From b1c5b3370653db6e495bbf6b3799cba592746cdd Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Mon, 22 Aug 2022 00:43:28 +0200
Subject: [PATCH] partial update in delayed_updating

+ syntax and substitution components updated
+ path_head parked and replaced by path_closed
+ balanced reductions introduced and main theorem proved
+ minor corrections to notation
---
 .../head/downarrowright_2.etc}                |   2 +-
 .../head/lift_path_head.etc}                  |   0
 .../head/lift_rmap_head.etc}                  |   0
 .../head/path_depth_labels.etc}               |   0
 .../path_head.ma => etc/head/path_head.etc}   |  32 +++-
 .../head/path_head_depth.etc}                 |   0
 .../head/path_head_structure.etc}             |   0
 .../head/path_height_labels.etc}              |   0
 .../head/path_labels.etc}                     |   0
 .../head/path_structure_labels.etc}           |   0
 .../power_2.ma => etc/head/power_2.etc}       |   0
 .../functions/black_downtriangle_2.ma         |   2 +-
 .../functions/black_righttriangle_2.ma        |   2 +-
 .../notation/functions/circled_times_1.ma     |   2 +-
 .../notation/functions/class_c_1.ma           |   2 +-
 .../notation/functions/flat_1.ma              |   2 +-
 .../notation/functions/tau_2.ma               |   2 +-
 .../notation/functions/uparrow_2.ma           |   2 +-
 .../notation/functions/uptriangle_1.ma        |   2 +-
 .../relations/black_rightarrow_dbf_3.ma       |  19 ++
 .../relations/black_rightarrow_df_3.ma        |   4 +-
 .../relations/black_rightarrow_ibf_3.ma       |  19 ++
 .../relations/black_rightarrow_if_3.ma        |   4 +-
 .../delayed_updating/reduction/dbfr.ma        |  38 ++++
 .../delayed_updating/reduction/dbfr_ibfr.ma   |  78 +++++++++
 .../delayed_updating/reduction/dbfr_lift.ma   |  51 ++++++
 .../delayed_updating/reduction/dfr.ma         |   1 -
 .../delayed_updating/reduction/dfr_lift.ma    |   2 -
 .../delayed_updating/reduction/ibfr.ma        |  39 +++++
 .../substitution/lift_path_closed.ma          |  33 ++++
 .../substitution/lift_rmap_closed.ma          |  36 ++++
 .../delayed_updating/syntax/path_closed.ma    | 162 ++++++++++++++++++
 .../syntax/path_closed_height.ma              |  27 +++
 .../syntax/path_closed_structure.ma           |  33 ++++
 .../unwind/unwind2_rmap_head.ma               |   2 +-
 35 files changed, 579 insertions(+), 19 deletions(-)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{notation/functions/downarrowright_2.ma => etc/head/downarrowright_2.etc} (95%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{substitution/lift_path_head.ma => etc/head/lift_path_head.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{substitution/lift_rmap_head.ma => etc/head/lift_rmap_head.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{syntax/path_depth_labels.ma => etc/head/path_depth_labels.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{syntax/path_head.ma => etc/head/path_head.etc} (78%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{syntax/path_head_depth.ma => etc/head/path_head_depth.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{syntax/path_head_structure.ma => etc/head/path_head_structure.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{syntax/path_height_labels.ma => etc/head/path_height_labels.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{syntax/path_labels.ma => etc/head/path_labels.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{syntax/path_structure_labels.ma => etc/head/path_structure_labels.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{notation/functions/power_2.ma => etc/head/power_2.etc} (100%)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_dbf_3.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_ibf_3.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_ibfr.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_lift.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_closed.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_closed.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_height.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_structure.ma

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/downarrowright_2.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/downarrowright_2.etc
similarity index 95%
rename from matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/downarrowright_2.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/head/downarrowright_2.etc
index 7531c6bcc..f086d72e2 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/downarrowright_2.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/downarrowright_2.etc
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
-notation "hvbox( ↳[ term 46 n ] break term 70 p )"
+notation "hvbox( ↳[ break term 46 n ] break term 70 p )"
   non associative with precedence 70
   for @{ 'DownArrowRight $n $p }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_head.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/lift_path_head.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_head.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/head/lift_path_head.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_head.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/lift_rmap_head.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_head.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/head/lift_rmap_head.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth_labels.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_depth_labels.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth_labels.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_depth_labels.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head.etc
similarity index 78%
rename from matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head.etc
index 4610e4f69..08054d900 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head.etc
@@ -96,7 +96,35 @@ qed-.
