From ad6182251b8192ee7d25c53156afbce35e3715b6 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Tue, 24 May 2022 00:39:34 +0200
Subject: [PATCH] update in delayed_updating

+ crux of the main proof completed
+ notation for path_reverse fixed
---
 .../notation/functions/nec_r_1.ma             |   2 +-
 .../delayed_updating/reduction/dfr.ma         |  15 +-
 .../delayed_updating/reduction/dfr_ifr.ma     | 144 ++++++------------
 .../delayed_updating/reduction/ifr.ma         |  15 +-
 .../substitution/fsubst_eq.ma                 |   6 +-
 .../syntax/path_depth_reverse.ma              |  32 ++++
 .../syntax/path_head_structure.ma             |  37 +++++
 .../delayed_updating/syntax/path_labels.ma    |   8 +
 .../delayed_updating/syntax/path_reverse.ma   |   9 ++
 .../syntax/path_structure_reverse.ma          |  29 ++++
 .../unwind/unwind2_path_eq.ma                 |   1 -
 .../unwind/unwind2_path_lift.ma               |   1 -
 .../unwind/unwind2_path_structure.ma          |   3 -
 .../unwind/unwind2_rmap_eq.ma                 |   9 ++
 .../unwind/unwind2_rmap_head.ma               |  73 +++++++--
 15 files changed, 247 insertions(+), 137 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth_reverse.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_structure.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure_reverse.ma

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/nec_r_1.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/nec_r_1.ma
index ec39b28d9..79df117c7 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/nec_r_1.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/nec_r_1.ma
@@ -15,5 +15,5 @@
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
 notation "hvbox( p á´¿ )"
-  non associative with precedence 70
+  non associative with precedence 71
   for @{ 'NEcR $p }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
index b7f900b6c..7ae112491 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
@@ -12,24 +12,21 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/unwind1/unwind.ma".
 include "delayed_updating/substitution/fsubst.ma".
 include "delayed_updating/syntax/prototerm_constructors.ma".
 include "delayed_updating/syntax/prototerm_equivalence.ma".
-include "delayed_updating/syntax/path_structure.ma".
-include "delayed_updating/syntax/path_balanced.ma".
+include "delayed_updating/syntax/path_head.ma".
 include "delayed_updating/syntax/path_depth.ma".
+include "delayed_updating/syntax/path_reverse.ma".
 include "delayed_updating/notation/relations/black_rightarrow_df_4.ma".
-include "ground/xoa/ex_1_2.ma".
-include "ground/xoa/and_4.ma".
 
 (* DELAYED FOCUSED REDUCTION ************************************************)
 
 definition dfr (p) (q): relation2 prototerm prototerm ≝
-           λt1,t2. ∃∃b,n.
-           let r ≝ p●𝗔◗b●𝗟◗q in
-           ∧∧ (⊗b ϵ 𝐁 ∧ 𝟎 = ♭b) & ↑♭q = (▼[r]𝐢)＠⧣❨n❩ & r◖𝗱n ϵ t1 &
-              t1[⋔r←𝛗(n+♭b).(t1⋔(p◖𝗦))] ⇔ t2
+           λt1,t2. ∃n:pnat.
+           let r ≝ p●𝗔◗𝗟◗q in
+           ∧∧ (𝗟◗q)ᴿ = ↳[n](rᴿ) & r◖𝗱n ϵ t1 &
+              t1[⋔r←𝛗n.(t1⋔(p◖𝗦))] ⇔ t2
 .
 
 interpretation
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma
index 29c44898d..b4ccad9d3 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma
@@ -15,119 +15,63 @@
 include "delayed_updating/reduction/dfr.ma".
 include "delayed_updating/reduction/ifr.ma".
 
-include "delayed_updating/unwind1/unwind_fsubst.ma".
-include "delayed_updating/unwind1/unwind_constructors.ma".
-include "delayed_updating/unwind1/unwind_preterm_eq.ma".
-include "delayed_updating/unwind1/unwind_structure_depth.ma".
-include "delayed_updating/unwind1/unwind_depth.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".
-include "ground/relocation/tr_uni_compose.ma".
-include "ground/relocation/tr_pap_pushs.ma".
+include "delayed_updating/syntax/path_structure_reverse.ma".
+include "delayed_updating/syntax/path_depth_reverse.ma".
 
