From e6482f7f499fbb88aff8a205d2a2c808158042c5 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <fguidi@maelstrom.helm.cs.unibo.it>
Date: Wed, 27 Apr 2022 19:24:30 +0200
Subject: [PATCH] update un delayed updating

+ some parked files
---
 .../etc/relocation/sbr_pap.etc                |  48 ++++
 .../etc/relocation/sbr_pap_eq.etc             |  30 ++
 .../relocation/sbr_pap_id.etc}                |  20 +-
 .../etc/relocation/sbr_pap_push.etc           |  31 ++
 .../etc/relocation/sbr_push.etc               |  30 ++
 .../relocation/sbr_push_eq.etc}               |  23 +-
 .../etc/relocation/sbr_push_uni.etc           |  24 ++
 .../etc/relocation/upspoonverticalbar_2.etc   |  19 ++
 .../etc/relocation/verticalbarat_2.etc        |  19 ++
 .../unwind.ma => etc/unwind2/unwind.etc}      |   0
 .../unwind2/unwind_constructors.etc}          |   0
 .../unwind2/unwind_depth.etc}                 |   0
 .../unwind2/unwind_eq.etc}                    |   0
 .../etc/unwind2/unwind_etc.etc                |  10 +
 .../unwind2/unwind_fsubst.etc}                |   0
 .../unwind2/unwind_height.etc}                |   0
 .../unwind2/unwind_preterm_eq.etc}            |   0
 .../unwind2/unwind_prototerm.etc}             |   0
 .../unwind2/unwind_prototerm_eq.etc}          |   0
 .../unwind2/unwind_structure.etc}             |   0
 .../unwind2/unwind_structure_depth.etc}       |   0
 .../delayed_updating/etc/unwind3/unwind.etc   | 189 ++++++++++++
 .../etc/unwind3/unwind_constructors.etc       |  44 +++
 .../etc/unwind3/unwind_depth.etc              |  31 ++
 .../etc/unwind3/unwind_eq.etc                 | 124 ++++++++
 .../etc/unwind3/unwind_fsubst.etc             |  61 ++++
 .../etc/unwind3/unwind_length.etc             |  35 +++
 .../etc/unwind3/unwind_lift.etc               |  32 +++
 .../etc/unwind3/unwind_preterm_eq.etc         |  68 +++++
 .../etc/unwind3/unwind_prototerm.etc          |  30 ++
 .../etc/unwind3/unwind_prototerm_eq.etc       |  41 +++
 .../etc/unwind3/unwind_structure.etc          | 269 ++++++++++++++++++
 .../etc/unwind3/unwind_structure_depth.etc    |  31 ++
 33 files changed, 1181 insertions(+), 28 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap_eq.etc
 rename matita/matita/contribs/lambdadelta/delayed_updating/{substitution/lift_id.ma => etc/relocation/sbr_pap_id.etc} (75%)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap_push.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_push.etc
 rename matita/matita/contribs/lambdadelta/delayed_updating/{substitution/lift_prototerm_id.ma => etc/relocation/sbr_push_eq.etc} (70%)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_push_uni.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/upspoonverticalbar_2.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/verticalbarat_2.etc
 rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind2/unwind.ma => etc/unwind2/unwind.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind2/unwind_constructors.ma => etc/unwind2/unwind_constructors.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind2/unwind_depth.ma => etc/unwind2/unwind_depth.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind2/unwind_eq.ma => etc/unwind2/unwind_eq.etc} (100%)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_etc.etc
 rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind2/unwind_fsubst.ma => etc/unwind2/unwind_fsubst.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind2/unwind_height.ma => etc/unwind2/unwind_height.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind2/unwind_preterm_eq.ma => etc/unwind2/unwind_preterm_eq.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind2/unwind_prototerm.ma => etc/unwind2/unwind_prototerm.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind2/unwind_prototerm_eq.ma => etc/unwind2/unwind_prototerm_eq.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind2/unwind_structure.ma => etc/unwind2/unwind_structure.etc} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind2/unwind_structure_depth.ma => etc/unwind2/unwind_structure_depth.etc} (100%)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_constructors.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_depth.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_eq.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_fsubst.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_length.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_lift.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_preterm_eq.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_prototerm.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_prototerm_eq.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_structure.etc
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_structure_depth.etc

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap.etc
new file mode 100644
index 000000000..c35244d88
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap.etc
@@ -0,0 +1,48 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/arith/pnat_plus.ma".
+include "ground/relocation/tr_map.ma".
+include "delayed_updating/notation/functions/verticalbarat_2.ma".
+
+(* POSITIVE APPLICATION FOR STACKS OF BASIC RELOCATION MAPS *****************)
+
+rec definition sbr_pap (i: pnat) on i: tr_map → pnat.
+* #p #f cases i -i
+[ @(p)
+| #i cases p -p
+  [ lapply (sbr_pap i f) -sbr_pap -i -f
+    #i @(↑i)
+  | #p @(i+↑p)
+  ]
+]
+defined.
+
+interpretation
+  "functional positive application (stack of relocation maps)"
+  'VerticalBarAt f i = (sbr_pap i f).
+
+(* Basic constructions ******************************************************)
+
+lemma sbr_pap_unfold_unit (f):
+      ∀p. p = (p⨮f)❘@❨𝟏❩.
+// qed.
+
+lemma sbr_pap_unfold_unit_succ (f):
+      ∀i. ↑(f❘@❨i❩) = (𝟏⨮f)❘@❨↑i❩.
+// qed.
+
+lemma sbr_pap_unfold_succ_bi (f):
+      ∀p,i. i+↑p = (↑p⨮f)❘@❨↑i❩.
+// qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap_eq.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap_eq.etc
new file mode 100644
index 000000000..10b0d939e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap_eq.etc
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/relocation/sbr_pap.ma".
+include "ground/relocation/tr_eq.ma".
+
+(* POSITIVE APPLICATION FOR STACKS OF BASIC RELOCATION MAPS *****************)
+
+(* Main constructions with stream_eq ****************************************)
+
+theorem sbr_pap_eq_repl_sn (i):
+        stream_eq_repl … (λf1,f2. f1❘@❨i❩ = f2❘@❨i❩).
