From cd2f5b59215ea771ac137b9a7b115a05175f45d5 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Mon, 14 Nov 2022 23:13:29 +0100
Subject: [PATCH] update in delayed_updating

+ example of unprotected balanced reduction continued
---
 .../reduction/dbfr_unprotected.ma             | 26 +++++++++++++++++++
 .../reduction/ibfr_constructors.ma            | 26 +++++++++++++++++--
 2 files changed, 50 insertions(+), 2 deletions(-)

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_unprotected.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_unprotected.ma
index a0e5a6944..7c9276021 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_unprotected.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_unprotected.ma
@@ -18,6 +18,12 @@ include "ground/arith/pnat_two.ma".
 
 (* DELAYED BALANCED FOCUSED REDUCTION ***************************************)
 
+(* Prerequisites ************************************************************)
+
+lemma list_rcons_prop_1 (A) (a1) (a2):
+      ⓔ ⨭ a1 ⨭{A} a2 = a1 ⨮ (ⓔ ⨭ a2).
+// qed.
+
 (* Example of unprotected balanced β-reduction ******************************)
 
 definition l3_t0: prototerm ≝
@@ -26,6 +32,9 @@ definition l3_t0: prototerm ≝
 definition l3_t1: prototerm ≝
            (𝛌.＠(⧣𝟏).＠(𝛌.＠(⧣𝟐).⧣𝟏).𝛌.(𝛌.＠(⧣↑𝟐).⧣𝟏)).
 
+definition l3_t2: prototerm ≝
+           (𝛌.＠(⧣𝟏).＠(𝛌.＠(⧣𝟐).⧣𝟏).𝛌.(𝛌.＠(⧣↑𝟐).⧣↑𝟐)).
+
 lemma l3_t01:
       l3_t0 ➡𝐢𝐛𝐟[𝗟◗𝗔◗𝗔◗𝗟◗𝐞] l3_t1.
 @ibfr_abst_hd
@@ -40,3 +49,20 @@ lemma l3_t01:
 @appl_eq_repl
 @(subset_eq_canc_sn … (lift_term_oref_pap … )) //
 qed.
+
+lemma l3_t12:
+      l3_t1 ➡𝐢𝐛𝐟[𝗟◗𝗔◗𝗔◗𝗟◗𝗟◗𝗔◗𝐞] l3_t2.
+@ibfr_abst_hd
+@ibfr_eq_trans
+[| @(ibfr_beta_1 … (𝟎)) [| <list_rcons_prop_1 ]
+   /2 width=3 by pcc_A_sn, in_comp_appl_hd/
+]
+@appl_eq_repl [ // ]
+@appl_eq_repl [ // ]
+@abst_eq_repl
+@abst_eq_repl
+@(subset_eq_canc_sn … (fsubst_appl_hd …))
+@appl_eq_repl [ // ]
+@(subset_eq_canc_sn … (fsubst_empty_rc …))
+@(subset_eq_canc_sn … (lift_term_oref_pap … )) //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr_constructors.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr_constructors.ma
index 2a98609ad..112a98c39 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr_constructors.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr_constructors.ma
@@ -68,8 +68,11 @@ lemma ibfr_beta_0 (v) (t) (q) (n):
       q ϵ 𝐂❨Ⓕ,n❩ → q◖𝗱↑n ϵ t →
       ＠v.𝛌.t ➡𝐢𝐛𝐟[𝗔◗𝗟◗q] ＠v.𝛌.(t[⋔q←🠡[𝐮❨↑n❩]v]).
 #v #t #q #n #Hn #Ht
-@(ex6_5_intro … (𝐞) (𝐞) q (𝟎) … Hn) -Hn //
-[ /3 width=1 by in_comp_appl_hd, in_comp_abst_hd/
+@(ex6_5_intro … (𝐞) (𝐞) q (𝟎) … Hn) -Hn
+[ //
+| //
+| //
+| /3 width=1 by in_comp_appl_hd, in_comp_abst_hd/
 | @(subset_eq_canc_sn … (fsubst_appl_hd …)) @appl_eq_repl [ // ]
   @(subset_eq_canc_sn … (fsubst_abst_hd …)) @abst_eq_repl
   @fsubst_eq_repl // <nplus_zero_sn @lift_term_eq_repl_dx
@@ -77,3 +80,22 @@ lemma ibfr_beta_0 (v) (t) (q) (n):
   @(subset_eq_canc_sn … (grafted_empty_dx … )) //
 ]
 qed.
+
+lemma ibfr_beta_1 (v) (v1) (t) (q) (n):
+      q ϵ 𝐂❨Ⓕ,n❩ → q◖𝗱↑n ϵ t →
+      ＠v.＠v1.𝛌.𝛌.t ➡𝐢𝐛𝐟[𝗔◗𝗔◗𝗟◗𝗟◗q] ＠v.＠v1.𝛌.𝛌.(t[⋔q←🠡[𝐮❨↑↑n❩]v]).
+#v #v1 #t #q #n #Hn #Ht
+@(ex6_5_intro … (𝐞) (𝗔◗𝗟◗𝐞) q (𝟏) … Hn) -Hn
+[ //
+| /2 width=1 by pbc_empty, pbc_redex/
+| /3 width=1 by pcc_A_sn, pcc_L_sn, pcc_empty/
+| /5 width=1 by in_comp_appl_hd, in_comp_abst_hd/
+| @(subset_eq_canc_sn … (fsubst_appl_hd …)) @appl_eq_repl [ // ]
+  @(subset_eq_canc_sn … (fsubst_appl_hd …)) @appl_eq_repl [ // ]
+  @(subset_eq_canc_sn … (fsubst_abst_hd …)) @abst_eq_repl
+  @(subset_eq_canc_sn … (fsubst_abst_hd …)) @abst_eq_repl
+  @fsubst_eq_repl // <nplus_unit_sn @lift_term_eq_repl_dx
+  >list_cons_comm @(subset_eq_canc_sn … (grafted_appl_sd … ))
+  @(subset_eq_canc_sn … (grafted_empty_dx … )) //
+]
+qed.
-- 
2.39.2