From 4ac2becfaa45abb18acb2bdf3db5d2587cadb6d4 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Tue, 14 Jun 2022 00:30:57 +0200
Subject: [PATCH] update in delayed_updating

+ final definition of lift
+ WIP on dfr_lift_bi
+ updates and corrections
+ old definition of lift parked
---
 .../etc/{lift_k => lift_i}/lift.etc           | 24 ++++++++-----
 .../{lift_k => lift_i}/lift_constructors.etc  | 20 +++++------
 .../etc/{lift_k => lift_i}/lift_eq.etc        | 35 +++++++++++--------
 .../{lift_k => lift_i}/lift_path_after.etc    |  9 ++---
 .../{lift_k => lift_i}/lift_path_proper.etc   |  0
 .../etc/{lift_k => lift_i}/lift_path_uni.etc  |  8 ++---
 .../etc/{lift_k => lift_i}/lift_prototerm.etc |  0
 .../{lift_k => lift_i}/lift_prototerm_eq.etc  | 17 +++++++++
 .../lift_prototerm_proper.etc                 |  0
 ...tarrow_f_4.ma => black_rightarrow_if_4.ma} |  4 +--
 .../delayed_updating/reduction/dfr_ifr.ma     |  2 +-
 .../delayed_updating/reduction/dfr_lift.ma    | 17 ++++++++-
 .../delayed_updating/reduction/ifr.ma         |  4 +--
 .../delayed_updating/reduction/ifr_unwind.ma  |  2 +-
 .../delayed_updating/substitution/lift.ma     | 24 +++++--------
 .../substitution/lift_constructors.ma         | 20 ++++++-----
 .../delayed_updating/substitution/lift_eq.ma  | 35 ++++++++-----------
 .../substitution/lift_path_after.ma           |  9 +++--
 .../substitution/lift_path_head.ma            | 31 ++++++++++++++++
 .../lift_path_id.ma}                          |  0
 .../substitution/lift_path_length.ma          |  5 +--
 .../substitution/lift_path_structure.ma       |  3 +-
 .../substitution/lift_path_uni.ma             |  8 +++--
 .../substitution/lift_prototerm_eq.ma         | 17 ---------
 .../lift_prototerm_id.ma}                     |  0
 .../delayed_updating/syntax/path_reverse.ma   |  2 +-
 26 files changed, 172 insertions(+), 124 deletions(-)
 rename matita/matita/contribs/lambdadelta/delayed_updating/etc/{lift_k => lift_i}/lift.etc (89%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/etc/{lift_k => lift_i}/lift_constructors.etc (77%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/etc/{lift_k => lift_i}/lift_eq.etc (89%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/etc/{lift_k => lift_i}/lift_path_after.etc (86%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/etc/{lift_k => lift_i}/lift_path_proper.etc (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/etc/{lift_k => lift_i}/lift_path_uni.etc (88%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/etc/{lift_k => lift_i}/lift_prototerm.etc (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/etc/{lift_k => lift_i}/lift_prototerm_eq.etc (85%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/etc/{lift_k => lift_i}/lift_prototerm_proper.etc (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/{black_rightarrow_f_4.ma => black_rightarrow_if_4.ma} (89%)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_head.ma
 rename matita/matita/contribs/lambdadelta/delayed_updating/{etc/lift_k/lift_path_id.etc => substitution/lift_path_id.ma} (100%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/{etc/lift_k/lift_prototerm_id.etc => substitution/lift_prototerm_id.ma} (100%)

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift.etc
similarity index 89%
rename from matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift.etc
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift.etc
index b873ed303..6933e9dbe 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift.etc
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift.etc
@@ -15,8 +15,7 @@
 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".
+include "ground/relocation/tr_id_pap.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
@@ -28,7 +27,7 @@ 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_d n ⇒ lift_gen (A) (λg,p. k g (𝗱(f＠⧣❨n❩)◗p)) (𝐢) 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
@@ -61,7 +60,7 @@ lemma lift_empty (A) (k) (f):
 // 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❩.
+      ↑❨(λg,p. k g (𝗱(f＠⧣❨n❩)◗p)), 𝐢, p❩ = ↑{A}❨k, f, 𝗱n◗p❩.
 // qed.
 