+#i elim i -i [| #i #IH ]
+* [ #p1 | * [| #p1 ]] #f1
+* #p2 #f2 #H
+elim (stream_eq_inv_cons_bi … H) -H [1,8,15: |*: // ] #Hp #Hf destruct //
+<sbr_pap_unfold_unit_succ <sbr_pap_unfold_unit_succ
+/3 width=1 by eq_f/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_id.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap_id.etc
similarity index 75%
rename from matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_id.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap_id.etc
index 1d8b45295..b7e30698a 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_id.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap_id.etc
@@ -12,19 +12,15 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_eq.ma".
-include "ground/relocation/tr_id_pap.ma".
-include "ground/relocation/tr_id_tls.ma".
+include "delayed_updating/relocation/sbr_pap.ma".
+include "ground/relocation/tr_id.ma".
 
-(* LIFT FOR PATH ***********************************************************)
+(* POSITIVE APPLICATION FOR STACKS OF BASIC RELOCATION MAPS *****************)
 
-(* Constructions with tr_id ************************************************)
+(* Constructions with tr_id *************************************************)
 
-lemma lift_path_id (p):
-      p = ↑[𝐢]p.
-#p elim p -p //
-* [ #n ] #p #IH //
-[ <lift_path_d_sn //
-| <lift_path_L_sn //
-]
+lemma sbr_pap_id (p):
+      p = 𝐢❘@❨p❩.
+#p elim p -p [| #p #IH ] <tr_id_unfold //
+<sbr_pap_unfold_unit_succ //
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap_push.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap_push.etc
new file mode 100644
index 000000000..bd2988e85
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_pap_push.etc
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "delayed_updating/relocation/sbr_pap.ma".
+include "delayed_updating/relocation/sbr_push.ma".
+
+(* POSITIVE APPLICATION FOR BASIC STACKS OF RELOCATION MAPS *****************)
+
+(* constructions with sbr_push **********************************************)
+
+lemma sbr_push_pap_unit (f) (p):
+      ↑p = (p⫯❘f)❘@❨𝟏❩.
+// qed.
+
+lemma sbr_push_pap (f) (q) (p):
+      q + p = (p⫯❘f)❘@❨q❩.
+#f * //
+#q * //
+#p <sbr_pap_unfold_succ_bi //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_push.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_push.etc
new file mode 100644
index 000000000..e1329185e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_push.etc
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/relocation/tr_map.ma".
+include "delayed_updating/notation/functions/upspoonverticalbar_2.ma".
+
+(* PUSH FOR STACKS OF BASIC RELOCATION MAPS *********************************)
+
+definition sbr_push (p:pnat) (f): tr_map ≝ (↑p)⨮f.
+
+interpretation
+  "push (stack of basic relocation maps)"
+  'UpSpoonVerticalBar n f = (sbr_push n f).
+
+(* Basic properties *********************************************************)
+
+lemma sbr_push_unfold (p) (f):
+      (↑p)⨮f = p⫯❘f.
+// qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_push_eq.etc
similarity index 70%
rename from matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_push_eq.etc
index cc7569284..65c9d387b 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_push_eq.etc
@@ -12,22 +12,13 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_prototerm_eq.ma".
-include "delayed_updating/substitution/lift_id.ma".
+include "delayed_updating/relocation/sbr_push.ma".
+include "ground/relocation/tr_eq.ma".
 
-lemma lift_term_id_sn (t):
-      t ⊆ ↑[𝐢]t.
-#t #p #Hp
->(lift_path_id p)
-/2 width=1 by in_comp_lift_bi/
-qed-.
+(* PUSH FOR STACKS OF BASIC RELOCATION MAPS *********************************)
 
-lemma lift_term_id_dx (t):
-      ↑[𝐢]t ⊆ t.
-#t #p * #q #Hq #H destruct //
-qed-.
+(* Main constructions with stream_eq ****************************************)
 
-lemma lift_term_id (t):
-      t ⇔ ↑[𝐢]t.
-/3 width=2 by lift_term_id_dx, lift_term_id_sn, conj/      
-qed.
+theorem sbr_push_eq_repl_dx (p):
+        stream_eq_repl … (λf1,f2. p⫯❘f1 ≗ p⫯❘f2).
+/2 width=1 by stream_eq_cons/ qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_push_uni.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_push_uni.etc
new file mode 100644
index 000000000..edfd82efd
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/sbr_push_uni.etc
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/relocation/sbr_push.ma".
+include "ground/relocation/tr_uni.ma".
+
+(* PUSH FOR STACKS OF BASIC RELOCATION MAPS *********************************)
+
+(* Constructions with tr_uni ************************************************)
+
+theorem sbr_push_id (p:pnat):
+        (𝐮❨p❩) = p⫯❘𝐢.
+// qed. 
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/upspoonverticalbar_2.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/upspoonverticalbar_2.etc
new file mode 100644
index 000000000..4b4866516
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/upspoonverticalbar_2.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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* GROUND NOTATION **********************************************************)
+
+notation "hvbox( hd ⫯❘ break tl )"
+  right associative with precedence 47
+  for @{ 'UpSpoonVerticalBar $hd $tl }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/verticalbarat_2.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/verticalbarat_2.etc
new file mode 100644
index 000000000..ce7cd1410
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/relocation/verticalbarat_2.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 DELAYED UPDATING ********************************************)
+
+notation "hvbox( f ❘@❨ break term 46 a ❩ )"
+  non associative with precedence 60
+  for @{ 'VerticalBarAt $f $a }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_constructors.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_constructors.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_constructors.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_constructors.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_depth.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_depth.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_depth.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_depth.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_eq.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_eq.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_eq.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_etc.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_etc.etc
new file mode 100644
index 000000000..a9e563db6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_etc.etc
@@ -0,0 +1,10 @@
+include "delayed_updating/unwind2/unwind_depth.ma".
+include "ground/relocation/tr_uni_compose.ma".
+
+lemma pippo (q) (p): 
+      ▼[p]𝐢 ≗ ⇂*[↑❘q❘]▼[p●𝗟◗q]𝐢.
