update in delayed_updating

commutations of balanced reduction completed

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_ibfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_ibfr.ma
index fd62cdd78d6686912a0d2633ce86e673d137a9ea..eee619916365c16cbad2af6094e405dd46e5c6fc 100644 (file)
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_ibfr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_ibfr.ma
@@ -45,7 +45,7 @@ theorem dbfr_des_ibfr (f) (t1) (t2) (r): t1 ϵ 𝐓 →
   /2 width=2 by path_closed_structure_depth/
 | lapply (in_comp_unwind2_path_term f … Ht1) -H0t1 -Hb -Hm -Ht2 -Ht1
   <unwind2_path_d_dx <tr_pap_succ_nap >list_append_rcons_dx >list_append_assoc
-  <unwind2_rmap_append_closed_Lq_dx_nap_depth //
+  <nap_unwind2_rmap_append_closed_Lq_dx_depth //
 | lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
   @(subset_eq_trans … Ht2) -t2
   @(subset_eq_trans … (unwind2_term_fsubst_ppc …))
@@ -59,7 +59,7 @@ theorem dbfr_des_ibfr (f) (t1) (t2) (r): t1 ϵ 𝐓 →
     <list_append_rcons_sn
     @(stream_eq_trans … (tr_compose_uni_dx_pap …)) <tr_pap_succ_nap
     @tr_compose_eq_repl
-    [ <unwind2_rmap_append_closed_bLq_dx_nap_plus //
+    [ <nap_plus_unwind2_rmap_append_closed_bLq_dx_depth //
     | >unwind2_rmap_A_dx
       /2 width=2 by tls_succ_plus_unwind2_rmap_append_closed_bLq_dx/
     ]

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_lift.ma
new file mode 100644 (file)
index 0000000..fc67c57
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_lift.ma
@@ -0,0 +1,56 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/dbfr.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_structure.ma".
+include "delayed_updating/substitution/lift_path_closed.ma".
+include "delayed_updating/substitution/lift_rmap_closed.ma".
+
+(* DELAYED BALANCED FOCUSED REDUCTION ***************************************)
+
+(* Constructions with lift **************************************************)
+
+theorem dbfr_lift_bi (f) (t1) (t2) (r):
+        t1 ➡𝐝𝐛𝐟[r] t2 → 🠡[f]t1 ➡𝐝𝐛𝐟[🠡[f]r] 🠡[f]t2.
+#f #t1 #t2 #r
+* #p #b #q #m #n #Hr #Hb #Hm #Hn #Ht1 #Ht2 destruct
+@(ex6_5_intro … (🠡[f]p) (🠡[🠢[f](p◖𝗔)]b) (🠡[🠢[f](p◖𝗔●b◖𝗟)]q) (🠢[f](p●𝗔◗b)＠❨m❩) (🠢[f](p●𝗔◗b●𝗟◗q)＠§❨n❩))
+[ -Hb -Hm -Hn -Ht1 -Ht2 //
+| -Hm -Hn -Ht1 -Ht2 //
+| -Hb -Hn -Ht1 -Ht2
+  /2 width=1 by lift_path_closed/
+| -Hb -Hm -Ht1 -Ht2
+  /2 width=1 by lift_path_rmap_closed_L/
+| lapply (in_comp_lift_path_term f … Ht1) -Ht2 -Ht1 -Hn
+  <lift_path_d_dx #Ht1 //
+| lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2 -Ht1
+  @(subset_eq_trans … Ht2) -t2
+  @(subset_eq_trans … (lift_term_fsubst …))
+  @fsubst_eq_repl [ // | // ]
+  @(subset_eq_trans … (lift_term_iref_nap …))
+  <list_append_rcons_sn <list_append_rcons_sn >list_append_assoc
+  >(nap_plus_lift_rmap_append_closed_Lq_dx … Hn)
+  @iref_eq_repl
+  @(subset_eq_canc_sn … (lift_term_grafted_S …))
+  @lift_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+  >lift_rmap_A_dx
+  /2 width=2 by tls_succ_plus_lift_rmap_append_closed_bLq_dx/
+(* Note: crux of the proof ends *)
+]
+qed.

