From 97ff918432e878ab8314c72fe2b948a253b26e21 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sat, 21 May 2022 20:23:22 +0200
Subject: [PATCH] update in delayed_updating
+ unwind completed with an important result on lift
---
.../unwind.ma => etc/unwind1/unwind.etc} | 0
.../unwind1/unwind_constructors.etc} | 0
.../unwind1/unwind_depth.etc} | 0
.../unwind1/unwind_eq.etc} | 0
.../unwind1/unwind_eq_etc.etc} | 0
.../unwind1/unwind_fsubst.etc} | 0
.../unwind1/unwind_preterm_eq.etc} | 0
.../unwind1/unwind_prototerm.etc} | 0
.../unwind1/unwind_prototerm_eq.etc} | 0
.../unwind1/unwind_structure.etc} | 0
.../unwind1/unwind_structure_depth.etc} | 0
.../unwind1/unwind_update.etc} | 0
.../delayed_updating/substitution/lift_eq.ma | 43 ++++++-
.../delayed_updating/syntax/bdd_term.ma | 2 +-
.../delayed_updating/syntax/path_proper.ma | 8 ++
.../delayed_updating/unwind/nap.ma | 1 -
.../unwind/unwind2_constructors.ma | 39 ++++++
.../delayed_updating/unwind/unwind2_path.ma | 84 +++++++++++++
.../unwind/unwind2_path_eq.ma | 40 ++++++
.../unwind/unwind2_path_lift.ma | 31 +++++
.../unwind/unwind2_path_structure.ma | 116 ++++++++++++++++++
.../unwind/unwind2_preterm_eq.ma | 67 ++++++++++
.../unwind/unwind2_preterm_fsubst.ma | 62 ++++++++++
.../unwind/unwind2_prototerm.ma | 30 +++++
.../unwind/unwind2_prototerm_eq.ma | 41 +++++++
.../unwind/unwind2_prototerm_lift.ma | 31 +++++
.../unwind/unwind2_rmap_head.ma | 55 +++++++--
.../delayed_updating/unwind/unwind_gen.ma | 4 +-
.../unwind/unwind_gen_after.ma | 3 +-
.../delayed_updating/unwind/unwind_gen_eq.ma | 12 +-
.../unwind/unwind_gen_structure.ma | 24 ++--
.../delayed_updating/unwind/xap.ma | 55 +++++++++
32 files changed, 710 insertions(+), 38 deletions(-)
rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind1/unwind.ma => etc/unwind1/unwind.etc} (100%)
rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind1/unwind_constructors.ma => etc/unwind1/unwind_constructors.etc} (100%)
rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind1/unwind_depth.ma => etc/unwind1/unwind_depth.etc} (100%)
rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind1/unwind_eq.ma => etc/unwind1/unwind_eq.etc} (100%)
rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind1/unwind_eq_etc.ma => etc/unwind1/unwind_eq_etc.etc} (100%)
rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind1/unwind_fsubst.ma => etc/unwind1/unwind_fsubst.etc} (100%)
rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind1/unwind_preterm_eq.ma => etc/unwind1/unwind_preterm_eq.etc} (100%)
rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind1/unwind_prototerm.ma => etc/unwind1/unwind_prototerm.etc} (100%)
rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind1/unwind_prototerm_eq.ma => etc/unwind1/unwind_prototerm_eq.etc} (100%)
rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind1/unwind_structure.ma => etc/unwind1/unwind_structure.etc} (100%)
rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind1/unwind_structure_depth.ma => etc/unwind1/unwind_structure_depth.etc} (100%)
rename matita/matita/contribs/lambdadelta/delayed_updating/{unwind1/unwind_update.ma => etc/unwind1/unwind_update.etc} (100%)
create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_constructors.ma
create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path.ma
create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_eq.ma
create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_lift.ma
create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_structure.ma
create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_preterm_eq.ma
create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_preterm_fsubst.ma
create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_prototerm.ma
create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_prototerm_eq.ma
create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_prototerm_lift.ma
create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/unwind/xap.ma
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_constructors.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_constructors.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_constructors.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_constructors.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_depth.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_depth.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_depth.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_depth.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_eq.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_eq.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_eq.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_eq_etc.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_eq_etc.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_eq_etc.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_eq_etc.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_fsubst.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_fsubst.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_fsubst.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_fsubst.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_preterm_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_preterm_eq.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_preterm_eq.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_preterm_eq.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_prototerm.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_prototerm.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_prototerm.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_prototerm.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_prototerm_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_prototerm_eq.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_prototerm_eq.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_prototerm_eq.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_structure.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_structure.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_structure.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_structure_depth.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_structure_depth.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_structure_depth.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_structure_depth.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_update.ma b/matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_update.etc
similarity index 100%
rename from matita/matita/contribs/lambdadelta/delayed_updating/unwind1/unwind_update.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/etc/unwind1/unwind_update.etc
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
index 68a4ceb36..5eac1985e 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
@@ -13,11 +13,6 @@
(**************************************************************************)
include "delayed_updating/substitution/lift.ma".
-(*
-include "ground/relocation/tr_uni_compose.ma".
-include "ground/relocation/tr_compose_compose.ma".
-include "ground/relocation/tr_compose_eq.ma".
-*)
include "ground/relocation/tr_pap_eq.ma".
include "ground/relocation/tr_pn_eq.ma".
include "ground/lib/stream_tls_eq.ma".
