include "basic_2/notation/relations/freestar_4.ma".
+include "basic_2/relocation/lift_neg.ma".
+include "basic_2/relocation/ldrop.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+inductive frees: relation4 ynat lenv term nat ≝
+| frees_eq: ∀L,U,d,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → frees d L U i
+| frees_be: ∀I,L,K,U,W,d,i,j. d ≤ yinj j → j < i →
+            (∀T. ⇧[j, 1] T ≡ U → ⊥) → ⇩[j]L ≡ K.ⓑ{I}W →
+            frees 0 K W (i-j-1) → frees d L U i.
+
+interpretation
+   "context-sensitive free variables (term)"
+   'FreeStar L i d U = (frees d L U i).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma frees_inv: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
+                 (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
+                 ∃∃I,K,W,j. d ≤ yinj j & j < i & (∀T. ⇧[j, 1] T ≡ U → ⊥) &
+                            ⇩[j]L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #U #d #i * -L -U -d -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
+qed-.
+
+lemma frees_inv_sort: ∀L,d,i,k. L ⊢ i ϵ 𝐅*[d]⦃⋆k⦄ → ⊥.
+#L #d #i #k #H elim (frees_inv … H) -H [|*] /2 width=2 by/
+qed-.
+
+lemma frees_inv_gref: ∀L,d,i,p. L ⊢ i ϵ 𝐅*[d]⦃§p⦄ → ⊥.
+#L #d #i #p #H elim (frees_inv … H) -H [|*] /2 width=2 by/
+qed-.
+
+lemma frees_inv_lref: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ →
+                      j = i ∨
+                      ∃∃I,K,W. d ≤ yinj j & j < i & ⇩[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #d #x #i #H elim (frees_inv … H) -H
+[ /4 width=2 by nlift_inv_lref_be_SO, or_introl/
+| * #I #K #W #j #Hdj #Hji #Hnx #HLK #HW
+  >(nlift_inv_lref_be_SO … Hnx) -x /3 width=5 by ex4_3_intro, or_intror/
+]
+qed-.
+
+lemma frees_inv_lref_free: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → |L| ≤ j → j = i.
+#L #d #j #i #H #Hj elim (frees_inv_lref … H) -H //
+* #I #K #W #_ #_ #HLK lapply (ldrop_fwd_length_lt2 … HLK) -I
+#H elim (lt_refl_false j) /2 width=3 by lt_to_le_to_lt/
+qed-.
+
+lemma frees_inv_lref_skip: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → yinj j < d → j = i.
+#L #d #j #i #H #Hjd elim (frees_inv_lref … H) -H //
+* #I #K #W #Hdj elim (ylt_yle_false … Hdj) -Hdj //
+qed-. 
+
+lemma frees_inv_lref_ge: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → i ≤ j → j = i.
+#L #d #j #i #H #Hij elim (frees_inv_lref … H) -H //
+* #I #K #W #_ #Hji elim (lt_refl_false j) -I -L -K -W -d /2 width=3 by lt_to_le_to_lt/
+qed-.
+
+lemma frees_inv_lref_lt: ∀L,d,j,i.L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → j < i →
+                         ∃∃I,K,W. d ≤ yinj j & ⇩[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #d #j #i #H #Hji elim (frees_inv_lref … H) -H
+[ #H elim (lt_refl_false j) //
+| * /2 width=5 by ex3_3_intro/
+]
+qed-.
+
+lemma frees_inv_bind: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
+                      L ⊢ i ϵ 𝐅*[d]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯d]⦃U⦄ .
+#a #J #L #V #U #d #i #H elim (frees_inv … H) -H
+[ #HnX elim (nlift_inv_bind … HnX) -HnX
+  /4 width=2 by frees_eq, or_intror, or_introl/
+| * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_bind … HnX) -HnX
+  [ /4 width=9 by frees_be, or_introl/
+  | #HnT @or_intror @(frees_be … HnT) -HnT
+    [4,5,6: /2 width=1 by ldrop_drop, yle_succ, lt_minus_to_plus/
+    |7: >minus_plus_plus_l //
+    |*: skip
+    ]
+  ]
+]
+qed-.
