From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sun, 21 Aug 2022 22:43:28 +0000 (+0200)
Subject: partial update in delayed_updating
X-Git-Tag: make_still_working~43
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=b1c5b3370653db6e495bbf6b3799cba592746cdd;p=helm.git
partial update in delayed_updating
+ syntax and substitution components updated
+ path_head parked and replaced by path_closed
+ balanced reductions introduced and main theorem proved
+ minor corrections to notation
---
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/downarrowright_2.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/downarrowright_2.etc
new file mode 100644
index 000000000..f086d72e2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/downarrowright_2.etc
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR DELAYED UPDATING ********************************************)
+
+notation "hvbox( â³[ break term 46 n ] break term 70 p )"
+ non associative with precedence 70
+ for @{ 'DownArrowRight $n $p }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/lift_path_head.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/lift_path_head.etc
new file mode 100644
index 000000000..21b70c57a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/lift_path_head.etc
@@ -0,0 +1,61 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/substitution/lift_path.ma".
+include "delayed_updating/syntax/path_head.ma".
+include "ground/relocation/xap.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+(* Constructions with head for path *****************************************)
+
+lemma lift_path_head_closed (f) (p) (q) (n):
+ q = â³[n]q â
+ â³[â[pâq]fï¼ â¨nâ©]â[â[p]f]q = â[â[p]f]q.
+#f #p #q elim q -q
+[ #n #H0
+ <(eq_inv_path_empty_head ⦠H0) -H0
+ <path_head_zero //
+| #l #q #IH #n @(nat_ind_succ â¦n) -n [| #n #_ ]
+ [ <path_head_zero #H0 destruct
+ | cases l [ #k ]
+ [ <path_head_d_dx #H0
+ elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
+ <lift_rmap_d_dx <lift_path_d_dx
+ <tr_xap_succ_nap <path_head_d_dx
+ <(IH ⦠H0) in ⢠(???%); -IH -H0 <tr_xap_plus //
+ | <path_head_m_dx #H0
+ elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
+ <lift_rmap_m_dx <lift_path_m_dx
+ <tr_xap_succ_nap <path_head_m_dx
+ <(IH ⦠H0) in ⢠(???%); -IH -H0 //
+ | <path_head_L_dx #H0
+ elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
+ <lift_rmap_L_dx <lift_path_L_dx
+ <tr_xap_succ_nap <path_head_L_dx
+ <(IH ⦠H0) in ⢠(???%); -IH -H0 //
+ | <path_head_A_dx #H0
+ elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
+ <lift_rmap_A_dx <lift_path_A_dx
+ <tr_xap_succ_nap <path_head_A_dx
+ <(IH ⦠H0) in ⢠(???%); -IH -H0 //
+ | <path_head_S_dx #H0
+ elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
+ <lift_rmap_S_dx <lift_path_S_dx
+ <tr_xap_succ_nap <path_head_S_dx
+ <(IH ⦠H0) in ⢠(???%); -IH -H0 //
+ ]
+ ]
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/lift_rmap_head.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/lift_rmap_head.etc
new file mode 100644
index 000000000..033a2c13e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/lift_rmap_head.etc
@@ -0,0 +1,57 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/substitution/lift_rmap_eq.ma".
+include "delayed_updating/syntax/path_head.ma".
+include "ground/lib/stream_eq_eq.ma".
+
+(* LIFT MAP FOR PATH ********************************************************)
+
+(* Constructions with path_head *********************************************)
+
+lemma tls_plus_lift_rmap_closed (f) (q) (n) (m):
+ q = â³[n]q â
+ â*[m]f â â*[n+m]â[q]f.
+#f #q elim q -q
+[ #n #m #Hq
+ <(eq_inv_path_empty_head ⦠Hq) -n //
+| #l #q #IH #n @(nat_ind_succ ⦠n) -n //
+ #n #_ #m cases l [ #k ]
+ [ <path_head_d_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq <nrplus_inj_sn
+ @(stream_eq_trans ⦠(tls_lift_rmap_d_dx â¦))
+ >nrplus_inj_dx >nrplus_inj_sn >nrplus_inj_sn <nplus_plus_comm_23
+ >nsucc_unfold /2 width=1 by/
+ | <path_head_m_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <lift_rmap_m_dx /2 width=1 by/
+ | <path_head_L_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <lift_rmap_L_dx <nplus_succ_sn /2 width=1 by/
+ | <path_head_A_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <lift_rmap_A_dx /2 width=2 by/
+ | <path_head_S_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <lift_rmap_S_dx /2 width=2 by/
+ ]
+]
+qed-.
+
+lemma tls_lift_rmap_closed (f) (q) (n):
+ q = â³[n]q â
+ f â â*[n]â[q]f.
+#f #q #n #H0
+/2 width=1 by tls_plus_lift_rmap_closed/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_depth_labels.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_depth_labels.etc
new file mode 100644
index 000000000..791a09a51
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_depth_labels.etc
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/path_depth.ma".
+include "delayed_updating/syntax/path_labels.ma".
+
+(* DEPTH FOR PATH ***********************************************************)
+
+(* Constructions with labels ************************************************)
+
+lemma depth_labels_L (n):
+ n = â(ðâân).
+#n @(nat_ind_succ ⦠n) -n //
+#n #IH <labels_succ <depth_L_dx //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head.etc
new file mode 100644
index 000000000..08054d900
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head.etc
@@ -0,0 +1,162 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/path_labels.ma".
+include "delayed_updating/notation/functions/downarrowright_2.ma".
+include "ground/arith/nat_plus.ma".
+include "ground/arith/nat_pred_succ.ma".
+
+(* HEAD FOR PATH ************************************************************)
+
+rec definition path_head (n) (p) on p: path â
+match n with
+[ nzero â ð
+| ninj m â
+ match p with
+ [ list_empty â ðâân
+ | list_lcons l q â
+ match l with
+ [ label_d k â (path_head (n+k) q)âl
+ | label_m â (path_head n q)âl
+ | label_L â (path_head (âm) q)âl
+ | label_A â (path_head n q)âl
+ | label_S â (path_head n q)âl
+ ]
+ ]
+].
+
+interpretation
+ "head (path)"
+ 'DownArrowRight n p = (path_head n p).
+
+(* basic constructions ****************************************************)
+
+lemma path_head_zero (p):
+ (ð) = â³[ð]p.
+* // qed.
+
+lemma path_head_empty (n):
+ (ðâân) = â³[n]ð.