@@ -110,6 +105,44 @@ lemma lift_path_A_sn (f) (p):
lemma lift_path_S_sn (f) (p):
(ð¦ââ[f]p) = â[f](ð¦âp).
// qed.
+
+lemma lift_path_append (p2) (p1) (f):
+ (â[f]p1)â(â[â[p1]f]p2) = â[f](p1âp2).
+#p2 #p1 elim p1 -p1 //
+* [ #n1 ] #p1 #IH #f
+[ <lift_path_d_sn <lift_path_d_sn <IH //
+| <lift_path_m_sn <lift_path_m_sn <IH //
+| <lift_path_L_sn <lift_path_L_sn <IH //
+| <lift_path_A_sn <lift_path_A_sn <IH //
+| <lift_path_S_sn <lift_path_S_sn <IH //
+]
+qed.
+
+lemma lift_path_d_dx (f) (p) (n):
+ (â[f]p)âð±((â[p]f)ï¼ â§£â¨nâ©) = â[f](pâð±n).
+#f #p #n <lift_path_append //
+qed.
+
+lemma lift_path_m_dx (f) (p):
+ (â[f]p)âðº = â[f](pâðº).
+#f #p <lift_path_append //
+qed.
+
+lemma lift_path_L_dx (f) (p):
+ (â[f]p)âð = â[f](pâð).
+#f #p <lift_path_append //
+qed.
+
+lemma lift_path_A_dx (f) (p):
+ (â[f]p)âð = â[f](pâð).
+#f #p <lift_path_append //
+qed.
+
+lemma lift_path_S_dx (f) (p):
+ (â[f]p)âð¦ = â[f](pâð¦).
+#f #p <lift_path_append //
+qed.
+
(* COMMENT
(* Advanced constructions with proj_rmap and stream_tls *********************)
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/bdd_term.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/bdd_term.ma
index 9cfe28ad9..8aaefad91 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/bdd_term.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/bdd_term.ma
@@ -27,7 +27,7 @@ inductive bdd: ð«â¨prototermâ© â
| bdd_oref: ân. bdd (⧣n)
| bdd_iref: ât,n. bdd t â bdd (ðn.t)
| bdd_abst: ât. bdd t â bdd (ð.t)
-| bdd_appl: âu,t. bdd u â bdd t â bdd (@u.t)
+| bdd_appl: âu,t. bdd u â bdd t â bdd (ï¼ u.t)
| bdd_conv: ât1,t2. t1 â t2 â bdd t1 â bdd t2
.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_proper.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_proper.ma
index 66de8a4f6..9cab60b87 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_proper.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_proper.ma
@@ -47,3 +47,11 @@ lemma ppc_inv_lcons (p):
| #l #q #_ /2 width=3 by ex1_2_intro/
]
qed-.
+
+lemma ppc_inv_rcons (p):
+ p ϵ ð â ââq,l. qâl = p.
+#p @(list_ind_rcons ⦠p) -p
+[ #H0 elim (ppc_inv_empty ⦠H0)
+| #q #l #_ #_ /2 width=3 by ex1_2_intro/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/nap.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/nap.ma
index bfa89a784..985b3ae26 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/nap.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/nap.ma
@@ -1,6 +1,5 @@
include "ground/relocation/tr_uni_pap.ma".
include "ground/relocation/tr_compose_pap.ma".
-include "ground/relocation/tr_pap_pn.ma".
include "ground/notation/functions/applysucc_2.ma".
include "ground/arith/nat_lt.ma".
include "ground/arith/nat_plus_rplus.ma".
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_constructors.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_constructors.ma
new file mode 100644
index 000000000..c70e09e1a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_constructors.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/unwind/unwind2_prototerm_eq.ma".
+include "delayed_updating/syntax/prototerm_constructors.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+(* Constructions with constructors ******************************************)
+
+lemma unwind2_term_iref_sn (f) (t) (n:pnat):
+ â¼[fâð®â¨nâ©]t â â¼[f](ðn.t).
+#f #t #n #p * #q #Hq #H0 destruct
+@(ex2_intro ⦠(ð±nâðºâq))
+/2 width=1 by in_comp_iref/
+qed-.
+
+lemma unwind2_term_iref_dx (f) (t) (n:pnat):
+ â¼[f](ðn.t) â â¼[fâð®â¨nâ©]t.
+#f #t #n #p * #q #Hq #H0 destruct
+elim (in_comp_inv_iref ⦠Hq) -Hq #p #Hp #Ht destruct
+/2 width=1 by in_comp_unwind2_path_term/
+qed-.
+
+lemma unwind2_term_iref (f) (t) (n:pnat):
+ â¼[fâð®â¨nâ©]t â â¼[f](ðn.t).
+/3 width=2 by conj, unwind2_term_iref_sn, unwind2_term_iref_dx/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path.ma
new file mode 100644
index 000000000..3288fc734
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path.ma
@@ -0,0 +1,84 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/unwind/unwind_gen.ma".
+include "delayed_updating/unwind/unwind2_rmap.ma".
+include "delayed_updating/syntax/path_reverse.ma".
+include "delayed_updating/notation/functions/black_downtriangle_2.ma".
+include "ground/lib/list_tl.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+definition unwind2_path (f) (p): path â
+ â[â¶[f]â(pá´¿)]p
+.