 (* DELAYED FOCUSED REDUCTION ************************************************)
 
-(* COMMENT
-axiom pippo (b) (q) (n):
-      ↑❘q❘ = (↑[q]𝐢)＠⧣❨n❩ →
-      ↑❘q❘+❘b❘= (↑[b●𝗟◗q]𝐢)＠⧣❨n+❘b❘❩.
-
-lemma unwind_rmap_tls_eq_id (p) (n):
-      ❘p❘ = ↑[p]𝐢＠⧣❨n❩ →
-      (𝐢) ≗ ⇂*[n]↑[p]𝐢.
-#p @(list_ind_rcons … p) -p
-[ #n <depth_empty #H destruct
-| #p * [ #m ] #IH #n
-  [ <depth_d_dx <unwind_rmap_pap_d_dx #H0
-    @(stream_eq_trans … (unwind_rmap_tls_d_dx …))
-    @(stream_eq_trans … (IH …)) -IH //
-  | /2 width=1 by/
-  | <depth_L_dx <unwind_rmap_L_dx
-    cases n -n [| #n ] #H0
-    [
-    | 
-    ]
-  | /2 width=1 by/
-  | /2 width=1 by/
-  ]
-]
-
-
-(*  (↑❘q❘+❘b❘=↑[b●𝗟◗q]𝐢＠⧣❨n+❘b❘❩ *)
-(* [↑[p]𝐢＠⧣❨n❩]⫯*[❘p❘]f∘⇂*[n]↑[p]𝐢) *)
-lemma unwind_rmap_tls_eq (f) (p) (n):
-      ❘p❘ = ↑[p]𝐢＠⧣❨n❩ →
-      f ≗ ⇂*[n]↑[p]f.
-#f #p #n #Hp
-@(stream_eq_canc_dx … (stream_tls_eq_repl …))
-[| @unwind_rmap_decompose | skip ]
-<tr_compose_tls <Hp
+(* Main destructions with ifr ***********************************************)
 