 lemma lift_m_sn (A) (k) (p) (f):
@@ -93,7 +92,7 @@ lemma lift_rmap_empty (f):
 // qed.
 
 lemma lift_rmap_d_sn (f) (p) (n):
-      ↑[p](⇂*[ninj n]f) = ↑[𝗱n◗p]f.
+      ↑[p]𝐢 = ↑[𝗱n◗p]f.
 // qed.
 
 lemma lift_rmap_m_sn (f) (p):
@@ -112,12 +111,21 @@ lemma lift_rmap_S_sn (f) (p):
       ↑[p]f = ↑[𝗦◗p]f.
 // qed.
 
+(* Advanced cinstructionswith proj_rmap and tr_id ***************************)
+
+lemma lift_rmap_id (p):
+      (𝐢) = ↑[p]𝐢.
+#p elim p -p //
+* [ #n ] #p #IH //
+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_d_sn <lift_rmap_d_sn //
+| <lift_rmap_m_sn <lift_rmap_m_sn //
 | <lift_rmap_A_sn <lift_rmap_A_sn //
 | <lift_rmap_S_sn <lift_rmap_S_sn //
 ]
@@ -126,7 +134,7 @@ qed.
 (* Advanced constructions with proj_rmap and path_rcons *********************)
 
 lemma lift_rmap_d_dx (f) (p) (n):
-      ⇂*[ninj n](↑[p]f) = ↑[p◖𝗱n]f.
+      (𝐢) = ↑[p◖𝗱n]f.
 // qed.
 
 lemma lift_rmap_m_dx (f) (p):
@@ -146,5 +154,5 @@ lemma lift_rmap_S_dx (f) (p):
 // qed.
 
 lemma lift_rmap_pap_d_dx (f) (p) (n) (m):
-      ↑[p]f＠⧣❨m+n❩ = ↑[p◖𝗱n]f＠⧣❨m❩+↑[p]f＠⧣❨n❩.
+      m = ↑[p◖𝗱n]f＠⧣❨m❩.
 // qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_constructors.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_constructors.etc
similarity index 77%
rename from matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_constructors.etc
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_constructors.etc
index 1cdcd4650..4eeb2ab6d 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_constructors.etc
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_constructors.etc
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_prototerm_id.ma".
+include "delayed_updating/substitution/lift_prototerm_eq.ma".
 include "delayed_updating/substitution/lift_path_uni.ma".
 include "delayed_updating/syntax/prototerm_constructors.ma".
 
@@ -24,28 +24,24 @@ lemma lift_iref_bi (t1) (t2) (n):
 qed.
 
 lemma lift_iref_sn (f) (t:prototerm) (n:pnat):
-      (𝛗f＠⧣❨n❩.↑[⇂*[n]f]t) ⊆ ↑[f](𝛗n.t).
-#f #t #n #p * #q * #r #Hr #H1 #H2 destruct
-@(ex2_intro … (𝗱n◗𝗺◗r))
+      (𝛗f＠⧣❨n❩.t) ⊆ ↑[f](𝛗n.t).
+#f #t #n #p * #q #Hq #H0 destruct
+@(ex2_intro … (𝗱n◗𝗺◗q))
 /2 width=1 by in_comp_iref/
 qed-.
 
 lemma lift_iref_dx (f) (t) (n:pnat):
-      ↑[f](𝛗n.t) ⊆ 𝛗f＠⧣❨n❩.↑[⇂*[n]f]t.
+      ↑[f](𝛗n.t) ⊆ 𝛗f＠⧣❨n❩.t.
 #f #t #n #p * #q #Hq #H0 destruct
 elim (in_comp_inv_iref … Hq) -Hq #p #H0 #Hp destruct
-/3 width=1 by in_comp_iref, in_comp_lift_bi/
+/2 width=1 by in_comp_iref/
 qed-.
 
 lemma lift_iref (f) (t) (n:pnat):
-      (𝛗f＠⧣❨n❩.↑[⇂*[n]f]t) ⇔ ↑[f](𝛗n.t).
+      (𝛗f＠⧣❨n❩.t) ⇔ ↑[f](𝛗n.t).
 /3 width=1 by conj, lift_iref_sn, lift_iref_dx/
 qed.
 