+
+lemma frees_inv_flat: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
+                      L ⊢ i ϵ 𝐅*[d]⦃W⦄ ∨ L ⊢ i ϵ 𝐅*[d]⦃U⦄ .
+#J #L #V #U #d #i #H elim (frees_inv … H) -H
+[ #HnX elim (nlift_inv_flat … HnX) -HnX
+  /4 width=2 by frees_eq, or_intror, or_introl/
+| * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_flat … HnX) -HnX
+  /4 width=9 by frees_be, or_intror, or_introl/
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma frees_lref_eq: ∀L,d,i. L ⊢ i ϵ 𝐅*[d]⦃#i⦄.
+/3 width=7 by frees_eq, lift_inv_lref2_be/ qed.
+
+lemma frees_lref_be: ∀I,L,K,W,d,i,j. d ≤ yinj j → j < i → ⇩[j]L ≡ K.ⓑ{I}W →
+                     K ⊢ i-j-1 ϵ 𝐅*[0]⦃W⦄ → L ⊢ i ϵ 𝐅*[d]⦃#j⦄.
+/3 width=9 by frees_be, lift_inv_lref2_be/ qed.
+
+lemma frees_bind_sn: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃W⦄ →
+                     L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄.
+#a #I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
+/4 width=9 by frees_be, frees_eq, nlift_bind_sn/
+qed.
+
+lemma frees_bind_dx: ∀a,I,L,W,U,d,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯d]⦃U⦄ →
+                     L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄.
+#a #J #L #V #U #d #i #H elim (frees_inv … H) -H
+[ /4 width=9 by frees_eq, nlift_bind_dx/
+| * #I #K #W #j #Hdj #Hji #HnU #HLK #HW
+  elim (yle_inv_succ1 … Hdj) -Hdj <yminus_SO2 #Hyj #H
+  lapply (ylt_O … H) -H #Hj
+  >(plus_minus_m_m j 1) in HnU; // <minus_le_minus_minus_comm in HW;
+  /4 width=9 by frees_be, nlift_bind_dx, ldrop_inv_drop1_lt, lt_plus_to_minus/
+]
+qed.
+
+lemma frees_flat_sn: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃W⦄ →
+                     L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄.
+#I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
+/4 width=9 by frees_be, frees_eq, nlift_flat_sn/
+qed.
+
+lemma frees_flat_dx: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
+                     L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄.
+#I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
+/4 width=9 by frees_be, frees_eq, nlift_flat_dx/
+qed.
+
+lemma frees_weak: ∀L,U,d1,i. L ⊢ i ϵ 𝐅*[d1]⦃U⦄ →
+                  ∀d2. d2 ≤ d1 → L ⊢ i ϵ 𝐅*[d2]⦃U⦄.
+#L #U #d1 #i #H elim H -L -U -d1 -i
+/3 width=9 by frees_be, frees_eq, yle_trans/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma frees_inv_bind_O: ∀a,I,L,W,U,i. L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄ →
+                        L ⊢ i ϵ 𝐅*[0]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[0]⦃U⦄ .
+#a #I #L #W #U #i #H elim (frees_inv_bind … H) -H
+/3 width=3 by frees_weak, or_intror, or_introl/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/frees_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/frees_lift.ma
new file mode 100644
index 000000000..77b62d071
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/frees_lift.ma
@@ -0,0 +1,73 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_ldrop.ma".
+include "basic_2/substitution/frees.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma frees_dec: ∀L,U,d,i. Decidable (frees d L U i).
