From 9084ed69b2ad9c9c2899def8a12dd907f9ce4ced Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Wed, 29 Jun 2022 14:22:26 +0200
Subject: [PATCH] update in delayed updating

+ two lemmas on path_head
+ invocation added to first file
---
 .../delayed_updating/syntax/label.ma          |  4 ++
 .../delayed_updating/syntax/path_head.ma      | 44 +++++++++++++++++++
 2 files changed, 48 insertions(+)

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/label.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/label.ma
index 74318f333..98e63f06b 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/label.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/label.ma
@@ -12,6 +12,10 @@
 (*                                                                        *)
 (**************************************************************************)
 
+(* A SYSTEM OF λ-CALCULUS WITH DELAYED UPDATING 
+ * Initial invocation: - Patience on me to gain peace and perfection! -
+ *)
+
 include "ground/arith/pnat.ma".
 include "delayed_updating/notation/functions/nodelabel_d_1.ma".
 include "delayed_updating/notation/functions/nodelabel_m_0.ma".
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head.ma
index 9cdfe620f..438c5e6a7 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head.ma
@@ -74,6 +74,11 @@ lemma path_head_S_sn (p) (n):
 
 (* Basic inversions *********************************************************)
 
+lemma eq_inv_path_head_zero_dx (q) (p):
+      q = ↳[𝟎]p → 𝐞 = q.
+#q * //
+qed-.
+
 lemma eq_inv_path_empty_head (p) (n):
       (𝐞) = ↳[n]p → 𝟎 = n.
 *
@@ -88,3 +93,42 @@ lemma eq_inv_path_empty_head (p) (n):
   ] #H0 destruct
 ]
 qed-.
+
+(* Constructions with list_append *******************************************)
+
+lemma path_head_refl_append (p) (q) (n):
+      q = ↳[n]q → q = ↳[n](q●p).
+#p #q elim q -q
+[ #n #Hn <(eq_inv_path_empty_head … Hn) -Hn //
+| #l #q #IH #n @(nat_ind_succ … n) -n
+  [ #Hq | #n #_ cases l [ #m ] ]
+  [ lapply (eq_inv_path_head_zero_dx … Hq) -Hq #Hq destruct
+  | <path_head_d_sn <path_head_d_sn
+  | <path_head_m_sn <path_head_m_sn
+  | <path_head_L_sn <path_head_L_sn
+  | <path_head_A_sn <path_head_A_sn
+  | <path_head_S_sn <path_head_S_sn
+  ] #Hq
+  elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+  /3 width=1 by eq_f/
+]
+qed-.
+
+(* Inversions with list_append **********************************************)
+
+lemma eq_inv_path_head_refl_append (p) (q) (n):
+      q = ↳[n](q●p) → q = ↳[n]q.
+#p #q elim q -q
+[ #n #Hn <(eq_inv_path_empty_head … Hn) -p //
+| #l #q #IH #n @(nat_ind_succ … n) -n //
+  #n #_ cases l [ #m ]
+  [ <path_head_d_sn <path_head_d_sn
+  | <path_head_m_sn <path_head_m_sn
+  | <path_head_L_sn <path_head_L_sn
+  | <path_head_A_sn <path_head_A_sn
+  | <path_head_S_sn <path_head_S_sn
+  ] #Hq
+  elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+  /3 width=1 by eq_f/
+]
+qed-.
-- 
2.39.2