 
 (* Constructions with path_append *******************************************)
 
-lemma path_head_refl_append (p) (q) (n):
+lemma path_head_refl_append_bi (p) (q) (m) (n):
+      p = ↳[m]p → q = ↳[n]q → p●q = ↳[n+m](p●q).
+#p #q elim q -q
+[ #m #n #Hp #H0
+  <(eq_inv_path_empty_head … H0) -n //
+| #l #q #IH #m #n #Hp
+  @(nat_ind_succ … n) -n // #n #_
+  cases l [ #k ]
+  <list_append_lcons_sn <nplus_succ_sn
+  [ <path_head_d_dx <path_head_d_dx #Hq
+    elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+    >(IH … Hp Hq) in ⊢ (??%?); -IH -Hp -Hq
+    <nplus_plus_comm_23 //
+  | <path_head_m_dx <path_head_m_dx #Hq
+    elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+    >(IH … Hp Hq) in ⊢ (??%?); -IH -Hp -Hq //
+  | <path_head_L_dx <path_head_L_dx #Hq
+    elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+    >(IH … Hp Hq) in ⊢ (??%?); -IH -Hp -Hq //
+  | <path_head_A_dx <path_head_A_dx #Hq
+    elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+    >(IH … Hp Hq) in ⊢ (??%?); -IH -Hp -Hq //
+  | <path_head_S_dx <path_head_S_dx #Hq
+    elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+    >(IH … Hp Hq) in ⊢ (??%?); -IH -Hp -Hq //
+  ]
+qed.
+
+lemma path_head_refl_append_sn (p) (q) (n):
       q = ↳[n]q → q = ↳[n](p●q).
 #p #q elim q -q
 [ #n #Hn <(eq_inv_path_empty_head … Hn) -Hn //
@@ -116,7 +144,7 @@ qed-.
 
 (* Inversions with path_append **********************************************)
 
-lemma eq_inv_path_head_refl_append (p) (q) (n):
+lemma eq_inv_path_head_refl_append_sn (p) (q) (n):
       q = ↳[n](p●q) → q = ↳[n]q.
 #p #q elim q -q
 [ #n #Hn <(eq_inv_path_empty_head … Hn) -p //
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_depth.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head_depth.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_depth.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head_depth.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head_structure.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_structure.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head_structure.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_height_labels.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_height_labels.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_height_labels.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_height_labels.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_labels.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_labels.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_labels.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_labels.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure_labels.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_structure_labels.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure_labels.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_structure_labels.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/power_2.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/power_2.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/power_2.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/head/power_2.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_downtriangle_2.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_downtriangle_2.ma
index ddb2bc56d..8299e9384 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_downtriangle_2.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_downtriangle_2.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
-notation "hvbox( ▼[ term 46 f ] break term 70 p )"
+notation "hvbox( ▼[ break term 46 f ] break term 70 p )"
   non associative with precedence 70
   for @{ 'BlackDownTriangle $f $p }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_righttriangle_2.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_righttriangle_2.ma
index 90b075806..020fc5538 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_righttriangle_2.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_righttriangle_2.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
-notation "hvbox( ▶[ term 46 f ] break term 70 p )"
+notation "hvbox( ▶[ break term 46 f ] break term 70 p )"
   non associative with precedence 70
   for @{ 'BlackRightTriangle $f $p }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/circled_times_1.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/circled_times_1.ma
index 8c39289b5..a489dc8da 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/circled_times_1.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/circled_times_1.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
-notation "hvbox( ⊗ term 70 p )"
+notation "hvbox( ⊗ break term 70 p )"
   non associative with precedence 70
   for @{ 'CircledTimes $p }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_c_1.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_c_1.ma
index 41a736d62..352144291 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_c_1.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_c_1.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
-notation "hvbox( 𝐂❨ term 46 n ❩ )"