-@(stream_eq_canc_dx) … (unwind_rmap_decompose …)) 
-
-*)
-lemma dfr_unwind_id_bi (p) (q) (t1) (t2): t1 ϵ 𝐓 →
-      t1 ➡𝐝𝐟[p,q] t2 → ▼[𝐢]t1 ➡𝐟[⊗p,⊗q] ▼[𝐢]t2.
-#p #q #t1 #t2 #H0t1
-* #b #n * #Hb #Hn #Ht1 #Ht2
-@(ex1_2_intro … (⊗b) (↑♭⊗q)) @and4_intro
-[ //
-| (*//*)
-| lapply (in_comp_unwind_bi (𝐢) … Ht1) -Ht1 -H0t1 -Hb -Ht2
-  <unwind_path_d_empty_dx <depth_structure //
-| lapply (unwind_term_eq_repl_dx (𝐢) … Ht2) -Ht2 #Ht2
+theorem dfr_des_ifr (f) (p) (q) (t1) (t2): t1 ϵ 𝐓 →
+        t1 ➡𝐝𝐟[p,q] t2 → ▼[f]t1 ➡𝐟[⊗p,⊗q] ▼[f]t2.
+#f #p #q #t1 #t2 #H0t1
+* #n * #H1n #Ht1 #Ht2
+@(ex_intro … (↑♭⊗q)) @and3_intro
+[ -H0t1 -H1n -Ht1 -Ht2
+  >list_append_rcons_sn <reverse_append
+  >nsucc_unfold >depth_reverse >depth_L_dx >reverse_lcons
+  >structure_L_sn >structure_reverse
+  <path_head_structure //
+| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H0t1
+  <unwind2_path_d_dx <depth_structure
+  >list_append_rcons_sn in H1n; <reverse_append #H1n
+  lapply (unwind2_rmap_append_pap_closed f … H1n)
+  <reverse_lcons <depth_L_dx #H2n
+  lapply (eq_inv_ninj_bi … H2n) -H2n #H2n <H2n -H2n -H1n #Ht1 //
+| lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
   @(subset_eq_trans … Ht2) -t2
-  @(subset_eq_trans … (unwind_fsubst …))
-  [ (*<unwind_rmap_append <unwind_rmap_A_sn <unwind_rmap_append <unwind_rmap_L_sn *)
-    <structure_append <structure_A_sn <structure_append <structure_L_sn
-    <depth_append <depth_L_sn <depth_structure <depth_structure
-    @fsubst_eq_repl [ // ]
-    @(subset_eq_trans … (unwind_iref …))
-
-    elim Hb -Hb #Hb #H0 <H0 -H0 <nrplus_zero_dx <nplus_zero_dx <nsucc_unfold
-    >Hn
+  @(subset_eq_trans … (unwind2_term_fsubst …))
+  [ @fsubst_eq_repl [ // | // ]
+    @(subset_eq_trans … (unwind2_term_iref …))
     @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
-    [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
-    <Hn <Hn
-(*    
-    @(subset_eq_trans … (lift_term_eq_repl_dx …))
-    [ @(unwind_term_eq_repl_sn … (tls_succ_unwind q …)) | skip ]
-*)
-(*
-    
-    @subset_eq_trans
-    [2: @unwind_term_eq_repl_dx
-    @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
-
-    @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
-    [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
-
-    @(subset_eq_trans … (unwind_term_after …))
-    @(subset_eq_canc_dx … (unwind_term_after …))
-    @unwind_term_eq_repl_sn -t1
-    @(stream_eq_trans … (tr_compose_uni_dx …))
-    lapply (Hn (𝐢)) -Hn >tr_id_unfold #Hn
-    lapply (pippo … b … Hn) -Hn #Hn
-    @tr_compose_eq_repl
-    [ <unwind_rmap_pap_le //
-      <Hn <nrplus_inj_sn //
-    |
-    ]
-*)
+    [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+    @(subset_eq_trans … (unwind2_lift_term_after …))
+    @unwind2_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+    @nstream_eq_inv_ext #m
+    <tr_compose_pap <tr_compose_pap
+    <tr_uni_pap <tr_uni_pap <tr_pap_plus
+    >list_append_rcons_sn in H1n; <reverse_append #H1n
+    lapply (unwind2_rmap_append_pap_closed f … H1n) #H2n
+    >nrplus_inj_dx in ⊢ (???%); <H2n -H2n
+    lapply (tls_unwind2_rmap_append_closed f … H1n) #H2n
+    <(tr_pap_eq_repl … H2n) -H2n -H1n //
+(* Note: crux of the proof ends *)
   | //
   | /2 width=2 by ex_intro/
   | //
   ]
 ]
-
-(*
-Hn : ↑❘q❘ = ↑[p●𝗔◗b●𝗟◗q]𝐢＠⧣❨n❩
----------------------------
-↑[𝐮❨↑❘q❘+❘b❘❩] ↑[↑[p]𝐢] t ⇔ ↑[𝐮❨↑[p●𝗔◗b●𝗟◗q]𝐢＠⧣❨n+❘b❘❩❩] t
-*)
-(*
-(↑[𝐮❨↑❘q❘❩]▼[⇂*[↑❘q❘]▼[p●𝗟◗q]𝐢](t1⋔(p◖𝗦))⇔▼[𝐮❨↑❘q❘❩∘▼[p●𝗔◗b●𝗟◗q]𝐢](t1⋔(p◖𝗦))
-*)
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma
index 6c7be675a..4e70a6c43 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma
@@ -12,24 +12,21 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/unwind1/unwind_prototerm.ma".
 include "delayed_updating/substitution/fsubst.ma".
 include "delayed_updating/substitution/lift_prototerm.ma".
 include "delayed_updating/syntax/prototerm_equivalence.ma".
+include "delayed_updating/syntax/path_head.ma".
 include "delayed_updating/syntax/path_depth.ma".
-include "delayed_updating/syntax/path_structure.ma".
-include "delayed_updating/syntax/path_balanced.ma".
+include "delayed_updating/syntax/path_reverse.ma".
 include "delayed_updating/notation/relations/black_rightarrow_f_4.ma".
-include "ground/xoa/ex_1_2.ma".
-include "ground/xoa/and_4.ma".
 
 (* IMMEDIATE FOCUSED REDUCTION ************************************************)
 
 definition ifr (p) (q): relation2 prototerm prototerm ≝
-           λt1,t2. ∃∃b,n.
-           let r ≝ p●𝗔◗b●𝗟◗q in
-           ∧∧ (⊗b ϵ 𝐁 ∧ 𝟎 = ♭b) & ↑♭q = (▼[r]𝐢)＠⧣❨n❩ & r◖𝗱n ϵ t1 &
-              t1[⋔r←↑[𝐮❨♭(b●𝗟◗q)❩](t1⋔(p◖𝗦))] ⇔ t2
+           λt1,t2. ∃n:pnat.
+           let r ≝ p●𝗔◗𝗟◗q in
+           ∧∧ (𝗟◗q)ᴿ = ↳[n](rᴿ) & r◖𝗱n ϵ t1 &
+              t1[⋔r←↑[𝐮❨♭(𝗟◗q)❩](t1⋔(p◖𝗦))] ⇔ t2
 .
 
 interpretation
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/fsubst_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/fsubst_eq.ma
index a6681b592..035a647a8 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/fsubst_eq.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/fsubst_eq.ma
@@ -26,8 +26,8 @@ lemma subset_inclusion_fsubst_bi (t1) (t2) (u1) (u2) (p):
 ]
 qed.
 
