From 5c2d38b46908f662cbb717156b29101ff30f8352 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Wed, 13 Jul 2022 20:19:56 +0200
Subject: [PATCH] partial commit in delayed_updating

+ reversing paths in component "substitution"
+ lift reformulated in terms of simpler functions
---
 .../delayed_updating/etc/lift_path.etc        |  69 ++++
 .../substitution/lift_constructors.ma         |  25 +-
 .../delayed_updating/substitution/lift_gen.ma | 150 ---------
 .../substitution/lift_gen_eq.ma               | 304 ------------------
 .../substitution/lift_path.ma                 | 145 +++++++++
 .../substitution/lift_path_after.ma           |  18 +-
 .../substitution/lift_path_eq.ma              |  32 ++
 .../substitution/lift_path_head.ma            |  49 ++-
 .../substitution/lift_path_id.ma              |  14 +-
 .../substitution/lift_path_length.ma          |  16 +-
 .../substitution/lift_path_proper.ma          |  15 +-
 .../substitution/lift_path_structure.ma       |  19 +-
 .../substitution/lift_path_uni.ma             |   7 +-
 .../substitution/lift_prototerm.ma            |   6 +-
 .../substitution/lift_prototerm_after.ma      |  29 ++
 .../substitution/lift_prototerm_eq.ma         |  15 +-
 .../substitution/lift_prototerm_id.ma         |   2 +-
 .../substitution/lift_rmap.ma                 |  91 ++++++
 ...ift_rmap_prelift.ma => lift_rmap_after.ma} |  19 +-
 .../substitution/lift_rmap_eq.ma              |  42 +++
 .../substitution/lift_rmap_head.ma            |  37 +--
 .../substitution/lift_rmap_id.ma              |   7 +-
 .../substitution/lift_rmap_pap.ma             |  24 ++
 .../substitution/prelift_label.ma             |  81 ++++-
 .../substitution/prelift_label_after.ma       |  25 ++
 .../substitution/prelift_label_eq.ma          |  28 ++
 .../substitution/prelift_label_id.ma          |  25 ++
 .../substitution/prelift_rmap.ma              |   6 +-
 ..._path_prelift.ma => prelift_rmap_after.ma} |  21 +-
 .../substitution/prelift_rmap_eq.ma           |  26 ++
 .../substitution/prelift_rmap_id.ma           |  25 ++
 31 files changed, 764 insertions(+), 608 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_path.etc
 delete mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_gen.ma
 delete mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_gen_eq.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_eq.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_after.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap.ma
 rename matita/matita/contribs/lambdadelta/delayed_updating/substitution/{lift_rmap_prelift.ma => lift_rmap_after.ma} (74%)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_eq.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_pap.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label_after.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label_eq.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label_id.ma
 rename matita/matita/contribs/lambdadelta/delayed_updating/substitution/{lift_path_prelift.ma => prelift_rmap_after.ma} (73%)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap_eq.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap_id.ma

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_path.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_path.etc
new file mode 100644
index 000000000..28e876c10
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_path.etc
@@ -0,0 +1,69 @@
+lemma lift_path_inv_d_sn (f) (p) (q) (k):
+      (𝗱k◗q) = ↑[f]p →
+      ∃∃r,h. k = f＠⧣❨h❩ & q = ↑[⇂*[h]f]r & 𝗱h◗r = p.
+#f * [| * [ #n ] #p ] #q #k
+[ <lift_path_empty
+| <lift_path_d_sn
+| <lift_path_m_sn
+| <lift_path_L_sn
+| <lift_path_A_sn
+| <lift_path_S_sn
+] #H destruct
+/2 width=5 by ex3_2_intro/
+qed-.
+
+lemma lift_path_inv_m_sn (f) (p) (q):
+      (𝗺◗q) = ↑[f]p →
+      ∃∃r. q = ↑[f]r & 𝗺◗r = p.
+#f * [| * [ #n ] #p ] #q
+[ <lift_path_empty
+| <lift_path_d_sn
+| <lift_path_m_sn
+| <lift_path_L_sn
+| <lift_path_A_sn
+| <lift_path_S_sn
+] #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lift_path_inv_L_sn (f) (p) (q):
+      (𝗟◗q) = ↑[f]p →
+      ∃∃r. q = ↑[⫯f]r & 𝗟◗r = p.
+#f * [| * [ #n ] #p ] #q
+[ <lift_path_empty
+| <lift_path_d_sn
+| <lift_path_m_sn
+| <lift_path_L_sn
+| <lift_path_A_sn
+| <lift_path_S_sn
+] #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lift_path_inv_A_sn (f) (p) (q):
+      (𝗔◗q) = ↑[f]p →
+      ∃∃r. q = ↑[f]r & 𝗔◗r = p.
+#f * [| * [ #n ] #p ] #q
+[ <lift_path_empty
+| <lift_path_d_sn
+| <lift_path_m_sn
+| <lift_path_L_sn
+| <lift_path_A_sn
+| <lift_path_S_sn
+] #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lift_path_inv_S_sn (f) (p) (q):
+      (𝗦◗q) = ↑[f]p →
+      ∃∃r. q = ↑[f]r & 𝗦◗r = p.
+#f * [| * [ #n ] #p ] #q
+[ <lift_path_empty
+| <lift_path_d_sn
+| <lift_path_m_sn
+| <lift_path_L_sn
+| <lift_path_A_sn
+| <lift_path_S_sn
+] #H destruct
+/2 width=3 by ex2_intro/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma
index d36698b5f..e87ccb21f 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma
@@ -18,28 +18,29 @@ include "delayed_updating/syntax/prototerm_constructors_eq.ma".
 
 (* LIFT FOR PROTOTERM *******************************************************)
 
-lemma lift_term_iref_sn (f) (t:prototerm) (n:pnat):
-      (𝛕f＠⧣❨n❩.↑[⇂*[n]f]t) ⊆ ↑[f](𝛕n.t).
-#f #t #n #p * #q * #r #Hr #H1 #H2 destruct
-@(ex2_intro … (𝗱n◗𝗺◗r))
+lemma lift_term_iref_sn (f) (t:prototerm) (k:pnat):
+      (𝛕f＠⧣❨k❩.↑[⇂*[k]f]t) ⊆ ↑[f](𝛕k.t).
+#f #t #k #p * #q * #r #Hr #H1 #H2 destruct
+@(ex2_intro … (𝗱k◗𝗺◗r))
 /2 width=1 by in_comp_iref/
 qed-.
 
-lemma lift_term_iref_dx (f) (t) (n:pnat):
-      ↑[f](𝛕n.t) ⊆ 𝛕f＠⧣❨n❩.↑[⇂*[n]f]t.
-#f #t #n #p * #q #Hq #H0 destruct
+lemma lift_term_iref_dx (f) (t) (k:pnat):
+      ↑[f](𝛕k.t) ⊆ 𝛕f＠⧣❨k❩.↑[⇂*[k]f]t.
+#f #t #k #p * #q #Hq #H0 destruct
 elim (in_comp_inv_iref … Hq) -Hq #p #H0 #Hp destruct
+<lift_path_d_sn <lift_path_m_sn
 /3 width=1 by in_comp_iref, in_comp_lift_path_term/
 qed-.
 
-lemma lift_term_iref (f) (t) (n:pnat):
-      (𝛕f＠⧣❨n❩.↑[⇂*[n]f]t) ⇔ ↑[f](𝛕n.t).
+lemma lift_term_iref (f) (t) (k:pnat):
+      (𝛕f＠⧣❨k❩.↑[⇂*[k]f]t) ⇔ ↑[f](𝛕k.t).
 /3 width=1 by conj, lift_term_iref_sn, lift_term_iref_dx/
 qed.
 