+
+interpretation
+ "unwind (path)"
+ 'BlackDownTriangle f p = (unwind2_path f p).
+
+(* Basic constructions ******************************************************)
+
+lemma unwind2_path_unfold (f) (p):
+ â[â¶[f]â(pá´¿)]p = â¼[f]p.
+// qed.
+
+lemma unwind2_path_empty (f):
+ (ð) = â¼[f]ð.
+// qed.
+
+lemma unwind2_path_d_empty (f) (n):
+ (ð±(fï¼ â§£â¨nâ©)âð) = â¼[f](ð±nâð).
+// qed.
+
+lemma unwind2_path_d_lcons (f) (p) (l) (n:pnat):
+ â¼[fâð®â¨nâ©](lâp) = â¼[f](ð±nâlâp).
+#f #p #l #n <unwind2_path_unfold <unwind2_path_unfold
+<unwind_gen_d_lcons <reverse_lcons
+@(list_ind_rcons ⦠p) -p // #p #l0 #_
+<reverse_rcons <reverse_lcons <reverse_lcons <reverse_rcons
+<list_tl_lcons <list_tl_lcons //
+qed.
+
+lemma unwind2_path_m_sn (f) (p):
+ â¼[f]p = â¼[f](ðºâp).
+#f #p <unwind2_path_unfold <unwind2_path_unfold
+<unwind_gen_m_sn <reverse_lcons
+@(list_ind_rcons ⦠p) -p // #p #l #_
+<reverse_rcons <list_tl_lcons <list_tl_lcons //
+qed.
+
+lemma unwind2_path_L_sn (f) (p):
+ (ðââ¼[⫯f]p) = â¼[f](ðâp).
+#f #p <unwind2_path_unfold <unwind2_path_unfold
+<unwind_gen_L_sn <reverse_lcons
+@(list_ind_rcons ⦠p) -p // #p #l #_
+<reverse_rcons <list_tl_lcons <list_tl_lcons //
+qed.
+
+lemma unwind2_path_A_sn (f) (p):
+ (ðââ¼[f]p) = â¼[f](ðâp).
+#f #p <unwind2_path_unfold <unwind2_path_unfold
+<unwind_gen_A_sn <reverse_lcons
+@(list_ind_rcons ⦠p) -p // #p #l #_
+<reverse_rcons <list_tl_lcons <list_tl_lcons //
+qed.
+
+lemma unwind2_path_S_sn (f) (p):
+ (ð¦ââ¼[f]p) = â¼[f](ð¦âp).
+#f #p <unwind2_path_unfold <unwind2_path_unfold
+<unwind_gen_S_sn <reverse_lcons
+@(list_ind_rcons ⦠p) -p // #p #l #_
+<reverse_rcons <list_tl_lcons <list_tl_lcons //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_eq.ma
new file mode 100644
index 000000000..cb0826a0b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_eq.ma
@@ -0,0 +1,40 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/unwind/unwind2_path.ma".
+include "delayed_updating/unwind/unwind2_rmap_eq.ma".
+include "delayed_updating/unwind/unwind_gen_eq.ma".
+include "ground/relocation/tr_compose_compose.ma".
+include "ground/relocation/tr_compose_pn.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+(* Constructions with stream_eq *********************************************)
+
+lemma unwind2_path_eq_repl (p):
+ stream_eq_repl ⦠(λf1,f2. â¼[f1]p = â¼[f2]p).
+/3 width=1 by unwind2_rmap_eq_repl, unwind_gen_eq_repl/
+qed-.
+
+(* Properties with tr_compose ***********************************************)
+
+lemma unwind2_path_after (p) (f1) (f2):
+ â¼[f2]â¼[f1]p = â¼[f2âf1]p.
+#p @(path_ind_unwind ⦠p) -p // [ #n #l #p | #p ] #IH #f1 #f2
+[ <unwind2_path_d_lcons <unwind2_path_d_lcons
+ >(unwind2_path_eq_repl ⦠(tr_compose_assoc â¦)) //
+| <(unwind2_path_L_sn ⦠f1) <unwind2_path_L_sn <unwind2_path_L_sn
+ >tr_compose_push_bi //
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_lift.ma
new file mode 100644
index 000000000..642056c89
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_lift.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/unwind/unwind2_path_eq.ma".
+include "delayed_updating/substitution/lift_eq.ma".
+include "ground/relocation/tr_uni_compose.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+(* Properties with lift_path ************************************************)
+
+lemma unwind2_lift_path_after (p) (f1) (f2):
+ â[f2]â¼[f1]p = â¼[f2âf1]p.
+#p @(path_ind_unwind ⦠p) -p // [ #n | #p ] #IH #f1 #f2
+[ <unwind2_path_d_empty <unwind2_path_d_empty
+ <lift_path_d_sn <lift_path_empty //
+| <unwind2_path_L_sn <unwind2_path_L_sn <lift_path_L_sn
+ >tr_compose_push_bi //
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_structure.ma
new file mode 100644
index 000000000..7cb6190b1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_path_structure.ma
@@ -0,0 +1,116 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/unwind/unwind2_path.ma".
+include "delayed_updating/unwind/unwind_gen_structure.ma".
+include "delayed_updating/syntax/path_inner.ma".
+include "delayed_updating/syntax/path_proper.ma".
+include "ground/xoa/ex_4_2.ma".