+#q @(list_ind_rcons … q) -q //
+#q * [ #n ] #IH #p //
+<depth_d_dx >list_cons_shift <list_append_assoc
+<unwind_rmap_d_dx
+check tr_tls_compose_uni_sn
\ No newline at end of file
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_fsubst.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_fsubst.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_fsubst.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_fsubst.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_height.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_height.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_height.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_height.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_preterm_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_preterm_eq.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_preterm_eq.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_preterm_eq.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_prototerm.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_prototerm.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_prototerm.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_prototerm.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_prototerm_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_prototerm_eq.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_prototerm_eq.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_prototerm_eq.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_structure.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_structure.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_structure.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_structure_depth.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_structure_depth.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind2/unwind_structure_depth.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind2/unwind_structure_depth.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind.etc
new file mode 100644
index 000000000..87790b370
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind.etc
@@ -0,0 +1,189 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/relocation/sbr_pap_push.ma".
+include "delayed_updating/syntax/path.ma".
+include "delayed_updating/notation/functions/black_downtriangle_4.ma".
+include "delayed_updating/notation/functions/black_downtriangle_2.ma".
+include "ground/relocation/tr_pn.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+definition unwind_continuation (A:Type[0]) ≝
+           tr_map → path → A.
+
+(* Note: f is a stack of update functions *)
+rec definition unwind_gen (A:Type[0]) (k:unwind_continuation A) (f:tr_map) (p) on p ≝
+match p with
+[ list_empty     ⇒ k f (𝐞)
+| list_lcons l q ⇒
+  match l with
+  [ label_d n ⇒
+    match q with
+    [ list_empty     ⇒ unwind_gen (A) (λg,p. k g (𝗱(f❘@❨n❩)◗p)) (f❘@❨n❩⫯❘f) q
+    | list_lcons _ _ ⇒ unwind_gen (A) k (f❘@❨n❩⫯❘f) q
+    ]
+  | label_m   ⇒ unwind_gen (A) k f q
+  | label_L   ⇒ unwind_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
+  | label_A   ⇒ unwind_gen (A) (λg,p. k g (𝗔◗p)) f q
+  | label_S   ⇒ unwind_gen (A) (λg,p. k g (𝗦◗p)) f q
+  ]
+].
+
+interpretation
+  "unwind (gneric)"
+  'BlackDownTriangle A k f p = (unwind_gen A k f p).
+
+definition proj_path: unwind_continuation … ≝
+           λf,p.p.
+
+definition proj_rmap: unwind_continuation … ≝
+           λf,p.f.
+
+interpretation
+  "unwind (path)"
+  'BlackDownTriangle f p = (unwind_gen ? proj_path f p).
+
+interpretation
+  "unwind (relocation map)"
+  'BlackDownTriangle p f = (unwind_gen ? proj_rmap f p).
+
+(* Basic constructions ******************************************************)
+
+lemma unwind_empty (A) (k) (f):
+      k f (𝐞) = ▼{A}❨k, f, 𝐞❩.
+// qed.
+
+lemma unwind_d_empty_sn (A) (k) (n) (f):
+      ▼❨(λg,p. k g (𝗱(f❘@❨n❩)◗p)), f❘@❨n❩⫯❘f, 𝐞❩ = ▼{A}❨k, f, 𝗱n◗𝐞❩.
+// qed.
+
+lemma unwind_d_lcons_sn (A) (k) (p) (l) (n) (f):
+      ▼❨k, f❘@❨n❩⫯❘f, l◗p❩ = ▼{A}❨k, f, 𝗱n◗l◗p❩.
+// qed.
+
+lemma unwind_m_sn (A) (k) (p) (f):
+      ▼❨k, f, p❩ = ▼{A}❨k, f, 𝗺◗p❩.
+// qed.
+
+lemma unwind_L_sn (A) (k) (p) (f):
+      ▼❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ▼{A}❨k, f, 𝗟◗p❩.
+// qed.
+
+lemma unwind_A_sn (A) (k) (p) (f):
+      ▼❨(λg,p. k g (𝗔◗p)), f, p❩ = ▼{A}❨k, f, 𝗔◗p❩.
+// qed.
+
+lemma unwind_S_sn (A) (k) (p) (f):
+      ▼❨(λg,p. k g (𝗦◗p)), f, p❩ = ▼{A}❨k, f, 𝗦◗p❩.
+// qed.
+
+(* Basic constructions with proj_path ***************************************)
+
+lemma unwind_path_empty (f):
+      (𝐞) = ▼[f]𝐞.
+// qed.
+
+lemma unwind_path_d_empty_sn (f) (n):
+      𝗱(f❘@❨n❩)◗𝐞 = ▼[f](𝗱n◗𝐞).
+// qed.
+
+lemma unwind_path_d_lcons_sn (f) (p) (l) (n):
+      ▼[f❘@❨n❩⫯❘f](l◗p) = ▼[f](𝗱n◗l◗p).
+// qed.
+
+lemma unwind_path_m_sn (f) (p):
+      ▼[f]p = ▼[f](𝗺◗p).
+// qed.
+
+(* Basic constructions with proj_rmap ***************************************)
+
+lemma unwind_rmap_empty (f):
+      f = ▼[𝐞]f.
+// qed.
+
+lemma unwind_rmap_d_sn (f) (p) (n):
+      ▼[p](f❘@❨n❩⫯❘f) = ▼[𝗱n◗p]f.
+#f * // qed.
+
+lemma unwind_rmap_m_sn (f) (p):
+      ▼[p]f = ▼[𝗺◗p]f.
+// qed.
+
+lemma unwind_rmap_L_sn (f) (p):
+      ▼[p](⫯f) = ▼[𝗟◗p]f.
+// qed.
+
+lemma unwind_rmap_A_sn (f) (p):
+      ▼[p]f = ▼[𝗔◗p]f.
+// qed.
+
+lemma unwind_rmap_S_sn (f) (p):
+      ▼[p]f = ▼[𝗦◗p]f.
+// qed.
+
+(* Advanced constructions with proj_rmap and path_append ********************)
+
+lemma unwind_rmap_append (p2) (p1) (f):
+      ▼[p2]▼[p1]f = ▼[p1●p2]f.
+#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
+[ <unwind_rmap_m_sn <unwind_rmap_m_sn //
+| <unwind_rmap_A_sn <unwind_rmap_A_sn //
+| <unwind_rmap_S_sn <unwind_rmap_S_sn //
+]
+qed.