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 c07a3d26a96d7c445cd8c4286c41ad953975511a..1cb62ed3b68759ce39c3d3ab2461f28ed62cab2c 100644 (file)
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma
@@ -42,7 +42,7 @@ theorem dfr_des_ifr (f) (t1) (t2) (r): t1 ϵ 𝐓 →
   /2 width=2 by path_closed_structure_depth/
 | lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H0t1
   <unwind2_path_d_dx <tr_pap_succ_nap <list_append_rcons_sn
-  <unwind2_rmap_append_closed_Lq_dx_nap_depth //
+  <nap_unwind2_rmap_append_closed_Lq_dx_depth //
 | lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
   @(subset_eq_trans … Ht2) -t2
   @(subset_eq_trans … (unwind2_term_fsubst_ppc …))
@@ -56,7 +56,7 @@ theorem dfr_des_ifr (f) (t1) (t2) (r): t1 ϵ 𝐓 →
     <list_append_rcons_sn
     @(stream_eq_trans … (tr_compose_uni_dx_pap …)) <tr_pap_succ_nap
     @tr_compose_eq_repl
-    [ <unwind2_rmap_append_closed_Lq_dx_nap_depth //
+    [ <nap_unwind2_rmap_append_closed_Lq_dx_depth //
     | /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 ed5778166badfc540f70ee0159523c4c3935b8e2..9478f78bcbb9d2e7791d3e1f5b448064f210825b 100644 (file)
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
@@ -43,7 +43,7 @@ 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=2 by tls_succ_lift_rmap_append_L_closed_dx/
+  /2 width=2 by tls_succ_lift_rmap_append_closed_Lq_dx/
 (* Note: crux of the proof ends *)
 ]
 qed.

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr_lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr_lift.ma
new file mode 100644 (file)
index 0000000..ac690a3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr_lift.ma
@@ -0,0 +1,63 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/ibfr.ma".
+
+include "delayed_updating/substitution/fsubst_lift.ma".
+include "delayed_updating/substitution/fsubst_eq.ma".
+include "delayed_updating/substitution/lift_prototerm_after.ma".
+include "delayed_updating/substitution/lift_path_structure.ma".
+include "delayed_updating/substitution/lift_path_closed.ma".
+include "delayed_updating/substitution/lift_rmap_closed.ma".
+
+include "ground/relocation/tr_uni_compose.ma".
+include "ground/relocation/tr_compose_eq.ma".
+
+(* IMMEDIATE BALANCED FOCUSED REDUCTION *************************************)
+
+(* Constructions with lift **************************************************)
+
+theorem ibfr_lift_bi (f) (t1) (t2) (r):
+        t1 ➡𝐢𝐛𝐟[r] t2 → 🠡[f]t1 ➡𝐢𝐛𝐟[🠡[f]r] 🠡[f]t2.
+#f #t1 #t2 #r
+* #p #b #q #m #n #Hr #Hb #Hm #Hn #Ht1 #Ht2 destruct
+@(ex6_5_intro … (🠡[f]p) (🠡[🠢[f](p◖𝗔)]b) (🠡[🠢[f](p◖𝗔●b◖𝗟)]q) (🠢[f](p●𝗔◗b)＠❨m❩) (🠢[f](p●𝗔◗b●𝗟◗q)＠§❨n❩))
+[ -Hb -Hm -Hn -Ht1 -Ht2 //
+| -Hm -Hn -Ht1 -Ht2 //
+| -Hb -Hn -Ht1 -Ht2
+  /2 width=1 by lift_path_closed/
+| -Hb -Hm -Ht1 -Ht2
+  /2 width=1 by lift_path_rmap_closed_L/
+| lapply (in_comp_lift_path_term f … Ht1) -Ht2 -Ht1 -Hn
+  <lift_path_d_dx #Ht1 //
+| lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2 -Ht1
+  @(subset_eq_trans … Ht2) -t2
+  @(subset_eq_trans … (lift_term_fsubst …))
+  @fsubst_eq_repl [ // | <lift_path_append // ]
+  @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
+  [ @lift_term_grafted_S | skip ]
+  @(subset_eq_trans … (lift_term_after …))
+  @(subset_eq_canc_dx … (lift_term_after …))
+  @lift_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+  @(stream_eq_trans … (tr_compose_uni_dx_pap …)) <tr_pap_succ_nap
+  @tr_compose_eq_repl
+  [ <list_append_rcons_sn <list_append_rcons_sn >list_append_assoc
+    >(nap_plus_lift_rmap_append_closed_Lq_dx … Hn) -Hn //
+  | >lift_rmap_A_dx >nsucc_unfold
+    /2 width=2 by tls_succ_plus_lift_rmap_append_closed_bLq_dx/
+  ]
+(* Note: crux of the proof ends *)
+]
+qed.