+notation "hvbox( 𝐂❨ break term 46 n ❩ )"
   non associative with precedence 70
   for @{ 'ClassC $n }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/flat_1.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/flat_1.ma
index 5d873910b..077255a35 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/flat_1.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/flat_1.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
-notation "hvbox( ♭ term 70 p )"
+notation "hvbox( ♭ break term 70 p )"
   non associative with precedence 70
   for @{ 'Flat $p }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/tau_2.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/tau_2.ma
index 099360cce..6d4dd8839 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/tau_2.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/tau_2.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
-notation "hvbox( 𝛕 break term 76 n. break term 70 t )"
+notation "hvbox( 𝛕 break term 76 n . break term 70 t )"
   non associative with precedence 70
   for @{ 'Tau $n $t }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uparrow_2.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uparrow_2.ma
index 2cbac4b24..b3123df7b 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uparrow_2.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uparrow_2.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
-notation "hvbox( ↑[ term 46 t1 ] break term 70 t2  )"
+notation "hvbox( ↑[ break term 46 t1 ] break term 70 t2  )"
   non associative with precedence 70
   for @{ 'UpArrow $t1 $t2 }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uptriangle_1.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uptriangle_1.ma
index df2bee52c..9a3516c7b 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uptriangle_1.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uptriangle_1.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
-notation "hvbox( ▵ term 70 t )"
+notation "hvbox( ▵ break term 70 t )"
   non associative with precedence 70
   for @{ 'UpTriangle $t }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_dbf_3.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_dbf_3.ma
new file mode 100644
index 000000000..390aabf5e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_dbf_3.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 DELAYED UPDATING ********************************************)
+
+notation "hvbox( t1 ➡𝐝𝐛𝐟[ break term 46 r ] break term 46 t2 )"
+  non associative with precedence 45
+  for @{ 'BlackRightArrowDBF $t1 $r $t2 }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_df_3.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_df_3.ma
index f50962063..1750f335b 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_df_3.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_df_3.ma
@@ -15,5 +15,5 @@
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
 notation "hvbox( t1 ➡𝐝𝐟[ break term 46 r ] break term 46 t2 )"
-   non associative with precedence 45
-   for @{ 'BlackRightArrowDF $t1 $r $t2 }.
+  non associative with precedence 45
+  for @{ 'BlackRightArrowDF $t1 $r $t2 }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_ibf_3.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_ibf_3.ma
new file mode 100644
index 000000000..12a1b0ff2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_ibf_3.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 DELAYED UPDATING ********************************************)
+
+notation "hvbox( t1 ➡𝐢𝐛𝐟[ break term 46 r ] break term 46 t2 )"
+  non associative with precedence 45
+  for @{ 'BlackRightArrowIBF $t1 $r $t2 }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_if_3.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_if_3.ma
index d2568773a..90fe54b75 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_if_3.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_if_3.ma
@@ -15,5 +15,5 @@
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
 notation "hvbox( t1 ➡𝐢𝐟[ break term 46 r ] break term 46 t2 )"
-   non associative with precedence 45
-   for @{ 'BlackRightArrowIF $t1 $r $t2 }.
+  non associative with precedence 45
+  for @{ 'BlackRightArrowIF $t1 $r $t2 }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr.ma
new file mode 100644
index 000000000..349195cb0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr.ma
@@ -0,0 +1,38 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "delayed_updating/substitution/fsubst.ma".
+include "delayed_updating/syntax/prototerm_constructors.ma".
+include "delayed_updating/syntax/prototerm_eq.ma".
+include "delayed_updating/syntax/path_head.ma".
+include "delayed_updating/syntax/path_balanced.ma".
+include "delayed_updating/syntax/path_structure.ma".
+include "delayed_updating/notation/relations/black_rightarrow_dbf_3.ma".
+include "ground/arith/nat_rplus.ma".
+include "ground/xoa/ex_6_5.ma".
+
+(* DELAYED BALANCED FOCUSED REDUCTION ***************************************)
+
+(**) (* explicit ninj because we cannot declare the expectd type of k *)
+definition dbfr (r): relation2 prototerm prototerm ≝
+           λt1,t2.
