From f5fa6554c93bec72a5bb098c0a2991fe294883b6 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Mon, 30 May 2022 23:50:58 +0200
Subject: [PATCH 1/1] update in delayed_updating

+ lift: additions and refactoring
+ reduction: WIP
---
 .../delayed_updating/reduction/dfr_lift.ma    | 70 +++++++++++++++++
 .../delayed_updating/reduction/ifr_unwind.ma  | 77 +++++++++++++++++++
 .../delayed_updating/substitution/lift.ma     |  2 +-
 .../substitution/lift_constructors.ma         |  2 +-
 .../delayed_updating/substitution/lift_eq.ma  |  3 +-
 .../{lift_after.ma => lift_path_after.ma}     |  4 +-
 .../{lift_id.ma => lift_path_id.ma}           |  4 +-
 .../substitution/lift_path_proper.ma          | 42 ++++++++++
 .../{lift_uni.ma => lift_path_uni.ma}         |  6 +-
 .../substitution/lift_prototerm_eq.ma         |  2 +-
 .../substitution/lift_prototerm_id.ma         |  6 +-
 .../substitution/lift_prototerm_proper.ma     | 36 +++++++++
 12 files changed, 242 insertions(+), 12 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_unwind.ma
 rename matita/matita/contribs/lambdadelta/delayed_updating/substitution/{lift_after.ma => lift_path_after.ma} (98%)
 rename matita/matita/contribs/lambdadelta/delayed_updating/substitution/{lift_id.ma => lift_path_id.ma} (97%)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_proper.ma
 rename matita/matita/contribs/lambdadelta/delayed_updating/substitution/{lift_uni.ma => lift_path_uni.ma} (94%)
 create mode 100644 matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_proper.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
new file mode 100644
index 000000000..d25ddc502
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_lift.ma
@@ -0,0 +1,70 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "delayed_updating/reduction/dfr.ma".
+include "delayed_updating/reduction/ifr.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".
+
+(* DELAYED FOCUSED REDUCTION ************************************************)
+
+(* Constructions with lift **************************************************)
+
+theorem dfr_lift_bi (f) (p) (q) (t1) (t2): (*t1 ϵ 𝐓 → *)
+        t1 ➡𝐝𝐟[p,q] t2 → ↑[f]t1 ➡𝐟[↑[f]p,↑[↑[p◖𝗔◖𝗟]f]q] ↑[f]t2.
+#f #p #q #t1 #t2
+* #n * #H1n #Ht1 #Ht2
+@(ex_intro … ((↑[p●𝗔◗𝗟◗q]f)＠⧣❨n❩)) @and3_intro
+[ -Ht1 -Ht2
+  >H1n >path_head_structure_depth <H1n -H1n //
+| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H0t1
+  <unwind2_path_d_dx <depth_structure
+  >list_append_rcons_sn in H1n; <reverse_append #H1n
+  lapply (unwind2_rmap_append_pap_closed f … H1n)
+  <reverse_lcons <depth_L_dx #H2n
+  lapply (eq_inv_ninj_bi … H2n) -H2n #H2n <H2n -H2n -H1n #Ht1 //
+| lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
+  @(subset_eq_trans … Ht2) -t2
+  @(subset_eq_trans … (unwind2_term_fsubst …))
+  [ @fsubst_eq_repl [ // | // ]
+    @(subset_eq_trans … (unwind2_term_iref …))
+    @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
+    [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+    @(subset_eq_trans … (unwind2_lift_term_after …))
+    @unwind2_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+    @nstream_eq_inv_ext #m
+    <tr_compose_pap <tr_compose_pap
+    <tr_uni_pap <tr_uni_pap <tr_pap_plus
+    >list_append_rcons_sn in H1n; <reverse_append #H1n
+    lapply (unwind2_rmap_append_pap_closed f … H1n) #H2n
+    >nrplus_inj_dx in ⊢ (???%); <H2n -H2n
+    lapply (tls_unwind2_rmap_append_closed f … H1n) #H2n
+    <(tr_pap_eq_repl … H2n) -H2n -H1n //
+(* Note: crux of the proof ends *)
+  | //
+  | /2 width=2 by ex_intro/
+  | //
+  ]
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_unwind.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_unwind.ma
new file mode 100644
index 000000000..7634974a5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr_unwind.ma
@@ -0,0 +1,77 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/ifr.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".
+include "delayed_updating/syntax/path_structure_reverse.ma".
+include "delayed_updating/syntax/path_depth_reverse.ma".
+
+(* IMMEDIATE FOCUSED REDUCTION **********************************************)
+
+(* Constructions with unwind ************************************************)
+
+theorem ifr_unwind_bi (f) (p) (q) (t1) (t2):
+        t1 ϵ 𝐓 → t1⋔(p◖𝗦) ϵ 𝐏 →
+        t1 ➡𝐟[p,q] t2 → ▼[f]t1 ➡𝐟[⊗p,⊗q] ▼[f]t2.
+#f #p #q #t1 #t2 #H1t1 #H2t1 
+* #n * #H1n #Ht1 #Ht2
+@(ex_intro … (↑♭⊗q)) @and3_intro
+[ -H0t1 -Ht1 -Ht2
+  >structure_L_sn >structure_reverse
+  >H1n >path_head_structure_depth <H1n -H1n //
+| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H0t1
+  <unwind2_path_d_dx <depth_structure
+  >list_append_rcons_sn in H1n; <reverse_append #H1n
+  lapply (unwind2_rmap_append_pap_closed f … H1n)
+  <reverse_lcons <depth_L_dx #H2n
+  lapply (eq_inv_ninj_bi … H2n) -H2n #H2n <H2n -H2n -H1n #Ht1 //
+| lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
+  @(subset_eq_trans … Ht2) -t2
+  @(subset_eq_trans … (unwind2_term_fsubst …))
+  [ @fsubst_eq_repl [ // | // ]
+(*  
+    @(subset_eq_trans … (unwind2_term_iref …))
+    @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
+    [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+    @(subset_eq_trans … (unwind2_lift_term_after …))
+    @unwind2_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+    @nstream_eq_inv_ext #m
+    <tr_compose_pap <tr_compose_pap
+    <tr_uni_pap <tr_uni_pap <tr_pap_plus
+    >list_append_rcons_sn in H1n; <reverse_append #H1n
+    lapply (unwind2_rmap_append_pap_closed f … H1n) #H2n
+    >nrplus_inj_dx in ⊢ (???%); <H2n -H2n
+    lapply (tls_unwind2_rmap_append_closed f … H1n) #H2n
+    <(tr_pap_eq_repl … H2n) -H2n -H1n //
+(* Note: crux of the proof ends *)
+*)
+  | //
+  | /2 width=2 by ex_intro/
+  | //
+  ]
+]
+qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift.ma
index ec7d9726f..b873ed303 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift.ma
@@ -18,7 +18,7 @@ include "delayed_updating/syntax/path.ma".
 include "ground/relocation/tr_uni.ma".
 include "ground/relocation/tr_pap_tls.ma".
 