+* // qed.
+
+lemma path_head_d_dx (p) (n) (k:pnat):
+ (â³[ân+k]p)âð±k = â³[ân](pâð±k).
+// qed.
+
+lemma path_head_m_dx (p) (n):
+ (â³[ân]p)âðº = â³[ân](pâðº).
+// qed.
+
+lemma path_head_L_dx (p) (n):
+ (â³[n]p)âð = â³[ân](pâð).
+#p #n
+whd in ⢠(???%); //
+qed.
+
+lemma path_head_A_dx (p) (n):
+ (â³[ân]p)âð = â³[ân](pâð).
+// qed.
+
+lemma path_head_S_dx (p) (n):
+ (â³[ân]p)âð¦ = â³[ân](pâð¦).
+// qed.
+
+(* Basic inversions *********************************************************)
+
+lemma eq_inv_path_head_zero_dx (q) (p):
+ q = â³[ð]p â ð = q.
+#q * //
+qed-.
+
+lemma eq_inv_path_empty_head (p) (n):
+ (ð) = â³[n]p â ð = n.
+*
+[ #n <path_head_empty #H0
+ <(eq_inv_empty_labels ⦠H0) -n //
+| * [ #k ] #p #n @(nat_ind_succ ⦠n) -n // #n #_
+ [ <path_head_d_dx
+ | <path_head_m_dx
+ | <path_head_L_dx
+ | <path_head_A_dx
+ | <path_head_S_dx
+ ] #H0 destruct
+]
+qed-.
+
+(* Constructions with path_append *******************************************)
+
+lemma path_head_refl_append_bi (p) (q) (m) (n):
+ p = â³[m]p â q = â³[n]q â pâq = â³[n+m](pâq).
+#p #q elim q -q
+[ #m #n #Hp #H0
+ <(eq_inv_path_empty_head ⦠H0) -n //
+| #l #q #IH #m #n #Hp
+ @(nat_ind_succ ⦠n) -n // #n #_
+ cases l [ #k ]
+ <list_append_lcons_sn <nplus_succ_sn
+ [ <path_head_d_dx <path_head_d_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ >(IH ⦠Hp Hq) in ⢠(??%?); -IH -Hp -Hq
+ <nplus_plus_comm_23 //
+ | <path_head_m_dx <path_head_m_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ >(IH ⦠Hp Hq) in ⢠(??%?); -IH -Hp -Hq //
+ | <path_head_L_dx <path_head_L_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ >(IH ⦠Hp Hq) in ⢠(??%?); -IH -Hp -Hq //
+ | <path_head_A_dx <path_head_A_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ >(IH ⦠Hp Hq) in ⢠(??%?); -IH -Hp -Hq //
+ | <path_head_S_dx <path_head_S_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ >(IH ⦠Hp Hq) in ⢠(??%?); -IH -Hp -Hq //
+ ]
+qed.
+
+lemma path_head_refl_append_sn (p) (q) (n):
+ q = â³[n]q â q = â³[n](pâq).
+#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 [ #k ] ]
+ [ lapply (eq_inv_path_head_zero_dx ⦠Hq) -Hq #Hq destruct
+ | <path_head_d_dx <path_head_d_dx
+ | <path_head_m_dx <path_head_m_dx
+ | <path_head_L_dx <path_head_L_dx
+ | <path_head_A_dx <path_head_A_dx
+ | <path_head_S_dx <path_head_S_dx
+ ] #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ /3 width=1 by eq_f/
+]
+qed-.
+
+(* Inversions with path_append **********************************************)
+
+lemma eq_inv_path_head_refl_append_sn (p) (q) (n):
+ q = â³[n](pâq) â q = â³[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 [ #k ]
+ [ <path_head_d_dx <path_head_d_dx
+ | <path_head_m_dx <path_head_m_dx
+ | <path_head_L_dx <path_head_L_dx
+ | <path_head_A_dx <path_head_A_dx
+ | <path_head_S_dx <path_head_S_dx
+ ] #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ /3 width=1 by eq_f/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head_depth.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head_depth.etc
new file mode 100644
index 000000000..6283e2c21
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head_depth.etc
@@ -0,0 +1,35 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/path_head.ma".
+include "delayed_updating/syntax/path_height_labels.ma".
+include "delayed_updating/syntax/path_depth_labels.ma".
+
+(* HEAD FOR PATH ************************************************************)
+
+(* Constructions with depth and height **************************************)
+
+lemma path_head_depth (p) (n):
+ n + â¯(â³[n]p) = ââ³[n]p.
+#p elim p -p // #l #p #IH
+#n @(nat_ind_succ ⦠n) -n // #n #_
+cases l [ #k ]
+[ <path_head_d_dx <height_d_dx <depth_d_dx <IH //
+| <path_head_m_dx <height_m_dx <depth_m_dx //
+| <path_head_L_dx <height_L_dx <depth_L_dx
+ <nplus_succ_sn //
+| <path_head_A_dx <height_A_dx <depth_A_dx //
+| <path_head_S_dx <height_S_dx <depth_S_dx //
+]
+qed-. (**) (* gets in the way with auto *)
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head_structure.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head_structure.etc
new file mode 100644
index 000000000..37ca7be2c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_head_structure.etc
@@ -0,0 +1,39 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/path_head_depth.ma".
+include "delayed_updating/syntax/path_structure_labels.ma".
+
+(* HEAD FOR PATH ************************************************************)
+
+(* Constructions with structure *********************************************)
+
+lemma path_head_structure_height (p) (n):
+ ââ³[n]p = â³[n+â¯â³[n]p]âp.
+#p elim p -p //
+#l #p #IH #n @(nat_ind_succ ⦠n) -n //
+#n #_ cases l [ #k ]
+[ <height_d_dx <structure_d_dx //
+| <structure_m_dx //
+| <structure_L_dx //
+| <height_A_dx <structure_A_dx <nplus_succ_sn <path_head_A_dx //
+| <height_S_dx <structure_S_dx <nplus_succ_sn <path_head_S_dx //
+]
+qed.
+
+lemma path_head_structure_depth (p) (n):
+ ââ³[n]p = â³[ââ³[n]p]âp.
+#p #n
+<path_head_depth //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_height_labels.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_height_labels.etc
new file mode 100644
index 000000000..fb55c3606
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_height_labels.etc
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/path_height.ma".
+include "delayed_updating/syntax/path_labels.ma".