+#L #U @(f2_ind … rfw … L U) -L -U
+#n #IH #L * *
+[ -IH /3 width=5 by frees_inv_sort, or_intror/
+| #j #Hn #d #i elim (lt_or_eq_or_gt i j) #Hji
+  [ -n @or_intror #H elim (lt_refl_false i)
+    lapply (frees_inv_lref_ge … H ?) -L -d /2 width=1 by lt_to_le/
+  | -n /2 width=1 by or_introl/
+  | elim (ylt_split j d) #Hdi
+    [ -n @or_intror #H elim (lt_refl_false i)
+      lapply (frees_inv_lref_skip … H ?) -L //
+    | elim (lt_or_ge j (|L|)) #Hj
+      [ elim (ldrop_O1_lt (Ⓕ) L j) // -Hj #I #K #W #HLK destruct
+        elim (IH K W … 0 (i-j-1)) -IH [1,3: /3 width=5 by frees_lref_be, ldrop_fwd_rfw, or_introl/ ] #HnW
+        @or_intror #H elim (frees_inv_lref_lt … H) // #Z #Y #X #_ #HLY -d
+        lapply (ldrop_mono … HLY … HLK) -L #H destruct /2 width=1 by/  
+      | -n @or_intror #H elim (lt_refl_false i)
+        lapply (frees_inv_lref_free … H ?) -d //
+      ]
+    ]
+  ]
+| -IH /3 width=5 by frees_inv_gref, or_intror/
+| #a #I #W #U #Hn #d #i destruct
+  elim (IH L W … d i) [1,3: /3 width=1 by frees_bind_sn, or_introl/ ] #HnW
+  elim (IH (L.ⓑ{I}W) U … (⫯d) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
+  @or_intror #H elim (frees_inv_bind … H) -H /2 width=1 by/
+| #I #W #U #Hn #d #i destruct
+  elim (IH L W … d i) [1,3: /3 width=1 by frees_flat_sn, or_introl/ ] #HnW
+  elim (IH L U … d i) -IH [1,3: /3 width=1 by frees_flat_dx, or_introl/ ] #HnU
+  @or_intror #H elim (frees_inv_flat … H) -H /2 width=1 by/
+]
+qed-.
+
+lemma frees_S: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[yinj d]⦃U⦄ → ∀I,K,W. ⇩[d] L ≡ K.ⓑ{I}W →
+               (K ⊢ i-d-1 ϵ 𝐅*[0]⦃W⦄ → ⊥) → L ⊢ i ϵ 𝐅*[⫯d]⦃U⦄.
+#L #U #d #i #H elim (frees_inv … H) -H /3 width=2 by frees_eq/
+* #I #K #W #j #Hdj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
+lapply (yle_inv_inj … Hdj) -Hdj #Hdj
+elim (le_to_or_lt_eq … Hdj) -Hdj
+[ -I0 -K0 -W0 /3 width=9 by frees_be, yle_inj/
+| -Hji -HnU #H destruct
+  lapply (ldrop_mono … HLK0 … HLK) #H destruct -I
+  elim HnW0 -L -U -HnW0 //
+]
+qed.
+
+(* Note: lemma 1250 *)
+lemma frees_bind_dx_O: ∀a,I,L,W,U,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[0]⦃U⦄ →
+                       L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄.
+#a #I #L #W #U #i #HU elim (frees_dec L W 0 i)
+/4 width=5 by frees_S, frees_bind_dx, frees_bind_sn/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma
index aa9bc1b27..a05a51d26 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma
@@ -20,7 +20,7 @@ include "basic_2/substitution/lleq.ma".
 (* Alternative definition (not recursive) ***********************************)
 
 theorem lleq_intro_alt: ∀L1,L2,T,d. |L1| = |L2| →
-                        (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (L1 ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥) →
+                        (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → L1 ⊢ i ϵ 𝐅*[d]⦃T⦄ →
                            ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
                            I1 = I2 ∧ V1 = V2
                         ) → L1 ≡[T, d] L2.
@@ -31,7 +31,7 @@ qed.
 
 theorem lleq_inv_alt: ∀L1,L2,T,d. L1 ≡[T, d] L2 →
                       |L1| = |L2| ∧
-                      ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (L1 ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥) →
+                      ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → L1 ⊢ i ϵ 𝐅*[d]⦃T⦄ →
                       ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
                       I1 = I2 ∧ V1 = V2.
 #L1 #L2 #T #d #H elim (llpx_sn_llpx_sn_alt … H) -H
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_llor.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_llor.ma
index 80bc596f2..8cbff4cd9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_llor.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_llor.ma
@@ -18,19 +18,17 @@ include "basic_2/substitution/lleq_alt.ma".