-(* LIFT FOR PATH ***********************************************************)
+(* LIFT FOR PATH ************************************************************)
 
 definition lift_continuation (A:Type[0]) ≝
            tr_map → path → A.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma
index 06e03b13a..1cdcd4650 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma
@@ -13,7 +13,7 @@
 (**************************************************************************)
 
 include "delayed_updating/substitution/lift_prototerm_id.ma".
-include "delayed_updating/substitution/lift_uni.ma".
+include "delayed_updating/substitution/lift_path_uni.ma".
 include "delayed_updating/syntax/prototerm_constructors.ma".
 
 (* LIFT FOR PROTOTERM *******************************************************)
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 5eac1985e..9c4b0f3b5 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_eq.ma
@@ -17,7 +17,7 @@ include "ground/relocation/tr_pap_eq.ma".
 include "ground/relocation/tr_pn_eq.ma".
 include "ground/lib/stream_tls_eq.ma".
 
-(* LIFT FOR PATH ***********************************************************)
+(* LIFT FOR PATH ************************************************************)
 
 definition lift_exteq (A): relation2 (lift_continuation A) (lift_continuation A) ≝
            λk1,k2. ∀f1,f2,p. f1 ≗ f2 → k1 f1 p = k2 f2 p.
@@ -153,4 +153,5 @@ lemma lift_rmap_tls_d_dx (f) (p) (m) (n):
 <lift_rmap_d_dx >nrplus_inj_dx
 /2 width=1 by tr_tls_compose_uni_dx/
 qed.
+
 *)
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_after.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_after.ma
similarity index 98%
rename from matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_after.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_after.ma
index 756b73587..869d680fe 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_after.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_after.ma
@@ -17,9 +17,9 @@ include "ground/relocation/tr_compose_pap.ma".
 include "ground/relocation/tr_compose_pn.ma".
 include "ground/relocation/tr_compose_tls.ma".
 
-(* LIFT FOR PATH ***********************************************************)
+(* LIFT FOR PATH ************************************************************)
 
-(* Constructions with tr_after *********************************************)
+(* Constructions with tr_after **********************************************)
 
 lemma lift_path_after (p) (f1) (f2):
       ↑[f2]↑[f1]p = ↑[f2∘f1]p.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_id.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_id.ma
similarity index 97%
rename from matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_id.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_id.ma
index 1d8b45295..1966c25f2 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_id.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_id.ma
@@ -16,9 +16,9 @@ include "delayed_updating/substitution/lift_eq.ma".
 include "ground/relocation/tr_id_pap.ma".
 include "ground/relocation/tr_id_tls.ma".
 