+           ∃∃p,b,q,h,k. p●𝗔◗b●𝗟◗q = r &
+           ⊗b ϵ 𝐁 & b = ↳[h]b &
+           (𝗟◗q) = ↳[ninj k](𝗟◗q) & r◖𝗱k ϵ t1 &
+           t1[⋔r←𝛕(k+h).(t1⋔(p◖𝗦))] ⇔ t2
+.
+
+interpretation
+  "balanced focused reduction with delayed updating (prototerm)"
+  'BlackRightArrowDBF t1 r t2 = (dbfr r t1 t2).
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_ibfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_ibfr.ma
new file mode 100644
index 000000000..043f61085
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_ibfr.ma
@@ -0,0 +1,78 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "delayed_updating/reduction/dbfr.ma".
+include "delayed_updating/reduction/ibfr.ma".
+
+include "delayed_updating/unwind/unwind2_constructors.ma".
+include "delayed_updating/unwind/unwind2_preterm_fsubst.ma".
+include "delayed_updating/unwind/unwind2_preterm_eq.ma".
+include "delayed_updating/unwind/unwind2_prototerm_lift.ma".
+include "delayed_updating/unwind/unwind2_rmap_head.ma".
+
+include "delayed_updating/substitution/fsubst_eq.ma".
+include "delayed_updating/substitution/lift_prototerm_eq.ma".
+
+include "delayed_updating/syntax/prototerm_proper_constructors.ma".
+include "delayed_updating/syntax/path_head_structure.ma".
+include "delayed_updating/syntax/path_structure_depth.ma".
+
+(* DELAYED BALANCED FOCUSED REDUCTION ***************************************)
+
+(* Main destructions with ibfr **********************************************)
+
+theorem dbfr_des_ibfr (f) (t1) (t2) (r): t1 ϵ 𝐓 →
+        t1 ➡𝐝𝐛𝐟[r] t2 → ▼[f]t1 ➡𝐢𝐛𝐟[⊗r] ▼[f]t2.
+#f #t1 #t2 #r #H0t1
+* #p #b #q #h #k #Hr #Hb #Hh #H1k #Ht1 #Ht2 destruct
+@(ex6_5_intro … (⊗p) (⊗b) (⊗q) (♭b) (↑♭q))
+[ -H0t1 -Hb -Hh -H1k -Ht1 -Ht2 //
+| -H0t1 -Hh -H1k -Ht1 -Ht2 //
+| -H0t1 -Hb -Ht1 -H1k -Ht2
+  >Hh in ⊢ (??%?); >path_head_structure_depth <Hh -Hh //
+| -H0t1 -Hb -Hh -Ht1 -Ht2
+  >structure_L_sn
+  >H1k in ⊢ (??%?); >path_head_structure_depth <H1k -H1k //
+| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H0t1 -Hb -Hh
+  <unwind2_path_d_dx >list_append_rcons_dx >list_append_assoc
+  lapply (unwind2_rmap_append_pap_closed f … (p●𝗔◗b) … H1k) -H1k
+  <depth_L_sn #H2k
+  lapply (eq_inv_ninj_bi … H2k) -H2k #H2k <H2k -H2k #Ht1 //
+| lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
+  lapply (path_head_refl_append_bi … Hh H1k) -Hh -H1k <nrplus_inj_sn #H1k
+  @(subset_eq_trans … Ht2) -t2
+  @(subset_eq_trans … (unwind2_term_fsubst_ppc …))
+  [ @fsubst_eq_repl [ // | // ]
+    @(subset_eq_trans … (unwind2_term_iref …))
+    @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
+    [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+    @(subset_eq_trans … (lift_unwind2_term_after …))
+    @unwind2_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+    <list_append_rcons_sn
+    @(stream_eq_trans … (tr_compose_uni_dx …))
+    @tr_compose_eq_repl
+    [ <unwind2_rmap_append_pap_closed //
+      <depth_append <depth_L_sn
+      <nplus_comm <nrplus_npsucc_sn <nplus_succ_sn //
+    | >unwind2_rmap_A_dx
+      /2 width=1 by tls_unwind2_rmap_closed/
+    ]
+(* Note: crux of the proof ends *)
+  | //
+  | /2 width=2 by ex_intro/
+  | //
+  ]
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_lift.ma
new file mode 100644
index 000000000..f23728916
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_lift.ma
@@ -0,0 +1,51 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "delayed_updating/reduction/dbfr.ma".
