From d06053844638d88936d711b66fddbcca2a9add1c Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Wed, 26 Oct 2022 00:02:00 +0200
Subject: [PATCH] update in delayed_updating and ground

+ single specification for protected and unprotected closed path
+ some parked files removed
---
 .../delayed_updating/etc/reduction/dfr.etc    |  37 ---
 .../etc/reduction/dfr_ifr.etc                 | 118 ----------
 .../delayed_updating/etc/reduction/ifr.etc    |  37 ---
 .../lambdadelta/delayed_updating/names.txt    |   1 +
 .../functions/{class_c_1.ma => class_c_2.ma}  |   4 +-
 .../delayed_updating/reduction/dfr.ma         |   2 +-
 .../delayed_updating/reduction/dfr_ifr.ma     |   2 +-
 .../delayed_updating/reduction/dfr_lift.ma    |   7 +-
 .../delayed_updating/reduction/ifr.ma         |   2 +-
 .../delayed_updating/reduction/ifr_lift.ma    |   2 +-
 .../delayed_updating/reduction/ifr_unwind.ma  |   2 +-
 .../substitution/lift_path_closed.ma          |  25 +-
 .../substitution/lift_rmap_closed.ma          |  22 +-
 .../delayed_updating/syntax/path_closed.ma    | 217 +++++++++++-------
 .../syntax/path_closed_structure.ma           |   6 +-
 .../unwind/unwind2_rmap_closed.ma             |  54 ++---
 .../unwind/unwind2_rmap_unprotected.ma        |   4 +-
 .../lambdadelta/ground/lib/bool_and.ma        |  21 +-
 .../lambdadelta/ground/relocation/xap.ma      |   6 +
 19 files changed, 230 insertions(+), 339 deletions(-)
 delete mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/dfr.etc
 delete mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/dfr_ifr.etc
 delete mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/ifr.etc
 rename matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/{class_c_1.ma => class_c_2.ma} (92%)

