From 2cc4eb5d0210be58286e028278852122dcb68052 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Thu, 20 Jan 2022 20:48:22 +0100
Subject: [PATCH] update in delayed_updating

+ a premise removed from lift_fsubst_sn
+ dfr and ifr restated to take equiprovability into account
+ notatation update for preterms
---
 .../class_t_0.ma}                               |  6 +++---
 .../delayed_updating/reduction/dfr.ma           | 10 ++++++----
 .../delayed_updating/reduction/dfr_ifr.ma       | 14 +++++++++++---
 .../delayed_updating/reduction/ifr.ma           |  9 +++++----
 .../substitution/fsubst_lift.ma                 | 17 ++++++++++-------
 .../substitution/lift_prototerm.ma              |  6 ++++++
 .../substitution/lift_structure.ma              | 10 +++++++++-
 .../delayed_updating/syntax/preterm.ma          | 13 ++++---------
 8 files changed, 54 insertions(+), 31 deletions(-)
 rename matita/matita/contribs/lambdadelta/delayed_updating/notation/{relations/predicate_t_hook_1.ma => functions/class_t_0.ma} (91%)

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/predicate_t_hook_1.ma b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_t_0.ma
similarity index 91%
rename from matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/predicate_t_hook_1.ma
rename to matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_t_0.ma
index 34af172fb..182544bf0 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/notation/relations/predicate_t_hook_1.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/notation/functions/class_t_0.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR DELAYED UPDATING ********************************************)
 
-notation "hvbox( Ƭ term 70 x )"
-  non associative with precedence 45
-  for @{ 'PredicateTHook $x }.
+notation "hvbox( 𝐓 )"
+  non associative with precedence 70
+  for @{ 'ClassT }.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
index ed077ae2c..5b5036593 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr.ma
@@ -16,15 +16,17 @@ include "delayed_updating/syntax/path_depth.ma".
 include "delayed_updating/syntax/path_structure.ma".
 include "delayed_updating/syntax/path_balanced.ma".
 include "delayed_updating/syntax/prototerm_constructors.ma".
+include "delayed_updating/syntax/prototerm_equivalence.ma".
 include "delayed_updating/substitution/fsubst.ma".
 include "delayed_updating/notation/relations/black_rightarrow_df_4.ma".
 
 (* DELAYED FOCUSED REDUCTION ************************************************)
 
-inductive dfr (p) (q) (t): predicate prototerm ≝
-| dfr_beta (b):
-  let r ≝ p●𝗔◗b●𝗟◗q in
-  r◖𝗱(↑❘q❘) ϵ t → ⊓(⊗b) → dfr p q t (t[⋔r←𝛗(↑❘q❘).t⋔(p◖𝗦)])
+definition dfr (p) (q): relation2 prototerm prototerm ≝
+           λt1,t2. ∃b.
+           let r ≝ p●𝗔◗b●𝗟◗q in
+           ∧∧ ⊓(⊗b) & r◖𝗱(↑❘q❘) ϵ t1 &
+              t2 ⇔ t1[⋔r←𝛗(↑❘q❘).t1⋔(p◖𝗦)]
 .
 
 interpretation
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma
index 9a713d1e5..5d6c3cfef 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/dfr_ifr.ma
@@ -18,7 +18,15 @@ include "delayed_updating/substitution/fsubst_lift.ma".
 
 (* DELAYED FOCUSED REDUCTION ************************************************)
 
-lemma dfr_lift_bi (f) (p) (q) (t1) (t2): Ƭt1 →
+lemma dfr_lift_bi (f) (p) (q) (t1) (t2): t1 ϵ 𝐓 →
       t1 ➡𝐝𝐟[p,q] t2 → ↑[f]t1 ➡𝐟[⊗p,⊗q] ↑[f]t2.
-#f #p #q #t1 #t2 #Ht1
-* -t2 #b #Hr #Hb
+#f #p #q #t1 #t2 #H0t1
+* #b * #Hb #Ht1 #Ht2
+@(ex_intro … (⊗b)) @and3_intro
+[ //
+| lapply (in_comp_lift_bi f … Ht1) -Ht1 #Ht1 
+
+
+
+
+
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma
index d087bdbb7..582f6ed57 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/reduction/ifr.ma
@@ -21,10 +21,11 @@ include "delayed_updating/notation/relations/black_rightarrow_f_4.ma".
 
 (* IMMEDIATE FOCUSED REDUCTION ************************************************)
 
-inductive ifr (p) (q) (t): predicate prototerm ≝
-| ifr_beta (b):
-  let r ≝ p●𝗔◗b●𝗟◗q in
-  r◖𝗱(↑❘q❘) ϵ t → ⊓(⊗b) → ifr p q t (t[⋔r←↑[𝐮❨↑❘q❘❩]t⋔(p◖𝗦)])
+definition ifr (p) (q): relation2 prototerm prototerm ≝
+           λt1,t2. ∃b.
+           let r ≝ p●𝗔◗b●𝗟◗q in
+           ∧∧ ⊓(⊗b) & r◖𝗱(↑❘q❘) ϵ t1 &
+              t2 ⇔ t1[⋔r←↑[𝐮❨↑❘q❘❩]t1⋔(p◖𝗦)]
 .
 
 interpretation
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/fsubst_lift.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/fsubst_lift.ma
index c9d656028..08ff3ebb3 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/fsubst_lift.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/fsubst_lift.ma
@@ -20,20 +20,23 @@ include "delayed_updating/syntax/prototerm_proper.ma".
 