-lemma fsubst_eq_repl (t1) (t2) (u1) (u2) (p):
-      t1 ⇔ t2 → u1 ⇔ u2 → t1[⋔p←u1] ⇔ t2[⋔p←u2].
-#t1 #t2 #u1 #u2 #p * #H1t #H2t * #H1u #H2u
+lemma fsubst_eq_repl (t1) (t2) (u1) (u2) (p1) (p2):
+      t1 ⇔ t2 → p1 = p2 → u1 ⇔ u2 → t1[⋔p1←u1] ⇔ t2[⋔p2←u2].
+#t1 #t2 #u1 #u2 #p1 #p2 * #H1t #H2t #Hp * #H1u #H2u
 /3 width=5 by conj, subset_inclusion_fsubst_bi/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth_reverse.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth_reverse.ma
new file mode 100644
index 000000000..33f678ade
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth_reverse.ma
@@ -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/syntax/path_depth.ma".
+include "delayed_updating/syntax/path_reverse.ma".
+
+(* DEPTH FOR PATH ***********************************************************)
+
+(* Constructions with reverse ***********************************************)
+
+lemma depth_reverse (p):
+      ♭p = ♭pᴿ.
+#p elim p -p //
+* [ #n ] #p #IH <reverse_lcons
+[ <depth_d_sn <depth_d_dx //
+| <depth_m_sn <depth_m_dx //
+| <depth_L_sn <depth_L_dx //
+| <depth_A_sn <depth_A_dx //
+| <depth_S_sn <depth_S_dx //
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_structure.ma
new file mode 100644
index 000000000..c97fbfb2e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_structure.ma
@@ -0,0 +1,37 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_head.ma".
+include "delayed_updating/syntax/path_structure.ma".
+include "delayed_updating/syntax/path_depth.ma".
+
+(* HEAD FOR PATH ************************************************************)
+
+(* Constructions with structure *********************************************)
+
+axiom path_head_structure (p) (q):
+      ⊗p = ↳[♭⊗p]((⊗p)●q).
+(*
+#p elim p -p
+[ #q <path_head_zero //
+| * [ #n ] #p #IH #q
+  [ <structure_d_sn //
+  | <structure_m_sn //
+  | <structure_L_sn
+  | <structure_A_sn <list_append_lcons_sn 
+    <path_head_A_sn //
+  | <structure_S_sn //
+  ]
+]
+*)
\ No newline at end of file
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_labels.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_labels.ma
index a01c225ee..e7e1391da 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_labels.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_labels.ma
@@ -40,3 +40,11 @@ lemma labels_succ (l) (n):
 #l #n
 <labels_unfold <labels_unfold <niter_succ //
 qed.
+
+(* Basic inversions *********************************************************)
+
+lemma eq_inv_empty_labels (l) (n):
+      (𝐞) = l∗∗n → 𝟎 = n.
+#l #n @(nat_ind_succ … n) -n //
+#n #_ <labels_succ #H0 destruct
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_reverse.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_reverse.ma
index 43d9d85e8..046404f03 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_reverse.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_reverse.ma
@@ -51,3 +51,12 @@ lemma reverse_rcons (p) (l):
 #p #l
 <reverse_append //
 qed.
+
+(* Main constructions *******************************************************)
+
+theorem reverse_revrse (p):
+        p = pá´¿á´¿.
+#p elim p -p //
+#l #p #IH
+<reverse_lcons <reverse_rcons //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure_reverse.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure_reverse.ma
new file mode 100644
index 000000000..1cc32777b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure_reverse.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_structure.ma".
+include "delayed_updating/syntax/path_reverse.ma".
+
+(* STRUCTURE FOR PATH *******************************************************)
+
+(* Constructions with reverse ***********************************************)
+
+lemma structure_reverse (p):
+      (⊗p)ᴿ = ⊗(pᴿ).
+#p elim p -p //
+* [ #n ] #p #IH <reverse_lcons //
+[ <structure_d_sn <structure_d_dx //
+| <structure_m_sn <structure_m_dx //
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_eq.ma
index cb0826a0b..ed75b0d7a 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_eq.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_eq.ma
@@ -16,7 +16,6 @@ include "delayed_updating/unwind/unwind2_path.ma".
 include "delayed_updating/unwind/unwind2_rmap_eq.ma".
 include "delayed_updating/unwind/unwind_gen_eq.ma".
 include "ground/relocation/tr_compose_compose.ma".
-include "ground/relocation/tr_compose_pn.ma".
 