-lemma lift_term_iref_uni (t) (m) (n):
-      (𝛕(n+m).t) ⇔ ↑[𝐮❨m❩](𝛕n.t).
-#t #m #n
+lemma lift_term_iref_uni (t) (n) (k):
+      (𝛕(k+n).t) ⇔ ↑[𝐮❨n❩](𝛕k.t).
+#t #n #k
 @(subset_eq_trans … (lift_term_iref …))
 <tr_uni_pap >nsucc_pnpred <tr_tls_succ_uni
 /3 width=1 by iref_eq_repl, lift_term_id/
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_gen.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_gen.ma
deleted file mode 100644
index b873ed303..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_gen.ma
+++ /dev/null
@@ -1,150 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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/notation/functions/uparrow_4.ma".
-include "delayed_updating/notation/functions/uparrow_2.ma".
-include "delayed_updating/syntax/path.ma".
-include "ground/relocation/tr_uni.ma".
-include "ground/relocation/tr_pap_tls.ma".
-
-(* LIFT FOR PATH ************************************************************)
-
-definition lift_continuation (A:Type[0]) ≝
-           tr_map → path → A.
-
-rec definition lift_gen (A:Type[0]) (k:lift_continuation A) (f) (p) on p ≝
-match p with
-[ list_empty     ⇒ k f (𝐞)
-| list_lcons l q ⇒
-  match l with
-  [ label_d n ⇒ lift_gen (A) (λg,p. k g (𝗱(f＠⧣❨n❩)◗p)) (⇂*[n]f) q
-  | label_m   ⇒ lift_gen (A) (λg,p. k g (𝗺◗p)) f q
-  | label_L   ⇒ lift_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
-  | label_A   ⇒ lift_gen (A) (λg,p. k g (𝗔◗p)) f q
-  | label_S   ⇒ lift_gen (A) (λg,p. k g (𝗦◗p)) f q
-  ]
-].
-
-interpretation
-  "lift (gneric)"
-  'UpArrow A k f p = (lift_gen A k f p).
-
-definition proj_path: lift_continuation … ≝
-           λf,p.p.
-
-definition proj_rmap: lift_continuation … ≝
-           λf,p.f.
-
-interpretation
-  "lift (path)"
-  'UpArrow f p = (lift_gen ? proj_path f p).
-
-interpretation
-  "lift (relocation map)"
-  'UpArrow p f = (lift_gen ? proj_rmap f p).
-
-(* Basic constructions ******************************************************)
-
-lemma lift_empty (A) (k) (f):
-      k f (𝐞) = ↑{A}❨k, f, 𝐞❩.
-// qed.
-
-lemma lift_d_sn (A) (k) (p) (n) (f):
-      ↑❨(λg,p. k g (𝗱(f＠⧣❨n❩)◗p)), ⇂*[n]f, p❩ = ↑{A}❨k, f, 𝗱n◗p❩.
-// qed.
-
-lemma lift_m_sn (A) (k) (p) (f):
-      ↑❨(λg,p. k g (𝗺◗p)), f, p❩ = ↑{A}❨k, f, 𝗺◗p❩.
-// qed.
-
-lemma lift_L_sn (A) (k) (p) (f):
-      ↑❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ↑{A}❨k, f, 𝗟◗p❩.
-// qed.
-
-lemma lift_A_sn (A) (k) (p) (f):
-      ↑❨(λg,p. k g (𝗔◗p)), f, p❩ = ↑{A}❨k, f, 𝗔◗p❩.
-// qed.
-
-lemma lift_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 lift_path_empty (f):
-      (𝐞) = ↑[f]𝐞.
-// qed.
-
-(* Basic constructions with proj_rmap ***************************************)
-
-lemma lift_rmap_empty (f):
-      f = ↑[𝐞]f.
-// qed.
-
-lemma lift_rmap_d_sn (f) (p) (n):
-      ↑[p](⇂*[ninj n]f) = ↑[𝗱n◗p]f.
-// qed.
-
-lemma lift_rmap_m_sn (f) (p):
-      ↑[p]f = ↑[𝗺◗p]f.
-// qed.
-
-lemma lift_rmap_L_sn (f) (p):
-      ↑[p](⫯f) = ↑[𝗟◗p]f.
-// qed.
-
-lemma lift_rmap_A_sn (f) (p):
-      ↑[p]f = ↑[𝗔◗p]f.
-// qed.
-
-lemma lift_rmap_S_sn (f) (p):
-      ↑[p]f = ↑[𝗦◗p]f.
-// qed.
-
-(* Advanced constructions with proj_rmap and path_append ********************)
-
-lemma lift_rmap_append (p2) (p1) (f):
-      ↑[p2]↑[p1]f = ↑[p1●p2]f.
-#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
-[ <lift_rmap_m_sn <lift_rmap_m_sn //
-| <lift_rmap_A_sn <lift_rmap_A_sn //
-| <lift_rmap_S_sn <lift_rmap_S_sn //
-]
-qed.
-
-(* Advanced constructions with proj_rmap and path_rcons *********************)
-
-lemma lift_rmap_d_dx (f) (p) (n):
-      ⇂*[ninj n](↑[p]f) = ↑[p◖𝗱n]f.
-// qed.
-
-lemma lift_rmap_m_dx (f) (p):
-      ↑[p]f = ↑[p◖𝗺]f.
-// qed.
-
-lemma lift_rmap_L_dx (f) (p):
-      (⫯↑[p]f) = ↑[p◖𝗟]f.
-// qed.
-
-lemma lift_rmap_A_dx (f) (p):
-      ↑[p]f = ↑[p◖𝗔]f.
-// qed.
-
-lemma lift_rmap_S_dx (f) (p):
-      ↑[p]f = ↑[p◖𝗦]f.
-// qed.
-
-lemma lift_rmap_pap_d_dx (f) (p) (n) (m):
-      ↑[p]f＠⧣❨m+n❩ = ↑[p◖𝗱n]f＠⧣❨m❩+↑[p]f＠⧣❨n❩.
-// qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_gen_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_gen_eq.ma
deleted file mode 100644
index e589276a8..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_gen_eq.ma
+++ /dev/null
@@ -1,304 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "delayed_updating/substitution/lift_gen.ma".
-include "ground/relocation/tr_pap_pap.ma".
-include "ground/relocation/tr_pap_eq.ma".
-include "ground/relocation/tr_pn_eq.ma".
-include "ground/lib/stream_tls_plus.ma".
-include "ground/lib/stream_tls_eq.ma".
-include "ground/arith/nat_plus_rplus.ma".
-include "ground/arith/nat_rplus_pplus.ma".
-
-(* LIFT FOR PATH ************************************************************)
-
-definition lift_exteq (A): relation2 (lift_continuation A) (lift_continuation A) ≝
-           λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
-
-interpretation
-  "extensional equivalence (lift continuation)"
-  'RingEq A k1 k2 = (lift_exteq A k1 k2).
-
-(* Constructions with lift_exteq ********************************************)
-
-lemma lift_eq_repl (A) (p) (k1) (k2):
-      k1 ≗{A} k2 → stream_eq_repl … (λf1,f2. ↑❨k1, f1, p❩ = ↑❨k2, f2, p❩).
-#A #p elim p -p [| * [ #n ] #q #IH ]
-#k1 #k2 #Hk #f1 #f2 #Hf
-[ <lift_empty <lift_empty /2 width=1 by/
-| <lift_d_sn <lift_d_sn <(tr_pap_eq_repl … Hf)
-  /3 width=3 by stream_tls_eq_repl, compose_repl_fwd_sn/
-| /3 width=1 by/
-| /3 width=1 by tr_push_eq_repl/
-| /3 width=1 by/
-| /3 width=1 by/
-]
-qed-.
-
-(* Advanced constructions ***************************************************)
-
-lemma lift_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
-@lift_eq_repl // #g1 #g2 #p2 #Hg @Hk -Hk // (**) (* auto fail *)
-qed.
-
-lemma lift_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
-@lift_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 lift_path_eq_repl (p):
-      stream_eq_repl … (λf1,f2. ↑[f1]p = ↑[f2]p).
-/2 width=1 by lift_eq_repl/ qed.
-
-lemma lift_path_append_sn (p) (f) (q):
-      q●↑[f]p = ↑❨(λg,p. proj_path g (q●p)), f, p❩.
-#p elim p -p // * [ #n ] #p #IH #f #q
-[ <lift_d_sn <lift_d_sn
-| <lift_m_sn <lift_m_sn
-| <lift_L_sn <lift_L_sn
-| <lift_A_sn <lift_A_sn
-| <lift_S_sn <lift_S_sn
-] 
->lift_lcons_alt // >lift_append_rcons_sn //
-<IH <IH -IH <list_append_rcons_sn //
-qed.
-
-lemma lift_path_lcons (f) (p) (l):
-      l◗↑[f]p = ↑❨(λg,p. proj_path g (l◗p)), f, p❩.
-#f #p #l
->lift_lcons_alt <lift_path_append_sn //
-qed.
-
-lemma lift_path_d_sn (f) (p) (n):
-      (𝗱(f＠⧣❨n❩)◗↑[⇂*[n]f]p) = ↑[f](𝗱n◗p).
-// qed.
-
-lemma lift_path_m_sn (f) (p):
-      (𝗺◗↑[f]p) = ↑[f](𝗺◗p).
-// qed.
-
-lemma lift_path_L_sn (f) (p):
-      (𝗟◗↑[⫯f]p) = ↑[f](𝗟◗p).
-// qed.
-
-lemma lift_path_A_sn (f) (p):
-      (𝗔◗↑[f]p) = ↑[f](𝗔◗p).
-// qed.
-
-lemma lift_path_S_sn (f) (p):
-      (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
-// qed.
-
-lemma lift_path_append (p2) (p1) (f):
-      (↑[f]p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
-#p2 #p1 elim p1 -p1 //
-* [ #n1 ] #p1 #IH #f
-[ <lift_path_d_sn <lift_path_d_sn <IH //
-| <lift_path_m_sn <lift_path_m_sn <IH //
-| <lift_path_L_sn <lift_path_L_sn <IH //
-| <lift_path_A_sn <lift_path_A_sn <IH //
-| <lift_path_S_sn <lift_path_S_sn <IH //
-]
-qed.
-
-lemma lift_path_d_dx (f) (p) (n):
-      (↑[f]p)◖𝗱((↑[p]f)＠⧣❨n❩) = ↑[f](p◖𝗱n).
-#f #p #n <lift_path_append //
-qed.
-
-lemma lift_path_m_dx (f) (p):
-      (↑[f]p)◖𝗺 = ↑[f](p◖𝗺).
-#f #p <lift_path_append //
-qed.
-
-lemma lift_path_L_dx (f) (p):
-      (↑[f]p)◖𝗟 = ↑[f](p◖𝗟).
-#f #p <lift_path_append //
-qed.
-
-lemma lift_path_A_dx (f) (p):
-      (↑[f]p)◖𝗔 = ↑[f](p◖𝗔).
-#f #p <lift_path_append //
-qed.
-
-lemma lift_path_S_dx (f) (p):
-      (↑[f]p)◖𝗦 = ↑[f](p◖𝗦).
-#f #p <lift_path_append //
-qed.
-
-(* Advanced constructions with proj_rmap ************************************)
-
-lemma lift_rmap_eq_repl (p):
-      stream_eq_repl … (λf1,f2. ↑[p]f1 ≗ ↑[p]f2).
-#p elim p -p //
-* [ #n ] #p #IH #f1 #f2 #Hf
-[ /3 width=1 by stream_tls_eq_repl/
-| /2 width=1 by/
-| /3 width=1 by tr_push_eq_repl/
-| /2 width=1 by/
-| /2 width=1 by/
-]
-qed.
-
-lemma tls_lift_rmap_d_dx (f) (p) (m) (n):
-      ⇂*[m+n]↑[p]f ≗ ⇂*[m]↑[p◖𝗱n]f.
-#f #p #m #n
-<lift_rmap_d_dx >nrplus_inj_dx >nrplus_inj_sn //
-qed.
-
-(* Advanced inversions with proj_path ***************************************)
-
-lemma lift_path_inv_empty (f) (p):
-      (𝐞) = ↑[f]p → 𝐞 = p.
-#f * // * [ #n ] #p
-[ <lift_path_d_sn
-| <lift_path_m_sn
-| <lift_path_L_sn
-| <lift_path_A_sn
-| <lift_path_S_sn
-] #H destruct
-qed-.
-
-lemma lift_path_inv_d_sn (f) (p) (q) (k):
-      (𝗱k◗q) = ↑[f]p →
-      ∃∃r,h. k = f＠⧣❨h❩ & q = ↑[⇂*[h]f]r & 𝗱h◗r = p.
-#f * [| * [ #n ] #p ] #q #k
-[ <lift_path_empty
-| <lift_path_d_sn
-| <lift_path_m_sn
-| <lift_path_L_sn
-| <lift_path_A_sn
-| <lift_path_S_sn
-] #H destruct
-/2 width=5 by ex3_2_intro/
-qed-.
-
-lemma lift_path_inv_m_sn (f) (p) (q):
-      (𝗺◗q) = ↑[f]p →
-      ∃∃r. q = ↑[f]r & 𝗺◗r = p.
-#f * [| * [ #n ] #p ] #q
-[ <lift_path_empty
-| <lift_path_d_sn
-| <lift_path_m_sn
-| <lift_path_L_sn
-| <lift_path_A_sn
-| <lift_path_S_sn
-] #H destruct
-/2 width=3 by ex2_intro/
-qed-.
-
-lemma lift_path_inv_L_sn (f) (p) (q):
-      (𝗟◗q) = ↑[f]p →
-      ∃∃r. q = ↑[⫯f]r & 𝗟◗r = p.
-#f * [| * [ #n ] #p ] #q
-[ <lift_path_empty
-| <lift_path_d_sn
-| <lift_path_m_sn
-| <lift_path_L_sn
-| <lift_path_A_sn
-| <lift_path_S_sn
-] #H destruct
-/2 width=3 by ex2_intro/
-qed-.
-
-lemma lift_path_inv_A_sn (f) (p) (q):
-      (𝗔◗q) = ↑[f]p →
-      ∃∃r. q = ↑[f]r & 𝗔◗r = p.
-#f * [| * [ #n ] #p ] #q
-[ <lift_path_empty
-| <lift_path_d_sn
-| <lift_path_m_sn
-| <lift_path_L_sn
-| <lift_path_A_sn
-| <lift_path_S_sn
-] #H destruct
-/2 width=3 by ex2_intro/
-qed-.
-
-lemma lift_path_inv_S_sn (f) (p) (q):
-      (𝗦◗q) = ↑[f]p →
-      ∃∃r. q = ↑[f]r & 𝗦◗r = p.
-#f * [| * [ #n ] #p ] #q
-[ <lift_path_empty
-| <lift_path_d_sn
-| <lift_path_m_sn
-| <lift_path_L_sn
-| <lift_path_A_sn
-| <lift_path_S_sn
-] #H destruct
-/2 width=3 by ex2_intro/
-qed-.
-
-lemma lift_path_inv_append_sn (q2) (q1) (p) (f):
-      q1●q2 = ↑[f]p →
-      ∃∃p1,p2. q1 = ↑[f]p1 & q2 = ↑[↑[p1]f]p2 & p1●p2 = p.
-#q2 #q1 elim q1 -q1
-[| * [ #n1 ] #q1 #IH ] #p #f
-[ <list_append_empty_sn #H0 destruct
-  /2 width=5 by ex3_2_intro/
-| <list_append_lcons_sn #H0
-  elim (lift_path_inv_d_sn … H0) -H0 #r1 #m1 #H1 #H0 #H2 destruct
-  elim (IH … H0) -IH -H0 #p1 #p2 #H1 #H2 #H3 destruct
-  /2 width=5 by ex3_2_intro/
-| <list_append_lcons_sn #H0
-  elim (lift_path_inv_m_sn … H0) -H0 #r1 #H0 #H1 destruct
-  elim (IH … H0) -IH -H0 #p1 #p2 #H1 #H2 #H3 destruct
-  /2 width=5 by ex3_2_intro/
-| <list_append_lcons_sn #H0
-  elim (lift_path_inv_L_sn … H0) -H0 #r1 #H0 #H1 destruct
-  elim (IH … H0) -IH -H0 #p1 #p2 #H1 #H2 #H3 destruct
-  /2 width=5 by ex3_2_intro/
-| <list_append_lcons_sn #H0
-  elim (lift_path_inv_A_sn … H0) -H0 #r1 #H0 #H1 destruct
-  elim (IH … H0) -IH -H0 #p1 #p2 #H1 #H2 #H3 destruct
-  /2 width=5 by ex3_2_intro/
-| <list_append_lcons_sn #H0
-  elim (lift_path_inv_S_sn … H0) -H0 #r1 #H0 #H1 destruct
-  elim (IH … H0) -IH -H0 #p1 #p2 #H1 #H2 #H3 destruct
-  /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Main inversions with proj_path *******************************************)
-
-theorem lift_path_inj (q:path) (p) (f):
-        ↑[f]q = ↑[f]p → q = p.
-#q elim q -q [| * [ #k ] #q #IH ] #p #f
-[ <lift_path_empty #H0
-  <(lift_path_inv_empty … H0) -H0 //
-| <lift_path_d_sn #H0
-  elim (lift_path_inv_d_sn … H0) -H0 #r #h #H0
-  >(tr_pap_inj ???? H0) -k [1,3: // ] #Hr #H0 destruct
-| <lift_path_m_sn #H0
-  elim (lift_path_inv_m_sn … H0) -H0 #r #Hr #H0 destruct
-| <lift_path_L_sn #H0
-  elim (lift_path_inv_L_sn … H0) -H0 #r #Hr #H0 destruct
-| <lift_path_A_sn #H0
-  elim (lift_path_inv_A_sn … H0) -H0 #r #Hr #H0 destruct
-| <lift_path_S_sn #H0
-  elim (lift_path_inv_S_sn … H0) -H0 #r #Hr #H0 destruct
-]
-<(IH … Hr) -r -IH //
-qed-.
-
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path.ma
new file mode 100644
index 000000000..1ffe94862
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path.ma
@@ -0,0 +1,145 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "delayed_updating/substitution/prelift_label.ma".
+include "delayed_updating/substitution/lift_rmap.ma".
+include "ground/xoa/ex_3_2.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+rec definition lift_path (f) (p) on p: path ≝
+match p with
+[ list_empty     ⇒ (𝐞)
+| list_lcons l q ⇒ (lift_path f q)◖↑[↑[q]f]l
+].
+
+interpretation
+  "lift (path)"
+  'UpArrow f l = (lift_path f l).
+
+(* Basic constructions ******************************************************)
+
+lemma lift_path_empty (f):
+      (𝐞) = ↑[f]𝐞.
+// qed.
+
+lemma lift_path_rcons (f) (p) (l):
+      (↑[f]p)◖↑[↑[p]f]l = ↑[f](p◖l).
+// qed.
+
+lemma lift_path_d_dx (f) (p) (k):
+      (↑[f]p)◖𝗱((↑[p]f)＠⧣❨k❩) = ↑[f](p◖𝗱k).
+// qed.
+
+lemma lift_path_m_dx (f) (p):
+      (↑[f]p)◖𝗺 = ↑[f](p◖𝗺).
+// qed.
+
+lemma lift_path_L_dx (f) (p):
+      (↑[f]p)◖𝗟 = ↑[f](p◖𝗟).
+// qed.
+
+lemma lift_path_A_dx (f) (p):
+      (↑[f]p)◖𝗔 = ↑[f](p◖𝗔).
+// qed.
+
+lemma lift_path_S_dx (f) (p):
+      (↑[f]p)◖𝗦 = ↑[f](p◖𝗦).
+// qed.
+
+(* Constructions with path_append *******************************************)
+
+lemma lift_path_append (f) (p) (q):
+      (↑[f]p)●(↑[↑[p]f]q) = ↑[f](p●q).
+#f #p #q elim q -q //
+#l #q #IH
+<lift_path_rcons <lift_path_rcons
+<list_append_lcons_sn //
+qed.
+
+(* Constructions with path_lcons ********************************************)
+
+lemma lift_path_lcons (f) (p) (l):
+      (↑[f]l)◗↑[↑[l]f]p = ↑[f](l◗p).
+#f #p #l
+<lift_path_append //
+qed.
+
+lemma lift_path_d_sn (f) (p) (k:pnat):
+      (𝗱(f＠⧣❨k❩)◗↑[⇂*[k]f]p) = ↑[f](𝗱k◗p).
+// qed.
+
+lemma lift_path_m_sn (f) (p):
+      (𝗺◗↑[f]p) = ↑[f](𝗺◗p).
+// qed.
+
+lemma lift_path_L_sn (f) (p):
+      (𝗟◗↑[⫯f]p) = ↑[f](𝗟◗p).
+// qed.
+
+lemma lift_path_A_sn (f) (p):
+      (𝗔◗↑[f]p) = ↑[f](𝗔◗p).
+// qed.
+
+lemma lift_path_S_sn (f) (p):
+      (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
+// qed.
+
+(* Basic inversions *********************************************************)
+
+lemma lift_path_inv_empty (f) (p):
+      (𝐞) = ↑[f]p → 𝐞 = p.
+#f * // #p #l
+<lift_path_rcons #H0 destruct
+qed-.
+
+lemma lift_path_inv_rcons (f) (p2) (q1) (l1):
+      q1◖l1 = ↑[f]p2 →
+      ∃∃q2,l2. q1 = ↑[f]q2 & l1 = ↑[↑[q2]f]l2 & q2◖l2 = p2.
+#f * [| #l2 #q2 ] #q1 #l1
+[ <lift_path_empty
+| <lift_path_rcons
+]
+#H0 destruct
+/2 width=5 by ex3_2_intro/
+qed-.
+
+(* Inversions with path_append **********************************************)
+
+lemma lift_path_inv_append_sn (f) (q1) (r1) (p2):
+      q1●r1 = ↑[f]p2 →
+      ∃∃q2,r2. q1 = ↑[f]q2 & r1 = ↑[↑[q2]f]r2 & q2●r2 = p2.
+#f #q1 #r1 elim r1 -r1 [| #l1 #r1 #IH ] #p2
+[ <list_append_empty_sn #H0 destruct
+  /2 width=5 by ex3_2_intro/
+| <list_append_lcons_sn #H0
+  elim (lift_path_inv_rcons … H0) -H0 #x2 #l2 #H0 #H1 #H2 destruct
+  elim (IH … H0) -IH -H0 #q2 #r2 #H1 #H2 #H3 destruct
+  /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Main inversions **********************************************************)
+
+theorem lift_path_inj (f) (p1) (p2):
+        ↑[f]p1 = ↑[f]p2 → p1 = p2.
+#f #p1 elim p1 -p1 [| #l1 #q1 #IH ] #p2
+[ <lift_path_empty #H0
+  <(lift_path_inv_empty … H0) -H0 //
+| <lift_path_rcons #H0
+  elim (lift_path_inv_rcons … H0) -H0 #q2 #l2 #Hq
+  <(IH … Hq) -IH -q2 #Hl #H0 destruct
+  <(prelift_label_inj … Hl) -l2 //
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_after.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_after.ma
index 1bdf3c592..38bded30e 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_after.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_after.ma
@@ -12,20 +12,16 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_gen_eq.ma".
-include "ground/relocation/tr_compose_pap.ma".
-include "ground/relocation/tr_compose_pn.ma".
-include "ground/relocation/tr_compose_tls.ma".
+include "delayed_updating/substitution/lift_rmap_after.ma".
+include "delayed_updating/substitution/prelift_label_after.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
 (* Constructions with tr_after **********************************************)
 