 lemma lift_iref_uni (t) (m) (n):
       (𝛗(n+m).t) ⇔ ↑[𝐮❨m❩](𝛗n.t).
-#t #m #n
-@(subset_eq_trans … (lift_iref …))
-<tr_uni_pap >nsucc_pnpred <tr_tls_succ_uni
-/3 width=1 by lift_iref_bi, lift_term_id/
-qed.
+// qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_eq.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_eq.etc
similarity index 89%
rename from matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_eq.etc
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_eq.etc
index 9c4b0f3b5..f6759bd1e 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_eq.etc
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_eq.etc
@@ -15,7 +15,6 @@
 include "delayed_updating/substitution/lift.ma".
 include "ground/relocation/tr_pap_eq.ma".
 include "ground/relocation/tr_pn_eq.ma".
-include "ground/lib/stream_tls_eq.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
@@ -34,7 +33,7 @@ lemma lift_eq_repl (A) (p) (k1) (k2):
 #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 stream_eq_refl/
 | /3 width=1 by/
 | /3 width=1 by tr_push_eq_repl/
 | /3 width=1 by/
@@ -87,7 +86,7 @@ lemma lift_path_lcons (f) (p) (l):
 qed.
 
 lemma lift_path_d_sn (f) (p) (n):
-      (𝗱(f＠⧣❨n❩)◗↑[⇂*[n]f]p) = ↑[f](𝗱n◗p).
+      (𝗱(f＠⧣❨n❩)◗↑[𝐢]p) = ↑[f](𝗱n◗p).
 // qed.
 
 lemma lift_path_m_sn (f) (p):
@@ -106,6 +105,15 @@ lemma lift_path_S_sn (f) (p):
       (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
 // qed.
 
+lemma lift_path_id (p):
+      p = ↑[𝐢]p.
+#p elim p -p //
+* [ #n ] #p #IH //
+[ <lift_path_d_sn //
+| <lift_path_L_sn //
+]
+qed.
+
 lemma lift_path_append (p2) (p1) (f):
       (↑[f]p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
 #p2 #p1 elim p1 -p1 //
@@ -118,29 +126,29 @@ lemma lift_path_append (p2) (p1) (f):
 ]
 qed.
 
-lemma lift_path_d_dx (f) (p) (n):
+lemma lift_path_d_dx (n) (p) (f):
       (↑[f]p)◖𝗱((↑[p]f)＠⧣❨n❩) = ↑[f](p◖𝗱n).
-#f #p #n <lift_path_append //
+#n #p #f <lift_path_append //
 qed.
 
-lemma lift_path_m_dx (f) (p):
+lemma lift_path_m_dx (p) (f):
       (↑[f]p)◖𝗺 = ↑[f](p◖𝗺).
-#f #p <lift_path_append //
+#p #f <lift_path_append //
 qed.
 
-lemma lift_path_L_dx (f) (p):
+lemma lift_path_L_dx (p) (f):
       (↑[f]p)◖𝗟 = ↑[f](p◖𝗟).
-#f #p <lift_path_append //
+#p #f <lift_path_append //
 qed.
 
-lemma lift_path_A_dx (f) (p):
+lemma lift_path_A_dx (p) (f):
       (↑[f]p)◖𝗔 = ↑[f](p◖𝗔).
-#f #p <lift_path_append //
+#p #f <lift_path_append //
 qed.
 
-lemma lift_path_S_dx (f) (p):
+lemma lift_path_S_dx (p) (f):
       (↑[f]p)◖𝗦 = ↑[f](p◖𝗦).
-#f #p <lift_path_append //
+#p #f <lift_path_append //
 qed.
 
 (* COMMENT 
@@ -153,5 +161,4 @@ lemma lift_rmap_tls_d_dx (f) (p) (m) (n):
 <lift_rmap_d_dx >nrplus_inj_dx
 /2 width=1 by tr_tls_compose_uni_dx/
 qed.
-
 *)
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_path_after.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_path_after.etc
similarity index 86%
rename from matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_path_after.etc
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_path_after.etc
index 869d680fe..938d21534 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_path_after.etc
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_path_after.etc
@@ -15,7 +15,6 @@
 include "delayed_updating/substitution/lift_eq.ma".
 include "ground/relocation/tr_compose_pap.ma".
 include "ground/relocation/tr_compose_pn.ma".
-include "ground/relocation/tr_compose_tls.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
@@ -23,9 +22,7 @@ include "ground/relocation/tr_compose_tls.ma".
 