 (* UNWIND FOR PATH **********************************************************)
 
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_lift.ma
index 642056c89..71a169dbf 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_lift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_lift.ma
@@ -14,7 +14,6 @@
 
 include "delayed_updating/unwind/unwind2_path_eq.ma".
 include "delayed_updating/substitution/lift_eq.ma".
-include "ground/relocation/tr_uni_compose.ma".
 
 (* UNWIND FOR PATH **********************************************************)
 
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_structure.ma
index 7cb6190b1..f01ac1de7 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_structure.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_structure.ma
@@ -14,9 +14,6 @@
 
 include "delayed_updating/unwind/unwind2_path.ma".
 include "delayed_updating/unwind/unwind_gen_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 **********************************************************)
 
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_eq.ma
index cc15e01b5..7ca7f175c 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_eq.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_eq.ma
@@ -13,8 +13,10 @@
 (**************************************************************************)
 
 include "delayed_updating/unwind/unwind2_rmap.ma".
+include "ground/relocation/tr_uni_compose.ma".
 include "ground/relocation/tr_compose_eq.ma".
 include "ground/relocation/tr_pn_eq.ma".
+include "ground/arith/nat_rplus_pplus.ma".
 
 (* UNWIND MAP FOR PATH ******************************************************)
 
@@ -31,3 +33,10 @@ lemma unwind2_rmap_eq_repl (p):
 | /2 width=1 by/
 ]
 qed-.
+
+lemma tls_unwind2_rmap_d_sn (f) (p) (m) (n):
+      ⇂*[m+n]▶[f]p ≗ ⇂*[m]▶[f](𝗱n◗p).
+#f #p #m #n
+<unwind2_rmap_d_sn >nrplus_inj_dx
+/2 width=1 by tr_tls_compose_uni_dx/
+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 6b159e8c5..b7fc5664c 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
@@ -13,55 +13,63 @@
 (**************************************************************************)
 
 include "delayed_updating/unwind/unwind2_rmap_labels.ma".
+include "delayed_updating/unwind/unwind2_rmap_eq.ma".
 include "delayed_updating/unwind/xap.ma".
 include "delayed_updating/syntax/path_head_depth.ma".
+include "ground/lib/stream_eq_eq.ma".
 include "ground/arith/nat_le_plus.ma".
 
 (* UNWIND MAP FOR PATH ******************************************************)
 
 (* Constructions with path_head *********************************************)
 
