From 7982e65abf88a2fbaa6422debb29121e4c884dd8 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Fri, 8 Mar 2013 13:33:29 +0000
Subject: [PATCH] improved "big treee" computation allows to close a case in
 the preservation proof

---
 .../lambdadelta/basic_2/computation/ygt.ma    | 112 ++++++------
 .../basic_2/computation/ygt_ygt.ma            |   4 +-
 .../lambdadelta/basic_2/computation/yprs.ma   |  77 ++++++++
 .../basic_2/computation/yprs_yprs.ma          |  20 ++
 .../lambdadelta/basic_2/dynamic/snv_cpcs.ma   | 171 ++++++++----------
 .../lambdadelta/basic_2/dynamic/snv_ltpr.ma   |  16 +-
 .../basic_2/dynamic/snv_ssta_ltpr.ma          |  28 ++-
 .../contribs/lambdadelta/basic_2/notation.ma  |  12 +-
 .../lambdadelta/basic_2/reducibility/ypr.ma   |  35 ++++
 .../lambdadelta/basic_2/reducibility/ysc.ma   |  24 ++-
 10 files changed, 319 insertions(+), 180 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/computation/yprs.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/computation/yprs_yprs.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/reducibility/ypr.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/ygt.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/ygt.ma
index e5140a8e7..708c1a8c7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/ygt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/ygt.ma
@@ -12,84 +12,84 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/unwind/sstas.ma".
 include "basic_2/reducibility/ysc.ma".
-include "basic_2/computation/cprs.ma".
+include "basic_2/computation/yprs.ma".
 
-(* "BIG TREE" ORDER FOR CLOSURES ********************************************)
+(* "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **********************)
 
-definition ygt: ∀h. sd h → bi_relation lenv term ≝
-                λh,g. bi_TC … (ysc h g).
+inductive ygt (h) (g) (L1) (T1): relation2 lenv term ≝
+| ygt_inj : ∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≻[g] ⦃L2, T2⦄ →
+            ygt h g L1 T1 L2 T2
+| ygt_step: ∀L,L2,T. ygt h g L1 T1 L T → L ➡ L2 → ygt h g L1 T1 L2 T
+.
 
-interpretation "'big tree' order (closure)"
-   'BTGreaterThan h g L1 T1 L2 T2 = (ygt h g L1 T1 L2 T2).
+interpretation "'big tree' proper parallel computation (closure)"
+   'BTPRedStarProper h g L1 T1 L2 T2 = (ygt h g L1 T1 L2 T2).
 
-(* Basic eliminators ********************************************************)
+(* Basic forvard lemmas *****************************************************)
 
-lemma ygt_ind: ∀h,g,L1,T1. ∀R:relation2 lenv term.
-               (∀L2,T2. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ → R L2 T2) →
-               (∀L,T,L2,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≻[g] ⦃L2, T2⦄ → R L T → R L2 T2) →
-               ∀L2,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄ → R L2 T2.
-#h #g #L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
-@(bi_TC_ind  … IH1 IH2 L2 T2 H)
-qed-. (**) (* /3 width=6 by bi_TC_ind/ fails *)
-
-lemma ygt_ind_dx: ∀h,g,L2,T2. ∀R:relation2 lenv term.
-                  (∀L1,T1. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ → R L1 T1) →
-                  (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → R L T → R L1 T1) →
-                  ∀L1,T1. h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄ → R L1 T1.
-/3 width=6 by bi_TC_ind_dx/ qed-.
+lemma ygt_fwd_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄ →
+                    h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+#h #g #L1 #L2 #T1 #T2 #H elim H -L2 -T2
+/3 width=4 by yprs_strap1, ysc_ypr, ypr_ltpr/
+qed-.
 
 (* Basic properties *********************************************************)
 
-lemma ygt_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
-                  h ⊢ ⦃L, T⦄ ≻[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
-/2 width=4/ qed-.
-
-lemma ygt_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L, T⦄ →
-                  h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
-/2 width=4/ qed-.
+lemma ysc_ygt: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ →
+               h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+/3 width=4/ qed.
 
-lemma ygt_cprs_trans: ∀h,g,L1,L,T1,T. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
-                      ∀T2. L ⊢ T ➡* T2 → h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T2⦄.
-#h #g #L1 #L #T1 #T #HLT1 #T2 #H @(cprs_ind … H) -T2 //
-#T0 #T2 #_ #HT02 #IHT0 -HLT1
-elim (term_eq_dec T0 T2) #HT02 destruct //
-@(ygt_strap1 … IHT0) /3 width=1/
+lemma ygt_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
+                  h ⊢ ⦃L, T⦄ ≽[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+#h #g #L1 #L #L2 #T1 #T #T2 #H1 #H2
+lapply (ygt_fwd_yprs … H1) #H0
+elim (ypr_inv_ysc … H2) -H2 [| * #HL2 #H destruct ] /2 width=4/
 qed-.
 