+
+(* Advanced constructions with proj_rmap and path_rcons *********************)
+
+lemma unwind_rmap_d_dx (f) (p) (n):
+      (▼[p]f)❘@❨n❩⫯❘▼[p]f = ▼[p◖𝗱n]f.
+// qed.
+
+lemma unwind_rmap_m_dx (f) (p):
+      ▼[p]f = ▼[p◖𝗺]f.
+// qed.
+
+lemma unwind_rmap_L_dx (f) (p):
+      (⫯▼[p]f) = ▼[p◖𝗟]f.
+// qed.
+
+lemma unwind_rmap_A_dx (f) (p):
+      ▼[p]f = ▼[p◖𝗔]f.
+// qed.
+
+lemma unwind_rmap_S_dx (f) (p):
+▼[p]f = ▼[p◖𝗦]f.
+// qed.
+
+lemma unwind_rmap_pap_d_dx (f) (p) (n) (m):
+      m+▼[p]f❘@❨n❩ = ▼[p◖𝗱n]f❘@❨m❩.
+#f #p #n #m
+<unwind_rmap_d_dx <sbr_push_pap //
+qed.
+
+(* Advanced eliminations with path ******************************************)
+
+lemma path_ind_unwind (Q:predicate …):
+      Q (𝐞) →
+      (∀n. Q (𝐞) → Q (𝗱n◗𝐞)) →
+      (∀n,l,p. Q (l◗p) → Q (𝗱n◗l◗p)) →
+      (∀p. Q p → Q (𝗺◗p)) →
+      (∀p. Q p → Q (𝗟◗p)) →
+      (∀p. Q p → Q (𝗔◗p)) →
+      (∀p. Q p → Q (𝗦◗p)) →
+      ∀p. Q p.
+#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #p
+elim p -p [| * [ #n * ] ]
+/2 width=1 by/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_constructors.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_constructors.etc
new file mode 100644
index 000000000..4849ecf6d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_constructors.etc
@@ -0,0 +1,44 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/unwind3/unwind_prototerm_eq.ma".
+include "delayed_updating/syntax/prototerm_constructors.ma".
+include "delayed_updating/relocation/sbr_push_uni.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+(* Constructions with constructors for prototerm ****************************)
+
+lemma unwind_iref_sn (f) (t:prototerm) (n:pnat):
+      ▼[f❘@❨n❩⫯❘f]t ⊆ ▼[f](𝛗n.t).
+#f #t #n #p * #q #Hq #H0 destruct
+@(ex2_intro … (𝗱n◗𝗺◗q))
+/2 width=1 by in_comp_iref/
+qed-.
+
+lemma unwind_iref_dx (f) (t) (n:pnat):
+      ▼[f](𝛗n.t) ⊆ ▼[f❘@❨n❩⫯❘f]t.
+#f #t #n #p * #q #Hq #H0 destruct
+elim (in_comp_inv_iref … Hq) -Hq #p #Hp #Ht destruct
+/2 width=1 by in_comp_unwind_bi/
+qed-.
+
+lemma unwind_iref (f) (t) (n:pnat):
+      ▼[f❘@❨n❩⫯❘f]t ⇔ ▼[f](𝛗n.t).
+/3 width=1 by conj, unwind_iref_sn, unwind_iref_dx/
+qed.
+
+lemma unwind_iref_id (t) (n:pnat):
+      ▼[𝐮❨n❩]t ⇔ ▼[𝐢](𝛗n.t).
+// qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_depth.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_depth.etc
new file mode 100644
index 000000000..920036093
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_depth.etc
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "delayed_updating/unwind3/unwind.ma".
+include "delayed_updating/syntax/path_depth.ma".
+include "ground/relocation/tr_uni_compose.ma".
+include "ground/lib/stream_tls_plus.ma".
+include "ground/lib/stream_eq_eq.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+(* Constructions with depth *************************************************)
+
+lemma tls_succ_unwind (q) (p) (f):
+      ▼[p]f ≗ ⇂*[↑❘q❘]▼[p●𝗟◗q]f.
+#q @(list_ind_rcons … q) -q //
+#q * [ #n ] #IH #p #f //
+<depth_d_dx >list_cons_shift <list_append_assoc <unwind_rmap_d_dx
+/3 width=3 by tr_tls_compose_uni_sn, stream_eq_trans/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_eq.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_eq.etc
new file mode 100644
index 000000000..78d755b50
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_eq.etc
@@ -0,0 +1,124 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/unwind3/unwind.ma".
+include "delayed_updating/relocation/sbr_pap_id.ma".
+include "delayed_updating/relocation/sbr_pap_eq.ma".
+include "delayed_updating/relocation/sbr_push_eq.ma".
+include "ground/relocation/tr_pn_eq.ma".
+(*
+include "ground/lib/stream_tls.ma".
+*)
+
+(* UNWIND FOR PATH **********************************************************)
+
+definition unwind_exteq (A): relation2 (unwind_continuation A) (unwind_continuation A) ≝
+           λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
+
+interpretation
+  "extensional equivalence (unwind continuation)"
+  'RingEq A k1 k2 = (unwind_exteq A k1 k2).
+
+(* Constructions with unwind_exteq ******************************************)
+
+lemma unwind_eq_repl (A) (p) (k1) (k2):
+      k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ▼❨k1, f1, p❩ = ▼❨k2, f2, p❩).
+#A #p @(path_ind_unwind … p) -p [| #n #IH | #n #l0 #q #IH |*: #q #IH ]
+#k1 #k2 #Hk #f1 #f2 #Hf
+[ <unwind_empty <unwind_empty /2 width=1 by/
+| <unwind_d_empty_sn <unwind_d_empty_sn <(sbr_pap_eq_repl_sn … Hf) 
+  /3 width=1 by sbr_push_eq_repl_dx/
+| <unwind_d_lcons_sn <unwind_d_lcons_sn <(sbr_pap_eq_repl_sn … Hf)
+  /3 width=1 by sbr_push_eq_repl_dx/
+| /2 width=1 by/
+| /3 width=1 by tr_push_eq_repl/
+| /3 width=1 by/
+| /3 width=1 by/
+]
+qed-.