-lemma unwind2_rmap_head_xap_closed (f) (p) (n) (k):
-      (∃q. p = (↳[n]p)●q) → k ≤ n →
+lemma unwind2_rmap_head_xap_le_closed (f) (p) (q) (n) (k):
+      p = (↳[n]p)●q → k ≤ n →
       (▶[f]p)＠❨k❩ = (▶[f]↳[n]p)＠❨k❩.
 #f #p elim p -p
-[ #n #k * #q #Hq #Hk
+[ #q #n #k #Hq #Hk
   elim (eq_inv_list_empty_append … Hq) -Hq * #_ //
-| #l #p #IH #n @(nat_ind_succ … n) -n
+| #l #p #IH #q #n @(nat_ind_succ … n) -n
   [ #k #_ #Hk <(nle_inv_zero_dx … Hk) -k -IH
    <path_head_zero <unwind2_rmap_empty //
-  | #n #_ #k cases l [ #m ] * #q
+  | #n #_ #k cases l [ #m ]
     [ <path_head_d_sn <list_append_lcons_sn #Hq #Hk
       elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
       <unwind2_rmap_d_sn <unwind2_rmap_d_sn
       <tr_compose_xap <tr_compose_xap
-      @IH [ /2 width=2 by ex_intro/ ] (**) (* auto too slow *)
+      @(IH … Hq) -IH -Hq (**) (* auto too slow *)
       @nle_trans [| @tr_uni_xap ]
       /2 width=1 by nle_plus_bi_dx/
     | <path_head_m_sn <list_append_lcons_sn #Hq #Hk
       elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
       <unwind2_rmap_m_sn <unwind2_rmap_m_sn
-      /3 width=2 by ex_intro/
+      /2 width=2 by/
     | <path_head_L_sn <list_append_lcons_sn #Hq
       @(nat_ind_succ … k) -k // #k #_ #Hk
       lapply (nle_inv_succ_bi … Hk) -Hk #Hk
       elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
       <unwind2_rmap_L_sn <unwind2_rmap_L_sn
       <tr_xap_push <tr_xap_push
-      /4 width=2 by ex_intro, eq_f/
+      /3 width=2 by eq_f/
     | <path_head_A_sn <list_append_lcons_sn #Hq #Hk
       elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
       <unwind2_rmap_A_sn <unwind2_rmap_A_sn
-      /3 width=2 by ex_intro/
+      /2 width=2 by/
     | <path_head_S_sn <list_append_lcons_sn #Hq #Hk
       elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
       <unwind2_rmap_S_sn <unwind2_rmap_S_sn
-      /3 width=2 by ex_intro/
+      /2 width=2 by/
     ]
   ]
 ]
 qed-.
 
+lemma unwind2_rmap_head_append_xap_closed (f) (p) (q) (n):
+      p = ↳[n](p●q) →
+      ▶[f](p●q)＠❨n❩ = ▶[f]↳[n](p●q)＠❨n❩.
+/2 width=2 by unwind2_rmap_head_xap_le_closed/
+qed-.
+
 lemma unwind2_rmap_head_xap (f) (p) (n):
       n + ♯(↳[n]p) = (▶[f]↳[n]p)＠❨n❩.
 #f #p elim p -p
@@ -79,3 +87,48 @@ lemma unwind2_rmap_head_xap (f) (p) (n):
   ]
 ]
 qed.
+
+lemma unwind2_rmap_append_pap_closed (f) (p) (q) (n:pnat):
+      p = ↳[n](p●q) →
+      ♭p = ninj (▶[f](p●q)＠⧣❨n❩).
+#f #p #q #n #Hn
+>tr_xap_ninj >Hn in ⊢ (??%?);
+>(unwind2_rmap_head_append_xap_closed … Hn) -Hn //
+qed.
+
+lemma tls_unwind2_rmap_plus_closed (f) (p) (q) (n) (k):
+      p = (↳[n]p)●q →
+      ⇂*[k]▶[f]q ≗ ⇂*[n+k]▶[f]p.
+#f #p elim p -p
+[ #q #n #k #Hq
+  elim (eq_inv_list_empty_append … Hq) -Hq #Hn #H0 destruct
+  <path_head_empty in Hn; #Hn
+  <(eq_inv_empty_labels … Hn) -n //
+| #l #p #IH #q #n @(nat_ind_succ … n) -n //
+  #n #_ #k cases l [ #m ]
+  [ <path_head_d_sn <list_append_lcons_sn #Hq
+    elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq <nrplus_inj_sn
+    @(stream_eq_trans … (tls_unwind2_rmap_d_sn …))
+    >nrplus_inj_dx >nrplus_inj_sn >nrplus_inj_sn <nplus_plus_comm_23
+    /2 width=1 by/
+  | <path_head_m_sn <list_append_lcons_sn #Hq
+    elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+    <unwind2_rmap_m_sn /2 width=1 by/
+  | <path_head_L_sn <list_append_lcons_sn #Hq
+    elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+    <unwind2_rmap_L_sn <nplus_succ_sn /2 width=1 by/
+  | <path_head_A_sn <list_append_lcons_sn #Hq
+    elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+    <unwind2_rmap_A_sn /2 width=2 by/
+  | <path_head_S_sn <list_append_lcons_sn #Hq
+    elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+    <unwind2_rmap_S_sn /2 width=2 by/
+  ]
+]
+qed-.
+
+lemma tls_unwind2_rmap_append_closed (f) (p) (q) (n):
+      p = ↳[n](p●q) →
+      ▶[f]q ≗ ⇂*[n]▶[f](p●q).
+/2 width=1 by tls_unwind2_rmap_plus_closed/
+qed.
-- 
2.39.2