 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 //
-]
+#p elim p -p [| * ] // #p #IH #f1 #f2
+<lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
+>tr_compose_push_bi //
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_path_proper.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_path_proper.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_path_proper.etc
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_path_proper.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_path_uni.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_path_uni.etc
similarity index 88%
rename from matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_path_uni.etc
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_path_uni.etc
index f68010b32..9a62cb28a 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_path_uni.etc
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_path_uni.etc
@@ -12,9 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_path_id.ma".
+include "delayed_updating/substitution/lift_eq.ma".
 include "ground/relocation/tr_uni_pap.ma".
-include "ground/relocation/tr_uni_tls.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
@@ -22,7 +21,4 @@ include "ground/relocation/tr_uni_tls.ma".
 
 lemma lift_path_d_sn_uni (p) (m) (n):
       (𝗱(n+m)◗p) = ↑[𝐮❨m❩](𝗱(n)◗p).
-#p #m #n
-<lift_path_d_sn <tr_uni_pap >nsucc_pnpred
-<tr_tls_succ_uni //
-qed.
+// qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_prototerm.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_prototerm.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_prototerm.etc
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_prototerm.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_prototerm_eq.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_prototerm_eq.etc
similarity index 85%
rename from matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_prototerm_eq.etc
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_prototerm_eq.etc
index 920a16cf0..0b3ba39b0 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_prototerm_eq.etc
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_prototerm_eq.etc
@@ -38,3 +38,20 @@ lemma lift_term_after (f1) (f2) (t):
 | @subset_equivalence_ext_f1_exteq /2 width=5/
 ]
 qed.
+
+lemma lift_term_id_sn (t):
+      t ⊆ ↑[𝐢]t.
+#t #p #Hp
+>(lift_path_id p)
+/2 width=1 by in_comp_lift_bi/
+qed-.
+
+lemma lift_term_id_dx (t):
+      ↑[𝐢]t ⊆ t.
+#t #p * #q #Hq #H destruct //
+qed-.
+
+lemma lift_term_id (t):
+      t ⇔ ↑[𝐢]t.
+/3 width=2 by lift_term_id_dx, lift_term_id_sn, conj/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_prototerm_proper.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_prototerm_proper.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_prototerm_proper.etc
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_i/lift_prototerm_proper.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_f_4.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_if_4.ma
similarity index 89%
rename from matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_f_4.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_if_4.ma
index c69d6c79b..2b35b91ea 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_f_4.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_if_4.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
-notation "hvbox( t1 ➡𝐟[ break term 46 p, break term 46 q ] break term 46 t2 )"
+notation "hvbox( t1 ➡𝐢𝐟[ break term 46 p, break term 46 q ] break term 46 t2 )"
    non associative with precedence 45
-   for @{ 'BlackRightArrowF $t1 $p $q $t2 }.
+   for @{ 'BlackRightArrowIF $t1 $p $q $t2 }.
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 be7f3b3d3..fd5f76b54 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma
@@ -33,7 +33,7 @@ include "delayed_updating/syntax/path_depth_reverse.ma".
 (* Main destructions with ifr ***********************************************)
 
 theorem dfr_des_ifr (f) (p) (q) (t1) (t2): t1 ϵ 𝐓 →
-        t1 ➡𝐝𝐟[p,q] t2 → ▼[f]t1 ➡𝐟[⊗p,⊗q] ▼[f]t2.
+        t1 ➡𝐝𝐟[p,q] t2 → ▼[f]t1 ➡𝐢𝐟[⊗p,⊗q] ▼[f]t2.
 #f #p #q #t1 #t2 #H0t1
 * #n * #H1n #Ht1 #Ht2
 @(ex_intro … (↑♭q)) @and3_intro
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
index d25ddc502..e33b454be 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
@@ -23,12 +23,23 @@ 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/substitution/lift_path_head.ma".
 
 include "delayed_updating/syntax/prototerm_proper_constructors.ma".
 