+
+(* HEIGHT FOR PATH **********************************************************)
+
+(* Constructions with labels ************************************************)
+
+lemma height_labels_L (n):
+ (ð) = â¯(ðâân).
+#n @(nat_ind_succ ⦠n) -n //
+#n #IH <labels_succ <height_L_dx //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_labels.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_labels.etc
new file mode 100644
index 000000000..7f0a71a8c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_labels.etc
@@ -0,0 +1,50 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/path.ma".
+include "delayed_updating/notation/functions/power_2.ma".
+include "ground/arith/nat_succ_iter.ma".
+
+(* *)
+
+definition labels (l) (n:nat): path â
+ ((list_lcons ? l)^n) (ð).
+
+interpretation
+ "labels (path)"
+ 'Power l n = (labels l n).
+
+(* Basic constructions ******************************************************)
+
+lemma labels_unfold (l) (n):
+ ((list_lcons ? l)^n) (ð) = lâân.
+// qed.
+
+lemma labels_zero (l):
+ (ð) = lââð.
+// qed.
+
+lemma labels_succ (l) (n):
+ (lâân)âl = lââ(ân).
+#l #n
+<labels_unfold <labels_unfold <niter_succ //
+qed.
+
+(* Basic inversions *********************************************************)
+
+lemma eq_inv_empty_labels (l) (n):
+ (ð) = lâân â ð = n.
+#l #n @(nat_ind_succ ⦠n) -n //
+#n #_ <labels_succ #H0 destruct
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_structure_labels.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_structure_labels.etc
new file mode 100644
index 000000000..0dcc367cd
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/path_structure_labels.etc
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/path_structure.ma".
+include "delayed_updating/syntax/path_labels.ma".
+
+(* STRUCTURE FOR PATH *******************************************************)
+
+(* Constructions with labels ************************************************)
+
+lemma structure_labels_L (n):
+ (ðâân) = â(ðâân).
+#n @(nat_ind_succ ⦠n) -n //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/power_2.etc b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/power_2.etc
new file mode 100644
index 000000000..bed0c7fe5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/etc/head/power_2.etc
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR DELAYED UPDATING ********************************************)
+
+notation "hvbox( a ââ break b )"
+ left associative with precedence 65
+ for @{ 'Power $a $b }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_downtriangle_2.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_downtriangle_2.ma
index ddb2bc56d..8299e9384 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_downtriangle_2.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_downtriangle_2.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR DELAYED UPDATING ********************************************)
-notation "hvbox( â¼[ term 46 f ] break term 70 p )"
+notation "hvbox( â¼[ break term 46 f ] break term 70 p )"
non associative with precedence 70
for @{ 'BlackDownTriangle $f $p }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_righttriangle_2.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_righttriangle_2.ma
index 90b075806..020fc5538 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_righttriangle_2.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/black_righttriangle_2.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR DELAYED UPDATING ********************************************)
-notation "hvbox( â¶[ term 46 f ] break term 70 p )"
+notation "hvbox( â¶[ break term 46 f ] break term 70 p )"
non associative with precedence 70
for @{ 'BlackRightTriangle $f $p }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/circled_times_1.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/circled_times_1.ma
index 8c39289b5..a489dc8da 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/circled_times_1.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/circled_times_1.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR DELAYED UPDATING ********************************************)
-notation "hvbox( â term 70 p )"
+notation "hvbox( â break term 70 p )"
non associative with precedence 70
for @{ 'CircledTimes $p }.
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_1.ma
index 41a736d62..352144291 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_1.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR DELAYED UPDATING ********************************************)
-notation "hvbox( ð⨠term 46 n â© )"
+notation "hvbox( ð⨠break term 46 n â© )"
non associative with precedence 70
for @{ 'ClassC $n }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/downarrowright_2.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/downarrowright_2.ma
deleted file mode 100644
index 7531c6bcc..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/downarrowright_2.ma
+++ /dev/null
@@ -1,19 +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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR DELAYED UPDATING ********************************************)
-
-notation "hvbox( â³[ term 46 n ] break term 70 p )"
- non associative with precedence 70
- for @{ 'DownArrowRight $n $p }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/flat_1.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/flat_1.ma
index 5d873910b..077255a35 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/flat_1.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/flat_1.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR DELAYED UPDATING ********************************************)
-notation "hvbox( â term 70 p )"
+notation "hvbox( â break term 70 p )"
non associative with precedence 70
for @{ 'Flat $p }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/power_2.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/power_2.ma
deleted file mode 100644
index bed0c7fe5..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/power_2.ma
+++ /dev/null
@@ -1,19 +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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR DELAYED UPDATING ********************************************)
-
-notation "hvbox( a ââ break b )"
- left associative with precedence 65
- for @{ 'Power $a $b }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/tau_2.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/tau_2.ma
index 099360cce..6d4dd8839 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/tau_2.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/tau_2.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR DELAYED UPDATING ********************************************)
-notation "hvbox( ð break term 76 n. break term 70 t )"
+notation "hvbox( ð break term 76 n . break term 70 t )"
non associative with precedence 70
for @{ 'Tau $n $t }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uparrow_2.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uparrow_2.ma
index 2cbac4b24..b3123df7b 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uparrow_2.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uparrow_2.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR DELAYED UPDATING ********************************************)
-notation "hvbox( â[ term 46 t1 ] break term 70 t2 )"
+notation "hvbox( â[ break term 46 t1 ] break term 70 t2 )"
non associative with precedence 70
for @{ 'UpArrow $t1 $t2 }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uptriangle_1.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uptriangle_1.ma
index df2bee52c..9a3516c7b 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uptriangle_1.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/uptriangle_1.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR DELAYED UPDATING ********************************************)
-notation "hvbox( âµ term 70 t )"
+notation "hvbox( âµ break term 70 t )"
non associative with precedence 70
for @{ 'UpTriangle $t }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_dbf_3.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_dbf_3.ma
new file mode 100644
index 000000000..390aabf5e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_dbf_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR DELAYED UPDATING ********************************************)
+
+notation "hvbox( t1 â¡ððð[ break term 46 r ] break term 46 t2 )"
+ non associative with precedence 45
+ for @{ 'BlackRightArrowDBF $t1 $r $t2 }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_df_3.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_df_3.ma
index f50962063..1750f335b 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_df_3.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_df_3.ma
@@ -15,5 +15,5 @@
(* NOTATION FOR DELAYED UPDATING ********************************************)
notation "hvbox( t1 â¡ðð[ break term 46 r ] break term 46 t2 )"
- non associative with precedence 45
- for @{ 'BlackRightArrowDF $t1 $r $t2 }.