 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
 
 (* Properties on poinwise union for local environments **********************)
-(*
-lemma llor_lleq_O: ∀R,L1,L2,T. llpx_sn R 0 T L1 L2 →
-                   ∀L. L1 ⩖[T] L2 ≡ L → L2 ≡[T, 0] L.
-#R #L1 #L2 #T #H1 #L #H2
+
+lemma llpx_sn_llor_dx: ∀R,L1,L2.
+                       (∀U,i. L2 ⊢ i ϵ 𝐅*[0]⦃U⦄ → L1 ⊢ i ϵ 𝐅*[0]⦃U⦄) →
+                       ∀T. llpx_sn R 0 T L1 L2 → ∀L. L1 ⩖[T] L2 ≡ L → L2 ≡[T, 0] L.
+#R #L1 #L2 #HR #T #H1 #L #H2
 elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1
 elim H2 -H2 #_ #HL1 #IH2
 @lleq_intro_alt // #I2 #I #K2 #K #V2 #V #i #Hi #HnT #HLK2 #HLK
 lapply (ldrop_fwd_length_lt2 … HLK) #HiL
 elim (ldrop_O1_lt (Ⓕ) L1 i) // -HiL #I1 #K1 #V1 #HLK1
-elim (IH1 … HLK1 HLK2) // -IH1
-[ #H #_ destruct
-  elim (IH2 … HLK1 HLK2 HLK) * /2 width=1 by conj/
-  #H elim HnT -HnT
-| #H @HnT -HnT
-]
-*)
+elim (IH1 … HLK1 HLK2) -IH1 /2 width=1 by/ #H #_ destruct
+elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK1 -HLK2 -HLK * /2 width=1 by conj/ #H
+elim H -H /2 width=1 by/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/llor.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/llor.ma
index 2c1b48ad9..94dfeeb25 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/llor.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/llor.ma
@@ -13,7 +13,7 @@
 (**************************************************************************)
 
 include "basic_2/notation/relations/lazyor_4.ma".
-include "basic_2/substitution/cofrees.ma".
+include "basic_2/substitution/frees.ma".
 
 (* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
 
@@ -21,8 +21,8 @@ definition llor: relation4 term lenv lenv lenv ≝ λT,L2,L1,L.
                  ∧∧ |L1| ≤ |L2| & |L1| = |L|
                   & (∀I1,I2,I,K1,K2,K,V1,V2,V,i.
                        ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → ⇩[i] L ≡ K.ⓑ{I}V →
-                       (∧∧ L1 ⊢ i ~ϵ 𝐅*[yinj 0]⦃T⦄ & I1 = I & V1 = V) ∨
-                       (∧∧ (L1 ⊢ i ~ϵ 𝐅*[yinj 0]⦃T⦄ → ⊥) & I1 = I & V2 = V)
+                       (∧∧ (L1 ⊢ i ϵ 𝐅*[yinj 0]⦃T⦄ → ⊥) & I1 = I & V1 = V) ∨
+                       (∧∧ L1 ⊢ i ϵ 𝐅*[yinj 0]⦃T⦄  & I1 = I & V2 = V)
                     ).
 
 interpretation
@@ -36,5 +36,3 @@ lemma llor_atom: ∀T,L2. ⋆ ⩖[T] L2 ≡ ⋆.
 #I1 #I2 #I #K1 #K2 #K #V1 #V2 #V #i #HLK1
 elim (ldrop_inv_atom1 … HLK1) -HLK1 #H destruct
 qed.
-
-axiom llor_total: ∀L1,L2,T. |L1| ≤ |L2| → ∃L. L1 ⩖[T] L2 ≡ L.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/llor_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/llor_ldrop.ma
new file mode 100644
index 000000000..07f88b0f4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/llor_ldrop.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/substitution/frees_lift.ma".
+include "basic_2/substitution/llor.ma".
+
+(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Advanced properties ******************************************************)
+
+axiom llor_total: ∀L1,L2,T. |L1| ≤ |L2| → ∃L. L1 ⩖[T] L2 ≡ L.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt.ma
index b2792cee2..23c48d45a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/llpx_sn_alt.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/substitution/cofrees_alt.ma".