+
 (* DELAYED FOCUSED REDUCTION ************************************************)
 
 (* Constructions with lift **************************************************)
+(*
+lemma pippo (f) (r):
+      ↑[↑[r]f](rᴿ) = (↑[f]r)ᴿ.
+#f #r @(list_ind_rcons … r) -r //
+#p * [ #n ] #IH
+[ <reverse_rcons <lift_path_d_sn <lift_rmap_d_dx
+  <lift_path_d_dx <reverse_rcons
+  <IH  
+*)
 
 theorem dfr_lift_bi (f) (p) (q) (t1) (t2): (*t1 ϵ 𝐓 → *)
         t1 ➡𝐝𝐟[p,q] t2 → ↑[f]t1 ➡𝐟[↑[f]p,↑[↑[p◖𝗔◖𝗟]f]q] ↑[f]t2.
@@ -36,7 +47,10 @@ theorem dfr_lift_bi (f) (p) (q) (t1) (t2): (*t1 ϵ 𝐓 → *)
 * #n * #H1n #Ht1 #Ht2
 @(ex_intro … ((↑[p●𝗔◗𝗟◗q]f)＠⧣❨n❩)) @and3_intro
 [ -Ht1 -Ht2
-  >H1n >path_head_structure_depth <H1n -H1n //
+  <lift_rmap_L_dx >lift_path_L_sn
+  >list_append_rcons_sn in H1n; <reverse_append #H1n
+  <(lift_path_head … H1n) -H1n //
+(*
 | 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
@@ -68,3 +82,4 @@ theorem dfr_lift_bi (f) (p) (q) (t1) (t2): (*t1 ϵ 𝐓 → *)
   ]
 ]
 qed.
+*)
\ No newline at end of file
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma
index 6469bb2a0..87d855281 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma
@@ -17,7 +17,7 @@ include "delayed_updating/substitution/fsubst.ma".
 include "delayed_updating/syntax/prototerm_equivalence.ma".
 include "delayed_updating/syntax/path_head.ma".
 include "delayed_updating/syntax/path_reverse.ma".
-include "delayed_updating/notation/relations/black_rightarrow_f_4.ma".
+include "delayed_updating/notation/relations/black_rightarrow_if_4.ma".
 
 (* IMMEDIATE FOCUSED REDUCTION ************************************************)
 
@@ -30,4 +30,4 @@ definition ifr (p) (q): relation2 prototerm prototerm ≝
 
 interpretation
   "focused reduction with immediate updating (prototerm)"
-  'BlackRightArrowF t1 p q t2 = (ifr p q t1 t2).
+  'BlackRightArrowIF t1 p q t2 = (ifr p q t1 t2).
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_unwind.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_unwind.ma
index 21e4ac27f..f05d86db3 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_unwind.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_unwind.ma
@@ -34,7 +34,7 @@ include "delayed_updating/syntax/path_depth_reverse.ma".
 
 lemma ifr_unwind_bi (f) (p) (q) (t1) (t2):
       t1 ϵ 𝐓 → t1⋔(p◖𝗦) ⧸≬ 𝐈 →
-      t1 ➡𝐟[p,q] t2 → ▼[f]t1 ➡𝐟[⊗p,⊗q] ▼[f]t2.
+      t1 ➡𝐢𝐟[p,q] t2 → ▼[f]t1 ➡𝐢𝐟[⊗p,⊗q] ▼[f]t2.
 #f #p #q #t1 #t2 #H1t1 #H2t1
 * #n * #H1n #Ht1 #Ht2
 @(ex_intro … (↑♭q)) @and3_intro
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift.ma
index 6933e9dbe..b873ed303 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift.ma
@@ -15,7 +15,8 @@
 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_id_pap.ma".
+include "ground/relocation/tr_uni.ma".
+include "ground/relocation/tr_pap_tls.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
@@ -27,7 +28,7 @@ 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)) (𝐢) q
+  [ 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
@@ -60,7 +61,7 @@ lemma lift_empty (A) (k) (f):
 // qed.
 
 lemma lift_d_sn (A) (k) (p) (n) (f):
-      ↑❨(λg,p. k g (𝗱(f＠⧣❨n❩)◗p)), 𝐢, p❩ = ↑{A}❨k, f, 𝗱n◗p❩.
+      ↑❨(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):
@@ -92,7 +93,7 @@ lemma lift_rmap_empty (f):
 // qed.
 
 lemma lift_rmap_d_sn (f) (p) (n):
-      ↑[p]𝐢 = ↑[𝗱n◗p]f.
+      ↑[p](⇂*[ninj n]f) = ↑[𝗱n◗p]f.
 // qed.
 
 lemma lift_rmap_m_sn (f) (p):
@@ -111,21 +112,12 @@ lemma lift_rmap_S_sn (f) (p):
       ↑[p]f = ↑[𝗦◗p]f.
 // qed.
 