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/dfr.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/dfr.etc
deleted file mode 100644
index 576df6ec6..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/dfr.etc
+++ /dev/null
@@ -1,37 +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/unwind1/unwind.ma".
-include "delayed_updating/substitution/fsubst.ma".
-include "delayed_updating/syntax/prototerm_constructors.ma".
-include "delayed_updating/syntax/prototerm_equivalence.ma".
-include "delayed_updating/syntax/path_structure.ma".
-include "delayed_updating/syntax/path_balanced.ma".
-include "delayed_updating/syntax/path_depth.ma".
-include "delayed_updating/notation/relations/black_rightarrow_df_4.ma".
-include "ground/xoa/ex_1_2.ma".
-include "ground/xoa/and_4.ma".
-
-(* DELAYED FOCUSED REDUCTION ************************************************)
-
-definition dfr (p) (q): relation2 prototerm prototerm ≝
-           λt1,t2. ∃∃b,n.
-           let r ≝ p●𝗔◗b●𝗟◗q in
-           ∧∧ (⊗b ϵ 𝐁 ∧ 𝟎 = ❘b❘) & ↑❘q❘ = (▼[r]𝐢)@❨n❩ & r◖𝗱n ϵ t1 &
-              t1[⋔r←𝛗(n+❘b❘).(t1⋔(p◖𝗦))] ⇔ t2
-.
-
-interpretation
-  "focused balanced reduction with delayed updating (prototerm)"
-  'BlackRightArrowDF t1 p q t2 = (dfr p q t1 t2).
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/dfr_ifr.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/dfr_ifr.etc
deleted file mode 100644
index dd24f8568..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/dfr_ifr.etc
+++ /dev/null
@@ -1,118 +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/reduction/dfr.ma".
-include "delayed_updating/reduction/ifr.ma".
-include "delayed_updating/unwind1/unwind_fsubst.ma".
-include "delayed_updating/unwind1/unwind_constructors.ma".
-include "delayed_updating/unwind1/unwind_preterm_eq.ma".
-include "delayed_updating/unwind1/unwind_structure_depth.ma".
-include "delayed_updating/unwind1/unwind_depth.ma".
-include "delayed_updating/substitution/fsubst_eq.ma".
-include "delayed_updating/substitution/lift_prototerm_eq.ma".
-include "delayed_updating/syntax/prototerm_proper_constructors.ma".
-include "delayed_updating/syntax/path_structure_depth.ma".
-include "ground/relocation/tr_uni_compose.ma".
-include "ground/relocation/tr_pap_pushs.ma".
-
-(* DELAYED FOCUSED REDUCTION ************************************************)
-
-(* COMMENT
-axiom pippo (b) (q) (n):
-      ↑❘q❘ = (↑[q]𝐢)@❨n❩ →
-      ↑❘q❘+❘b❘= (↑[b●𝗟◗q]𝐢)@❨n+❘b❘❩.
-
-lemma unwind_rmap_tls_eq_id (p) (n):
-      ❘p❘ = ↑[p]𝐢@❨n❩ →
-      (𝐢) ≗ ⇂*[n]↑[p]𝐢.
-#p @(list_ind_rcons … p) -p
-[ #n <depth_empty #H destruct
-| #p * [ #m ] #IH #n
-  [ <depth_d_dx <unwind_rmap_pap_d_dx #H0
-    @(stream_eq_trans … (unwind_rmap_tls_d_dx …))
-    @(stream_eq_trans … (IH …)) -IH //
-  | /2 width=1 by/
-  | <depth_L_dx <unwind_rmap_L_dx
-    cases n -n [| #n ] #H0
-    [
-    | 
-    ]
-  | /2 width=1 by/
-  | /2 width=1 by/
-  ]
-]
-
-
-(*  (↑❘q❘+❘b❘=↑[b●𝗟◗q]𝐢@❨n+❘b❘❩ *)
-(* [↑[p]𝐢@❨n❩]⫯*[❘p❘]f∘⇂*[n]↑[p]𝐢) *)
-lemma unwind_rmap_tls_eq (f) (p) (n):
-      ❘p❘ = ↑[p]𝐢@❨n❩ →
-      f ≗ ⇂*[n]↑[p]f.
-#f #p #n #Hp
-@(stream_eq_canc_dx … (stream_tls_eq_repl …))
-[| @unwind_rmap_decompose | skip ]
-<tr_compose_tls <Hp
-
-@(stream_eq_canc_dx) … (unwind_rmap_decompose …)) 
-
-*)
-lemma dfr_unwind_id_bi (p) (q) (t1) (t2): t1 ϵ 𝐓 →
-      t1 ➡𝐝𝐟[p,q] t2 → ▼[𝐢]t1 ➡𝐟[⊗p,⊗q] ▼[𝐢]t2.
-#p #q #t1 #t2 #H0t1
-* #b #n * #Hb #Hn #Ht1 #Ht2
-@(ex1_2_intro … (⊗b) (↑❘⊗q❘)) @and4_intro
-[ //
-| //
-| lapply (in_comp_unwind_bi (𝐢) … Ht1) -Ht1 -H0t1 -Hb -Ht2
-  <unwind_path_d_empty_dx <depth_structure //
-| lapply (unwind_term_eq_repl_dx (𝐢) … Ht2) -Ht2 #Ht2
-  @(subset_eq_trans … Ht2) -t2
-  @(subset_eq_trans … (unwind_fsubst …))
-  [ (*<unwind_rmap_append <unwind_rmap_A_sn <unwind_rmap_append <unwind_rmap_L_sn *)
-    <structure_append <structure_A_sn <structure_append <structure_L_sn
-    <depth_append <depth_L_sn <depth_structure <depth_structure
-    @fsubst_eq_repl [ // ]
-    @(subset_eq_trans … (unwind_iref …))
-
-    elim Hb -Hb #Hb #H0 <H0 -H0 <nrplus_zero_dx <nplus_zero_dx <Hn
-    @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
-    [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
-
-(*
-    @(subset_eq_canc_sn … (unwind_term_eq_repl_dx …))
-    [ @unwind_grafted_S /2 width=2 by ex_intro/ | skip ]
-
-    @(subset_eq_trans … (unwind_term_after …))
-    @(subset_eq_canc_dx … (unwind_term_after …))
-    @unwind_term_eq_repl_sn -t1
-    @(stream_eq_trans … (tr_compose_uni_dx …))
-    lapply (Hn (𝐢)) -Hn >tr_id_unfold #Hn
-    lapply (pippo … b … Hn) -Hn #Hn
-    @tr_compose_eq_repl
-    [ <unwind_rmap_pap_le //
-      <Hn <nrplus_inj_sn //
-    |
-    ]
-*)
-  | //
-  | /2 width=2 by ex_intro/
-  | //
-  ]
-]
-
-(*
-Hn : ↑❘q❘ = ↑[p●𝗔◗b●𝗟◗q]𝐢@❨n❩
----------------------------
-↑[𝐮❨↑❘q❘+❘b❘❩] ↑[↑[p]𝐢] t ⇔ ↑[𝐮❨↑[p●𝗔◗b●𝗟◗q]𝐢@❨n+❘b❘❩❩] t
-*)
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/ifr.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/ifr.etc
deleted file mode 100644
index 8d0cfc52d..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/etc/reduction/ifr.etc
+++ /dev/null
@@ -1,37 +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/unwind1/unwind_prototerm.ma".
-include "delayed_updating/substitution/fsubst.ma".
-include "delayed_updating/substitution/lift_prototerm.ma".
-include "delayed_updating/syntax/prototerm_equivalence.ma".
-include "delayed_updating/syntax/path_depth.ma".
-include "delayed_updating/syntax/path_structure.ma".
-include "delayed_updating/syntax/path_balanced.ma".
-include "delayed_updating/notation/relations/black_rightarrow_f_4.ma".
-include "ground/xoa/ex_1_2.ma".
-include "ground/xoa/and_4.ma".
-
-(* IMMEDIATE FOCUSED REDUCTION ************************************************)
-
-definition ifr (p) (q): relation2 prototerm prototerm ≝
-           λt1,t2. ∃∃b,n.
-           let r ≝ p●𝗔◗b●𝗟◗q in
-           ∧∧ (⊗b ϵ 𝐁 ∧ 𝟎 = ❘b❘) & ↑❘q❘ = (▼[r]𝐢)@❨n❩ & r◖𝗱n ϵ t1 &
-              t1[⋔r←↑[𝐮❨❘b●𝗟◗q❘❩](t1⋔(p◖𝗦))] ⇔ t2
-.
-
-interpretation
-  "focused balanced reduction with immediate updating (prototerm)"
-  'BlackRightArrowF t1 p q t2 = (ifr p q t1 t2).
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/names.txt b/matita/matita/contribs/lambdadelta/delayed_updating/names.txt
index cd980b568..715cb268e 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/names.txt
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/names.txt
@@ -6,5 +6,6 @@ METAVARIABLES
   h, k      : reference index by depth
   l         : label
   m, n      : natural number
+  o         : option
   p, q, r, s: path
   t, u, v, w: term
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_c_1.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_c_2.ma
similarity index 92%
rename from matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_c_1.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_c_2.ma
index 352144291..95e3eeb35 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_c_1.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_c_2.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
-notation "hvbox( 𝐂❨ break term 46 n ❩ )"
+notation "hvbox( 𝐂❨ break term 46 b, break term 46 n ❩ )"
   non associative with precedence 70
-  for @{ 'ClassC $n }.
+  for @{ 'ClassC $b $n }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
index 27468022a..8e26b62b9 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
@@ -24,7 +24,7 @@ include "ground/xoa/ex_4_3.ma".
 definition dfr (r): relation2 prototerm prototerm ≝
            λt1,t2.
            ∃∃p,q,n. p●𝗔◗𝗟◗q = r &
-           q ϵ 𝐂❨n❩ & r◖𝗱↑n ϵ t1 &
+           q ϵ 𝐂❨Ⓕ,n❩ & r◖𝗱↑n ϵ t1 &
            t1[⋔r←𝛕↑n.(t1⋔(p◖𝗦))] ⇔ 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 d64067123..c07a3d26a 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma
@@ -57,7 +57,7 @@ theorem dfr_des_ifr (f) (t1) (t2) (r): t1 ϵ 𝐓 →
     @(stream_eq_trans … (tr_compose_uni_dx_pap …)) <tr_pap_succ_nap
     @tr_compose_eq_repl
     [ <unwind2_rmap_append_closed_Lq_dx_nap_depth //
-    | /2 width=1 by tls_succ_unwind2_rmap_append_closed_Lq_dx/
+    | /2 width=2 by tls_succ_unwind2_rmap_append_closed_Lq_dx/
     ]
 (* Note: crux of the proof ends *)
   | //
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 830dbd640..28b21e5e0 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
@@ -12,14 +12,15 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/reduction/dfr.ma".
-
 include "delayed_updating/substitution/fsubst_lift.ma".
 include "delayed_updating/substitution/fsubst_eq.ma".
 include "delayed_updating/substitution/lift_constructors.ma".
 include "delayed_updating/substitution/lift_path_closed.ma".
 include "delayed_updating/substitution/lift_rmap_closed.ma".
 