+
+(* UNWIND FOR PATH **********************************************************)
+
+(* Constructions with list_rcons ********************************************)
+
+lemma unwind2_path_d_dx (f) (p) (n) :
+ (âp)âð±((â¶[f](pá´¿))ï¼ â§£â¨nâ©) = â¼[f](pâð±n).
+#f #p #n <unwind2_path_unfold
+<unwind_gen_d_dx //
+qed.
+
+lemma unwind2_path_m_dx (f) (p):
+ âp = â¼[f](pâðº).
+#f #p <unwind2_path_unfold //
+qed.
+
+lemma unwind2_path_L_dx (f) (p):
+ (âp)âð = â¼[f](pâð).
+#f #p <unwind2_path_unfold //
+qed.
+
+lemma unwind2_path_A_dx (f) (p):
+ (âp)âð = â¼[f](pâð).
+#f #p <unwind2_path_unfold //
+qed.
+
+lemma unwind2_path_S_dx (f) (p):
+ (âp)âð¦ = â¼[f](pâð¦).
+#f #p <unwind2_path_unfold //
+qed.
+
+lemma unwind2_path_root (f) (p):
+ ââr. ð = âr & âpâr = â¼[f]p.
+#f #p
+elim (unwind_gen_root)
+/2 width=3 by ex2_intro/
+qed-.
+
+(* Constructions with proper condition for path *****************************)
+
+lemma unwind2_path_append_proper_dx (f) (p1) (p2): p2 ϵ ð â
+ (âp1)â(â¼[â¶[f]p1á´¿]p2) = â¼[f](p1âp2).
+#f #p1 #p2 #Hp2 <unwind2_path_unfold <unwind2_path_unfold
+<unwind_gen_append_proper_dx // -Hp2 <reverse_append
+@(list_ind_rcons ⦠p2) -p2 // #q2 #l2 #_
+<reverse_rcons <list_tl_lcons <list_tl_lcons //
+qed-.
+
+(* Constructions with inner condition for path ******************************)
+
+lemma unwind2_path_append_inner_sn (f) (p1) (p2): p1 ϵ ð â
+ (âp1)â(â¼[â¶[f]p1á´¿]p2) = â¼[f](p1âp2).
+#f #p1 #p2 #Hp1 <unwind2_path_unfold <unwind2_path_unfold
+<unwind_gen_append_inner_sn // -Hp1 <reverse_append
+@(list_ind_rcons ⦠p2) -p2 // #q2 #l2 #_
+<reverse_rcons <list_tl_lcons <list_tl_lcons //
+qed-.
+
+(* Inversions with list_lcons ***********************************************)
+
+lemma unwind2_path_inv_S_sn (f) (p) (q):
+ (ð¦âq) = â¼[f]p â
+ ââr1,r2. ð = âr1 & q = â¼[â¶[f]r1á´¿]r2 & r1âð¦âr2 = p.
+#f #p #q #H0
+elim (unwind_gen_inv_S_sn ⦠H0) -H0 #r1 #r2 #Hr1 #H1 #H2 destruct
+<reverse_append <reverse_lcons
+@(list_ind_rcons ⦠r2) -r2 [ /2 width=5 by ex3_2_intro/ ] #r2 #l2 #_
+<reverse_rcons <list_append_lcons_sn <list_append_rcons_sn
+<list_tl_lcons <unwind2_rmap_append <unwind2_rmap_S_sn
+/2 width=5 by ex3_2_intro/
+qed-.
+
+(* Inversions with proper condition for path ********************************)
+
+lemma unwind2_path_inv_append_proper_dx (f) (p) (q1) (q2):
+ q2 ϵ ð â q1âq2 = â¼[f]p â
+ ââp1,p2. âp1 = q1 & â¼[â¶[f]p1á´¿]p2 = q2 & p1âp2 = p.
+#f #p #q1 #q2 #Hq2 #H0
+elim (unwind_gen_inv_append_proper_dx ⦠Hq2 H0) -Hq2 -H0
+#p1 #p2 #H1 #H2 #H3 destruct <reverse_append
+@(list_ind_rcons ⦠p2) -p2 [ /2 width=5 by ex3_2_intro/ ] #q2 #l2 #_
+<reverse_rcons <list_tl_lcons <unwind2_rmap_append
+@ex3_2_intro [4: |*: // ] <unwind2_path_unfold // (**) (* auto fails *)
+qed-.
+
+(* Inversions with inner condition for path *********************************)
+
+lemma unwind2_path_inv_append_inner_sn (f) (p) (q1) (q2):
+ q1 ϵ ð â q1âq2 = â¼[f]p â
+ ââp1,p2. âp1 = q1 & â¼[â¶[f]p1á´¿]p2 = q2 & p1âp2 = p.
+#f #p #q1 #q2 #Hq1 #H0
+elim (unwind_gen_inv_append_inner_sn ⦠Hq1 H0) -Hq1 -H0
+#p1 #p2 #H1 #H2 #H3 destruct <reverse_append
+@(list_ind_rcons ⦠p2) -p2 [ /2 width=5 by ex3_2_intro/ ] #q2 #l2 #_
+<reverse_rcons <list_tl_lcons <unwind2_rmap_append
+@ex3_2_intro [4: |*: // ] <unwind2_path_unfold // (**) (* auto fails *)
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_preterm_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_preterm_eq.ma
new file mode 100644
index 000000000..9ceccc0e2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_preterm_eq.ma
@@ -0,0 +1,67 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/unwind/unwind2_prototerm.ma".