-(* Advanced cinstructionswith proj_rmap and tr_id ***************************)
-
-lemma lift_rmap_id (p):
-      (𝐢) = ↑[p]𝐢.
-#p elim p -p //
-* [ #n ] #p #IH //
-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_d_sn <lift_rmap_d_sn //
-| <lift_rmap_m_sn <lift_rmap_m_sn //
+[ <lift_rmap_m_sn <lift_rmap_m_sn //
 | <lift_rmap_A_sn <lift_rmap_A_sn //
 | <lift_rmap_S_sn <lift_rmap_S_sn //
 ]
@@ -134,7 +126,7 @@ qed.
 (* Advanced constructions with proj_rmap and path_rcons *********************)
 
 lemma lift_rmap_d_dx (f) (p) (n):
-      (𝐢) = ↑[p◖𝗱n]f.
+      ⇂*[ninj n](↑[p]f) = ↑[p◖𝗱n]f.
 // qed.
 
 lemma lift_rmap_m_dx (f) (p):
@@ -154,5 +146,5 @@ lemma lift_rmap_S_dx (f) (p):
 // qed.
 
 lemma lift_rmap_pap_d_dx (f) (p) (n) (m):
-      m = ↑[p◖𝗱n]f＠⧣❨m❩.
+      ↑[p]f＠⧣❨m+n❩ = ↑[p◖𝗱n]f＠⧣❨m❩+↑[p]f＠⧣❨n❩.
 // 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 4eeb2ab6d..1cdcd4650 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_prototerm_eq.ma".
+include "delayed_updating/substitution/lift_prototerm_id.ma".
 include "delayed_updating/substitution/lift_path_uni.ma".
 include "delayed_updating/syntax/prototerm_constructors.ma".
 
@@ -24,24 +24,28 @@ lemma lift_iref_bi (t1) (t2) (n):
 qed.
 
 lemma lift_iref_sn (f) (t:prototerm) (n:pnat):
-      (𝛗f＠⧣❨n❩.t) ⊆ ↑[f](𝛗n.t).
-#f #t #n #p * #q #Hq #H0 destruct
-@(ex2_intro … (𝗱n◗𝗺◗q))
+      (𝛗f＠⧣❨n❩.↑[⇂*[n]f]t) ⊆ ↑[f](𝛗n.t).
+#f #t #n #p * #q * #r #Hr #H1 #H2 destruct
+@(ex2_intro … (𝗱n◗𝗺◗r))
 /2 width=1 by in_comp_iref/
 qed-.
 
 lemma lift_iref_dx (f) (t) (n:pnat):
-      ↑[f](𝛗n.t) ⊆ 𝛗f＠⧣❨n❩.t.
+      ↑[f](𝛗n.t) ⊆ 𝛗f＠⧣❨n❩.↑[⇂*[n]f]t.
 #f #t #n #p * #q #Hq #H0 destruct
 elim (in_comp_inv_iref … Hq) -Hq #p #H0 #Hp destruct
-/2 width=1 by in_comp_iref/
+/3 width=1 by in_comp_iref, in_comp_lift_bi/
 qed-.
 
 lemma lift_iref (f) (t) (n:pnat):
-      (𝛗f＠⧣❨n❩.t) ⇔ ↑[f](𝛗n.t).
+      (𝛗f＠⧣❨n❩.↑[⇂*[n]f]t) ⇔ ↑[f](𝛗n.t).
 /3 width=1 by conj, lift_iref_sn, lift_iref_dx/
 qed.
 