+(**) (* reverse include *)
+include "delayed_updating/reduction/dfr.ma".
+
 (* DELAYED FOCUSED REDUCTION ************************************************)
 
 (* Constructions with lift **************************************************)
@@ -43,7 +44,7 @@ theorem dfr_lift_bi (f) (t1) (t2) (r):
   @(subset_eq_canc_sn … (lift_term_grafted_S …))
   @lift_term_eq_repl_sn
 (* Note: crux of the proof begins *)
-  /2 width=1 by tls_succ_lift_rmap_append_L_closed_dx/
+  /2 width=2 by tls_succ_lift_rmap_append_L_closed_dx/
 (* Note: crux of the proof ends *)
 ]
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma
index 92512b342..9bc8d994d 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma
@@ -25,7 +25,7 @@ include "ground/xoa/ex_4_3.ma".
 definition ifr (r): relation2 prototerm prototerm ≝
            λt1,t2.
            ∃∃p,q,n. p●𝗔◗𝗟◗q = r &
-           q ϵ 𝐂❨n❩ & r◖𝗱↑n ϵ t1 &
+           q ϵ 𝐂❨Ⓕ,n❩ & r◖𝗱↑n ϵ t1 &
            t1[⋔r←↑[𝐮❨↑n❩](t1⋔(p◖𝗦))] ⇔ t2
 .
 
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_lift.ma
index 1d8f5855a..365447714 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_lift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_lift.ma
@@ -49,7 +49,7 @@ theorem ifr_lift_bi (f) (t1) (t2) (r):
 (* Note: crux of the proof begins *)
   @(stream_eq_trans … (tr_compose_uni_dx_pap …)) <tr_pap_succ_nap
   @tr_compose_eq_repl // >nsucc_unfold
-  /2 width=1 by tls_succ_lift_rmap_append_L_closed_dx/
+  /2 width=2 by tls_succ_lift_rmap_append_L_closed_dx/
 (* Note: crux of the proof ends *)
 ]
 qed.
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 d3b567b72..2e048c7a5 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_unwind.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_unwind.ma
@@ -55,7 +55,7 @@ lemma ifr_unwind_bi (f) (t1) (t2) (r):
     @(stream_eq_trans … (tr_compose_uni_dx_pap …)) <tr_pap_succ_nap
     @tr_compose_eq_repl
     [ <unwind2_rmap_append_closed_Lq_dx_nap_depth //
-    | /2 width=1 by tls_succ_unwind2_rmap_append_closed_Lq_dx/
+    | /2 width=2 by tls_succ_unwind2_rmap_append_closed_Lq_dx/
     ]
 (* Note: crux of the proof ends *)
   | //
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_closed.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_closed.ma
index 79443ca3b..0c2aa6219 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_closed.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_closed.ma
@@ -20,24 +20,25 @@ include "ground/relocation/xap.ma".
 
 (* Constructions with pcc ***************************************************)
 
-lemma lift_path_closed (f) (q) (n):
-      q ϵ 𝐂❨n❩ → ↑[f]q ϵ 𝐂❨↑[q]f＠❨n❩❩.
-#f #q #n #Hq elim Hq -Hq //
-#q #n [ #k ] #_ #IH
-/2 width=1 by pcc_d_dx, pcc_m_dx, pcc_L_dx, pcc_A_dx, pcc_S_dx/
+lemma lift_path_closed (o) (f) (q) (n):
+      q ϵ 𝐂❨o,n❩ → ↑[f]q ϵ 𝐂❨o,↑[q]f＠❨n❩❩.
+#o #f #q #n #H0 elim H0 -q -n //
+#q #n [ #k #Ho ] #_ #IH
+/2 width=1 by pcc_m_dx, pcc_L_dx, pcc_A_dx, pcc_S_dx/
+/4 width=1 by pcc_d_dx, tr_xap_pos/
 qed.
 
-lemma lift_path_rmap_closed (f) (p) (q) (n):
-      q ϵ 𝐂❨n❩ → ↑[↑[p]f]q ϵ 𝐂❨↑[p●q]f＠❨n❩❩.
+lemma lift_path_rmap_closed (o) (f) (p) (q) (n):
+      q ϵ 𝐂❨o,n❩ → ↑[↑[p]f]q ϵ 𝐂❨o,↑[p●q]f＠❨n❩❩.
 /2 width=1 by lift_path_closed/
 qed.
 