+include "delayed_updating/unwind/unwind2_path_structure.ma".
+include "delayed_updating/syntax/preterm.ma".
+include "delayed_updating/syntax/path_structure_inner.ma".
+include "ground/lib/subset_equivalence.ma".
+
+(* UNWIND FOR PRETERM *******************************************************)
+
+(* Constructions with subset_equivalence ************************************)
+
+lemma unwind2_term_grafted_sn (f) (t) (p): p ϵ ð â
+ â¼[â¶[f]pá´¿](tâp) â (â¼[f]t)â(âp).
+#f #t #p #Hp #q * #r #Hr #H0 destruct
+@(ex2_intro ⦠Hr) -Hr
+<unwind2_path_append_inner_sn //
+qed-.
+
+lemma unwind2_term_grafted_dx (f) (t) (p): p ϵ ð â p ϵ âµt â t ϵ ð â
+ (â¼[f]t)â(âp) â â¼[â¶[f]pá´¿](tâp).
+#f #t #p #H1p #H2p #Ht #q * #r #Hr #H0
+elim (unwind2_path_inv_append_inner_sn ⦠(sym_eq ⦠H0)) -H0 //
+#p0 #q0 #Hp0 #Hq0 #H0 destruct
+<(Ht ⦠Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+/2 width=1 by in_comp_unwind2_path_term/
+qed-.
+
+lemma unwind2_term_grafted (f) (t) (p): p ϵ ð â p ϵ âµt â t ϵ ð â
+ â¼[â¶[f]pá´¿](tâp) â (â¼[f]t)â(âp).
+/3 width=1 by unwind2_term_grafted_sn, unwind2_term_grafted_dx, conj/ qed.
+
+lemma unwind2_term_grafted_S_dx (f) (t) (p): p ϵ âµt â t ϵ ð â
+ (â¼[f]t)â((âp)âð¦) â â¼[â¶[f]pá´¿](tâ(pâð¦)).
+#f #t #p #Hp #Ht #q * #r #Hr
+<list_append_rcons_sn #H0
+elim (unwind2_path_inv_append_proper_dx ⦠(sym_eq ⦠H0)) -H0 //
+#p0 #q0 #Hp0 #Hq0 #H0 destruct
+<(Ht ⦠Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+elim (unwind2_path_inv_S_sn ⦠(sym_eq ⦠Hq0)) -Hq0
+#r1 #r2 #Hr1 #Hr2 #H0 destruct
+lapply (preterm_in_root_append_inv_structure_empty_dx ⦠p0 ⦠Ht Hr1)
+[ /2 width=2 by ex_intro/ ] -Hr1 #Hr1 destruct
+/2 width=1 by in_comp_unwind2_path_term/
+qed-.
+
+lemma unwind2_term_grafted_S (f) (t) (p): p ϵ âµt â t ϵ ð â
+ â¼[â¶[f]pá´¿](tâ(pâð¦)) â (â¼[f]t)â((âp)âð¦).
+#f #t #p #Hp #Ht
+@conj
+[ >unwind2_rmap_S_sn >reverse_rcons >structure_S_dx
+ @unwind2_term_grafted_sn // (**) (* auto fails *)
+| /2 width=1 by unwind2_term_grafted_S_dx/
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_preterm_fsubst.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_preterm_fsubst.ma
new file mode 100644
index 000000000..4245a6638
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_preterm_fsubst.ma
@@ -0,0 +1,62 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/unwind/unwind2_prototerm_eq.ma".
+include "delayed_updating/unwind/unwind2_path_structure.ma".
+include "delayed_updating/substitution/fsubst.ma".
+include "delayed_updating/syntax/preterm.ma".
+include "delayed_updating/syntax/prototerm_proper.ma".
+
+(* UNWIND FOR PRETERM ******************************************************)
+
+(* Constructions with fsubst ************************************************)
+
+lemma unwind2_term_fsubst_sn (f) (t) (u) (p): u ϵ ð â
+ (â¼[f]t)[â(âp)ââ¼[â¶[f]pá´¿]u] â â¼[f](t[âpâu]).
+#f #t #u #p #Hu #ql * *
+[ #rl * #r #Hr #H1 #H2 destruct
+ >unwind2_path_append_proper_dx
+ /4 width=5 by in_comp_unwind2_path_term, in_ppc_comp_trans, or_introl, ex2_intro/
+| * #q #Hq #H1 #H0
+ @(ex2_intro ⦠H1) @or_intror @conj // *
+ [ <list_append_empty_dx #H2 destruct
+ elim (unwind2_path_root f q) #r #_ #Hr /2 width=2 by/
+ | #l #r #H2 destruct
+ @H0 -H0 [| <unwind2_path_append_proper_dx /2 width=3 by ppc_lcons/ ]
+ ]
+]
+qed-.
+
+lemma unwind2_term_fsubst_dx (f) (t) (u) (p): u ϵ ð â p ϵ âµt â t ϵ ð â
+ â¼[f](t[âpâu]) â (â¼[f]t)[â(âp)ââ¼[â¶[f]pá´¿]u].