+
+(* Advanced constructions ***************************************************)
+
+lemma unwind_lcons_alt (A) (k) (f) (p) (l): k ≗ k →
+      ▼❨λg,p2. k g (l◗p2), f, p❩ = ▼{A}❨λg,p2. k g ((l◗𝐞)●p2), f, p❩.
+#A #k #f #p #l #Hk
+@unwind_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
+qed.
+
+lemma unwind_append_rcons_sn (A) (k) (f) (p1) (p) (l): k ≗ k →
+      ▼❨λg,p2. k g (p1●l◗p2), f, p❩ = ▼{A}❨λg,p2. k g (p1◖l●p2), f, p❩.
+#A #k #f #p1 #p #l #Hk
+@unwind_eq_repl // #g1 #g2 #p2 #Hg
+<list_append_rcons_sn @Hk -Hk // (**) (* auto fail *)
+qed.
+
+(* Advanced constructions with proj_path ************************************)
+
+lemma proj_path_proper:
+      proj_path ≗ proj_path.
+// qed.
+
+lemma unwind_path_eq_repl (p):
+      stream_eq_repl … (λf1,f2. ▼[f1]p = ▼[f2]p).
+/2 width=1 by unwind_eq_repl/ qed.
+
+lemma unwind_path_append_sn (p) (f) (q):
+      q●▼[f]p = ▼❨(λg,p. proj_path g (q●p)), f, p❩.
+#p @(path_ind_unwind … p) -p // [ #n #l #p |*: #p ] #IH #f #q
+[ <unwind_d_lcons_sn <unwind_d_lcons_sn <IH -IH //
+| <unwind_m_sn <unwind_m_sn //
+| <unwind_L_sn <unwind_L_sn >unwind_lcons_alt // >unwind_append_rcons_sn //
+  <IH <IH -IH <list_append_rcons_sn //
+| <unwind_A_sn <unwind_A_sn >unwind_lcons_alt >unwind_append_rcons_sn //
+  <IH <IH -IH <list_append_rcons_sn //
+| <unwind_S_sn <unwind_S_sn >unwind_lcons_alt >unwind_append_rcons_sn //
+  <IH <IH -IH <list_append_rcons_sn //
+]
+qed.
+
+lemma unwind_path_lcons (f) (p) (l):
+      l◗▼[f]p = ▼❨(λg,p. proj_path g (l◗p)), f, p❩.
+#f #p #l
+>unwind_lcons_alt <unwind_path_append_sn //
+qed.
+
+lemma unwind_path_L_sn (f) (p):
+      (𝗟◗▼[⫯f]p) = ▼[f](𝗟◗p).
+// qed.
+
+lemma unwind_path_A_sn (f) (p):
+      (𝗔◗▼[f]p) = ▼[f](𝗔◗p).
+// qed.
+
+lemma unwind_path_S_sn (f) (p):
+      (𝗦◗▼[f]p) = ▼[f](𝗦◗p).
+// qed.
+
+lemma unwind_path_after_id_sn (p) (f):
+      ▼[𝐢]▼[f]p = ▼[f]p.
+#p @(path_ind_unwind … p) -p // #p #IH #f
+[ <unwind_path_d_empty_sn <unwind_path_d_empty_sn //
+| <unwind_path_L_sn <unwind_path_L_sn //
+]
+qed.
+
+(* Advanced constructions with proj_rmap and stream_tls *********************)
+(* COMMENT
+lemma unwind_rmap_tls_d_dx (f) (p) (m:pnat) (n):
+      ⇂*[m]▼[p]f ≗ ⇂*[m]▼[p◖𝗱n]f.
+#f #p #m #n
+<unwind_rmap_d_dx
+/2 width=1 by tr_tls_compose_uni_sn/
+qed.
+*)
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_fsubst.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_fsubst.etc
new file mode 100644
index 000000000..411c5621a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_fsubst.etc
@@ -0,0 +1,61 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/unwind3/unwind_prototerm_eq.ma".
+include "delayed_updating/unwind3/unwind_structure.ma".
+include "delayed_updating/substitution/fsubst.ma".
+include "delayed_updating/syntax/preterm.ma".
+include "delayed_updating/syntax/prototerm_proper.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+(* Constructions with fsubst ************************************************)
+
+lemma unwind_fsubst_sn (f) (t) (u) (p): u ϵ 𝐏 →
+      (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u] ⊆ ▼[f](t[⋔p←u]).
+#f #t #u #p #Hu #ql * *
+[ #rl * #r #Hr #H1 #H2 destruct
+  >unwind_append_proper_dx
+  /4 width=5 by in_comp_unwind_bi, in_ppc_comp_trans, or_introl, ex2_intro/
+| * #q #Hq #H1 #H0
+  @(ex2_intro … H1) @or_intror @conj // *
+  [ <list_append_empty_dx #H2 destruct
+    elim (unwind_path_root f q) #r #_ #Hr /2 width=2 by/
+  | #l #r #H2 destruct
+    @H0 -H0 [| <unwind_append_proper_dx /2 width=3 by ppc_lcons/ ]
+  ]
+]
+qed-.
+
+lemma unwind_fsubst_dx (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
+      ▼[f](t[⋔p←u]) ⊆ (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u].
+#f #t #u #p #Hu #H1p #H2p #ql * #q * *
+[ #r #Hu #H1 #H2 destruct
+  @or_introl @ex2_intro
+  [|| <unwind_append_proper_dx /2 width=5 by in_ppc_comp_trans/ ]
+  /2 width=3 by ex2_intro/
+| #Hq #H0 #H1 destruct
+  @or_intror @conj [ /2 width=1 by in_comp_unwind_bi/ ] *
+  [ <list_append_empty_dx #Hr @(H0 … (𝐞)) -H0
+    <list_append_empty_dx @H2p -H2p /2 width=1 by prototerm_in_comp_root/
+  | #l #r #Hr
+    elim (unwind_inv_append_proper_dx … Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
+    lapply (H2p … Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
+  ]
+]
+qed-.
+
+lemma unwind_fsubst (f) (t) (u) (p): u ϵ 𝐏 → p ϵ ▵t → t ϵ 𝐓 →
+      (▼[f]t)[⋔(⊗p)←▼[▼[p]f]u] ⇔ ▼[f](t[⋔p←u]).