-lemma lift_path_rmap_closed_L (f) (p) (q) (n):
-      q ϵ 𝐂❨n❩ → ↑[↑[p◖𝗟]f]q ϵ 𝐂❨↑[p●𝗟◗q]f＠§❨n❩❩.
-#f #p #q #n #Hq
-lapply (lift_path_closed (↑[p◖𝗟]f) … Hq) #Hq0
+lemma lift_path_rmap_closed_L (o) (f) (p) (q) (n):
+      q ϵ 𝐂❨o,n❩ → ↑[↑[p◖𝗟]f]q ϵ 𝐂❨o,↑[p●𝗟◗q]f＠§❨n❩❩.
+#o #f #p #q #n #Hq
+lapply (lift_path_closed … (↑[p◖𝗟]f) … Hq) #Hq0
 lapply (pcc_L_sn … Hq) -Hq #Hq1
-lapply (lift_path_rmap_closed f p … Hq1) -Hq1
+lapply (lift_path_rmap_closed … f p … Hq1) -Hq1
 <lift_path_L_sn >lift_rmap_L_dx #Hq1
 elim (pcc_inv_L_sn … Hq1 Hq0) -Hq1 #H0 #_
 <H0 in Hq0; -H0 //
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_closed.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_closed.ma
index cc3e61bf7..56abc4707 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_closed.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_closed.ma
@@ -20,24 +20,24 @@ include "ground/lib/stream_eq_eq.ma".
 
 (* Destructions with cpp ****************************************************)
 
-lemma tls_plus_lift_rmap_closed (f) (q) (n):
-      q ϵ 𝐂❨n❩ →
+lemma tls_plus_lift_rmap_closed (o) (f) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
       ∀m. ⇂*[m]f ≗ ⇂*[m+n]↑[q]f.
-#f #q #n #Hq elim Hq -q -n //
+#o #f #q #n #Hq elim Hq -q -n //
 #q #n #_ #IH #m
 <nplus_succ_dx <stream_tls_swap //
 qed-.
 
-lemma tls_lift_rmap_closed (f) (q) (n):
-      q ϵ 𝐂❨n❩ →
+lemma tls_lift_rmap_closed (o) (f) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
       f ≗ ⇂*[n]↑[q]f.
-#f #q #n #H0
-/2 width=1 by tls_plus_lift_rmap_closed/
+#o #f #q #n #H0
+/2 width=2 by tls_plus_lift_rmap_closed/
 qed-.
 
-lemma tls_succ_lift_rmap_append_L_closed_dx (f) (p) (q) (n):
-      q ϵ 𝐂❨n❩ →
+lemma tls_succ_lift_rmap_append_L_closed_dx (o) (f) (p) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
       ↑[p]f ≗ ⇂*[↑n]↑[p●𝗟◗q]f.
-#f #p #q #n #Hq
-/3 width=1 by tls_lift_rmap_closed, pcc_L_sn/
+#o #f #p #q #n #Hq
+/3 width=2 by tls_lift_rmap_closed, pcc_L_sn/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma
index 620db955c..37a2b2618 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma
@@ -13,148 +13,211 @@
 (**************************************************************************)
 
 include "delayed_updating/syntax/path.ma".
-include "delayed_updating/notation/functions/class_c_1.ma".
+include "delayed_updating/notation/functions/class_c_2.ma".
 include "ground/arith/nat_plus.ma".
 include "ground/arith/nat_pred_succ.ma".
 include "ground/lib/subset.ma".
+include "ground/lib/bool_and.ma".
 include "ground/generated/insert_eq_1.ma".
 
 (* CLOSED CONDITION FOR PATH ************************************************)
 
-inductive pcc: relation2 nat path ≝
+inductive pcc (o): relation2 nat path ≝
 | pcc_empty:
-  pcc (𝟎) (𝐞)
+  pcc o (𝟎) (𝐞)
 | pcc_d_dx (p) (n) (k):
-  pcc (n+ninj k) p → pcc n (p◖𝗱k)
+  (Ⓣ = o → n = ↑↓n) →
+  pcc o (n+ninj k) p → pcc o n (p◖𝗱k)
 | pcc_m_dx (p) (n):
-  pcc n p → pcc n (p◖𝗺)
+  pcc o n p → pcc o n (p◖𝗺)
 | pcc_L_dx (p) (n):
-  pcc n p → pcc (↑n) (p◖𝗟)
+  pcc o n p → pcc o (↑n) (p◖𝗟)
 | pcc_A_dx (p) (n):
-  pcc n p → pcc n (p◖𝗔)
+  pcc o n p → pcc o n (p◖𝗔)
 | pcc_S_dx (p) (n):
-  pcc n p → pcc n (p◖𝗦)
+  pcc o n p → pcc o n (p◖𝗦)
 .
 
 interpretation
   "closed condition (path)"
-  'ClassC n = (pcc n).
+  'ClassC o n = (pcc o n).
+
+(* Advanced constructions ***************************************************)
+
+lemma pcc_false_d_dx (p) (n) (k:pnat):
+      p ϵ 𝐂❨Ⓕ,n+k❩ → p◖𝗱k ϵ 𝐂❨Ⓕ,n❩.
+#p #n #k #H0
+@pcc_d_dx [| // ]
+#H0 destruct
+qed.
+
+lemma pcc_true_d_dx (p) (n:pnat) (k:pnat):
+      p ϵ 𝐂❨Ⓣ,n+k❩ → p◖𝗱k ϵ 𝐂❨Ⓣ,n❩.
+/2 width=1 by pcc_d_dx/
+qed.
 
 (* Basic inversions ********************************************************)
 
-lemma pcc_inv_empty (n):
-      (𝐞) ϵ 𝐂❨n❩ → 𝟎 = n.
-#n @(insert_eq_1 … (𝐞))
+lemma pcc_inv_empty (o) (n):
+      (𝐞) ϵ 𝐂❨o,n❩ → 𝟎 = n.
+#o #n @(insert_eq_1 … (𝐞))
 #x * -n //
-#p #n [ #k ] #_ #H0 destruct
+#p #n [ #k #_ ] #_ #H0 destruct
 qed-.
 