+ non associative with precedence 45
+ for @{ 'BlackRightArrowDF $t1 $r $t2 }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_ibf_3.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_ibf_3.ma
new file mode 100644
index 000000000..12a1b0ff2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_ibf_3.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR DELAYED UPDATING ********************************************)
+
+notation "hvbox( t1 â¡ð¢ðð[ break term 46 r ] break term 46 t2 )"
+ non associative with precedence 45
+ for @{ 'BlackRightArrowIBF $t1 $r $t2 }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_if_3.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_if_3.ma
index d2568773a..90fe54b75 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_if_3.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/black_rightarrow_if_3.ma
@@ -15,5 +15,5 @@
(* NOTATION FOR DELAYED UPDATING ********************************************)
notation "hvbox( t1 â¡ð¢ð[ break term 46 r ] break term 46 t2 )"
- non associative with precedence 45
- for @{ 'BlackRightArrowIF $t1 $r $t2 }.
+ non associative with precedence 45
+ for @{ 'BlackRightArrowIF $t1 $r $t2 }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr.ma
new file mode 100644
index 000000000..349195cb0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr.ma
@@ -0,0 +1,38 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/substitution/fsubst.ma".
+include "delayed_updating/syntax/prototerm_constructors.ma".
+include "delayed_updating/syntax/prototerm_eq.ma".
+include "delayed_updating/syntax/path_head.ma".
+include "delayed_updating/syntax/path_balanced.ma".
+include "delayed_updating/syntax/path_structure.ma".
+include "delayed_updating/notation/relations/black_rightarrow_dbf_3.ma".
+include "ground/arith/nat_rplus.ma".
+include "ground/xoa/ex_6_5.ma".
+
+(* DELAYED BALANCED FOCUSED REDUCTION ***************************************)
+
+(**) (* explicit ninj because we cannot declare the expectd type of k *)
+definition dbfr (r): relation2 prototerm prototerm â
+ λt1,t2.
+ ââp,b,q,h,k. pâðâbâðâq = r &
+ âb ϵ ð & b = â³[h]b &
+ (ðâq) = â³[ninj k](ðâq) & râð±k ϵ t1 &
+ t1[ârâð(k+h).(t1â(pâð¦))] â t2
+.
+
+interpretation
+ "balanced focused reduction with delayed updating (prototerm)"
+ 'BlackRightArrowDBF t1 r t2 = (dbfr r t1 t2).
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_ibfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_ibfr.ma
new file mode 100644
index 000000000..043f61085
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_ibfr.ma
@@ -0,0 +1,78 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/reduction/ibfr.ma".
+
+include "delayed_updating/unwind/unwind2_constructors.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_head.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_head_structure.ma".
+include "delayed_updating/syntax/path_structure_depth.ma".
+
+(* DELAYED BALANCED FOCUSED REDUCTION ***************************************)
+
+(* Main destructions with ibfr **********************************************)
+
+theorem dbfr_des_ibfr (f) (t1) (t2) (r): t1 ϵ ð â
+ t1 â¡ððð[r] t2 â â¼[f]t1 â¡ð¢ðð[âr] â¼[f]t2.
+#f #t1 #t2 #r #H0t1
+* #p #b #q #h #k #Hr #Hb #Hh #H1k #Ht1 #Ht2 destruct
+@(ex6_5_intro ⦠(âp) (âb) (âq) (âb) (ââq))
+[ -H0t1 -Hb -Hh -H1k -Ht1 -Ht2 //
+| -H0t1 -Hh -H1k -Ht1 -Ht2 //
+| -H0t1 -Hb -Ht1 -H1k -Ht2
+ >Hh in ⢠(??%?); >path_head_structure_depth <Hh -Hh //
+| -H0t1 -Hb -Hh -Ht1 -Ht2
+ >structure_L_sn
+ >H1k in ⢠(??%?); >path_head_structure_depth <H1k -H1k //
+| lapply (in_comp_unwind2_path_term f ⦠Ht1) -Ht2 -Ht1 -H0t1 -Hb -Hh
+ <unwind2_path_d_dx >list_append_rcons_dx >list_append_assoc
+ lapply (unwind2_rmap_append_pap_closed f ⦠(pâðâb) ⦠H1k) -H1k
+ <depth_L_sn #H2k
+ lapply (eq_inv_ninj_bi ⦠H2k) -H2k #H2k <H2k -H2k #Ht1 //
+| lapply (unwind2_term_eq_repl_dx f ⦠Ht2) -Ht2 #Ht2
+ lapply (path_head_refl_append_bi ⦠Hh H1k) -Hh -H1k <nrplus_inj_sn #H1k
+ @(subset_eq_trans ⦠Ht2) -t2
+ @(subset_eq_trans ⦠(unwind2_term_fsubst_ppc â¦))
+ [ @fsubst_eq_repl [ // | // ]
+ @(subset_eq_trans ⦠(unwind2_term_iref â¦))
+ @(subset_eq_canc_sn ⦠(lift_term_eq_repl_dx â¦))
+ [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+ @(subset_eq_trans ⦠(lift_unwind2_term_after â¦))
+ @unwind2_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+ <list_append_rcons_sn
+ @(stream_eq_trans ⦠(tr_compose_uni_dx â¦))
+ @tr_compose_eq_repl
+ [ <unwind2_rmap_append_pap_closed //
+ <depth_append <depth_L_sn
+ <nplus_comm <nrplus_npsucc_sn <nplus_succ_sn //
+ | >unwind2_rmap_A_dx
+ /2 width=1 by tls_unwind2_rmap_closed/
+ ]
+(* Note: crux of the proof ends *)
+ | //
+ | /2 width=2 by ex_intro/
+ | //
+ ]
+]
+qed.
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
index 000000000..f23728916
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dbfr_lift.ma
@@ -0,0 +1,51 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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_head.ma".
+include "delayed_updating/substitution/lift_rmap_head.ma".
+
+(* DELAYED BALANCED FOCUSED REDUCTION ***************************************)
+
+(* Constructions with lift **************************************************)
+
+theorem dfr_lift_bi (f) (t1) (t2) (r):
+ t1 â¡ððð[r] t2 â â[f]t1 â¡ððð[â[f]r] â[f]t2.