+
+include "delayed_updating/substitution/fsubst_lift.ma".
+include "delayed_updating/substitution/fsubst_eq.ma".
+include "delayed_updating/substitution/lift_constructors.ma".
+include "delayed_updating/substitution/lift_path_head.ma".
+include "delayed_updating/substitution/lift_rmap_head.ma".
+
+(* DELAYED BALANCED FOCUSED REDUCTION ***************************************)
+
+(* Constructions with lift **************************************************)
+
+theorem dfr_lift_bi (f) (t1) (t2) (r):
+        t1 ➡𝐝𝐛𝐟[r] t2 → ↑[f]t1 ➡𝐝𝐛𝐟[↑[f]r] ↑[f]t2.
+#f #t1 #t2 #r
+* #p #q #k #Hr #H1k #Ht1 #Ht2 destruct
+@(ex4_3_intro … (↑[f]p) (↑[↑[p◖𝗔◖𝗟]f]q) ((↑[p●𝗔◗𝗟◗q]f)＠⧣❨k❩))
+[ -H1k -Ht1 -Ht2 //
+| -Ht1 -Ht2
+  <lift_rmap_L_dx >lift_path_L_sn
+  <(lift_path_head_closed … H1k) in ⊢ (??%?); -H1k //
+| lapply (in_comp_lift_path_term f … Ht1) -Ht2 -Ht1 -H1k
+  <lift_path_d_dx #Ht1 //
+| lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2 -Ht1
+  @(subset_eq_trans … Ht2) -t2
+  @(subset_eq_trans … (lift_term_fsubst …))
+  @fsubst_eq_repl [ // | // ]
+  @(subset_eq_trans … (lift_term_iref …))
+  @iref_eq_repl
+  @(subset_eq_canc_sn … (lift_term_grafted_S …))
+  @lift_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+  >list_append_rcons_sn in H1k; #H1k >lift_rmap_A_dx
+  /2 width=1 by tls_lift_rmap_closed/
+(* Note: crux of the proof ends *)
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
index a610cd323..d0dcf4eee 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
@@ -16,7 +16,6 @@ include "delayed_updating/substitution/fsubst.ma".
 include "delayed_updating/syntax/prototerm_constructors.ma".
 include "delayed_updating/syntax/prototerm_eq.ma".
 include "delayed_updating/syntax/path_head.ma".
-include "delayed_updating/syntax/path_depth.ma".
 include "delayed_updating/notation/relations/black_rightarrow_df_3.ma".
 include "ground/xoa/ex_4_3.ma".
 
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
index 71a599326..b60bd708c 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
@@ -24,8 +24,6 @@ include "delayed_updating/substitution/lift_rmap_head.ma".
 
 (* Constructions with lift **************************************************)
 
-(* ↑[↑[p◖𝗔◖𝗟]f]q *)
-
 theorem dfr_lift_bi (f) (t1) (t2) (r):
         t1 ➡𝐝𝐟[r] t2 → ↑[f]t1 ➡𝐝𝐟[↑[f]r] ↑[f]t2.
 #f #t1 #t2 #r
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr.ma
new file mode 100644
index 000000000..b235cb481
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr.ma
@@ -0,0 +1,39 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "delayed_updating/substitution/fsubst.ma".
+include "delayed_updating/substitution/lift_prototerm.ma".
+include "delayed_updating/syntax/prototerm_eq.ma".
+include "delayed_updating/syntax/path_head.ma".
+include "delayed_updating/syntax/path_balanced.ma".
+include "delayed_updating/syntax/path_structure.ma".
+include "delayed_updating/notation/relations/black_rightarrow_ibf_3.ma".
+include "ground/relocation/tr_uni.ma".
+include "ground/arith/nat_rplus.ma".
+include "ground/xoa/ex_6_5.ma".