-lemma pcc_inv_d_dx (p) (n) (k):
-      p◖𝗱k ϵ 𝐂❨n❩ → p ϵ 𝐂❨n+k❩.
-#p #n #h @(insert_eq_1 … (p◖𝗱h))
+(**) (* alias *)
+alias symbol "DownArrow" (instance 4) = "predecessor (non-negative integers)".
+alias symbol "UpArrow" (instance 3) = "successor (non-negative integers)".
+alias symbol "and" (instance 1) = "logical and".
+
+lemma pcc_inv_d_dx (o) (p) (n) (k):
+      p◖𝗱k ϵ 𝐂❨o, n❩ →
+      ∧∧ (Ⓣ = o → n = ↑↓n)
+       & p ϵ 𝐂❨o, n+k❩.
+#o #p #n #h @(insert_eq_1 … (p◖𝗱h))
 #x * -x -n
-[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+[|*: #x #n [ #k #Ho ] #Hx ] #H0 destruct
+/3 width=1 by conj/
 qed-.
 
-lemma pcc_inv_m_dx (p) (n):
-      p◖𝗺 ϵ 𝐂❨n❩ → p ϵ 𝐂❨n❩.
-#p #n @(insert_eq_1 … (p◖𝗺))
+lemma pcc_inv_m_dx (o) (p) (n):
+      p◖𝗺 ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨o,n❩.
+#o #p #n @(insert_eq_1 … (p◖𝗺))
 #x * -x -n
-[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+[|*: #x #n [ #k #_ ] #Hx ] #H0 destruct //
 qed-.
 
-lemma pcc_inv_L_dx (p) (n):
-      p◖𝗟 ϵ 𝐂❨n❩ →
-      ∧∧ p ϵ 𝐂❨↓n❩ & n = ↑↓n.
-#p #n @(insert_eq_1 … (p◖𝗟))
+lemma pcc_inv_L_dx (o) (p) (n):
+      p◖𝗟 ϵ 𝐂❨o,n❩ →
+      ∧∧ p ϵ 𝐂❨o,↓n❩ & n = ↑↓n.
+#o #p #n @(insert_eq_1 … (p◖𝗟))
 #x * -x -n
-[|*: #x #n [ #k ] #Hx ] #H0 destruct
+[|*: #x #n [ #k #_ ] #Hx ] #H0 destruct
 <npred_succ /2 width=1 by conj/
 qed-.
 
-lemma pcc_inv_A_dx (p) (n):
-      p◖𝗔 ϵ 𝐂❨n❩ → p ϵ 𝐂❨n❩.
-#p #n @(insert_eq_1 … (p◖𝗔))
+lemma pcc_inv_A_dx (o) (p) (n):
+      p◖𝗔 ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨o,n❩.
+#o #p #n @(insert_eq_1 … (p◖𝗔))
 #x * -x -n
-[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+[|*: #x #n [ #k #_ ] #Hx ] #H0 destruct //
 qed-.
 
-lemma pcc_inv_S_dx (p) (n):
-      p◖𝗦 ϵ 𝐂❨n❩ → p ϵ 𝐂❨n❩.
-#p #n @(insert_eq_1 … (p◖𝗦))
+lemma pcc_inv_S_dx (o) (p) (n):
+      p◖𝗦 ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨o,n❩.
+#o #p #n @(insert_eq_1 … (p◖𝗦))
 #x * -x -n
-[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+[|*: #x #n [ #k #_ ] #Hx ] #H0 destruct //
+qed-.
+
+(* Advanced destructions ****************************************************)
+
+lemma pcc_des_d_dx (o) (p) (n) (k):
+      p◖𝗱k ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨o,n+k❩.
+#o #p #n #k #H0
+elim (pcc_inv_d_dx … H0) -H0 #H1 #H2 //
+qed-.
+
+lemma pcc_des_gen (o) (p) (n):
+      p ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨Ⓕ,n❩.
+#o #p #n #H0 elim H0 -p -n //
+#p #n [ #k #Ho ] #_ #IH
+/2 width=1 by pcc_m_dx, pcc_L_dx, pcc_A_dx, pcc_S_dx, pcc_false_d_dx/
 qed-.
 
 (* Advanced inversions ******************************************************)
 
-lemma pcc_inv_empty_succ (n):
-      (𝐞) ϵ 𝐂❨↑n❩ → ⊥.
-#n #H0
+lemma pcc_inv_empty_succ (o) (n):
+      (𝐞) ϵ 𝐂❨o,↑n❩ → ⊥.
+#o #n #H0
 lapply (pcc_inv_empty … H0) -H0 #H0
 /2 width=7 by eq_inv_zero_nsucc/
 qed-.
 
-lemma pcc_inv_L_dx_zero (p):
-      p◖𝗟 ϵ 𝐂❨𝟎❩ → ⊥.
-#p #H0
+lemma pcc_true_inv_d_dx_zero (p) (k):
+      p◖𝗱k ϵ 𝐂❨Ⓣ,𝟎❩ → ⊥.
+#p #k #H0
+elim (pcc_inv_d_dx … H0) -H0 #H0 #_
+elim (eq_inv_zero_nsucc … (H0 ?)) -H0 //
+qed-.
+
+lemma pcc_inv_L_dx_zero (o) (p):
+      p◖𝗟 ϵ 𝐂❨o,𝟎❩ → ⊥.
+#o #p #H0
 elim (pcc_inv_L_dx … H0) -H0 #_ #H0
 /2 width=7 by eq_inv_zero_nsucc/
 qed-.
 
-lemma pcc_inv_L_dx_succ (p) (n):
-      p◖𝗟 ϵ 𝐂❨↑n❩ → p ϵ 𝐂❨n❩.
-#p #n #H0
+lemma pcc_inv_L_dx_succ (o) (p) (n):
+      p◖𝗟 ϵ 𝐂❨o,↑n❩ → p ϵ 𝐂❨o,n❩.
+#o #p #n #H0
 elim (pcc_inv_L_dx … H0) -H0 //
 qed-.
 