+#f #t1 #t2 #r
+* #p #q #k #Hr #H1k #Ht1 #Ht2 destruct
+@(ex4_3_intro ⦠(â[f]p) (â[â[pâðâð]f]q) ((â[pâðâðâq]f)ï¼ â§£â¨kâ©))
+[ -H1k -Ht1 -Ht2 //
+| -Ht1 -Ht2
+ <lift_rmap_L_dx >lift_path_L_sn
+ <(lift_path_head_closed ⦠H1k) in ⢠(??%?); -H1k //
+| lapply (in_comp_lift_path_term f ⦠Ht1) -Ht2 -Ht1 -H1k
+ <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 â¦))
+ @fsubst_eq_repl [ // | // ]
+ @(subset_eq_trans ⦠(lift_term_iref â¦))
+ @iref_eq_repl
+ @(subset_eq_canc_sn ⦠(lift_term_grafted_S â¦))
+ @lift_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+ >list_append_rcons_sn in H1k; #H1k >lift_rmap_A_dx
+ /2 width=1 by tls_lift_rmap_closed/
+(* Note: crux of the proof ends *)
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
index a610cd323..d0dcf4eee 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
@@ -16,7 +16,6 @@ include "delayed_updating/substitution/fsubst.ma".
include "delayed_updating/syntax/prototerm_constructors.ma".
include "delayed_updating/syntax/prototerm_eq.ma".
include "delayed_updating/syntax/path_head.ma".
-include "delayed_updating/syntax/path_depth.ma".
include "delayed_updating/notation/relations/black_rightarrow_df_3.ma".
include "ground/xoa/ex_4_3.ma".
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 71a599326..b60bd708c 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
@@ -24,8 +24,6 @@ include "delayed_updating/substitution/lift_rmap_head.ma".
(* Constructions with lift **************************************************)
-(* â[â[pâðâð]f]q *)
-
theorem dfr_lift_bi (f) (t1) (t2) (r):
t1 â¡ðð[r] t2 â â[f]t1 â¡ðð[â[f]r] â[f]t2.
#f #t1 #t2 #r
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr.ma
new file mode 100644
index 000000000..b235cb481
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ibfr.ma
@@ -0,0 +1,39 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/substitution/fsubst.ma".
+include "delayed_updating/substitution/lift_prototerm.ma".
+include "delayed_updating/syntax/prototerm_eq.ma".
+include "delayed_updating/syntax/path_head.ma".
+include "delayed_updating/syntax/path_balanced.ma".
+include "delayed_updating/syntax/path_structure.ma".
+include "delayed_updating/notation/relations/black_rightarrow_ibf_3.ma".
+include "ground/relocation/tr_uni.ma".
+include "ground/arith/nat_rplus.ma".
+include "ground/xoa/ex_6_5.ma".
+
+(* IMMEDIATE BALANCED FOCUSED REDUCTION *************************************)
+
+(**) (* explicit ninj because we cannot declare the expectd type of k *)
+definition ibfr (r): relation2 prototerm prototerm â
+ λt1,t2.
+ ââp,b,q,h,k. pâðâbâðâq = r &
+ âb ϵ ð & b = â³[h]b &
+ (ðâq) = â³[ninj k](ðâq) & râð±k ϵ t1 &
+ t1[ârââ[ð®â¨k+hâ©](t1â(pâð¦))] â t2
+.
+
+interpretation
+ "balanced focused reduction with immediate updating (prototerm)"
+ 'BlackRightArrowIBF t1 r t2 = (ibfr r t1 t2).
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
new file mode 100644
index 000000000..b332b841f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_closed.ma
@@ -0,0 +1,33 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/substitution/lift_path.ma".
+include "delayed_updating/syntax/path_closed.ma".
+include "ground/relocation/xap.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+(* Constructions with pcc ***************************************************)
+
+lemma lift_path_closed (f) (q) (n):
+ q ϵ ðâ¨nâ© â â[f]q ϵ ðâ¨â[q]fï¼ â¨nâ©â©.
+#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/
+qed.
+
+lemma lift_path_rmap_closed (f) (p) (q) (n):
+ q ϵ ðâ¨nâ© â â[â[p]f]q ϵ ðâ¨â[pâq]fï¼ â¨nâ©â©.
+/2 width=1 by lift_path_closed/
+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
deleted file mode 100644
index 21b70c57a..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_head.ma
+++ /dev/null
@@ -1,61 +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/substitution/lift_path.ma".
-include "delayed_updating/syntax/path_head.ma".
-include "ground/relocation/xap.ma".
-
-(* LIFT FOR PATH ************************************************************)
-
-(* Constructions with head for path *****************************************)
-
-lemma lift_path_head_closed (f) (p) (q) (n):
- q = â³[n]q â
- â³[â[pâq]fï¼ â¨nâ©]â[â[p]f]q = â[â[p]f]q.
-#f #p #q elim q -q
-[ #n #H0
- <(eq_inv_path_empty_head ⦠H0) -H0
- <path_head_zero //
-| #l #q #IH #n @(nat_ind_succ â¦n) -n [| #n #_ ]
- [ <path_head_zero #H0 destruct
- | cases l [ #k ]
- [ <path_head_d_dx #H0
- elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
- <lift_rmap_d_dx <lift_path_d_dx
- <tr_xap_succ_nap <path_head_d_dx
- <(IH ⦠H0) in ⢠(???%); -IH -H0 <tr_xap_plus //
- | <path_head_m_dx #H0
- elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
- <lift_rmap_m_dx <lift_path_m_dx
- <tr_xap_succ_nap <path_head_m_dx
- <(IH ⦠H0) in ⢠(???%); -IH -H0 //
- | <path_head_L_dx #H0
- elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
- <lift_rmap_L_dx <lift_path_L_dx
- <tr_xap_succ_nap <path_head_L_dx
- <(IH ⦠H0) in ⢠(???%); -IH -H0 //
- | <path_head_A_dx #H0
- elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
- <lift_rmap_A_dx <lift_path_A_dx
- <tr_xap_succ_nap <path_head_A_dx
- <(IH ⦠H0) in ⢠(???%); -IH -H0 //
- | <path_head_S_dx #H0
- elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
- <lift_rmap_S_dx <lift_path_S_dx
- <tr_xap_succ_nap <path_head_S_dx
- <(IH ⦠H0) in ⢠(???%); -IH -H0 //
- ]
- ]
-]
-qed.