-lemma lift_path_after (p) (f1) (f2):
-      ↑[f2]↑[f1]p = ↑[f2∘f1]p.
-#p elim p -p [| * ] // [ #n ] #p #IH #f1 #f2
-[ <lift_path_d_sn <lift_path_d_sn <lift_path_d_sn //
-| <lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
-  >tr_compose_push_bi //
-]
+lemma lift_path_after (g) (f) (p):
+      ↑[g]↑[f]p = ↑[g∘f]p.
+#g #f #p elim p -p //
+#l #p #IH
+<lift_path_rcons <lift_path_rcons  <IH -IH //
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_eq.ma
new file mode 100644
index 000000000..b9574e479
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_eq.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/substitution/lift_path.ma".
+include "delayed_updating/substitution/lift_rmap_eq.ma".
+include "delayed_updating/substitution/prelift_label_eq.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+(* Constructions with path_eq ***********************************************)
+
+lemma lift_path_eq_repl (p):
+      stream_eq_repl … (λf1,f2. ↑[f1]p = ↑[f2]p).
+#p elim p -p //
+#l #p #IH #f1 #f2 #Hf
+<lift_path_rcons <lift_path_rcons
+@eq_f2 (**) (* auto fails *)
+[ /3 width=1 by lift_rmap_eq_repl, prelift_label_eq_repl/
+| @IH -IH //
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_head.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_head.ma
index 6bbffaa77..21b70c57a 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_head.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_head.ma
@@ -12,49 +12,48 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_gen_eq.ma".
+include "delayed_updating/substitution/lift_path.ma".
 include "delayed_updating/syntax/path_head.ma".
-include "delayed_updating/syntax/path_reverse.ma".
 include "ground/relocation/xap.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
 (* Constructions with head for path *****************************************)
 