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr_unwind.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr_unwind.ma
new file mode 100644 (file)
index 0000000..f3d7d16
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr_unwind.ma
@@ -0,0 +1,70 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/ibfr.ma".
+
+include "delayed_updating/unwind/unwind2_preterm_fsubst.ma".
+include "delayed_updating/unwind/unwind2_preterm_eq.ma".
+include "delayed_updating/unwind/unwind2_prototerm_lift.ma".
+include "delayed_updating/unwind/unwind2_rmap_closed.ma".
+
+include "delayed_updating/substitution/fsubst_eq.ma".
+include "delayed_updating/substitution/lift_prototerm_eq.ma".
+
+include "delayed_updating/syntax/path_closed_structure.ma".
+include "delayed_updating/syntax/path_structure_depth.ma".
+
+(* IMMEDIATE BALANCED FOCUSED REDUCTION *************************************)
+
+(* Constructions with unwind2 ***********************************************)
+
+lemma ibfr_unwind_bi (f) (t1) (t2) (r):
+      t1 ϵ 𝐓 → r ϵ 𝐈 →
+      t1 ➡𝐢𝐛𝐟[r] t2 → ▼[f]t1 ➡𝐢𝐛𝐟[⊗r] ▼[f]t2.
+#f #t1 #t2 #r #H1t1 #H2r
+* #p #b #q #m #n #Hr #Hb #Hm #Hn #Ht1 #Ht2 destruct
+@(ex6_5_intro … (⊗p) (⊗b) (⊗q) (♭b) (♭q))
+[ -H1t1 -H2r -Hb -Hm -Hn -Ht1 -Ht2 //
+| -H1t1 -H2r -Hm -Hn -Ht1 -Ht2 //
+| -H1t1 -H2r -Hb -Hn -Ht1 -Ht2
+  /2 width=2 by path_closed_structure_depth/
+| -H1t1 -H2r -Hb -Hm -Ht1 -Ht2
+  /2 width=2 by path_closed_structure_depth/
+| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H1t1 -H2r -Hb
+  <unwind2_path_d_dx <tr_pap_succ_nap <list_append_rcons_sn >list_append_assoc
+  <nap_unwind2_rmap_append_closed_Lq_dx_depth //
+| lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
+  @(subset_eq_trans … Ht2) -t2
+  @(subset_eq_trans … (unwind2_term_fsubst_pic …))
+  [ @fsubst_eq_repl [ // | // ]
+    @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
+    [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+    @(subset_eq_trans … (lift_unwind2_term_after …))
+    @(subset_eq_canc_dx … (unwind2_lift_term_after …))
+    @unwind2_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+    <list_append_rcons_sn
+    @(stream_eq_trans … (tr_compose_uni_dx_pap …)) <tr_pap_succ_nap
+    @tr_compose_eq_repl
+    [ <nap_plus_unwind2_rmap_append_closed_bLq_dx_depth //
+    | >unwind2_rmap_A_dx
+      /2 width=2 by tls_succ_plus_unwind2_rmap_append_closed_bLq_dx/
+    ]
+(* Note: crux of the proof ends *)
+  | //
+  | /2 width=2 by ex_intro/
+  | //
+  ]
+]
+qed.

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 35a00a3682458f55d2109805b88b744ed31ef8bf..cd597c2bbdbf56b23f94edcc1d62b63f8557546b 100644 (file)
--- 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=2 by tls_succ_lift_rmap_append_L_closed_dx/
+  /2 width=2 by tls_succ_lift_rmap_append_closed_Lq_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 2e048c7a5842612865434c3e385ed34e7666ff62..d931a1a7b035ed229a6e8bef688d31357d885fd7 100644 (file)
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_unwind.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_unwind.ma
@@ -40,7 +40,7 @@ lemma ifr_unwind_bi (f) (t1) (t2) (r):
   /2 width=2 by path_closed_structure_depth/
 | lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H1t1 -H2r
   <unwind2_path_d_dx <tr_pap_succ_nap <list_append_rcons_sn
-  <unwind2_rmap_append_closed_Lq_dx_nap_depth //
+  <nap_unwind2_rmap_append_closed_Lq_dx_depth //
 | lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
   @(subset_eq_trans … Ht2) -t2
   @(subset_eq_trans … (unwind2_term_fsubst_pic …))
@@ -54,7 +54,7 @@ lemma ifr_unwind_bi (f) (t1) (t2) (r):
     <list_append_rcons_sn
     @(stream_eq_trans … (tr_compose_uni_dx_pap …)) <tr_pap_succ_nap
     @tr_compose_eq_repl
-    [ <unwind2_rmap_append_closed_Lq_dx_nap_depth //
+    [ <nap_unwind2_rmap_append_closed_Lq_dx_depth //
     | /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_rmap_closed.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_closed.ma
index 1fd176270d332f83b81c716b3c99649e6d9d4c79..ec386108eded9b3ab11c994a8d6b8fd0fa6d70d4 100644 (file)
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_closed.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_closed.ma
@@ -14,6 +14,7 @@
 
 include "delayed_updating/substitution/lift_rmap_eq.ma".
 include "delayed_updating/syntax/path_closed.ma".
+include "ground/relocation/xap.ma".
 include "ground/lib/stream_eq_eq.ma".
 