-(* LIFT FOR PATH ***********************************************************)
+(* LIFT FOR PATH ************************************************************)
 
-(* Constructions with tr_id ************************************************)
+(* Constructions with tr_id *************************************************)
 
 lemma lift_path_id (p):
       p = ↑[𝐢]p.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_proper.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_proper.ma
new file mode 100644
index 000000000..0eef97896
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_proper.ma
@@ -0,0 +1,42 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_eq.ma".
+include "delayed_updating/syntax/path_proper.ma".
+
+(* LIFT FOR PATH ************************************************************)
+
+(* Constructions with proper condition for path *****************************)
+
+lemma lift_path_proper (f) (p):
+      p ϵ 𝐏 → ↑[f]p ϵ 𝐏.
+#f *
+[ #H0 elim (ppc_inv_empty … H0)
+| * [ #n ] #p #_
+  [ <lift_path_d_sn /2 width=3 by ppc_lcons/
+  | <lift_path_m_sn /2 width=3 by ppc_lcons/
+  | <lift_path_L_sn /2 width=3 by ppc_lcons/
+  | <lift_path_A_sn /2 width=3 by ppc_lcons/
+  | <lift_path_S_sn /2 width=3 by ppc_lcons/
+  ]
+]
+qed.
+
+(* Inversions with proper condition for path ********************************)
+
+lemma lift_path_inv_proper (f) (p):
+      ↑[f]p ϵ 𝐏 → p ϵ 𝐏.
+#f * //
+#H0 elim (ppc_inv_empty … H0)
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_uni.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_uni.ma
similarity index 94%
rename from matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_uni.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_uni.ma
index cd6c79533..f68010b32 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_uni.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_path_uni.ma
@@ -12,13 +12,13 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_id.ma".
+include "delayed_updating/substitution/lift_path_id.ma".
 include "ground/relocation/tr_uni_pap.ma".
 include "ground/relocation/tr_uni_tls.ma".
 
-(* LIFT FOR PATH ***********************************************************)
+(* LIFT FOR PATH ************************************************************)
 
-(* Constructions with tr_uni ***********************************************)
+(* Constructions with tr_uni ************************************************)
 
 lemma lift_path_d_sn_uni (p) (m) (n):
       (𝗱(n+m)◗p) = ↑[𝐮❨m❩](𝗱(n)◗p).
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_eq.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_eq.ma
index fc3fb3fcf..920a16cf0 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_eq.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_eq.ma
@@ -13,7 +13,7 @@
 (**************************************************************************)
 
 include "ground/lib/subset_ext_equivalence.ma".
-include "delayed_updating/substitution/lift_after.ma".
+include "delayed_updating/substitution/lift_path_after.ma".
 include "delayed_updating/substitution/lift_prototerm.ma".
 
 (* LIFT FOR PROTOTERM *******************************************************)
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma
index cc7569284..b66b63f46 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_id.ma
@@ -13,7 +13,11 @@
 (**************************************************************************)
 
 include "delayed_updating/substitution/lift_prototerm_eq.ma".
-include "delayed_updating/substitution/lift_id.ma".
+include "delayed_updating/substitution/lift_path_id.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+(* Constructions with tr_id *************************************************)
 
 lemma lift_term_id_sn (t):
       t ⊆ ↑[𝐢]t.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_proper.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_proper.ma
new file mode 100644
index 000000000..a9b16219f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm_proper.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_prototerm.ma".
+include "delayed_updating/substitution/lift_path_proper.ma".
+include "delayed_updating/syntax/prototerm_proper.ma".
+
+(* LIFT FOR PROTOTERM *******************************************************)
+
+(* Constructions with proper condition for path *****************************)
+
+lemma lift_term_proper (f) (t):
+      t ϵ 𝐏 → ↑[f]t ϵ 𝐏.
+#f #t #Ht #p * #q #Hq #H0 destruct
+@lift_path_proper @Ht -Ht // (**) (* auto fails *)
+qed.
+
+(* Inversions with proper condition for path ********************************)
+
+lemma lift_term_inv_proper (f) (t):
+      ↑[f]t ϵ 𝐏 → t ϵ 𝐏.
+#f #t #Ht #p #Hp
+@(lift_path_inv_proper f)
+@Ht -Ht @in_comp_lift_bi // (**) (* auto fails *)
+qed-.
-- 
2.39.2