-lemma lift_path_head (f) (p) (q) (n):
-      pᴿ = ↳[n](pᴿ) →
-      ↳[↑[q●p]f＠❨n❩](↑[↑[q]f]p)ᴿ = (↑[↑[q]f]p)ᴿ.
-#f #p @(list_ind_rcons … p) -p
-[ #q #n #H0
+lemma lift_path_head_closed (f) (p) (q) (n):
+      q = ↳[n]q →
+      ↳[↑[p●q]f＠❨n❩]↑[↑[p]f]q = ↑[↑[p]f]q.
+#f #p #q elim q -q
+[ #n #H0
   <(eq_inv_path_empty_head … H0) -H0
   <path_head_zero //
-| #p #l #IH #q #n @(nat_ind_succ …n) -n [| #m #_ ]
-  [ <reverse_rcons <path_head_zero #H0 destruct
-  | cases l [ #n ]
-    [ <reverse_rcons <path_head_d_sn #H0
+| #l #q #IH #n @(nat_ind_succ …n) -n [| #n #_ ]
+  [ <path_head_zero #H0 destruct
+  | cases l [ #k ]
+    [ <path_head_d_dx #H0
       elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
-      <list_append_assoc <lift_rmap_d_dx <lift_path_d_dx <reverse_rcons
-      <tr_xap_succ_nap <path_head_d_sn
+      <lift_rmap_d_dx <lift_path_d_dx
+      <tr_xap_succ_nap <path_head_d_dx
       <(IH … H0) in ⊢ (???%); -IH -H0 <tr_xap_plus //
-    | <reverse_rcons <path_head_m_sn #H0
+    | <path_head_m_dx #H0
       elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
-      <list_append_assoc <lift_rmap_m_dx <lift_path_m_dx <reverse_rcons
-      <tr_xap_succ_nap <path_head_m_sn
+      <lift_rmap_m_dx <lift_path_m_dx
+      <tr_xap_succ_nap <path_head_m_dx
       <(IH … H0) in ⊢ (???%); -IH -H0 //
-    | <reverse_rcons <path_head_L_sn #H0
+    | <path_head_L_dx #H0
       elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
-      <list_append_assoc <lift_rmap_L_dx <lift_path_L_dx <reverse_rcons
-      <tr_xap_succ_nap <path_head_L_sn
+      <lift_rmap_L_dx <lift_path_L_dx
+      <tr_xap_succ_nap <path_head_L_dx
       <(IH … H0) in ⊢ (???%); -IH -H0 //
-    | <reverse_rcons <path_head_A_sn #H0
+    | <path_head_A_dx #H0
       elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
-      <list_append_assoc <lift_rmap_A_dx <lift_path_A_dx <reverse_rcons
-      <tr_xap_succ_nap <path_head_A_sn
+      <lift_rmap_A_dx <lift_path_A_dx
+      <tr_xap_succ_nap <path_head_A_dx
       <(IH … H0) in ⊢ (???%); -IH -H0 //
-    | <reverse_rcons <path_head_S_sn #H0
+    | <path_head_S_dx #H0
       elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
-      <list_append_assoc <lift_rmap_S_dx <lift_path_S_dx <reverse_rcons
-      <tr_xap_succ_nap <path_head_S_sn
+      <lift_rmap_S_dx <lift_path_S_dx
+      <tr_xap_succ_nap <path_head_S_dx
       <(IH … H0) in ⊢ (???%); -IH -H0 //
     ]
   ]
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_id.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_id.ma
index cac770811..665b5f9a6 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_id.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_id.ma
@@ -12,19 +12,17 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_gen_eq.ma".
-include "ground/relocation/tr_id_pap.ma".
-include "ground/relocation/tr_id_tls.ma".
+include "delayed_updating/substitution/lift_path.ma".
+include "delayed_updating/substitution/lift_rmap_id.ma".
+include "delayed_updating/substitution/prelift_label_id.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
-(* Constructions with proj_path and 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 //
-]
+#l #p #IH
+<lift_path_rcons //
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_length.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_length.ma
index 755f8d16e..635a13e1d 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_length.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_length.ma
@@ -12,21 +12,21 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_gen_eq.ma".
+include "delayed_updating/substitution/lift_path.ma".
 include "ground/lib/list_length.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
 (* Constructions with list_length *******************************************)
 