+/4 width=3 by unwind_fsubst_sn, conj, unwind_fsubst_dx/ qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_length.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_length.etc
new file mode 100644
index 000000000..7514a68cf
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_length.etc
@@ -0,0 +1,35 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/unwind3/unwind.ma".
+include "delayed_updating/syntax/path_depth.ma".
+include "ground/lib/stream_tls_plus.ma".
+include "ground/lib/stream_eq.ma".
+(*
+include "ground/relocation/tr_uni_compose.ma".
+include "ground/lib/stream_eq_eq.ma".
+*)
+(* UNWIND FOR PATH **********************************************************)
+
+(* Constructions with list_length ********************************************)
+
+axiom tls_unwind (p) (f):
+      f ≗ ⇂*[❘p❘]▼[p]f.
+(* COMMENT
+#q @(list_ind_rcons … q) -q //
+#q * [ #n ] #IH #f //
+<depth_d_dx >list_cons_shift <list_append_assoc <unwind_rmap_d_dx
+/3 width=3 by tr_tls_compose_uni_sn, stream_eq_trans/
+qed.
+*)
\ No newline at end of file
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_lift.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_lift.etc
new file mode 100644
index 000000000..9530004ca
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_lift.etc
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/unwind3/unwind_structure.ma".
+include "delayed_updating/substitution/lift_structure.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+(* Constructions with lift **************************************************)
+
+lemma pippo (p) (n:pnat) (f):
+      ↑[𝐮❨n❩]▼[⇂*[n]f]p = ▼[𝐮❨n❩∘f]p.
+#p @(list_ind_rcons … p) -p //
+#p * [ #m ] #IH #n #f //
+[
+| <unwind_path_m_dx
+
+ //
+<depth_d_dx >list_cons_shift <list_append_assoc <unwind_rmap_d_dx
+/3 width=3 by tr_tls_compose_uni_sn, stream_eq_trans/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_preterm_eq.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_preterm_eq.etc
new file mode 100644
index 000000000..d245cb1bf
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_preterm_eq.etc
@@ -0,0 +1,68 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/lib/subset_equivalence.ma".
+include "delayed_updating/syntax/path_structure_inner.ma".
+include "delayed_updating/syntax/preterm.ma".
+include "delayed_updating/unwind3/unwind_structure.ma".
+include "delayed_updating/unwind3/unwind_prototerm.ma".
+
+(* UNWIND FOR PRETERM *******************************************************)
+
+(* Constructions with subset_equivalence ************************************)
+
+lemma unwind_grafted_sn (f) (t) (p): p ϵ 𝐈 →
+      ▼[▼[p]f](t⋔p) ⊆ (▼[f]t)⋔(⊗p).
+#f #t #p #Hp #q * #r #Hr #H0 destruct
+@(ex2_intro … Hr) -Hr
+<unwind_append_inner_sn //
+qed-.
+
+lemma unwind_grafted_dx (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
+      (▼[f]t)⋔(⊗p) ⊆ ▼[▼[p]f](t⋔p).
+#f #t #p #H1p #H2p #Ht #q * #r #Hr #H0
+elim (unwind_inv_append_inner_sn … (sym_eq … H0)) -H0 //
+#p0 #q0 #Hp0 #Hq0 #H0 destruct
+<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+/2 width=1 by in_comp_unwind_bi/
+qed-.
+
+lemma unwind_grafted (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
+      ▼[▼[p]f](t⋔p) ⇔ (▼[f]t)⋔(⊗p).
+/3 width=1 by unwind_grafted_sn, conj, unwind_grafted_dx/ qed.
+
+
+lemma unwind_grafted_S_dx (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
+      (▼[f]t)⋔((⊗p)◖𝗦) ⊆ ▼[▼[p]f](t⋔(p◖𝗦)).
+#f #t #p #Hp #Ht #q * #r #Hr
+<list_append_rcons_sn #H0
+elim (unwind_inv_append_proper_dx … (sym_eq … H0)) -H0 //
+#p0 #q0 #Hp0 #Hq0 #H0 destruct
+<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+elim (unwind_path_inv_S_sn … (sym_eq … Hq0)) -Hq0
+#r1 #r2 #Hr1 #Hr2 #H0 destruct
+lapply (preterm_in_root_append_inv_structure_empty_dx … p0 … Ht Hr1)
+[ /2 width=2 by ex_intro/ ] -Hr1 #Hr1 destruct
+/2 width=1 by in_comp_unwind_bi/
+qed-.
+
+lemma unwind_grafted_S (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
+      ▼[▼[p]f](t⋔(p◖𝗦)) ⇔ (▼[f]t)⋔((⊗p)◖𝗦).
+#f #t #p #Hp #Ht
+@conj
+[ >unwind_rmap_S_dx >structure_S_dx
+  @unwind_grafted_sn //
+| /2 width=1 by unwind_grafted_S_dx/
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_prototerm.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_prototerm.etc
new file mode 100644
index 000000000..c4a103b61
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_prototerm.etc
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/unwind3/unwind.ma".
+include "delayed_updating/syntax/prototerm.ma".
+include "ground/lib/subset_ext.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+interpretation
+  "unwind (prototerm)"
+  'BlackDownTriangle f t = (subset_ext_f1 ? ? (unwind_gen ? proj_path f) t).
+
+(* Basic constructions ******************************************************)
+
+lemma in_comp_unwind_bi (f) (p) (t):
+      p ϵ t → ▼[f]p ϵ ▼[f]t.
+/2 width=1 by subset_in_ext_f1_dx/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_prototerm_eq.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_prototerm_eq.etc
new file mode 100644
index 000000000..01d302f3e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_prototerm_eq.etc
@@ -0,0 +1,41 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/lib/subset_ext_equivalence.ma".
+include "delayed_updating/unwind3/unwind_eq.ma".
+include "delayed_updating/unwind3/unwind_prototerm.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+(* Constructions with subset_equivalence ************************************)
+
+lemma unwind_term_eq_repl_sn (f1) (f2) (t):
+      f1 ≗ f2 → ▼[f1]t ⇔ ▼[f2]t.
+/3 width=1 by subset_equivalence_ext_f1_exteq, unwind_path_eq_repl/
+qed.