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
new file mode 100644
index 000000000..58bef8153
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_closed.ma
@@ -0,0 +1,36 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/substitution/lift_rmap_eq.ma".
+include "delayed_updating/syntax/path_closed.ma".
+include "ground/lib/stream_eq_eq.ma".
+
+(* LIFT MAP FOR PATH ********************************************************)
+
+(* Destructions with cpp ****************************************************)
+
+lemma tls_plus_lift_rmap_closed (f) (q) (n):
+ q ϵ ðâ¨nâ© â
+ âm. â*[m]f â â*[m+n]â[q]f.
+#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 ðâ¨nâ© â
+ f â â*[n]â[q]f.
+#f #q #n #H0
+/2 width=1 by tls_plus_lift_rmap_closed/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_head.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_head.ma
deleted file mode 100644
index 033a2c13e..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_rmap_head.ma
+++ /dev/null
@@ -1,57 +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/substitution/lift_rmap_eq.ma".
-include "delayed_updating/syntax/path_head.ma".
-include "ground/lib/stream_eq_eq.ma".
-
-(* LIFT MAP FOR PATH ********************************************************)
-
-(* Constructions with path_head *********************************************)
-
-lemma tls_plus_lift_rmap_closed (f) (q) (n) (m):
- q = â³[n]q â
- â*[m]f â â*[n+m]â[q]f.
-#f #q elim q -q
-[ #n #m #Hq
- <(eq_inv_path_empty_head ⦠Hq) -n //
-| #l #q #IH #n @(nat_ind_succ ⦠n) -n //
- #n #_ #m cases l [ #k ]
- [ <path_head_d_dx #Hq
- elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq <nrplus_inj_sn
- @(stream_eq_trans ⦠(tls_lift_rmap_d_dx â¦))
- >nrplus_inj_dx >nrplus_inj_sn >nrplus_inj_sn <nplus_plus_comm_23
- >nsucc_unfold /2 width=1 by/
- | <path_head_m_dx #Hq
- elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <lift_rmap_m_dx /2 width=1 by/
- | <path_head_L_dx #Hq
- elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <lift_rmap_L_dx <nplus_succ_sn /2 width=1 by/
- | <path_head_A_dx #Hq
- elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <lift_rmap_A_dx /2 width=2 by/
- | <path_head_S_dx #Hq
- elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- <lift_rmap_S_dx /2 width=2 by/
- ]
-]
-qed-.
-
-lemma tls_lift_rmap_closed (f) (q) (n):
- q = â³[n]q â
- f â â*[n]â[q]f.
-#f #q #n #H0
-/2 width=1 by tls_plus_lift_rmap_closed/
-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
new file mode 100644
index 000000000..fd7a9954f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma
@@ -0,0 +1,162 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/path.ma".
+include "delayed_updating/notation/functions/class_c_1.ma".
+include "ground/arith/nat_plus.ma".
+include "ground/arith/nat_pred_succ.ma".
+include "ground/lib/subset.ma".
+include "ground/generated/insert_eq_1.ma".
+
+(* CLOSED CONDITION FOR PATH ************************************************)
+
+inductive pcc: relation2 nat path â
+| pcc_empty:
+ pcc (ð) (ð)
+| pcc_d_dx (p) (n) (k):
+ pcc (n+ninj k) p â pcc n (pâð±k)
+| pcc_m_dx (p) (n):
+ pcc n p â pcc n (pâðº)
+| pcc_L_dx (p) (n):
+ pcc n p â pcc (ân) (pâð)
+| pcc_A_dx (p) (n):
+ pcc n p â pcc n (pâð)
+| pcc_S_dx (p) (n):
+ pcc n p â pcc n (pâð¦)
+.
+
+interpretation
+ "closed condition (path)"
+ 'ClassC n = (pcc n).
+
+(* Basic inversions ********************************************************)
+
+lemma pcc_inv_empty (n):
+ (ð) ϵ ðâ¨nâ© â ð = n.
+#n @(insert_eq_1 ⦠(ð))
+#x * -n //
+#p #n [ #k ] #_ #H0 destruct
+qed-.
+
+lemma pcc_inv_d_dx (p) (n) (k):
+ pâð±k ϵ ðâ¨nâ© â p ϵ ðâ¨n+kâ©.
+#p #n #h @(insert_eq_1 ⦠(pâð±h))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+qed-.
+
+lemma pcc_inv_m_dx (p) (n):
+ pâðº ϵ ðâ¨nâ© â p ϵ ðâ¨nâ©.
+#p #n @(insert_eq_1 ⦠(pâðº))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+qed-.
+
+lemma pcc_inv_L_dx (p) (n):
+ pâð ϵ ðâ¨nâ© â
+ â§â§ p ϵ ðâ¨ânâ© & âân = n.
+#p #n @(insert_eq_1 ⦠(pâð))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct
+<npred_succ /2 width=1 by conj/
+qed-.
+
+lemma pcc_inv_A_dx (p) (n):
+ pâð ϵ ðâ¨nâ© â p ϵ ðâ¨nâ©.
+#p #n @(insert_eq_1 ⦠(pâð))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+qed-.
+
+lemma pcc_inv_S_dx (p) (n):
+ pâð¦ ϵ ðâ¨nâ© â p ϵ ðâ¨nâ©.
+#p #n @(insert_eq_1 ⦠(pâð¦))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+qed-.
+
+(* Advanced inversions ******************************************************)
+
+lemma pcc_inv_empty_succ (n):
+ (ð) ϵ ðâ¨ânâ© â â¥.
+#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):
+ pâð ϵ ðâ¨ðâ© â â¥.
+#p #H0
+elim (pcc_inv_L_dx ⦠H0) -H0 #_ #H0
+/2 width=7 by eq_inv_nsucc_zero/
+qed-.
+
+lemma pcc_inv_L_dx_succ (p) (n):
+ pâð ϵ ðâ¨ânâ© â p ϵ ðâ¨nâ©.
+#p #n #H0
+elim (pcc_inv_L_dx ⦠H0) -H0 //
+qed-.
+
+(* Main constructions with path_append **************************************)
+
+theorem pcc_append_bi (p) (q) (m) (n):
+ p ϵ ðâ¨mâ© â q ϵ ðâ¨nâ© â pâq ϵ ðâ¨m+nâ©.
+#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/
+qed.
+
+(* Main inversions **********************************************************)
+
+theorem ppc_mono (q) (n1):
+ q ϵ ðâ¨n1â© â ân2. q ϵ ðâ¨n2â© â n1 = n2.