-lemma lift_path_length (p) (f):
+lemma lift_path_length (f) (p):
       ❘p❘ = ❘↑[f]p❘.
-#p elim p -p // * [ #n ] #p #IH #f
-[ <lift_path_d_sn
-| <lift_path_m_sn
-| <lift_path_L_sn
-| <lift_path_A_sn
-| <lift_path_S_sn
+#f #p elim p -p // * [ #k ] #p #IH
+[ <lift_path_d_dx
+| <lift_path_m_dx
+| <lift_path_L_dx
+| <lift_path_A_dx
+| <lift_path_S_dx
 ]
 >list_length_lcons >list_length_lcons //
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_proper.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_proper.ma
index 1473e819f..c87020d2f 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_proper.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_proper.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_gen_eq.ma".
+include "delayed_updating/substitution/lift_path.ma".
 include "delayed_updating/syntax/path_proper.ma".
 
 (* LIFT FOR PATH ************************************************************)
@@ -23,13 +23,14 @@ lemma lift_path_proper (f) (p):
       p ϵ 𝐏 → ↑[f]p ϵ 𝐏.
 #f *
 [ #H0 elim (ppc_inv_empty … H0)
-| * [ #n ] #p #_
-  [ <lift_path_d_sn /2 width=3 by ppc_lcons/
-  | <lift_path_m_sn /2 width=3 by ppc_lcons/
-  | <lift_path_L_sn /2 width=3 by ppc_lcons/
-  | <lift_path_A_sn /2 width=3 by ppc_lcons/
-  | <lift_path_S_sn /2 width=3 by ppc_lcons/
+| * [ #k ] #p #_
+  [ <lift_path_d_dx
+  | <lift_path_m_dx
+  | <lift_path_L_dx
+  | <lift_path_A_dx
+  | <lift_path_S_dx
   ]
+  /2 width=3 by ppc_rcons/
 ]
 qed.
 