+
+lemma unwind_term_eq_repl_dx (f) (t1) (t2):
+      t1 ⇔ t2 → ▼[f]t1 ⇔ ▼[f]t2.
+/2 width=1 by subset_equivalence_ext_f1_bi/
+qed.
+
+lemma unwind_term_after_id_sn (f) (t):
+      ▼[𝐢]▼[f]t ⇔ ▼[f]t.
+#f #t @subset_eq_trans
+[
+| @subset_inclusion_ext_f1_compose
+| @subset_equivalence_ext_f1_exteq
+  #p @unwind_path_after_id_sn
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_structure.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_structure.etc
new file mode 100644
index 000000000..c52a43871
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_structure.etc
@@ -0,0 +1,269 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/unwind3/unwind_eq.ma".
+include "delayed_updating/syntax/path_structure.ma".
+include "delayed_updating/syntax/path_inner.ma".
+include "delayed_updating/syntax/path_proper.ma".
+include "ground/xoa/ex_4_2.ma".
+
+(* UNWIND FOR PATH *********************************************************)
+
+(* Basic constructions with structure **************************************)
+
+lemma structure_unwind (p) (f):
+      ⊗p = ⊗▼[f]p.
+#p @(path_ind_unwind … p) -p // #p #IH #f
+<unwind_path_L_sn //
+qed.
+
+lemma unwind_structure (p) (f):
+      ⊗p = ▼[f]⊗p.
+#p @(path_ind_unwind … p) -p //
+qed.
+
+(* Destructions with structure **********************************************)
+
+lemma unwind_des_structure (q) (p) (f):
+      ⊗q = ▼[f]p → ⊗q = ⊗p.
+// qed-.
+
+(* Constructions with proper condition for path *****************************)
+
+lemma unwind_append_proper_dx (p2) (p1) (f): p2 ϵ 𝐏 →
+      (⊗p1)●(▼[▼[p1]f]p2) = ▼[f](p1●p2).
+#p2 #p1 @(path_ind_unwind … p1) -p1 //
+[ #n | #n #l #p1 |*: #p1 ] #IH #f #Hp2
+[ elim (ppc_inv_lcons … Hp2) -Hp2 #l #q #H destruct //
+| <unwind_path_d_lcons_sn <IH //
+| <unwind_path_m_sn <IH //
+| <unwind_path_L_sn <IH //
+| <unwind_path_A_sn <IH //
+| <unwind_path_S_sn <IH //
+]
+qed-.
+
+(* Constructions with inner condition for path ******************************)
+
+lemma unwind_append_inner_sn (p1) (p2) (f): p1 ϵ 𝐈 →
+      (⊗p1)●(▼[▼[p1]f]p2) = ▼[f](p1●p2).
+#p1 @(list_ind_rcons … p1) -p1 // #p1 *
+[ #n ] #_ #p2 #f #Hp1
+[ elim (pic_inv_d_dx … Hp1)
+| <list_append_rcons_sn <unwind_append_proper_dx //
+| <list_append_rcons_sn <unwind_append_proper_dx //
+  <structure_L_dx <list_append_rcons_sn //
+| <list_append_rcons_sn <unwind_append_proper_dx //
+  <structure_A_dx <list_append_rcons_sn //
+| <list_append_rcons_sn <unwind_append_proper_dx //
+  <structure_S_dx <list_append_rcons_sn //
+]
+qed-.
+
+(* Advanced constructions with proj_path ************************************)
+
+lemma unwind_path_d_empty_dx (n) (p) (f):
+      (⊗p)◖𝗱((▼[p]f)❘@❨n❩) = ▼[f](p◖𝗱n).
+#n #p #f <unwind_append_proper_dx // 
+qed.
+
+lemma unwind_path_m_dx (p) (f):
+      ⊗p = ▼[f](p◖𝗺).
+#p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_L_dx (p) (f):
+      (⊗p)◖𝗟 = ▼[f](p◖𝗟).
+#p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_A_dx (p) (f):
+      (⊗p)◖𝗔 = ▼[f](p◖𝗔).
+#p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_S_dx (p) (f):
+      (⊗p)◖𝗦 = ▼[f](p◖𝗦).
+#p #f <unwind_append_proper_dx //
+qed.
+
+lemma unwind_path_root (f) (p):
+      ∃∃r. 𝐞 = ⊗r & ⊗p●r = ▼[f]p.
+#f #p @(list_ind_rcons … p) -p
+[ /2 width=3 by ex2_intro/
+| #p * [ #n ] /2 width=3 by ex2_intro/
+]
+qed-.
+
+(* Advanced inversions with proj_path ***************************************)
+
+lemma unwind_path_inv_d_sn (k) (q) (p) (f):
+      (𝗱k◗q) = ▼[f]p →
+      ∃∃r,h. 𝐞 = ⊗r & (▼[r]f)❘@❨h❩ = k & 𝐞 = q & r◖𝗱h = p.
+#k #q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct -IH
+  /2 width=5 by ex4_2_intro/
+| <unwind_path_d_lcons_sn #H
+  elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
+  /2 width=5 by ex4_2_intro/
+| <unwind_path_m_sn #H
+  elim (IH … H) -IH -H #r #h #Hr #Hh #Hq #Hp destruct
+  /2 width=5 by ex4_2_intro/
+| <unwind_path_L_sn #H destruct
+| <unwind_path_A_sn #H destruct
+| <unwind_path_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_path_inv_m_sn (q) (p) (f):
+      (𝗺◗q) = ▼[f]p → ⊥.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct
+| <unwind_path_d_lcons_sn #H /2 width=2 by/
+| <unwind_path_m_sn #H /2 width=2 by/
+| <unwind_path_L_sn #H destruct
+| <unwind_path_A_sn #H destruct
+| <unwind_path_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_path_inv_L_sn (q) (p) (f):
+      (𝗟◗q) = ▼[f]p →
+      ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[⫯▼[r1]f]r2 & r1●𝗟◗r2 = p.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct
+| <unwind_path_d_lcons_sn #H
+  elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+  /2 width=5 by ex3_2_intro/
+| <unwind_path_m_sn #H
+  elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+  /2 width=5 by ex3_2_intro/
+| <unwind_path_L_sn #H destruct -IH
+  /2 width=5 by ex3_2_intro/
+| <unwind_path_A_sn #H destruct
+| <unwind_path_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_path_inv_A_sn (q) (p) (f):
+      (𝗔◗q) = ▼[f]p →
+      ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▼[r1]f]r2 & r1●𝗔◗r2 = p.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct
+| <unwind_path_d_lcons_sn #H
+  elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+  /2 width=5 by ex3_2_intro/
+| <unwind_path_m_sn #H
+  elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+  /2 width=5 by ex3_2_intro/
+| <unwind_path_L_sn #H destruct
+| <unwind_path_A_sn #H destruct -IH
+  /2 width=5 by ex3_2_intro/
+| <unwind_path_S_sn #H destruct
+]
+qed-.