+(* Constructions with land **************************************************)
+
+lemma pcc_land_dx (o1) (o2) (p) (n):
+      p ϵ 𝐂❨o1,n❩ → p ϵ 𝐂❨o1∧o2,n❩.
+#o1 * /2 width=2 by pcc_des_gen/
+qed.
+
+lemma pcc_land_sn (o1) (o2) (p) (n):
+      p ϵ 𝐂❨o2,n❩ → p ϵ 𝐂❨o1∧o2,n❩.
+* /2 width=2 by pcc_des_gen/
+qed.
+
 (* Main constructions with path_append **************************************)
 
-theorem pcc_append_bi (p) (q) (m) (n):
-        p ϵ 𝐂❨m❩ → q ϵ 𝐂❨n❩ → p●q ϵ 𝐂❨m+n❩.
-#p #q #m #n #Hm #Hm elim Hm -Hm // -Hm
-#p #n [ #k ] #_ #IH [3: <nplus_succ_dx ]
-/2 width=1 by pcc_d_dx, pcc_m_dx, pcc_L_dx, pcc_A_dx, pcc_S_dx/
+theorem pcc_append_bi (o1) (o2) (p) (q) (m) (n):
+        p ϵ 𝐂❨o1,m❩ → q ϵ 𝐂❨o2,n❩ → p●q ϵ 𝐂❨o1∧o2,m+n❩.
+#o1 #o2 #p #q #m #n #Hm #Hn elim Hn -q -n
+/2 width=1 by pcc_m_dx, pcc_A_dx, pcc_S_dx, pcc_land_dx/
+#q #n [ #k #Ho2 ] #_ #IH
+[ @pcc_d_dx // #H0
+  elim (andb_inv_true_sn … H0) -H0 #_ #H0 >Ho2 //
+  <nplus_succ_dx <npred_succ //
+| <nplus_succ_dx /2 width=1 by pcc_L_dx/
+]
 qed.
 
 (* Constructions with path_lcons ********************************************)
 
-lemma pcc_m_sn (q) (n):
-      q ϵ 𝐂❨n❩ → (𝗺◗q) ϵ 𝐂❨n❩.
-#q #n #Hq
-lapply (pcc_append_bi (𝐞◖𝗺) … Hq) -Hq
+lemma pcc_m_sn (o) (q) (n):
+      q ϵ 𝐂❨o,n❩ → (𝗺◗q) ϵ 𝐂❨o,n❩.
+#o #q #n #Hq
+lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗺) … Hq) -Hq
 /2 width=3 by pcc_m_dx/
 qed.
 
-lemma pcc_L_sn (q) (n):
-      q ϵ 𝐂❨n❩ → (𝗟◗q) ϵ 𝐂❨↑n❩.
-#q #n #Hq
-lapply (pcc_append_bi (𝐞◖𝗟) … Hq) -Hq
+lemma pcc_L_sn (o) (q) (n):
+      q ϵ 𝐂❨o,n❩ → (𝗟◗q) ϵ 𝐂❨o,↑n❩.
+#o #q #n #Hq
+lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗟) … Hq) -Hq
 /2 width=3 by pcc_L_dx/
 qed.
 
-lemma pcc_A_sn (q) (n):
-      q ϵ 𝐂❨n❩ → (𝗔◗q) ϵ 𝐂❨n❩.
-#q #n #Hq
-lapply (pcc_append_bi (𝐞◖𝗔) … Hq) -Hq
+lemma pcc_A_sn (o) (q) (n):
+      q ϵ 𝐂❨o,n❩ → (𝗔◗q) ϵ 𝐂❨o,n❩.
+#o #q #n #Hq
+lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗔) … Hq) -Hq
 /2 width=3 by pcc_A_dx/
 qed.
 
-lemma pcc_S_sn (q) (n):
-      q ϵ 𝐂❨n❩ → (𝗦◗q) ϵ 𝐂❨n❩.
-#q #n #Hq
-lapply (pcc_append_bi (𝐞◖𝗦) … Hq) -Hq
+lemma pcc_S_sn (o) (q) (n):
+      q ϵ 𝐂❨o,n❩ → (𝗦◗q) ϵ 𝐂❨o,n❩.
+#o #q #n #Hq
+lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗦) … Hq) -Hq
 /2 width=3 by pcc_S_dx/
 qed.
 
 (* Main inversions **********************************************************)
 
-theorem pcc_mono (q) (n1):
-        q ϵ 𝐂❨n1❩ → ∀n2. q ϵ 𝐂❨n2❩ → n1 = n2.
-#q1 #n1 #Hn1 elim Hn1 -q1 -n1
-[|*: #q1 #n1 [ #k1 ] #_ #IH ] #n2 #Hn2
+theorem pcc_mono (o1) (o2) (q) (n1):
+        q ϵ 𝐂❨o1,n1❩ → ∀n2. q ϵ 𝐂❨o2,n2❩ → n1 = n2.
+#o1 #o2 #q1 #n1 #Hn1 elim Hn1 -q1 -n1
+[|*: #q1 #n1 [ #k1 #_ ] #_ #IH ] #n2 #Hn2
 [ <(pcc_inv_empty … Hn2) -n2 //
-| lapply (pcc_inv_d_dx … Hn2) -Hn2 #Hn2
+| lapply (pcc_des_d_dx … Hn2) -Hn2 #Hn2
   lapply (IH … Hn2) -q1 #H0
   /2 width=2 by eq_inv_nplus_bi_dx/
 | lapply (pcc_inv_m_dx … Hn2) -Hn2 #Hn2
@@ -168,16 +231,16 @@ theorem pcc_mono (q) (n1):
 ]
 qed-.
 