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_structure.ma
index ece3e93e6..f4b7bdd0e 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_structure.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_structure.ma
@@ -12,25 +12,24 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_gen_eq.ma".
+include "delayed_updating/substitution/lift_path.ma".
 include "delayed_updating/syntax/path_structure.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
 (* Constructions with structure *********************************************)
 
-lemma structure_lift_path (p) (f):
+lemma structure_lift_path (f) (p):
       ⊗p = ⊗↑[f]p.
-#p elim p -p //
-* [ #n ] #p #IH #f //
-[ <lift_path_d_sn <structure_d_sn <structure_d_sn //
-| <lift_path_m_sn <structure_m_sn <structure_m_sn //
-| <lift_path_L_sn <structure_L_sn <structure_L_sn //
+#f #p elim p -p //
+* [ #k ] #p #IH //
+[ <lift_path_d_dx <structure_d_dx <structure_d_dx //
+| <lift_path_m_dx <structure_m_dx <structure_m_dx //
 ]
 qed.
 
-lemma lift_path_structure (p) (f):
+lemma lift_path_structure (f) (p):
       ⊗p = ↑[f]⊗p.
-#p elim p -p //
-* [ #n ] #p #IH #f //
+#f #p elim p -p //
+* [ #k ] #p #IH //
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_uni.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_uni.ma
index f68010b32..094330658 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_uni.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_uni.ma
@@ -15,14 +15,15 @@
 include "delayed_updating/substitution/lift_path_id.ma".
 include "ground/relocation/tr_uni_pap.ma".
 include "ground/relocation/tr_uni_tls.ma".
+include "ground/arith/nat_pred_succ.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
 (* Constructions with tr_uni ************************************************)
 
-lemma lift_path_d_sn_uni (p) (m) (n):
-      (𝗱(n+m)◗p) = ↑[𝐮❨m❩](𝗱(n)◗p).
-#p #m #n
+lemma lift_path_d_sn_uni (p) (n) (k):
+      (𝗱(k+n)◗p) = ↑[𝐮❨n❩](𝗱k◗p).
+#p #n #k
 <lift_path_d_sn <tr_uni_pap >nsucc_pnpred
 <tr_tls_succ_uni //
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm.ma
index 85aa8a0bb..094cf676f 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_gen.ma".
+include "delayed_updating/substitution/lift_path.ma".
 include "delayed_updating/syntax/prototerm.ma".
 include "ground/lib/subset_ext.ma".
 
@@ -20,11 +20,11 @@ include "ground/lib/subset_ext.ma".
 
 interpretation
   "lift (prototerm)"
-  'UpArrow f t = (subset_ext_f1 ? ? (lift_gen ? proj_path f) t).
+  'UpArrow f t = (subset_ext_f1 ? ? (lift_path f) t).
 
 (* Basic constructions ******************************************************)
 
-lemma in_comp_lift_path_term (f) (p) (t):
+lemma in_comp_lift_path_term (f) (t) (p):
       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/substitution/lift_prototerm_after.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_after.ma
new file mode 100644
index 000000000..862e2d4cb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_after.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/substitution/lift_prototerm_eq.ma".
+include "delayed_updating/substitution/lift_path_after.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+(* Constructions with tr_after **********************************************)
+
+lemma lift_term_after (f) (g) (t):
+      ↑[g]↑[f]t ⇔ ↑[g∘f]t.
+#f #g #t @subset_eq_trans
+[
+| @subset_inclusion_ext_f1_compose
+| @subset_equivalence_ext_f1_exteq /2 width=5/
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_eq.ma
index afe2ddd01..d23c7f058 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_eq.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_eq.ma
@@ -12,8 +12,9 @@
 (*                                                                        *)
 (**************************************************************************)
 
+(**) (* reverse include *)
 include "ground/lib/subset_ext_equivalence.ma".
-include "delayed_updating/substitution/lift_path_after.ma".
+include "delayed_updating/substitution/lift_path_eq.ma".
 include "delayed_updating/substitution/lift_prototerm.ma".
 
 (* LIFT FOR PROTOTERM *******************************************************)
@@ -30,20 +31,10 @@ lemma lift_term_eq_repl_dx (f) (t1) (t2):
 /2 width=1 by subset_equivalence_ext_f1_bi/
 qed.
 
-lemma lift_term_after (f1) (f2) (t):
-      ↑[f2]↑[f1]t ⇔ ↑[f2∘f1]t.
-#f1 #f2 #t @subset_eq_trans
-[
-| @subset_inclusion_ext_f1_compose
-| @subset_equivalence_ext_f1_exteq /2 width=5/
-]
-qed.
-
 lemma lift_term_grafted_sn (f) (t) (p):
       ↑[↑[p]f](t⋔p) ⊆ (↑[f]t)⋔(↑[f]p).
 #f #t #p #q * #r #Hr #H0 destruct
-@(ex2_intro … Hr) -Hr
-<lift_path_append_sn //
+/2 width=3 by ex2_intro/
 qed-.
 
 lemma lift_term_grafted_dx (f) (t) (p):
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma
index 11e39983b..95aecceb7 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma
@@ -33,5 +33,5 @@ qed-.
 
 lemma lift_term_id (t):
       t ⇔ ↑[𝐢]t.
-/3 width=2 by lift_term_id_dx, lift_term_id_sn, conj/      
+/3 width=2 by lift_term_id_dx, lift_term_id_sn, conj/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap.ma
new file mode 100644
index 000000000..85f41d489
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap.ma
@@ -0,0 +1,91 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "delayed_updating/substitution/prelift_rmap.ma".
+include "delayed_updating/syntax/path.ma".
+
+(* LIFT FOR RELOCATION MAP **************************************************)
+
+rec definition lift_rmap (f) (p) on p: tr_map ≝
+match p with
+[ list_empty     ⇒ f
+| list_lcons l q ⇒ ↑[l](lift_rmap f q)
+].
+
+interpretation
+  "lift (relocation map)"
+  'UpArrow p f = (lift_rmap f p).
+
+(* Basic constructions ******************************************************)
+
+lemma lift_rmap_empty (f):
+      f = ↑[𝐞]f.
+// qed.
+
+lemma lift_rmap_rcons (f) (p) (l):
+      ↑[l]↑[p]f = ↑[p◖l]f.
+// qed.
+
+lemma lift_rmap_d_dx (f) (p) (k:pnat):
+      ⇂*[k](↑[p]f) = ↑[p◖𝗱k]f.
+// qed.
+
+lemma lift_rmap_m_dx (f) (p):
+      ↑[p]f = ↑[p◖𝗺]f.
+// qed.
+
+lemma lift_rmap_L_dx (f) (p):
+      (⫯↑[p]f) = ↑[p◖𝗟]f.
+// qed.
+
+lemma lift_rmap_A_dx (f) (p):
+      ↑[p]f = ↑[p◖𝗔]f.
+// qed.
+
+lemma lift_rmap_S_dx (f) (p):
+      ↑[p]f = ↑[p◖𝗦]f.
+// qed.
+
+(* Constructions with path_append *******************************************)
+
+lemma lift_rmap_append (p) (q) (f):
+      ↑[q]↑[p]f = ↑[p●q]f.
+#p #q elim q -q //
+qed.
+
+(* Constructions with path_lcons ********************************************)
+
+lemma lift_rmap_lcons (f) (p) (l):
+      ↑[p]↑[l]f = ↑[l◗p]f.
+// qed.
+
+lemma lift_rmap_d_sn (f) (p) (k:pnat):
+      ↑[p](⇂*[k]f) = ↑[𝗱k◗p]f.
+// qed.
+
+lemma lift_rmap_m_sn (f) (p):
+      ↑[p]f = ↑[𝗺◗p]f.
+// qed.
+
+lemma lift_rmap_L_sn (f) (p):
+      ↑[p](⫯f) = ↑[𝗟◗p]f.
+// qed.
+
+lemma lift_rmap_A_sn (f) (p):
+      ↑[p]f = ↑[𝗔◗p]f.
+// qed.
+
+lemma lift_rmap_S_sn (f) (p):
+      ↑[p]f = ↑[𝗦◗p]f.
+// qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_prelift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_after.ma
similarity index 74%
rename from matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_prelift.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_after.ma
index ba6f96fcf..e209f97a7 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_prelift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_after.ma
@@ -12,19 +12,16 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_gen.ma".
-include "delayed_updating/substitution/prelift_rmap.ma".
+include "delayed_updating/substitution/prelift_rmap_after.ma".
+include "delayed_updating/substitution/lift_path.ma".
 