+
+(* IMMEDIATE BALANCED FOCUSED REDUCTION *************************************)
+
+(**) (* explicit ninj because we cannot declare the expectd type of k *)
+definition ibfr (r): relation2 prototerm prototerm ≝
+           λt1,t2.
+           ∃∃p,b,q,h,k. p●𝗔◗b●𝗟◗q = r &
+           ⊗b ϵ 𝐁 & b = ↳[h]b &
+           (𝗟◗q) = ↳[ninj k](𝗟◗q) & r◖𝗱k ϵ t1 &
+           t1[⋔r←↑[𝐮❨k+h❩](t1⋔(p◖𝗦))] ⇔ t2
+.
+
+interpretation
+  "balanced focused reduction with immediate updating (prototerm)"
+  'BlackRightArrowIBF t1 r t2 = (ibfr r t1 t2).
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_closed.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_closed.ma
new file mode 100644
index 000000000..b332b841f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_closed.ma
@@ -0,0 +1,33 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "delayed_updating/substitution/lift_path.ma".
+include "delayed_updating/syntax/path_closed.ma".
+include "ground/relocation/xap.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+(* Constructions with pcc ***************************************************)
+
+lemma lift_path_closed (f) (q) (n):
+      q ϵ 𝐂❨n❩ → ↑[f]q ϵ 𝐂❨↑[q]f＠❨n❩❩.
+#f #q #n #Hq elim Hq -Hq //
+#q #n [ #k ] #_ #IH
+/2 width=1 by pcc_d_dx, pcc_m_dx, pcc_L_dx, pcc_A_dx, pcc_S_dx/
+qed.
+
+lemma lift_path_rmap_closed (f) (p) (q) (n):
+      q ϵ 𝐂❨n❩ → ↑[↑[p]f]q ϵ 𝐂❨↑[p●q]f＠❨n❩❩.
+/2 width=1 by lift_path_closed/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_closed.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_closed.ma
new file mode 100644
index 000000000..58bef8153
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_closed.ma
@@ -0,0 +1,36 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "delayed_updating/substitution/lift_rmap_eq.ma".
+include "delayed_updating/syntax/path_closed.ma".
+include "ground/lib/stream_eq_eq.ma".
+
+(* LIFT MAP FOR PATH ********************************************************)
+
+(* Destructions with cpp ****************************************************)
+
+lemma tls_plus_lift_rmap_closed (f) (q) (n):
+      q ϵ 𝐂❨n❩ →
+      ∀m. ⇂*[m]f ≗ ⇂*[m+n]↑[q]f.
+#f #q #n #Hq elim Hq -q -n //
+#q #n #_ #IH #m
+<nplus_succ_dx <stream_tls_swap //
+qed-.
+
+lemma tls_lift_rmap_closed (f) (q) (n):
+      q ϵ 𝐂❨n❩ →
+      f ≗ ⇂*[n]↑[q]f.
+#f #q #n #H0
+/2 width=1 by tls_plus_lift_rmap_closed/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma
new file mode 100644
index 000000000..fd7a9954f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma
@@ -0,0 +1,162 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "delayed_updating/syntax/path.ma".
+include "delayed_updating/notation/functions/class_c_1.ma".
+include "ground/arith/nat_plus.ma".
+include "ground/arith/nat_pred_succ.ma".
+include "ground/lib/subset.ma".
+include "ground/generated/insert_eq_1.ma".
+
+(* CLOSED CONDITION FOR PATH ************************************************)
+
+inductive pcc: relation2 nat path ≝
+| pcc_empty:
+  pcc (𝟎) (𝐞)
+| pcc_d_dx (p) (n) (k):
+  pcc (n+ninj k) p → pcc n (p◖𝗱k)
+| pcc_m_dx (p) (n):
+  pcc n p → pcc n (p◖𝗺)
+| pcc_L_dx (p) (n):
+  pcc n p → pcc (↑n) (p◖𝗟)
+| pcc_A_dx (p) (n):
+  pcc n p → pcc n (p◖𝗔)
+| pcc_S_dx (p) (n):
+  pcc n p → pcc n (p◖𝗦)
+.
+
+interpretation
+  "closed condition (path)"
+  'ClassC n = (pcc n).