+
+lemma unwind_path_inv_S_sn (q) (p) (f):
+      (𝗦◗q) = ▼[f]p →
+      ∃∃r1,r2. 𝐞 = ⊗r1 & q = ▼[▼[r1]f]r2 & r1●𝗦◗r2 = p.
+#q #p @(path_ind_unwind … p) -p
+[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
+[ <unwind_path_empty #H destruct
+| <unwind_path_d_empty_sn #H destruct
+| <unwind_path_d_lcons_sn #H
+  elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+  /2 width=5 by ex3_2_intro/
+| <unwind_path_m_sn #H
+  elim (IH … H) -IH -H #r1 #r2 #Hr1 #Hq #Hp destruct
+  /2 width=5 by ex3_2_intro/| <unwind_path_L_sn #H destruct
+| <unwind_path_A_sn #H destruct
+| <unwind_path_S_sn #H destruct -IH
+  /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Inversions with proper condition for path ********************************)
+
+lemma unwind_inv_append_proper_dx (q2) (q1) (p) (f):
+      q2 ϵ 𝐏 → q1●q2 = ▼[f]p →
+      ∃∃p1,p2. ⊗p1 = q1 & ▼[▼[p1]f]p2 = q2 & p1●p2 = p.
+#q2 #q1 elim q1 -q1
+[ #p #f #Hq2 <list_append_empty_sn #H destruct
+  /2 width=5 by ex3_2_intro/
+| * [ #n1 ] #q1 #IH #p #f #Hq2 <list_append_lcons_sn #H
+  [ elim (unwind_path_inv_d_sn … H) -H #r1 #m1 #_ #_ #H0 #_ -IH
+    elim (eq_inv_list_empty_append … H0) -H0 #_ #H0 destruct
+    elim Hq2 -Hq2 //
+  | elim (unwind_path_inv_m_sn … H)
+  | elim (unwind_path_inv_L_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
+    elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
+    @(ex3_2_intro … (r1●𝗟◗p1)) //
+    <structure_append <Hr1 -Hr1 //
+  | elim (unwind_path_inv_A_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
+    elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
+    @(ex3_2_intro … (r1●𝗔◗p1)) //
+    <structure_append <Hr1 -Hr1 //
+  | elim (unwind_path_inv_S_sn … H) -H #r1 #s1 #Hr1 #Hs1 #H0 destruct
+    elim (IH … Hs1) -IH -Hs1 // -Hq2 #p1 #p2 #H1 #H2 #H3 destruct
+    @(ex3_2_intro … (r1●𝗦◗p1)) //
+    <structure_append <Hr1 -Hr1 //
+  ]
+]
+qed-.
+
+(* Inversions with inner condition for path *********************************)
+
+lemma unwind_inv_append_inner_sn (q1) (q2) (p) (f):
+      q1 ϵ 𝐈 → q1●q2 = ▼[f]p →
+      ∃∃p1,p2. ⊗p1 = q1 & ▼[▼[p1]f]p2 = q2 & p1●p2 = p.
+#q1 @(list_ind_rcons … q1) -q1
+[ #q2 #p #f #Hq1 <list_append_empty_sn #H destruct
+  /2 width=5 by ex3_2_intro/
+| #q1 * [ #n1 ] #_ #q2 #p #f #Hq2
+  [ elim (pic_inv_d_dx … Hq2)
+  | <list_append_rcons_sn #H0
+    elim (unwind_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+    elim (unwind_path_inv_m_sn … (sym_eq … H2))
+  | <list_append_rcons_sn #H0
+    elim (unwind_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+    elim (unwind_path_inv_L_sn … (sym_eq … H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
+    @(ex3_2_intro … (p1●r2◖𝗟)) [1,3: // ]
+    [ <structure_append <structure_L_dx <Hr2 -Hr2 //
+    | <list_append_assoc <list_append_rcons_sn //
+    ]
+  | <list_append_rcons_sn #H0
+    elim (unwind_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+    elim (unwind_path_inv_A_sn … (sym_eq … H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
+    @(ex3_2_intro … (p1●r2◖𝗔)) [1,3: // ]
+    [ <structure_append <structure_A_dx <Hr2 -Hr2 //
+    | <list_append_assoc <list_append_rcons_sn //
+    ]
+  | <list_append_rcons_sn #H0
+    elim (unwind_inv_append_proper_dx … H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+    elim (unwind_path_inv_S_sn … (sym_eq … H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
+    @(ex3_2_intro … (p1●r2◖𝗦)) [1,3: // ]
+    [ <structure_append <structure_S_dx <Hr2 -Hr2 //
+    | <list_append_assoc <list_append_rcons_sn //
+    ]
+  ]
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_structure_depth.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_structure_depth.etc
new file mode 100644
index 000000000..fdb7d781b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind3/unwind_structure_depth.etc
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "delayed_updating/unwind3/unwind.ma".
+include "delayed_updating/syntax/path_structure.ma".
+include "delayed_updating/syntax/path_depth.ma".
+include "ground/arith/nat_pred_succ.ma".
+
+(* UNWIND FOR PATH *********************************************************)
+
+(* Basic constructions with structure and depth ****************************)
+
+lemma unwind_rmap_structure (p) (f):
+      (⫯*[❘p❘]f) = ▼[⊗p]f.
+#p elim p -p //
+* [ #n ] #p #IH #f //
+[ <unwind_rmap_A_sn //
+| <unwind_rmap_S_sn //
+]
+qed.
-- 
2.39.2