 (* LIFT FOR RELOCATION MAP **************************************************)
 
-(* Constructions with prelift_rmap ******************************************)
+(* Constructions with tr_after **********************************************)
 
-lemma lift_rmap_lcons_prelift (f) (p) (l):
-      ↑[p]↑[l]f = ↑[l◗p]f.
-#f #p * [ #n ] //
-qed.
-
-lemma lift_rmap_rcons_prelift (f) (p) (l):
-      ↑[l]↑[p]f = ↑[p◖l]f.
-#f #p * [ #n ] //
+lemma lift_rmap_after (g) (f) (p:path):
+      ↑[↑[f]p]g∘↑[p]f = ↑[p](g∘f).
+#g #f #p elim p -p //
+#l #p #IH
+<lift_rmap_rcons //
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_eq.ma
new file mode 100644
index 000000000..a2f841e43
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_eq.ma
@@ -0,0 +1,42 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "delayed_updating/substitution/lift_rmap.ma".
+include "ground/relocation/tr_pn_eq.ma".
+include "ground/lib/stream_tls_plus.ma".
+include "ground/lib/stream_tls_eq.ma".
+include "ground/arith/nat_plus_rplus.ma".
+include "ground/arith/nat_rplus_pplus.ma".
+
+(* LIFT FOR RELOCATION MAP **************************************************)
+
+(* Constructions with lift_eq ***********************************************)
+
+lemma lift_rmap_eq_repl (p):
+      stream_eq_repl … (λf1,f2. ↑[p]f1 ≗ ↑[p]f2).
+#p elim p -p //
+* [ #k ] #p #IH #f1 #f2 #Hf
+[ /3 width=1 by stream_tls_eq_repl/
+| /2 width=1 by/
+| /3 width=1 by tr_push_eq_repl/
+| /2 width=1 by/
+| /2 width=1 by/
+]
+qed.
+
+lemma tls_lift_rmap_d_dx (f) (p) (n) (k):
+      ⇂*[n+k]↑[p]f ≗ ⇂*[n]↑[p◖𝗱k]f.
+#f #p #n #k
+<lift_rmap_d_dx >nrplus_inj_dx >nrplus_inj_sn //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_head.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_head.ma
index 66209d837..033a2c13e 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_head.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_head.ma
@@ -12,49 +12,46 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_gen_eq.ma".
+include "delayed_updating/substitution/lift_rmap_eq.ma".
 include "delayed_updating/syntax/path_head.ma".
-include "delayed_updating/syntax/path_reverse.ma".
 include "ground/lib/stream_eq_eq.ma".
 
 (* LIFT MAP FOR PATH ********************************************************)
 
 (* Constructions with path_head *********************************************)
 
-lemma tls_plus_lift_rmap_reverse_closed (f) (q) (n) (k):
+lemma tls_plus_lift_rmap_closed (f) (q) (n) (m):
       q = ↳[n]q →
-      ⇂*[k]f ≗ ⇂*[n+k]↑[qᴿ]f.
+      ⇂*[m]f ≗ ⇂*[n+m]↑[q]f.
 #f #q elim q -q
-[ #n #k #Hq
+[ #n #m #Hq
   <(eq_inv_path_empty_head … Hq) -n //
 | #l #q #IH #n @(nat_ind_succ … n) -n //
-  #n #_ #k cases l [ #m ]
-  [ <path_head_d_sn #Hq
-    elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hp <nrplus_inj_sn
-    <reverse_lcons
+  #n #_ #m cases l [ #k ]
+  [ <path_head_d_dx #Hq
+    elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq <nrplus_inj_sn
     @(stream_eq_trans … (tls_lift_rmap_d_dx …))
     >nrplus_inj_dx >nrplus_inj_sn >nrplus_inj_sn <nplus_plus_comm_23
     >nsucc_unfold /2 width=1 by/
-  | <path_head_m_sn #Hq
+  | <path_head_m_dx #Hq
     elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
-    <reverse_lcons <lift_rmap_m_dx /2 width=1 by/
-  | <path_head_L_sn #Hq
+    <lift_rmap_m_dx /2 width=1 by/
+  | <path_head_L_dx #Hq
     elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
-    <reverse_lcons <lift_rmap_L_dx <nplus_succ_sn /2 width=1 by/
-  | <path_head_A_sn #Hq
+    <lift_rmap_L_dx <nplus_succ_sn /2 width=1 by/
+  | <path_head_A_dx #Hq
     elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
-    <reverse_lcons <lift_rmap_A_dx /2 width=2 by/
-  | <path_head_S_sn #Hq
+    <lift_rmap_A_dx /2 width=2 by/
+  | <path_head_S_dx #Hq
     elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
-    <reverse_lcons <lift_rmap_S_dx /2 width=2 by/
+    <lift_rmap_S_dx /2 width=2 by/
   ]
 ]
 qed-.
 
 lemma tls_lift_rmap_closed (f) (q) (n):
-      qᴿ = ↳[n](qᴿ) →
+      q = ↳[n]q →
       f ≗ ⇂*[n]↑[q]f.
 #f #q #n #H0
->(reverse_reverse q)
-/2 width=1 by tls_plus_lift_rmap_reverse_closed/
+/2 width=1 by tls_plus_lift_rmap_closed/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_id.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_id.ma
index 8787ecbb9..98d4ab395 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_id.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_id.ma
@@ -12,15 +12,14 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_gen.ma".
-include "ground/relocation/tr_id_tls.ma".
+include "delayed_updating/substitution/lift_rmap.ma".
+include "delayed_updating/substitution/prelift_rmap_id.ma".
 
 (* LIFT FOR RELOCATION MAP **************************************************)
 
-(* Constructions with proj_rmap and tr_id ***********************************)
+(* Constructions with tr_id *************************************************)
 
 lemma lift_rmap_id (p):
       (𝐢) = ↑[p]𝐢.
 #p elim p -p //
-* [ #n ] #p #IH //
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_pap.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_pap.ma
new file mode 100644
index 000000000..6e1b39255
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_pap.ma
@@ -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/substitution/lift_rmap.ma".
+include "ground/relocation/tr_pap_tls.ma".
+
+(* LIFT FOR RELOCATION MAP **************************************************)
+
+(* Constructions with tr_pap ************************************************)
+
+lemma lift_rmap_pap_d_dx (f) (p) (k) (h):
+      ↑[p]f＠⧣❨h+k❩ = ↑[p◖𝗱k]f＠⧣❨h❩+↑[p]f＠⧣❨k❩.
+// qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label.ma
index e4a9c0596..be57d8601 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label.ma
@@ -14,13 +14,13 @@
 
 include "delayed_updating/notation/functions/uparrow_2.ma".
 include "delayed_updating/syntax/label.ma".
-include "ground/relocation/tr_pap.ma".
+include "ground/relocation/tr_pap_pap.ma".
 