 lemma lift_iref_uni (t) (m) (n):
       (𝛗(n+m).t) ⇔ ↑[𝐮❨m❩](𝛗n.t).
-// qed.
+#t #m #n
+@(subset_eq_trans … (lift_iref …))
+<tr_uni_pap >nsucc_pnpred <tr_tls_succ_uni
+/3 width=1 by lift_iref_bi, lift_term_id/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
index f6759bd1e..9c4b0f3b5 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
@@ -15,6 +15,7 @@
 include "delayed_updating/substitution/lift.ma".
 include "ground/relocation/tr_pap_eq.ma".
 include "ground/relocation/tr_pn_eq.ma".
+include "ground/lib/stream_tls_eq.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
@@ -33,7 +34,7 @@ lemma lift_eq_repl (A) (p) (k1) (k2):
 #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=1 by stream_eq_refl/
+  /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/
@@ -86,7 +87,7 @@ lemma lift_path_lcons (f) (p) (l):
 qed.
 
 lemma lift_path_d_sn (f) (p) (n):
-      (𝗱(f＠⧣❨n❩)◗↑[𝐢]p) = ↑[f](𝗱n◗p).
+      (𝗱(f＠⧣❨n❩)◗↑[⇂*[n]f]p) = ↑[f](𝗱n◗p).
 // qed.
 
 lemma lift_path_m_sn (f) (p):
@@ -105,15 +106,6 @@ lemma lift_path_S_sn (f) (p):
       (𝗦◗↑[f]p) = ↑[f](𝗦◗p).
 // qed.
 
-lemma lift_path_id (p):
-      p = ↑[𝐢]p.
-#p elim p -p //
-* [ #n ] #p #IH //
-[ <lift_path_d_sn //
-| <lift_path_L_sn //
-]
-qed.
-
 lemma lift_path_append (p2) (p1) (f):
       (↑[f]p1)●(↑[↑[p1]f]p2) = ↑[f](p1●p2).
 #p2 #p1 elim p1 -p1 //
@@ -126,29 +118,29 @@ lemma lift_path_append (p2) (p1) (f):
 ]
 qed.
 
-lemma lift_path_d_dx (n) (p) (f):
+lemma lift_path_d_dx (f) (p) (n):
       (↑[f]p)◖𝗱((↑[p]f)＠⧣❨n❩) = ↑[f](p◖𝗱n).
-#n #p #f <lift_path_append //
+#f #p #n <lift_path_append //
 qed.
 
-lemma lift_path_m_dx (p) (f):
+lemma lift_path_m_dx (f) (p):
       (↑[f]p)◖𝗺 = ↑[f](p◖𝗺).
-#p #f <lift_path_append //
+#f #p <lift_path_append //
 qed.
 
-lemma lift_path_L_dx (p) (f):
+lemma lift_path_L_dx (f) (p):
       (↑[f]p)◖𝗟 = ↑[f](p◖𝗟).
-#p #f <lift_path_append //
+#f #p <lift_path_append //
 qed.
 
-lemma lift_path_A_dx (p) (f):
+lemma lift_path_A_dx (f) (p):
       (↑[f]p)◖𝗔 = ↑[f](p◖𝗔).
-#p #f <lift_path_append //
+#f #p <lift_path_append //
 qed.
 
-lemma lift_path_S_dx (p) (f):
+lemma lift_path_S_dx (f) (p):
       (↑[f]p)◖𝗦 = ↑[f](p◖𝗦).
-#p #f <lift_path_append //
+#f #p <lift_path_append //
 qed.
 
 (* COMMENT 
@@ -161,4 +153,5 @@ lemma lift_rmap_tls_d_dx (f) (p) (m) (n):
 <lift_rmap_d_dx >nrplus_inj_dx
 /2 width=1 by tr_tls_compose_uni_dx/
 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 938d21534..869d680fe 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
@@ -15,6 +15,7 @@
 include "delayed_updating/substitution/lift_eq.ma".
 include "ground/relocation/tr_compose_pap.ma".
 include "ground/relocation/tr_compose_pn.ma".
+include "ground/relocation/tr_compose_tls.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
@@ -22,7 +23,9 @@ include "ground/relocation/tr_compose_pn.ma".
 