+#f #t #u #p #Hu #H1p #H2p #ql * #q * *
+[ #r #Hu #H1 #H2 destruct
+ @or_introl @ex2_intro
+ [|| <unwind2_path_append_proper_dx /2 width=5 by in_ppc_comp_trans/ ]
+ /2 width=3 by ex2_intro/
+| #Hq #H0 #H1 destruct
+ @or_intror @conj [ /2 width=1 by in_comp_unwind2_path_term/ ] *
+ [ <list_append_empty_dx #Hr @(H0 ⦠(ð)) -H0
+ <list_append_empty_dx @H2p -H2p
+ /2 width=2 by unwind_gen_des_structure, prototerm_in_comp_root/
+ | #l #r #Hr
+ elim (unwind2_path_inv_append_proper_dx ⦠Hr) -Hr // #s1 #s2 #Hs1 #_ #H1 destruct
+ lapply (H2p ⦠Hs1) -H2p -Hs1 /2 width=2 by ex_intro/
+ ]
+]
+qed-.
+
+lemma unwind2_term_fsubst (f) (t) (u) (p): u ϵ ð â p ϵ âµt â t ϵ ð â
+ (â¼[f]t)[â(âp)ââ¼[â¶[f]pá´¿]u] â â¼[f](t[âpâu]).
+/4 width=3 by unwind2_term_fsubst_sn, conj, unwind2_term_fsubst_dx/ qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_prototerm.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_prototerm.ma
new file mode 100644
index 000000000..e0515c7f0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_prototerm.ma
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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/unwind/unwind2_path.ma".
+include "delayed_updating/syntax/prototerm.ma".
+include "ground/lib/subset_ext.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+interpretation
+ "unwind (prototerm)"
+ 'BlackDownTriangle f t = (subset_ext_f1 ? ? (unwind2_path f) t).
+
+(* Basic constructions ******************************************************)
+
+lemma in_comp_unwind2_path_term (f) (p) (t):
+ p ϵ t â â¼[f]p ϵ â¼[f]t.
+/2 width=1 by subset_in_ext_f1_dx/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_prototerm_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_prototerm_eq.ma
new file mode 100644
index 000000000..dbde59810
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_prototerm_eq.ma
@@ -0,0 +1,41 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+(**) (* reverse include *)
+include "ground/lib/subset_ext_equivalence.ma".
+include "delayed_updating/unwind/unwind2_path_eq.ma".
+include "delayed_updating/unwind/unwind2_prototerm.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+(* Constructions with subset_equivalence ************************************)
+
+lemma unwind2_term_eq_repl_sn (f1) (f2) (t):
+ f1 â f2 â â¼[f1]t â â¼[f2]t.
+/3 width=1 by subset_equivalence_ext_f1_exteq, unwind2_path_eq_repl/
+qed.
+
+lemma unwind2_term_eq_repl_dx (f) (t1) (t2):
+ t1 â t2 â â¼[f]t1 â â¼[f]t2.
+/2 width=1 by subset_equivalence_ext_f1_bi/
+qed.
+
+lemma unwind2_term_after (f1) (f2) (t):
+ â¼[f2]â¼[f1]t â â¼[f2âf1]t.
+#f1 #f2 #t @subset_eq_trans
+[
+| @subset_inclusion_ext_f1_compose
+| @subset_equivalence_ext_f1_exteq /2 width=5/
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_prototerm_lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_prototerm_lift.ma
new file mode 100644
index 000000000..8e3d2cf2f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind2_prototerm_lift.ma
@@ -0,0 +1,31 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+(**) (* reverse include *)
+include "ground/lib/subset_ext_equivalence.ma".
+include "delayed_updating/substitution/lift_prototerm.ma".
+include "delayed_updating/unwind/unwind2_path_lift.ma".
+include "delayed_updating/unwind/unwind2_prototerm.ma".
+
+(* UNWIND FOR PROTOTERM *****************************************************)
+
+(* Constructions with lift_prototerm ****************************************)
+
+lemma unwind2_lift_term_after (f1) (f2) (t):
+ â[f2]â¼[f1]t â â¼[f2âf1]t.
+#f1 #f2 #t @subset_eq_trans
+[| @subset_inclusion_ext_f1_compose ]
+@subset_equivalence_ext_f1_exteq #p
+@unwind2_lift_path_after
+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 579cbd514..6b159e8c5 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
@@ -13,26 +13,67 @@
(**************************************************************************)
include "delayed_updating/unwind/unwind2_rmap_labels.ma".
-include "delayed_updating/unwind/nap.ma".
+include "delayed_updating/unwind/xap.ma".
include "delayed_updating/syntax/path_head_depth.ma".
-include "delayed_updating/syntax/path_height.ma".
+include "ground/arith/nat_le_plus.ma".
(* UNWIND MAP FOR PATH ******************************************************)
(* Constructions with path_head *********************************************)
-lemma unwind2_rmap_head_push (f) (p) (n):
- n + â¯(â³[n]p) = (â¶[⫯f]â³[n]p)@ââ¨nâ©.
+lemma unwind2_rmap_head_xap_closed (f) (p) (n) (k):
+ (âq. p = (â³[n]p)âq) â k ⤠n â
+ (â¶[f]p)ï¼ â¨kâ© = (â¶[f]â³[n]p)ï¼ â¨kâ©.