+
+(* Basic inversions ********************************************************)
+
+lemma pcc_inv_empty (n):
+      (𝐞) ϵ 𝐂❨n❩ → 𝟎 = n.
+#n @(insert_eq_1 … (𝐞))
+#x * -n //
+#p #n [ #k ] #_ #H0 destruct
+qed-.
+
+lemma pcc_inv_d_dx (p) (n) (k):
+      p◖𝗱k ϵ 𝐂❨n❩ → p ϵ 𝐂❨n+k❩.
+#p #n #h @(insert_eq_1 … (p◖𝗱h))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+qed-.
+
+lemma pcc_inv_m_dx (p) (n):
+      p◖𝗺 ϵ 𝐂❨n❩ → p ϵ 𝐂❨n❩.
+#p #n @(insert_eq_1 … (p◖𝗺))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+qed-.
+
+lemma pcc_inv_L_dx (p) (n):
+      p◖𝗟 ϵ 𝐂❨n❩ →
+      ∧∧ p ϵ 𝐂❨↓n❩ & ↑↓n = n.
+#p #n @(insert_eq_1 … (p◖𝗟))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct
+<npred_succ /2 width=1 by conj/
+qed-.
+
+lemma pcc_inv_A_dx (p) (n):
+      p◖𝗔 ϵ 𝐂❨n❩ → p ϵ 𝐂❨n❩.
+#p #n @(insert_eq_1 … (p◖𝗔))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+qed-.
+
+lemma pcc_inv_S_dx (p) (n):
+      p◖𝗦 ϵ 𝐂❨n❩ → p ϵ 𝐂❨n❩.
+#p #n @(insert_eq_1 … (p◖𝗦))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+qed-.
+
+(* Advanced inversions ******************************************************)
+
+lemma pcc_inv_empty_succ (n):
+      (𝐞) ϵ 𝐂❨↑n❩ → ⊥.
+#n #H0
+lapply (pcc_inv_empty … H0) -H0 #H0
+/2 width=7 by eq_inv_zero_nsucc/
+qed-.
+
+lemma pcc_inv_L_dx_zero (p):
+      p◖𝗟 ϵ 𝐂❨𝟎❩ → ⊥.
+#p #H0
+elim (pcc_inv_L_dx … H0) -H0 #_ #H0
+/2 width=7 by eq_inv_nsucc_zero/
+qed-.
+
+lemma pcc_inv_L_dx_succ (p) (n):
+      p◖𝗟 ϵ 𝐂❨↑n❩ → p ϵ 𝐂❨n❩.
+#p #n #H0
+elim (pcc_inv_L_dx … H0) -H0 //
+qed-.
+
+(* Main constructions with path_append **************************************)
+
+theorem pcc_append_bi (p) (q) (m) (n):
+        p ϵ 𝐂❨m❩ → q ϵ 𝐂❨n❩ → p●q ϵ 𝐂❨m+n❩.
+#p #q #m #n #Hm #Hm elim Hm -Hm // -Hm
+#p #n [ #k ] #_ #IH [3: <nplus_succ_dx ]
+/2 width=1 by pcc_d_dx, pcc_m_dx, pcc_L_dx, pcc_A_dx, pcc_S_dx/
+qed.
+
+(* Main inversions **********************************************************)
+
+theorem ppc_mono (q) (n1):
+        q ϵ 𝐂❨n1❩ → ∀n2. q ϵ 𝐂❨n2❩ → n1 = n2.
+#q1 #n1 #Hn1 elim Hn1 -q1 -n1
+[|*: #q1 #n1 [ #k1 ] #_ #IH ] #n2 #Hn2
+[ <(pcc_inv_empty … Hn2) -n2 //
+| lapply (pcc_inv_d_dx … Hn2) -Hn2 #Hn2
+  lapply (IH … Hn2) -q1 #H0
+  /2 width=2 by eq_inv_nplus_bi_dx/
+| lapply (pcc_inv_m_dx … Hn2) -Hn2 #Hn2
+  <(IH … Hn2) -q1 -n2 //
+| elim (pcc_inv_L_dx … Hn2) -Hn2 #Hn2 #H0
+  >(IH … Hn2) -q1 //
+| lapply (pcc_inv_A_dx … Hn2) -Hn2 #Hn2
+  <(IH … Hn2) -q1 -n2 //
+| lapply (pcc_inv_S_dx … Hn2) -Hn2 #Hn2
+  <(IH … Hn2) -q1 -n2 //
+]
+qed-.