-theorem pcc_inj_L_sn (p1) (p2) (q1) (n):
-        q1 ϵ 𝐂❨n❩ → ∀q2. q2 ϵ 𝐂❨n❩ →
+theorem pcc_inj_L_sn (o1) (o2) (p1) (p2) (q1) (n):
+        q1 ϵ 𝐂❨o1,n❩ → ∀q2. q2 ϵ 𝐂❨o2,n❩ →
         p1●𝗟◗q1 = p2●𝗟◗q2 → q1 = q2.
-#p1 #p2 #q1 #n #Hq1 elim Hq1 -q1 -n
-[|*: #q1 #n1 [ #k1 ] #_ #IH ] * //
+#o1 #o2 #p1 #p2 #q1 #n #Hq1 elim Hq1 -q1 -n
+[|*: #q1 #n1 [ #k1 #_ ] #_ #IH ] * //
 [1,3,5,7,9,11: #l2 #q2 ] #Hq2
 <list_append_lcons_sn <list_append_lcons_sn #H0
 elim (eq_inv_list_lcons_bi ????? H0) -H0 #H0 #H1 destruct
 [ elim (pcc_inv_L_dx_zero … Hq2)
-| lapply (pcc_inv_d_dx … Hq2) -Hq2 #Hq2
+| lapply (pcc_des_d_dx … Hq2) -Hq2 #Hq2
   <(IH … Hq2) //
 | lapply (pcc_inv_m_dx … Hq2) -Hq2 #Hq2
   <(IH … Hq2) //
@@ -191,10 +254,10 @@ elim (eq_inv_list_lcons_bi ????? H0) -H0 #H0 #H1 destruct
 ]
 qed-.
 
-theorem pcc_inv_L_sn (q) (n) (m):
-        (𝗟◗q) ϵ 𝐂❨n❩ → q ϵ 𝐂❨m❩ →
+theorem pcc_inv_L_sn (o) (q) (n) (m):
+        (𝗟◗q) ϵ 𝐂❨o,n❩ → q ϵ 𝐂❨o,m❩ →
         ∧∧ ↓n = m & n = ↑↓n.
-#q #n #m #H1q #H2q
+#o #q #n #m #H1q #H2q
 lapply (pcc_L_sn … H2q) -H2q #H2q
 <(pcc_mono … H2q … H1q) -q -n
 /2 width=1 by conj/
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_structure.ma
index 99b0a9235..44554b79e 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_structure.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_structure.ma
@@ -20,8 +20,8 @@ include "delayed_updating/syntax/path_depth.ma".
 
 (* Constructions with structure *********************************************)
 
-lemma path_closed_structure_depth (p):
-      ⊗p ϵ 𝐂❨♭p❩.
-#p elim p -p // *
+lemma path_closed_structure_depth (o) (p):
+      ⊗p ϵ 𝐂❨o,♭p❩.
+#o #p elim p -p // *
 /2 width=1 by pcc_L_dx, pcc_A_dx, pcc_S_dx/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_closed.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_closed.ma
index 9879b54fb..9dbc9ec50 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_closed.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_closed.ma
@@ -24,11 +24,11 @@ include "ground/arith/nat_le_pred.ma".
 
 (* Destructions with cpp ****************************************************)
 
-lemma unwind2_rmap_append_closed_dx_xap_le (f) (p) (q) (n):
-      q ϵ 𝐂❨n❩ → ∀m. m ≤ n →
+lemma unwind2_rmap_append_closed_dx_xap_le (o) (f) (p) (q) (n):
+      q ϵ 𝐂❨o,n❩ → ∀m. m ≤ n →
       ▶[f]q＠❨m❩ = ▶[f](p●q)＠❨m❩.
-#f #p #q #n #Hq elim Hq -q -n
-[|*: #q #n [ #k ] #_ #IH ] #m #Hm
+#o #f #p #q #n #Hq elim Hq -q -n
+[|*: #q #n [ #k #_ ] #_ #IH ] #m #Hm
 [ <(nle_inv_zero_dx … Hm) -m //
 | <unwind2_rmap_d_dx <unwind2_rmap_d_dx
   <tr_compose_xap <tr_compose_xap
@@ -48,55 +48,55 @@ lemma unwind2_rmap_append_closed_dx_xap_le (f) (p) (q) (n):
 ]
 qed-.
 
-lemma unwind2_rmap_append_closed_Lq_dx_nap (f) (p) (q) (n):
-      q ϵ 𝐂❨n❩ →
+lemma unwind2_rmap_append_closed_Lq_dx_nap (o) (f) (p) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
       ▶[f](𝗟◗q)＠§❨n❩ = ▶[f](p●𝗟◗q)＠§❨n❩.
-#f #p #q #n #Hq
+#o #f #p #q #n #Hq
 lapply (pcc_L_sn … Hq) -Hq #Hq
-lapply (unwind2_rmap_append_closed_dx_xap_le f p … Hq (↑n) ?) -Hq //
+lapply (unwind2_rmap_append_closed_dx_xap_le o f p … Hq (↑n) ?) -Hq //
 <tr_xap_succ_nap <tr_xap_succ_nap #Hq
 /2 width=1 by eq_inv_nsucc_bi/
 qed-.
 
-lemma unwind2_rmap_push_closed_nap (f) (q) (n):
-      q ϵ 𝐂❨n❩ →
+lemma unwind2_rmap_push_closed_nap (o) (f) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
       ♭q = ▶[⫯f]q＠§❨n❩.
-#f #q #n #Hq elim Hq -q -n
-[|*: #q #n [ #k ] #_ #IH ] //
+#o #f #q #n #Hq elim Hq -q -n
+[|*: #q #n [ #k #_ ] #_ #IH ] //
 <unwind2_rmap_d_dx <tr_compose_nap //
 qed-.
 