 (* LIFT MAP FOR PATH ********************************************************)
@@ -24,8 +25,6 @@ lemma tls_plus_lift_rmap_closed (o) (f) (q) (n):
       q ϵ 𝐂❨o,n❩ →
       ∀m. ⇂*[m]f ≗ ⇂*[m+n]🠢[f]q.
 #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 (o) (f) (q) (n):
@@ -35,9 +34,32 @@ lemma tls_lift_rmap_closed (o) (f) (q) (n):
 /2 width=2 by tls_plus_lift_rmap_closed/
 qed-.
 
-lemma tls_succ_lift_rmap_append_L_closed_dx (o) (f) (p) (q) (n):
+lemma tls_lift_rmap_append_closed_dx (o) (f) (p) (q) (n):
+      q ϵ 𝐂❨o,n❩ →
+      🠢[f]p ≗ ⇂*[n]🠢[f](p●q).
+#o #f #p #q #n #Hq
+/2 width=2 by tls_lift_rmap_closed/
+qed-.
+
+lemma tls_succ_lift_rmap_append_closed_Lq_dx (o) (f) (p) (q) (n):
       q ϵ 𝐂❨o,n❩ →
       🠢[f]p ≗ ⇂*[↑n]🠢[f](p●𝗟◗q).
 #o #f #p #q #n #Hq
-/3 width=2 by tls_lift_rmap_closed, pcc_L_sn/
+/3 width=2 by tls_lift_rmap_append_closed_dx, pcc_L_sn/
+qed-.
+
+lemma tls_succ_plus_lift_rmap_append_closed_bLq_dx (o1) (o2) (f) (p) (b) (q) (m) (n):
+      b ϵ 𝐂❨o1,m❩ → q ϵ 𝐂❨o2,n❩ →
+      🠢[f]p ≗ ⇂*[↑(m+n)]🠢[f](p●b●𝗟◗q).
+#o1 #o2 #f #p #b #q #m #n #Hm #Hn
+>nplus_succ_dx <stream_tls_plus
+@(stream_eq_trans … (tls_lift_rmap_append_closed_dx … Hm))
+/3 width=2 by stream_tls_eq_repl, tls_succ_lift_rmap_append_closed_Lq_dx/
+qed-.
+
+lemma nap_plus_lift_rmap_append_closed_Lq_dx (o) (f) (p) (q) (m) (n):
+      q ϵ 𝐂❨o,n❩ →
+      🠢[f](p)＠❨m❩+🠢[f](p●𝗟◗q)＠§❨n❩ = 🠢[f](p●𝗟◗q)＠§❨m+n❩.
+#o #f #p #q #m #n #Hq
+/4 width=2 by eq_f2, tr_xap_eq_repl, tls_succ_lift_rmap_append_closed_Lq_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 0afdb349d8d324d103ebc3d0ff61d91f099c239d..b0d062de63b8dc18ef33d422531c793d28cee4d2 100644 (file)
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_closed.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_closed.ma
@@ -25,7 +25,7 @@ include "ground/arith/nat_le_pred.ma".
 
 (* Destructions with cpp ****************************************************)
 
-lemma unwind2_rmap_append_closed_dx_xap_le (o) (f) (p) (q) (n):
+lemma xap_le_unwind2_rmap_append_closed_dx (o) (f) (p) (q) (n):
       q ϵ 𝐂❨o,n❩ → ∀m. m ≤ n →
       ▶[f]q＠❨m❩ = ▶[f](p●q)＠❨m❩.
 #o #f #p #q #n #Hq elim Hq -q -n
@@ -49,17 +49,17 @@ lemma unwind2_rmap_append_closed_dx_xap_le (o) (f) (p) (q) (n):
 ]
 qed-.
 