 (* FOCALIZED SUBSTITUTION ***************************************************)
 
-lemma lift_fsubst_sn (f) (t) (u) (p): Ꝕu → p ⧸ϵ t →
+lemma lift_fsubst_sn (f) (t) (u) (p): Ꝕu →
       (↑[f]t)[⋔(⊗p)←↑[↑[p]f]u] ⊆ ↑[f](t[⋔p←u]).
-#f #t #u #p #Hu #Hp #ql * *
+#f #t #u #p #Hu #ql * *
 [ #rl * #r #Hr #H1 #H2 destruct
   >lift_append_proper_dx
   /4 width=1 by subset_in_ext_f1_dx, or_introl/
 | * #q #Hq #H1 #H0
-  @(ex2_intro … H1) @or_intror @conj //
-  #r #H2 destruct
-  @H0 -H0 [| <lift_append_proper_dx /2 width=1 by/ ]
+  @(ex2_intro … H1) @or_intror @conj // *
+  [ <list_append_empty_dx #H2 destruct
+    elim (lift_root f q) #r #_ #Hr /2 width=2 by/
+  | #l #r #H2 destruct
+    @H0 -H0 [| <lift_append_proper_dx /2 width=3 by ppc_lcons/ ]
+  ]
 ]
 qed-.
 
-lemma lift_fsubst_dx (f) (t) (u) (p): Ꝕu → p ϵ ▵t → structure_injective t →
+lemma lift_fsubst_dx (f) (t) (u) (p): Ꝕu → p ϵ ▵t → t ϵ 𝐓 →
       ↑[f](t[⋔p←u]) ⊆ (↑[f]t)[⋔(⊗p)←↑[↑[p]f]u].
 #f #t #u #p #Hu #H1p #H2p #ql * #q * *
 [ #r #Hu #H1 #H2 destruct
@@ -51,6 +54,6 @@ lemma lift_fsubst_dx (f) (t) (u) (p): Ꝕu → p ϵ ▵t → structure_injective
 ]
 qed-.
 
-lemma lift_fsubst (f) (t) (u) (p): Ꝕu → p ⧸ϵ t → p ϵ ▵t → structure_injective t →
+lemma lift_fsubst (f) (t) (u) (p): Ꝕu → p ϵ ▵t → t ϵ 𝐓 →
       (↑[f]t)[⋔(⊗p)←↑[↑[p]f]u] ⇔ ↑[f](t[⋔p←u]).
 /4 width=3 by lift_fsubst_sn, conj, lift_fsubst_dx/ qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm.ma
index 708e8a82c..919326045 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_prototerm.ma
@@ -21,3 +21,9 @@ include "ground/lib/subset_ext_equivalence.ma".
 interpretation
   "lift (prototerm)"
   'UpArrow f t = (subset_ext_f1 ? ? (lift_gen ? proj_path f) t).
+
+(* Basic constructions ******************************************************)
+
+lemma in_comp_lift_bi (f) (p) (t):
+      p ϵ t → ↑[f]p ϵ ↑[f]t.
+/2 width=1 by subset_in_ext_f1_dx/ qed.
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_structure.ma
index 5cc0f1f48..4db66c064 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_structure.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_structure.ma
@@ -75,11 +75,19 @@ lemma lift_S_dx (p) (f):
 #p #f <lift_append_proper_dx //
 qed.
 
+lemma lift_root (f) (p):
+      ∃∃r. 𝐞 = ⊗r & ⊗p●r = ↑[f]p.
+#f #p @(list_ind_rcons … p) -p
+[ /2 width=3 by ex2_intro/
+| #p * [ #n ] /2 width=3 by ex2_intro/
+]
+qed-.
+
 (* Advanced inversions with proj_path ***************************************)
 
 lemma lift_path_inv_d_sn (k) (q) (p) (f):
       (𝗱k◗q) = ↑[f]p →
-      ∃∃r,h. 𝐞 = ⊗r & (↑[r]f)@❨h❩ = k & 𝐞  = q & r◖𝗱h = p.
+      ∃∃r,h. 𝐞 = ⊗r & (↑[r]f)@❨h❩ = k & 𝐞 = q & r◖𝗱h = p.
 #k #q #p @(path_ind_lift … p) -p
 [| #n | #n #l #p |*: #p ] [|*: #IH ] #f
 [ <lift_path_empty #H destruct
diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/preterm.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/preterm.ma
index 9da0aa838..b7b90a4cf 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/preterm.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/preterm.ma
@@ -14,21 +14,16 @@
 
 include "delayed_updating/syntax/prototerm.ma".
 include "delayed_updating/syntax/path_structure.ma".
-include "delayed_updating/notation/relations/predicate_t_hook_1.ma".
+include "delayed_updating/notation/functions/class_t_0.ma".
 
 (* PRETERM ******************************************************************)
 
+(* Note: a preterm is a prototerm satislying the condition below *)
 (* Note: different root paths have different structure *)
 definition structure_injective: predicate prototerm ≝
            λt. ∀p1,p2. p1 ϵ ▵t → p2 ϵ ▵t → ⊗p1 = ⊗p2 → p1 = p2
 .
 
-(* Note: a preterm is a prototerm satislying the conditions above *)
-record is_preterm (t): Prop ≝
-{
-  is_pt_injective: structure_injective t
-}.
-
 interpretation
-  "preterm condition (prototerm)"
-  'PredicateTHook t = (is_preterm t).
+  "preterm (prototerm)"
+  'ClassT = (structure_injective).
-- 
2.39.2