 (* PRELIFT FOR LABEL ********************************************************)
 
 definition prelift_label (f) (l): label ≝
 match l with
-[ label_d n ⇒ 𝗱(f＠⧣❨n❩)
+[ label_d k ⇒ 𝗱(f＠⧣❨k❩)
 | label_m   ⇒ 𝗺
 | label_L   ⇒ 𝗟
 | label_A   ⇒ 𝗔
@@ -33,8 +33,8 @@ interpretation
 
 (* Basic constructions ******************************************************)
 
-lemma prelift_label_d (f) (n):
-      (𝗱(f＠⧣❨n❩)) = ↑[f]𝗱n.
+lemma prelift_label_d (f) (k):
+      (𝗱(f＠⧣❨k❩)) = ↑[f]𝗱k.
 // qed.
 
 lemma prelift_label_m (f):
@@ -52,3 +52,76 @@ lemma prelift_label_A (f):
 lemma prelift_label_S (f):
       (𝗦) = ↑[f]𝗦.
 // qed.
+
+(* Basic inversions *********************************************************)
+
+lemma prelift_label_inv_d_sn (f) (l) (k1):
+      (𝗱k1) = ↑[f]l →
+      ∃∃k2. k1 = f＠⧣❨k2❩ & 𝗱k2 = l.
+#f * [ #k ] #k1
+[ <prelift_label_d
+| <prelift_label_m
+| <prelift_label_L
+| <prelift_label_A
+| <prelift_label_S
+] #H0 destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma prelift_label_inv_m_sn (f) (l):
+      (𝗺) = ↑[f]l → 𝗺 = l.
+#f * [ #k ]
+[ <prelift_label_d
+| <prelift_label_m
+| <prelift_label_L
+| <prelift_label_A
+| <prelift_label_S
+] #H0 destruct //
+qed-.
+
+lemma prelift_label_inv_L_sn (f) (l):
+      (𝗟) = ↑[f]l → 𝗟 = l.
+#f * [ #k ]
+[ <prelift_label_d
+| <prelift_label_m
+| <prelift_label_L
+| <prelift_label_A
+| <prelift_label_S
+] #H0 destruct //
+qed-.
+
+lemma prelift_label_inv_A_sn (f) (l):
+      (𝗔) = ↑[f]l → 𝗔 = l.
+#f * [ #k ]
+[ <prelift_label_d
+| <prelift_label_m
+| <prelift_label_L
+| <prelift_label_A
+| <prelift_label_S
+] #H0 destruct //
+qed-.
+
+lemma prelift_label_inv_S_sn (f) (l):
+      (𝗦) = ↑[f]l → 𝗦 = l.
+#f * [ #k ]
+[ <prelift_label_d
+| <prelift_label_m
+| <prelift_label_L
+| <prelift_label_A
+| <prelift_label_S
+] #H0 destruct //
+qed-.
+
+(* Main inversions **********************************************************)
+
+theorem prelift_label_inj (f) (l1) (l2):
+        ↑[f]l1 = ↑[f]l2 → l1 = l2.
+#f * [ #k1 ] #l2 #Hl
+[ elim (prelift_label_inv_d_sn … Hl) -Hl #k2 #Hk #H0 destruct
+  >(tr_pap_inj ???? Hk) -Hk //
+| <(prelift_label_inv_m_sn … Hl) -l2 //
+| <(prelift_label_inv_L_sn … Hl) -l2 //
+| <(prelift_label_inv_A_sn … Hl) -l2 //
+| <(prelift_label_inv_S_sn … Hl) -l2 //
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label_after.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label_after.ma
new file mode 100644
index 000000000..24720288b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label_after.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "delayed_updating/substitution/prelift_label.ma".
+include "ground/relocation/tr_compose_pap.ma".
+
+(* PRELIFT FOR LABEL ********************************************************)
+
+(* Constructions with tr_after **********************************************)
+
+lemma prelift_label_after (g) (f) (l):
+      ↑[g]↑[f]l = ↑[g∘f]l.
+#g #f * [ #k ] //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label_eq.ma
new file mode 100644
index 000000000..bccba6675
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label_eq.ma
@@ -0,0 +1,28 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "delayed_updating/substitution/prelift_label.ma".
+include "ground/relocation/tr_pap_eq.ma".
+
+(* PRELIFT FOR LABEL ********************************************************)
+
+(* constructions with tr_map_eq *********************************************)
+
+lemma prelift_label_eq_repl (l):
+      stream_eq_repl … (λf1,f2. ↑[f1]l = ↑[f2]l).
+* //
+#k #f1 #f2 #Hf
+<prelift_label_d <prelift_label_d
+/3 width=1 by tr_pap_eq_repl, eq_f/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label_id.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label_id.ma
new file mode 100644
index 000000000..8378a0f39
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_label_id.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "delayed_updating/substitution/prelift_label.ma".
+include "ground/relocation/tr_id_pap.ma".
+
+(* PRELIFT FOR LABEL ********************************************************)
+
+(* Constructions with tr_id *************************************************)
+
+lemma prelift_label_id (l):
+      l = ↑[𝐢]l.
+* [ #k ] //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap.ma
index de7f40321..dcec9b57a 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap.ma
@@ -21,7 +21,7 @@ include "ground/lib/stream_tls.ma".
 
 definition prelift_rmap (f) (l): tr_map ≝
 match l with
-[ label_d n ⇒ ⇂*[n]f
+[ label_d k ⇒ ⇂*[k]f
 | label_m   ⇒ f
 | label_L   ⇒ ⫯f
 | label_A   ⇒ f
@@ -34,8 +34,8 @@ interpretation
 
 (* Basic constructions ******************************************************)
 
-lemma prelift_rmap_d (f) (n:pnat):
-      ⇂*[n]f = ↑[𝗱n]f.
+lemma prelift_rmap_d (f) (k:pnat):
+      ⇂*[k]f = ↑[𝗱k]f.
 // qed.
 
 lemma prelift_rmap_m (f):
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_prelift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap_after.ma
similarity index 73%
rename from matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_prelift.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap_after.ma
index 345f437ab..3e83fbbf8 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_prelift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap_after.ma
@@ -12,20 +12,17 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_gen_eq.ma".
-include "delayed_updating/substitution/prelift_label.ma".
 include "delayed_updating/substitution/prelift_rmap.ma".
+include "delayed_updating/substitution/prelift_label.ma".
+include "ground/relocation/tr_compose_pn.ma".
+include "ground/relocation/tr_compose_tls.ma".
 
-(* LIFT FOR PATH ************************************************************)
-
-(* Constructions with prelift_rmap and prelift_path *************************)
+(* PRELIFT FOR RELOCATION MAP ***********************************************)
 
-lemma lift_path_lcons_prelift (f) (p) (l):
-      (↑[f]l)◗↑[↑[l]f]p = ↑[f](l◗p).
-#f #p * [ #n ] //
-qed.
+(* Constructions with tr_after **********************************************)
 
-lemma lift_path_rcons_prelift (f) (p) (l):
-      (↑[f]p)◖↑[↑[p]f]l = ↑[f](p◖l).
-#f #p * [ #n ] //
+lemma prelift_rmap_after (g) (f) (l):
+      ↑[↑[f]l]g∘↑[l]f = ↑[l](g∘f).
+#g #f * [ #k ] //
+<prelift_rmap_d //
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap_eq.ma
new file mode 100644
index 000000000..c23c1a0a6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap_eq.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "delayed_updating/substitution/prelift_rmap.ma".
+include "ground/relocation/tr_pn_eq.ma".
+include "ground/lib/stream_tls_eq.ma".
+
+(* PRELIFT FOR RELOCATION MAP ***********************************************)
+
+(* constructions with tr_map_eq *********************************************)
+
+lemma prelift_rmap_eq_repl (l):
+      stream_eq_repl … (λf1,f2. ↑[l]f1 ≗ ↑[l]f2).
+* //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap_id.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap_id.ma
new file mode 100644
index 000000000..0842044d3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/prelift_rmap_id.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "delayed_updating/substitution/prelift_rmap.ma".
+include "ground/relocation/tr_id_tls.ma".
+
+(* PRELIFT FOR RELOCATION MAP ***********************************************)
+
+(* Constructions with tr_id *************************************************)
+
+lemma prelift_rmap_id (l):
+      (𝐢) = ↑[l]𝐢.
+* [ #k ] //
+qed.
-- 
2.39.2