 lemma lift_path_after (p) (f1) (f2):
       ↑[f2]↑[f1]p = ↑[f2∘f1]p.
-#p elim p -p [| * ] // #p #IH #f1 #f2
-<lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
->tr_compose_push_bi //
+#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 //
+]
 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
new file mode 100644
index 000000000..126b691c6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_head.ma
@@ -0,0 +1,31 @@
+include "delayed_updating/substitution/lift_eq.ma".
+include "delayed_updating/syntax/path_head.ma".
+include "delayed_updating/syntax/path_reverse.ma".
+include "ground/relocation/xap.ma".
+
+axiom tr_xap_succ_pos (f) (n):
+      ↑↓(f＠❨↑n❩) = f＠❨↑n❩.
+
+axiom tr_xap_plus (n1) (n2) (f):
+      (⇂*[n2]f)＠❨n1❩+f＠❨n2❩ = f＠❨n1+n2❩.
+
+axiom eq_inv_path_empty_head (p) (n):
+      (𝐞) = ↳[n]p → 𝟎 = n.
+
+lemma lift_path_head (f) (p) (q) (n):
+      pᴿ = ↳[n](pᴿ●qᴿ) →
+      ↳[↑[q●p]f＠❨n❩](↑[f](q●p))ᴿ = (↑[↑[q]f]p)ᴿ.
+#f #p @(list_ind_rcons … p) -p
+[ #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
+      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_pos <path_head_d_sn >tr_xap_succ_pos
+      <lift_path_d_dx >lift_rmap_append <reverse_rcons  
+      @eq_f2 // <(IH … H0) -IH -H0
+      @eq_f2 // <tr_xap_plus //
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_path_id.etc b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_id.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_path_id.etc
rename to matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_id.ma
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 f384cf746..a4b81aeeb 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
@@ -21,8 +21,9 @@ include "ground/lib/list_length.ma".
 
 lemma lift_path_length (p) (f):
       ❘p❘ = ❘↑[f]p❘.
-#p elim p -p // * [ #n ] #p #IH #f //
-[ <lift_path_m_sn
+#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
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 15d6e0933..773749b63 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
@@ -23,7 +23,8 @@ lemma structure_lift_path (p) (f):
       ⊗p = ⊗↑[f]p.
 #p elim p -p //
 * [ #n ] #p #IH #f //
-[ <lift_path_m_sn <structure_m_sn <structure_m_sn //
+[ <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 //
 ]
 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 9a62cb28a..f68010b32 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
@@ -12,8 +12,9 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_eq.ma".
+include "delayed_updating/substitution/lift_path_id.ma".
 include "ground/relocation/tr_uni_pap.ma".
+include "ground/relocation/tr_uni_tls.ma".
 
 (* LIFT FOR PATH ************************************************************)
 
@@ -21,4 +22,7 @@ include "ground/relocation/tr_uni_pap.ma".
 
 lemma lift_path_d_sn_uni (p) (m) (n):
       (𝗱(n+m)◗p) = ↑[𝐮❨m❩](𝗱(n)◗p).
-// qed.
+#p #m #n
+<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_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_eq.ma
index 0b3ba39b0..920a16cf0 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
@@ -38,20 +38,3 @@ lemma lift_term_after (f1) (f2) (t):
 | @subset_equivalence_ext_f1_exteq /2 width=5/
 ]
 qed.
-
-lemma lift_term_id_sn (t):
-      t ⊆ ↑[𝐢]t.
-#t #p #Hp
->(lift_path_id p)
-/2 width=1 by in_comp_lift_bi/
-qed-.
-
-lemma lift_term_id_dx (t):
-      ↑[𝐢]t ⊆ t.
-#t #p * #q #Hq #H destruct //
-qed-.
-
-lemma lift_term_id (t):
-      t ⇔ ↑[𝐢]t.
-/3 width=2 by lift_term_id_dx, lift_term_id_sn, conj/
-qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_prototerm_id.etc b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/etc/lift_k/lift_prototerm_id.etc
rename to matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma
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 046404f03..7fa03a41e 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_reverse.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_reverse.ma
@@ -54,7 +54,7 @@ qed.
 
 (* Main constructions *******************************************************)
 
-theorem reverse_revrse (p):
+theorem reverse_reverse (p):
         p = pá´¿á´¿.
 #p elim p -p //
 #l #p #IH
-- 
2.39.2