+#f #p elim p -p
+[ #n #k * #q #Hq #Hk
+ elim (eq_inv_list_empty_append ⦠Hq) -Hq * #_ //
+| #l #p #IH #n @(nat_ind_succ ⦠n) -n
+ [ #k #_ #Hk <(nle_inv_zero_dx ⦠Hk) -k -IH
+ <path_head_zero <unwind2_rmap_empty //
+ | #n #_ #k cases l [ #m ] * #q
+ [ <path_head_d_sn <list_append_lcons_sn #Hq #Hk
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_d_sn <unwind2_rmap_d_sn
+ <tr_compose_xap <tr_compose_xap
+ @IH [ /2 width=2 by ex_intro/ ] (**) (* auto too slow *)
+ @nle_trans [| @tr_uni_xap ]
+ /2 width=1 by nle_plus_bi_dx/
+ | <path_head_m_sn <list_append_lcons_sn #Hq #Hk
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_m_sn <unwind2_rmap_m_sn
+ /3 width=2 by ex_intro/
+ | <path_head_L_sn <list_append_lcons_sn #Hq
+ @(nat_ind_succ ⦠k) -k // #k #_ #Hk
+ lapply (nle_inv_succ_bi ⦠Hk) -Hk #Hk
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_L_sn <unwind2_rmap_L_sn
+ <tr_xap_push <tr_xap_push
+ /4 width=2 by ex_intro, eq_f/
+ | <path_head_A_sn <list_append_lcons_sn #Hq #Hk
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_A_sn <unwind2_rmap_A_sn
+ /3 width=2 by ex_intro/
+ | <path_head_S_sn <list_append_lcons_sn #Hq #Hk
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_S_sn <unwind2_rmap_S_sn
+ /3 width=2 by ex_intro/
+ ]
+ ]
+]
+qed-.
+
+lemma unwind2_rmap_head_xap (f) (p) (n):
+ n + â¯(â³[n]p) = (â¶[f]â³[n]p)ï¼ â¨nâ©.
#f #p elim p -p
[ #n <path_head_empty <unwind2_rmap_labels_L <height_labels_L
- >tr_pushs_swap <tr_nap_pushs_lt //
+ <tr_xap_pushs_le //
| #l #p #IH #n @(nat_ind_succ ⦠n) -n //
#n #_ cases l [ #m ]
[ <unwind2_rmap_d_sn <path_head_d_sn <height_d_sn
- <nplus_assoc >IH -IH <tr_compose_nap //
+ <nplus_assoc >IH -IH <tr_compose_xap <tr_uni_xap_succ //
| <unwind2_rmap_m_sn <path_head_m_sn <height_m_sn //
| <unwind2_rmap_L_sn <path_head_L_sn <height_L_sn
- <tr_nap_push <npred_succ //
+ <tr_xap_push <npred_succ //
| <unwind2_rmap_A_sn <path_head_A_sn <height_A_sn //
| <unwind2_rmap_S_sn <path_head_S_sn <height_S_sn //
]
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen.ma
index 6eb8b3ae8..0ea3ab8ac 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen.ma
@@ -25,7 +25,7 @@ match p with
match l with
[ label_d n â
match q with
- [ list_empty â ð±((f n)ï¼ â§£â¨nâ©)â(unwind_gen f q)
+ [ list_empty â ð±(fï¼ â§£â¨nâ©)â(unwind_gen f q)
| list_lcons _ _ â unwind_gen f q
]
| label_m â unwind_gen f q
@@ -46,7 +46,7 @@ lemma unwind_gen_empty (f):
// qed.
lemma unwind_gen_d_empty (f) (n):
- ð±((f n)ï¼ â§£â¨nâ©)âð = â[f](ð±nâð).
+ ð±(fï¼ â§£â¨nâ©)âð = â[f](ð±nâð).
// qed.
lemma unwind_gen_d_lcons (f) (p) (l) (n):
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen_after.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen_after.ma
index a9b2683b6..2954be831 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen_after.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen_after.ma
@@ -20,7 +20,6 @@ include "ground/relocation/tr_compose_pap.ma".
(* Properties with tr_compose ***********************************************)
lemma unwind_gen_after (f2) (f1) (p):
- â[f2]â[f1]p = â[λn.(f2 ((f1 n)ï¼ â§£â¨nâ©))â(f1 n)]p.
+ â[f2]â[f1]p = â[f2âf1]p.
#f2 #f1 #p @(path_ind_unwind ⦠p) -p //
-#n #_ <unwind_gen_d_empty <unwind_gen_d_empty //
qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen_eq.ma
index 34cb91a5a..11ddd2d03 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen_eq.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen_eq.ma
@@ -19,15 +19,11 @@ include "ground/relocation/tr_pap_eq.ma".
(* Constructions with stream_eq *********************************************)
-lemma unwind_gen_eq_repl (p) (f1) (f2):
- (ân. f1 n â f2 n) â â[f1]p = â[f2]p.
-#p @(path_ind_unwind ⦠p) -p // [ #n | #n #l #p |*: #p ] #IH #f1 #f2 #Hf
+lemma unwind_gen_eq_repl (p):
+ stream_eq_repl ⦠(λf1,f2. â[f1]p = â[f2]p).