+#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 (IH ⦠Hn2) -q1 #H0
+ /2 width=2 by eq_inv_nplus_bi_dx/
+| lapply (pcc_inv_m_dx ⦠Hn2) -Hn2 #Hn2
+ <(IH ⦠Hn2) -q1 -n2 //
+| elim (pcc_inv_L_dx ⦠Hn2) -Hn2 #Hn2 #H0
+ >(IH ⦠Hn2) -q1 //
+| lapply (pcc_inv_A_dx ⦠Hn2) -Hn2 #Hn2
+ <(IH ⦠Hn2) -q1 -n2 //
+| lapply (pcc_inv_S_dx ⦠Hn2) -Hn2 #Hn2
+ <(IH ⦠Hn2) -q1 -n2 //
+]
+qed-.
+
+theorem pcc_inj_L_sn (p1) (p2) (q1) (n):
+ q1 ϵ ðâ¨nâ© â âq2. q2 ϵ ðâ¨nâ© â
+ p1âðâq1 = p2âðâq2 â q1 = q2.
+#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
+ <(IH ⦠Hq2) //
+| lapply (pcc_inv_m_dx ⦠Hq2) -Hq2 #Hq2
+ <(IH ⦠Hq2) //
+| lapply (pcc_inv_L_dx_succ ⦠Hq2) -Hq2 #Hq2
+ <(IH ⦠Hq2) //
+| lapply (pcc_inv_A_dx ⦠Hq2) -Hq2 #Hq2
+ <(IH ⦠Hq2) //
+| lapply (pcc_inv_S_dx ⦠Hq2) -Hq2 #Hq2
+ <(IH ⦠Hq2) //
+| elim (pcc_inv_empty_succ ⦠Hq2)
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_height.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_height.ma
new file mode 100644
index 000000000..88b0e413f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_height.ma
@@ -0,0 +1,27 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/path_closed.ma".
+include "delayed_updating/syntax/path_height.ma".
+include "delayed_updating/syntax/path_depth.ma".
+
+(* CLOSED CONDITION FOR PATH ************************************************)
+
+(* Constructions with height and depth **************************************)
+
+lemma path_closed_depth (p) (n):
+ p ϵ ðâ¨nâ© â â¯p + n = âp.
+#p #n #Hn elim Hn -Hn //
+#p #n #_ #IH <nplus_succ_dx //
+qed.
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
new file mode 100644
index 000000000..90522f905
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed_structure.ma
@@ -0,0 +1,33 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/syntax/path_closed_height.ma".
+include "delayed_updating/syntax/path_structure.ma".
+
+(* CLOSED CONDITION FOR PATH ************************************************)
+
+(* Constructions with structure *********************************************)
+
+lemma path_closed_structure_height (p) (n):
+ p ϵ ðâ¨nâ© â âp ϵ ðâ¨â¯p+nâ©.
+#p #n #Hn elim Hn -Hn //
+#p #n #_ #IH [ <nplus_succ_dx ]
+/2 width=1 by pcc_L_dx, pcc_A_dx, pcc_S_dx/
+qed.
+
+lemma path_closed_structure_depth (p) (n):
+ p ϵ ðâ¨nâ© â âp ϵ ðâ¨âpâ©.
+#p #n #Hp <path_closed_depth
+/2 width=3 by path_closed_structure_height/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth_labels.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth_labels.ma
deleted file mode 100644
index 791a09a51..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_depth_labels.ma
+++ /dev/null
@@ -1,26 +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/syntax/path_depth.ma".
-include "delayed_updating/syntax/path_labels.ma".
-
-(* DEPTH FOR PATH ***********************************************************)
-
-(* Constructions with labels ************************************************)
-
-lemma depth_labels_L (n):
- n = â(ðâân).
-#n @(nat_ind_succ ⦠n) -n //
-#n #IH <labels_succ <depth_L_dx //
-qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head.ma
deleted file mode 100644
index 4610e4f69..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head.ma
+++ /dev/null
@@ -1,134 +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/syntax/path_labels.ma".
-include "delayed_updating/notation/functions/downarrowright_2.ma".
-include "ground/arith/nat_plus.ma".
-include "ground/arith/nat_pred_succ.ma".
-
-(* HEAD FOR PATH ************************************************************)
-
-rec definition path_head (n) (p) on p: path â
-match n with
-[ nzero â ð
-| ninj m â
- match p with
- [ list_empty â ðâân
- | list_lcons l q â
- match l with
- [ label_d k â (path_head (n+k) q)âl
- | label_m â (path_head n q)âl
- | label_L â (path_head (âm) q)âl
- | label_A â (path_head n q)âl
- | label_S â (path_head n q)âl
- ]
- ]
-].
-
-interpretation
- "head (path)"
- 'DownArrowRight n p = (path_head n p).
-
-(* basic constructions ****************************************************)
-
-lemma path_head_zero (p):
- (ð) = â³[ð]p.
-* // qed.
-
-lemma path_head_empty (n):
- (ðâân) = â³[n]ð.
-* // qed.
-
-lemma path_head_d_dx (p) (n) (k:pnat):
- (â³[ân+k]p)âð±k = â³[ân](pâð±k).
-// qed.
-
-lemma path_head_m_dx (p) (n):
- (â³[ân]p)âðº = â³[ân](pâðº).
-// qed.
-
-lemma path_head_L_dx (p) (n):
- (â³[n]p)âð = â³[ân](pâð).
-#p #n
-whd in ⢠(???%); //
-qed.
-
-lemma path_head_A_dx (p) (n):
- (â³[ân]p)âð = â³[ân](pâð).
-// qed.
-
-lemma path_head_S_dx (p) (n):
- (â³[ân]p)âð¦ = â³[ân](pâð¦).
-// qed.
-
-(* Basic inversions *********************************************************)
-
-lemma eq_inv_path_head_zero_dx (q) (p):
- q = â³[ð]p â ð = q.
-#q * //
-qed-.
-
-lemma eq_inv_path_empty_head (p) (n):
- (ð) = â³[n]p â ð = n.
-*
-[ #n <path_head_empty #H0
- <(eq_inv_empty_labels ⦠H0) -n //
-| * [ #k ] #p #n @(nat_ind_succ ⦠n) -n // #n #_
- [ <path_head_d_dx
- | <path_head_m_dx
- | <path_head_L_dx
- | <path_head_A_dx
- | <path_head_S_dx
- ] #H0 destruct
-]
-qed-.