-lemma unwind2_rmap_append_closed_Lq_dx_nap_depth (f) (p) (q) (n):
-      q ϵ 𝐂❨n❩ →
+lemma unwind2_rmap_append_closed_Lq_dx_nap_depth (o) (f) (p) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
       ♭q = ▶[f](p●𝗟◗q)＠§❨n❩.
-#f #p #q #n #Hq
+#o #f #p #q #n #Hq
 <unwind2_rmap_append_closed_Lq_dx_nap //
-/2 width=1 by unwind2_rmap_push_closed_nap/
+/2 width=2 by unwind2_rmap_push_closed_nap/
 qed-.
 
-lemma tls_succ_plus_unwind2_rmap_push_closed (f) (q) (n):
-      q ϵ 𝐂❨n❩ →
+lemma tls_succ_plus_unwind2_rmap_push_closed (o) (f) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
       ∀m. ⇂*[m]f ≗ ⇂*[↑(m+n)]▶[⫯f]q.
-#f #q #n #Hq elim Hq -q -n //
-#q #n [ #k ] #_ #IH #m
+#o #f #q #n #Hq elim Hq -q -n //
+#q #n [ #k #_ ] #_ #IH #m
 [ @(stream_eq_trans … (tls_unwind2_rmap_d_dx …))
   >nrplus_inj_dx >nrplus_inj_sn >nsucc_unfold //
 | <unwind2_rmap_L_dx <nplus_succ_dx //
 ]
 qed-.
 
-lemma tls_succ_unwind2_rmap_append_closed_Lq_dx (f) (p) (q) (n):
-      q ϵ 𝐂❨n❩ →
+lemma tls_succ_unwind2_rmap_append_closed_Lq_dx (o) (f) (p) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
       ▶[f]p ≗ ⇂*[↑n]▶[f](p●𝗟◗q).
-/2 width=1 by tls_succ_plus_unwind2_rmap_push_closed/
+/2 width=2 by tls_succ_plus_unwind2_rmap_push_closed/
 qed-.
 
-lemma unwind2_rmap_append_closed_Lq_dx_nap_plus (f) (p) (q) (m) (n):
-      q ϵ 𝐂❨n❩ →
+lemma unwind2_rmap_append_closed_Lq_dx_nap_plus (o) (f) (p) (q) (m) (n):
+      q ϵ 𝐂❨o,n❩ →
       ▶[f]p＠❨m❩+♭q = ▶[f](p●𝗟◗q)＠§❨m+n❩.
-#f #p #q #m #n #Hq
+#o #f #p #q #m #n #Hq
 <tr_nap_plus @eq_f2
 [ <(tr_xap_eq_repl … (tls_succ_unwind2_rmap_append_closed_Lq_dx …)) //
-| /2 width=1 by unwind2_rmap_append_closed_Lq_dx_nap_depth/
+| /2 width=2 by unwind2_rmap_append_closed_Lq_dx_nap_depth/
 ]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_unprotected.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_unprotected.ma
index 1f2a9d530..b8cb1d2cf 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_unprotected.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_unprotected.ma
@@ -30,8 +30,8 @@ lemma b_balanced: ⊗b ϵ 𝐁.
 /2 width=1 by pbc_empty, pbc_redex/
 qed.
 
-lemma b_closed: b ϵ 𝐂❨𝟎❩.
-/4 width=1 by pcc_A_sn, pcc_empty, pcc_d_dx, pcc_L_dx/
+lemma b_closed: b ϵ 𝐂❨Ⓕ,𝟎❩.
+/4 width=1 by pcc_A_sn, pcc_empty, pcc_false_d_dx, pcc_L_dx/
 qed.
 
 lemma b_unwind2_rmap_unfold (f):
diff --git a/matita/matita/contribs/lambdadelta/ground/lib/bool_and.ma b/matita/matita/contribs/lambdadelta/ground/lib/bool_and.ma
index b7a8bf964..680ead787 100644
--- a/matita/matita/contribs/lambdadelta/ground/lib/bool_and.ma
+++ b/matita/matita/contribs/lambdadelta/ground/lib/bool_and.ma
@@ -23,16 +23,27 @@ lemma commutative_andb:
 * * // qed.
 
 lemma andb_false_dx (b):
-      (b ∧ Ⓕ) = Ⓕ.
+      Ⓕ = (b ∧ Ⓕ).
 * // qed.
 
 lemma andb_false_sn (b):
-      (Ⓕ ∧ b) = Ⓕ.
+      Ⓕ = (Ⓕ ∧ b).
+// qed.
+
+lemma andb_true_dx (b):
+      b = (b ∧ Ⓣ).
+* // qed.
+
+lemma andb_true_sn (b):
+      b = (Ⓣ ∧ b).
 // qed.
 
 (* Advanced inversions ******************************************************)
 
-lemma andb_inv_true_dx (b1) (b2):
-      (b1 ∧ b2) = Ⓣ → ∧∧ b1 = Ⓣ & b2 = Ⓣ.
-* normalize /2 width=1 by conj/ #b2 #H destruct
+lemma andb_inv_true_sn (b1) (b2):
+      Ⓣ = (b1 ∧ b2) → ∧∧ Ⓣ = b1 & Ⓣ = b2.
+* normalize
+[ /2 width=1 by conj/
+| #b2 #H destruct
+]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/xap.ma b/matita/matita/contribs/lambdadelta/ground/relocation/xap.ma
index 0f367b218..0835ec53f 100644
--- a/matita/matita/contribs/lambdadelta/ground/relocation/xap.ma
+++ b/matita/matita/contribs/lambdadelta/ground/relocation/xap.ma
@@ -82,3 +82,9 @@ lemma tr_nap_plus (f) (m) (n):
       ⇂*[↑n]f＠❨m❩+f＠§❨n❩ = f＠§❨m+n❩.
 /2 width=1 by eq_inv_nsucc_bi/
 qed.
+
+lemma tr_xap_pos (f) (n):
+      n = ↑↓n → f＠❨n❩=↑↓(f＠❨n❩).
+#f #n #H0 >H0 -H0
+<tr_xap_ninj <nsucc_pnpred //
+qed.
-- 
2.39.2