+#p @(path_ind_unwind ⦠p) -p // [ #n |*: #p ] #IH #f1 #f2 #Hf
[ <unwind_gen_d_empty <unwind_gen_d_empty
- <(tr_pap_eq_repl ⦠(Hf n)) -f2 -IH //
-| <unwind_gen_d_lcons <unwind_gen_d_lcons
- <(IH ⦠Hf) -f2 -IH //
-| <unwind_gen_m_sn <unwind_gen_m_sn
- <(IH ⦠Hf) -f2 -IH //
+ <(tr_pap_eq_repl ⦠Hf) -f2 -IH //
| <unwind_gen_L_sn <unwind_gen_L_sn
<(IH ⦠Hf) -f2 -IH //
| <unwind_gen_A_sn <unwind_gen_A_sn
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen_structure.ma
index fcc04282b..d8a44810c 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen_structure.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/unwind_gen_structure.ma
@@ -73,29 +73,29 @@ qed-.
(* Advanced constructions with list_rcons ***********************************)
-lemma unwind_gen_d_empty_dx (n) (p) (f):
- (âp)âð±((f n)ï¼ â§£â¨nâ©) = â[f](pâð±n).
-#n #p #f <unwind_gen_append_proper_dx //
+lemma unwind_gen_d_dx (f) (p) (n):
+ (âp)âð±(fï¼ â§£â¨nâ©) = â[f](pâð±n).
+#f #p #n <unwind_gen_append_proper_dx //
qed.
-lemma unwind_gen_m_dx (p) (f):
+lemma unwind_gen_m_dx (f) (p):
âp = â[f](pâðº).
-#p #f <unwind_gen_append_proper_dx //
+#f #p <unwind_gen_append_proper_dx //
qed.
-lemma unwind_gen_L_dx (p) (f):
+lemma unwind_gen_L_dx (f) (p):
(âp)âð = â[f](pâð).
-#p #f <unwind_gen_append_proper_dx //
+#f #p <unwind_gen_append_proper_dx //
qed.
-lemma unwind_gen_A_dx (p) (f):
+lemma unwind_gen_A_dx (f) (p):
(âp)âð = â[f](pâð).
-#p #f <unwind_gen_append_proper_dx //
+#f #p <unwind_gen_append_proper_dx //
qed.
-lemma unwind_gen_S_dx (p) (f):
+lemma unwind_gen_S_dx (f) (p):
(âp)âð¦ = â[f](pâð¦).
-#p #f <unwind_gen_append_proper_dx //
+#f #p <unwind_gen_append_proper_dx //
qed.
lemma unwind_gen_root (f) (p):
@@ -110,7 +110,7 @@ qed-.
lemma unwind_gen_inv_d_sn (k) (q) (p) (f):
(ð±kâq) = â[f]p â
- ââr,h. ð = âr & (f h)ï¼ â§£â¨hâ© = k & ð = q & râð±h = p.
+ ââr,h. ð = âr & fï¼ â§£â¨hâ© = k & ð = q & râð±h = p.
#k #q #p @(path_ind_unwind ⦠p) -p
[| #n | #n #l #p |*: #p ] [|*: #IH ] #f
[ <unwind_gen_empty #H destruct
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/unwind/xap.ma b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/xap.ma
new file mode 100644
index 000000000..f10ced5d6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/unwind/xap.ma
@@ -0,0 +1,55 @@
+include "delayed_updating/unwind/nap.ma".
+include "ground/notation/functions/apply_2.ma".
+include "ground/relocation/tr_compose_pn.ma".
+
+definition tr_xap (f) (l:nat): nat â
+ (⫯f)@ââ¨lâ©.
+
+interpretation
+ "functional extended application (total relocation maps)"
+ 'Apply f l = (tr_xap f l).
+
+lemma tr_xap_unfold (f) (l):
+ (⫯f)@ââ¨lâ© = fï¼ â¨lâ©.
+// qed.
+
+lemma tr_xap_zero (f):
+ (ð) = fï¼ â¨ðâ©.
+// qed.
+
+lemma tr_xap_ninj (f) (p):
+ ninj (fï¼ â§£â¨pâ©) = fï¼ â¨ninj pâ©.
+// qed.
+
+lemma tr_compose_xap (f2) (f1) (l):
+ f2ï¼ â¨f1ï¼ â¨lâ©â© = (f2âf1)ï¼ â¨lâ©.
+#f2 #f1 #l
+<tr_xap_unfold <tr_xap_unfold <tr_xap_unfold
+>tr_compose_nap >tr_compose_push_bi //
+qed.
+
+lemma tr_uni_xap_succ (n) (m):
+ âm + n = ð®â¨nâ©ï¼ â¨âmâ©.
+#n #m
+<tr_xap_unfold
+<tr_nap_push <tr_uni_nap //
+qed.
+
+lemma tr_uni_xap (n) (m):
+ ð®â¨nâ©ï¼ â¨m⩠⤠m+n.
+#n #m @(nat_ind_succ ⦠m) -m //
+qed.
+
+lemma tr_xap_push (f) (l):
+ â(fï¼ â¨lâ©) = (⫯f)ï¼ â¨âlâ©.
+#f #l
+<tr_xap_unfold <tr_xap_unfold
+<tr_nap_push //
+qed.
+
+lemma tr_xap_pushs_le (f) (n) (m):
+ m ⤠n â m = (⫯*[n]f)ï¼ â¨mâ©.
+#f #n #m #Hmn
+<tr_xap_unfold >tr_pushs_succ <tr_nap_pushs_lt //
+/2 width=1 by nlt_succ_dx/
+qed-.
--
2.39.2