-
-(* Constructions with path_append *******************************************)
-
-lemma path_head_refl_append (p) (q) (n):
- q = â³[n]q â q = â³[n](pâq).
-#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 [ #k ] ]
- [ lapply (eq_inv_path_head_zero_dx ⦠Hq) -Hq #Hq destruct
- | <path_head_d_dx <path_head_d_dx
- | <path_head_m_dx <path_head_m_dx
- | <path_head_L_dx <path_head_L_dx
- | <path_head_A_dx <path_head_A_dx
- | <path_head_S_dx <path_head_S_dx
- ] #Hq
- elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- /3 width=1 by eq_f/
-]
-qed-.
-
-(* Inversions with path_append **********************************************)
-
-lemma eq_inv_path_head_refl_append (p) (q) (n):
- q = â³[n](pâq) â q = â³[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 [ #k ]
- [ <path_head_d_dx <path_head_d_dx
- | <path_head_m_dx <path_head_m_dx
- | <path_head_L_dx <path_head_L_dx
- | <path_head_A_dx <path_head_A_dx
- | <path_head_S_dx <path_head_S_dx
- ] #Hq
- elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
- /3 width=1 by eq_f/
-]
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_depth.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_depth.ma
deleted file mode 100644
index 6283e2c21..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_depth.ma
+++ /dev/null
@@ -1,35 +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/syntax/path_head.ma".
-include "delayed_updating/syntax/path_height_labels.ma".
-include "delayed_updating/syntax/path_depth_labels.ma".
-
-(* HEAD FOR PATH ************************************************************)
-
-(* Constructions with depth and height **************************************)
-
-lemma path_head_depth (p) (n):
- n + â¯(â³[n]p) = ââ³[n]p.
-#p elim p -p // #l #p #IH
-#n @(nat_ind_succ ⦠n) -n // #n #_
-cases l [ #k ]
-[ <path_head_d_dx <height_d_dx <depth_d_dx <IH //
-| <path_head_m_dx <height_m_dx <depth_m_dx //
-| <path_head_L_dx <height_L_dx <depth_L_dx
- <nplus_succ_sn //
-| <path_head_A_dx <height_A_dx <depth_A_dx //
-| <path_head_S_dx <height_S_dx <depth_S_dx //
-]
-qed-. (**) (* gets in the way with auto *)
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_structure.ma
deleted file mode 100644
index 37ca7be2c..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_head_structure.ma
+++ /dev/null
@@ -1,39 +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/syntax/path_head_depth.ma".
-include "delayed_updating/syntax/path_structure_labels.ma".
-
-(* HEAD FOR PATH ************************************************************)
-
-(* Constructions with structure *********************************************)
-
-lemma path_head_structure_height (p) (n):
- ââ³[n]p = â³[n+â¯â³[n]p]âp.
-#p elim p -p //
-#l #p #IH #n @(nat_ind_succ ⦠n) -n //
-#n #_ cases l [ #k ]
-[ <height_d_dx <structure_d_dx //
-| <structure_m_dx //
-| <structure_L_dx //
-| <height_A_dx <structure_A_dx <nplus_succ_sn <path_head_A_dx //
-| <height_S_dx <structure_S_dx <nplus_succ_sn <path_head_S_dx //
-]
-qed.
-
-lemma path_head_structure_depth (p) (n):
- ââ³[n]p = â³[ââ³[n]p]âp.
-#p #n
-<path_head_depth //
-qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_height_labels.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_height_labels.ma
deleted file mode 100644
index fb55c3606..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_height_labels.ma
+++ /dev/null
@@ -1,26 +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/syntax/path_height.ma".
-include "delayed_updating/syntax/path_labels.ma".
-
-(* HEIGHT FOR PATH **********************************************************)
-
-(* Constructions with labels ************************************************)
-
-lemma height_labels_L (n):
- (ð) = â¯(ðâân).
-#n @(nat_ind_succ ⦠n) -n //
-#n #IH <labels_succ <height_L_dx //
-qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_labels.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_labels.ma
deleted file mode 100644
index 7f0a71a8c..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_labels.ma
+++ /dev/null
@@ -1,50 +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/syntax/path.ma".
-include "delayed_updating/notation/functions/power_2.ma".
-include "ground/arith/nat_succ_iter.ma".
-
-(* *)
-
-definition labels (l) (n:nat): path â
- ((list_lcons ? l)^n) (ð).
-
-interpretation
- "labels (path)"
- 'Power l n = (labels l n).
-
-(* Basic constructions ******************************************************)
-
-lemma labels_unfold (l) (n):
- ((list_lcons ? l)^n) (ð) = lâân.
-// qed.
-
-lemma labels_zero (l):
- (ð) = lââð.
-// qed.
-
-lemma labels_succ (l) (n):
- (lâân)âl = lââ(ân).
-#l #n
-<labels_unfold <labels_unfold <niter_succ //
-qed.
-
-(* Basic inversions *********************************************************)
-
-lemma eq_inv_empty_labels (l) (n):
- (ð) = lâân â ð = n.
-#l #n @(nat_ind_succ ⦠n) -n //
-#n #_ <labels_succ #H0 destruct
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure_labels.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure_labels.ma
deleted file mode 100644
index 0dcc367cd..000000000
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure_labels.ma
+++ /dev/null
@@ -1,25 +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/syntax/path_structure.ma".
-include "delayed_updating/syntax/path_labels.ma".
-
-(* STRUCTURE FOR PATH *******************************************************)
-
-(* Constructions with labels ************************************************)
-
-lemma structure_labels_L (n):
- (ðâân) = â(ðâân).
-#n @(nat_ind_succ ⦠n) -n //
-qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_head.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_head.ma
index 7b568dc86..416c6e792 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_head.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_rmap_head.ma
@@ -93,7 +93,7 @@ lemma unwind2_rmap_append_pap_closed (f) (p) (q) (h:pnat):
q = â³[h]q â
âq = ninj (â¶[f](pâq)ï¼ â§£â¨hâ©).
#f #p #q #h #Hh
->tr_xap_ninj >(path_head_refl_append p ⦠Hh) in ⢠(??%?);
+>tr_xap_ninj >(path_head_refl_append_sn p ⦠Hh) in ⢠(??%?);
>(unwind2_rmap_head_xap_closed ⦠Hh) -Hh
<path_head_depth //
qed.