+
+theorem pcc_inj_L_sn (p1) (p2) (q1) (n):
+        q1 ϵ 𝐂❨n❩ → ∀q2. q2 ϵ 𝐂❨n❩ →
+        p1●𝗟◗q1 = p2●𝗟◗q2 → q1 = q2.
+#p1 #p2 #q1 #n #Hq1 elim Hq1 -q1 -n
+[|*: #q1 #n1 [ #k1 ] #_ #IH ] * //
+[1,3,5,7,9,11: #l2 #q2 ] #Hq2
+<list_append_lcons_sn <list_append_lcons_sn #H0
+elim (eq_inv_list_lcons_bi ????? H0) -H0 #H0 #H1 destruct
+[ elim (pcc_inv_L_dx_zero … Hq2)
+| lapply (pcc_inv_d_dx … Hq2) -Hq2 #Hq2
+  <(IH … Hq2) //
+| lapply (pcc_inv_m_dx … Hq2) -Hq2 #Hq2
+  <(IH … Hq2) //
+| lapply (pcc_inv_L_dx_succ … Hq2) -Hq2 #Hq2
+  <(IH … Hq2) //
+| lapply (pcc_inv_A_dx … Hq2) -Hq2 #Hq2
+  <(IH … Hq2) //
+| lapply (pcc_inv_S_dx … Hq2) -Hq2 #Hq2
+  <(IH … Hq2) //
+| elim (pcc_inv_empty_succ … Hq2)
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_height.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_height.ma
new file mode 100644
index 000000000..88b0e413f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_height.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "delayed_updating/syntax/path_closed.ma".
+include "delayed_updating/syntax/path_height.ma".
+include "delayed_updating/syntax/path_depth.ma".
+
+(* CLOSED CONDITION FOR PATH ************************************************)
+
+(* Constructions with height and depth **************************************)
+
+lemma path_closed_depth (p) (n):
+      p ϵ 𝐂❨n❩ → ♯p + n = ♭p.
+#p #n #Hn elim Hn -Hn //
+#p #n #_ #IH <nplus_succ_dx //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_structure.ma
new file mode 100644
index 000000000..90522f905
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_structure.ma
@@ -0,0 +1,33 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "delayed_updating/syntax/path_closed_height.ma".
+include "delayed_updating/syntax/path_structure.ma".
+
+(* CLOSED CONDITION FOR PATH ************************************************)
+
+(* Constructions with structure *********************************************)
+
+lemma path_closed_structure_height (p) (n):
+      p ϵ 𝐂❨n❩ → ⊗p ϵ 𝐂❨♯p+n❩.
+#p #n #Hn elim Hn -Hn //
+#p #n #_ #IH [ <nplus_succ_dx ]
+/2 width=1 by pcc_L_dx, pcc_A_dx, pcc_S_dx/
+qed.
+
+lemma path_closed_structure_depth (p) (n):
+      p ϵ 𝐂❨n❩ → ⊗p ϵ 𝐂❨♭p❩.
+#p #n #Hp <path_closed_depth
+/2 width=3 by path_closed_structure_height/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_head.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_head.ma
index 7b568dc86..416c6e792 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_head.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_head.ma
@@ -93,7 +93,7 @@ lemma unwind2_rmap_append_pap_closed (f) (p) (q) (h:pnat):
       q = ↳[h]q →
       ♭q = ninj (▶[f](p●q)＠⧣❨h❩).
 #f #p #q #h #Hh
->tr_xap_ninj >(path_head_refl_append p … Hh) in ⊢ (??%?);
+>tr_xap_ninj >(path_head_refl_append_sn p … Hh) in ⊢ (??%?);
 >(unwind2_rmap_head_xap_closed … Hh) -Hh
 <path_head_depth //
 qed.
-- 
2.39.2