-lemma unwind2_rmap_append_closed_Lq_dx_nap (o) (f) (p) (q) (n):
+lemma nap_unwind2_rmap_append_closed_Lq_dx (o) (f) (p) (q) (n):
       q ϵ 𝐂❨o,n❩ →
       ▶[f](𝗟◗q)＠§❨n❩ = ▶[f](p●𝗟◗q)＠§❨n❩.
 #o #f #p #q #n #Hq
 lapply (pcc_L_sn … Hq) -Hq #Hq
-lapply (unwind2_rmap_append_closed_dx_xap_le o f p … Hq (↑n) ?) -Hq //
+lapply (xap_le_unwind2_rmap_append_closed_dx 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 (o) (f) (q) (n):
+lemma nap_unwind2_rmap_push_closed_depth (o) (f) (q) (n):
       q ϵ 𝐂❨o,n❩ →
       ♭q = ▶[⫯f]q＠§❨n❩.
 #o #f #q #n #Hq elim Hq -q -n
@@ -67,15 +67,15 @@ lemma unwind2_rmap_push_closed_nap (o) (f) (q) (n):
 <unwind2_rmap_d_dx <tr_compose_nap //
 qed-.
 
-lemma unwind2_rmap_append_closed_Lq_dx_nap_depth (o) (f) (p) (q) (n):
+lemma nap_unwind2_rmap_append_closed_Lq_dx_depth (o) (f) (p) (q) (n):
       q ϵ 𝐂❨o,n❩ →
       ♭q = ▶[f](p●𝗟◗q)＠§❨n❩.
 #o #f #p #q #n #Hq
-<unwind2_rmap_append_closed_Lq_dx_nap //
-/2 width=2 by unwind2_rmap_push_closed_nap/
+<nap_unwind2_rmap_append_closed_Lq_dx //
+/2 width=2 by nap_unwind2_rmap_push_closed_depth/
 qed-.
 
-lemma unwind2_rmap_append_closed_true_dx_xap_depth (f) (p) (q) (n):
+lemma xap_unwind2_rmap_append_closed_true_dx_depth (f) (p) (q) (n):
       q ϵ 𝐂❨Ⓣ,n❩ → ♭q = ▶[f](p●q)＠❨n❩.
 #f #p #q #n #Hq elim Hq -q -n //
 #q #n #k #Ho #_ #IH
@@ -115,30 +115,29 @@ lemma tls_unwind2_rmap_append_closed_true_dx (f) (p) (q) (n):
 /2 width=1 by tls_plus_unwind2_rmap_closed_true/
 qed-.
 
-lemma unwind2_rmap_append_closed_Lq_dx_nap_plus (o) (f) (p) (q) (m) (n):
+lemma nap_plus_unwind2_rmap_append_closed_Lq_dx_depth (o) (f) (p) (q) (m) (n):
       q ϵ 𝐂❨o,n❩ →
       ▶[f]p＠❨m❩+♭q = ▶[f](p●𝗟◗q)＠§❨m+n❩.
 #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=2 by unwind2_rmap_append_closed_Lq_dx_nap_depth/
+| /2 width=2 by nap_unwind2_rmap_append_closed_Lq_dx_depth/
 ]
 qed-.
 
-lemma unwind2_rmap_append_closed_bLq_dx_nap_plus (o) (f) (p) (b) (q) (m) (n):
+lemma nap_plus_unwind2_rmap_append_closed_bLq_dx_depth (o) (f) (p) (b) (q) (m) (n):
       b ϵ 𝐂❨Ⓣ,m❩ → q ϵ 𝐂❨o,n❩ →
       ♭b+♭q = ▶[f](p●b●𝗟◗q)＠§❨m+n❩.
 #o #f #p #b #q #m #n #Hb #Hq
->(unwind2_rmap_append_closed_true_dx_xap_depth f p … Hb) -Hb
->(unwind2_rmap_append_closed_Lq_dx_nap_plus … Hq) -Hq //
+>(xap_unwind2_rmap_append_closed_true_dx_depth f p … Hb) -Hb
+>(nap_plus_unwind2_rmap_append_closed_Lq_dx_depth … Hq) -Hq //
 qed-.
 
 lemma tls_succ_plus_unwind2_rmap_append_closed_bLq_dx (o) (f) (p) (b) (q) (m) (n):
       b ϵ 𝐂❨Ⓣ,m❩ → q ϵ 𝐂❨o,n❩ →
       ▶[f]p ≗ ⇂*[↑(m+n)]▶[f](p●b●𝗟◗q).
 #o #f #p #b #q #m #n #Hb #Hq
->nplus_succ_dx <stream_tls_plus
+>nplus_succ_dx <stream_tls_plus >list_append_assoc
 @(stream_eq_trans … (tls_unwind2_rmap_append_closed_true_dx … Hb)) -Hb
-@stream_tls_eq_repl
-@(stream_eq_trans … (tls_succ_unwind2_rmap_append_closed_Lq_dx … Hq)) -Hq //
+/3 width=2 by stream_tls_eq_repl, tls_succ_unwind2_rmap_append_closed_Lq_dx/
 qed-.