+include "basic_2/substitution/frees.ma".
 include "basic_2/substitution/llpx_sn_alt_rec.ma".
 
 (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
@@ -20,7 +20,7 @@ include "basic_2/substitution/llpx_sn_alt_rec.ma".
 (* alternative definition of llpx_sn (not recursive) *)
 definition llpx_sn_alt: relation3 lenv term term → relation4 ynat term lenv lenv ≝
                         λR,d,T,L1,L2. |L1| = |L2| ∧
-                        (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (L1 ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥) →
+                        (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → L1 ⊢ i ϵ 𝐅*[d]⦃T⦄ →
                            ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
                            I1 = I2 ∧ R K1 V1 V2
                         ).
@@ -31,9 +31,9 @@ theorem llpx_sn_llpx_sn_alt: ∀R,T,L1,L2,d. llpx_sn R d T L1 L2 → llpx_sn_alt
 #R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
 #n #IHn #L1 #U #Hn #L2 #d #H elim (llpx_sn_inv_alt_r … H) -H
 #HL12 #IHU @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2 elim (frees_inv_ge … H) -H //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2 elim (frees_inv … H) -H
 [ -n #HnU elim (IHU … HnU HLK1 HLK2) -IHU -HnU -HLK1 -HLK2 /2 width=1 by conj/
-| * #J1 #K10 #W10 #j #Hdj #Hji #HLK10 #HnW10 #HnU destruct
+| * #J1 #K10 #W10 #j #Hdj #Hji #HnU #HLK10 #HnW10 destruct
   lapply (ldrop_fwd_drop2 … HLK10) #H
   lapply (ldrop_conf_ge … H … HLK1 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK10
   elim (ldrop_O1_lt (Ⓕ) L2 j) [2: <HL12 /2 width=5 by ldrop_fwd_length_lt2/ ] #J2 #K20 #W20 #HLK20
@@ -48,7 +48,7 @@ theorem llpx_sn_alt_inv_llpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt R d T L1 L2 → llp
 #R #U #L1 @(f2_ind … rfw … L1 U) -L1 -U
 #n #IHn #L1 #U #Hn #L2 #d * #HL12 #IHU @llpx_sn_intro_alt_r //
 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU #HLK1 #HLK2 destruct
-elim (IHU … HLK1 HLK2) /3 width=6 by nlift_frees/
+elim (IHU … HLK1 HLK2) /3 width=2 by frees_eq/
 #H #HV12 @and3_intro // @IHn -IHn /3 width=6 by ldrop_fwd_rfw/
 lapply (ldrop_fwd_drop2 … HLK1) #H1
 lapply (ldrop_fwd_drop2 … HLK2) -HLK2 #H2
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 b97ddd853..40a36882b 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
@@ -132,7 +132,7 @@ table {
           }
         ]
         [ { "context-sensitive extended reduction" * } {
-             [ "lpx ( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lpx_ldrop" + "lpx_lleq" + "lpx_aaa" * ]
+             [ "lpx ( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lpx_ldrop" + "lpx_frees" "lpx_lleq" + "lpx_aaa" * ]
              [ "cpx ( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpx_leq" + "cpx_lift" + "cpx_llpx_sn" + "cpx_lleq" + "cpx_cix" * ]
           }
         ]
@@ -219,11 +219,11 @@ table {
           }
         ]
         [ { "pointwise union for local environments" * } {
-             [ "llor ( ? ⩖[?] ? ≡ ? )" * ]
+             [ "llor ( ? ⩖[?] ? ≡ ? )" "llor_ldrop" * ]
           }
         ]
         [ { "context-sensitive exclusion from free variables" * } {
-             [ "cofrees ( ? ⊢ ? ~ϵ 𝐅*[?]⦃?⦄ )" "cofrees_alt" + "cofrees_lift" * ]
+             [ "frees ( ? ⊢ ? ϵ 𝐅*[?]⦃?⦄ )" "frees_lift" * ]
           }
         ]
         [ { "contxt-sensitive extended multiple substitution" * } {
-- 
2.39.2