-lemma ygt_sstas_trans: ∀h,g,L1,L,T1,T. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
-                       ∀T2. ⦃h, L⦄ ⊢ T •*[g] T2 → h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T2⦄.
-#h #g #L1 #L #T1 #T #HLT1 #T2 #H @(sstas_ind … H) -T2 //
-#T0 #T2 #l #_ #HT02 #IHT0 -HLT1
-@(ygt_strap1 … IHT0) -IHT0 /2 width=2/
+lemma ygt_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L, T⦄ →
+                  h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+#h #g #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim H2 -L2 -T2
+[ /3 width=4 by ygt_inj, yprs_strap2/ | /2 width=3/ ]
 qed-.
 
-lemma cprs_ygt_trans: ∀h,g,L,T1,T. L ⊢ T1 ➡* T → 
-                      ∀L2,T2. h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L, T1⦄ >[g] ⦃L2, T2⦄.
-#h #g #L #T1 #T #H @(cprs_ind … H) -T //
-#T0 #T #_ #HT0 #IHT10 #L2 #T2 #HLT2
-elim (term_eq_dec T0 T) #HT0 destruct /2 width=1/
-@IHT10 -IHT10 @(ygt_strap2 … HLT2) /3 width=1/
+lemma ygt_yprs_trans: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
+                      h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+#h #g #L1 #L #L2 #T1 #T #T2 #HT1 #HT2 @(yprs_ind … HT2) -L2 -T2 //
+/2 width=4 by ygt_strap1/
 qed-.
 
-lemma sstas_ygt_trans: ∀h,g,L,T1,T.  ⦃h, L⦄ ⊢ T1 •*[g] T →
-                       ∀L2,T2. h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L, T1⦄ >[g] ⦃L2, T2⦄.
-#h #g #L #T1 #T #H @(sstas_ind … H) -T //
-#T0 #T #l #_ #HT0 #IHT10 #L2 #T2 #HLT2
-@IHT10 -IHT10 @(ygt_strap2 … HLT2) /2 width=2/
+lemma yprs_ygt_trans: ∀h,g,L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ →
+                      ∀L2,T2. h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+#h #g #L1 #L #T1 #T #HT1 @(yprs_ind … HT1) -L -T //
+/3 width=4 by ygt_strap2/
 qed-.
 
 lemma fw_ygt: ∀h,g,L1,L2,T1,T2. ♯{L2, T2} < ♯{L1, T1} → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
 /3 width=1/ qed.
 
-lemma cprs_ygt: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) → h ⊢ ⦃L, T1⦄ >[g] ⦃L, T2⦄.
+lemma cprs_ygt: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
+                h ⊢ ⦃L, T1⦄ >[g] ⦃L, T2⦄.
 #h #g #L #T1 #T2 #H @(cprs_ind … H) -T2
-[ #H elim H -H //
-| #T #T2 #_ #HT2 #IHT1 #H
-  elim (term_eq_dec T1 T) #HT1 destruct
-  [ -IHT1 /4 width=1 by ysc_cpr, bi_inj/ (**) (* auto too slow without trace *)
-  | -H /4 width=3 by inj, ygt_cprs_trans/
+[ #H elim H //
+| #T #T2 #_ #HT2 #IHT1 #HT12
+  elim (term_eq_dec T1 T) #H destruct
+  [ -IHT1 /4 width=1/
+  | lapply (IHT1 … H) -IHT1 -H -HT12 #HT1
+    @(ygt_strap1 … HT1) -HT1 /2 width=1/
+  ]
+]
+qed.
+
+lemma sstas_ygt: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 → (T1 = T2 → ⊥) →
+                 h ⊢ ⦃L, T1⦄ >[g] ⦃L, T2⦄.
+#h #g #L #T1 #T2 #H @(sstas_ind … H) -T2
+[ #H elim H //
+| #T #T2 #l #_ #HT2 #IHT1 #HT12 -HT12
+  elim (term_eq_dec T1 T) #H destruct
+  [ -IHT1 /3 width=2/
+  | lapply (IHT1 … H) -IHT1 -H #HT1
+    @(ygt_strap1 … HT1) -HT1 /2 width=2/
   ]
 ]
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/ygt_ygt.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/ygt_ygt.ma
index 0c2ae2995..31ab7a917 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/ygt_ygt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/ygt_ygt.ma
@@ -16,5 +16,7 @@ include "basic_2/computation/ygt.ma".
 
 (* "BIG TREE" ORDER FOR CLOSURES ********************************************)
 
+(* Main properties **********************************************************)
+
 theorem ygt_trans: ∀h,g. bi_transitive … (ygt h g).
-/2 width=4/ qed-.
+/3 width=4 by ygt_fwd_yprs, ygt_yprs_trans/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/yprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/yprs.ma
new file mode 100644
index 000000000..8b7e4cc98
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/yprs.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 "basic_2/unwind/sstas.ma".
+include "basic_2/reducibility/ypr.ma".
+include "basic_2/computation/ltprs.ma".
+include "basic_2/computation/cprs.ma".
+
+(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
+
+definition yprs: ∀h. sd h → bi_relation lenv term ≝
+                 λh,g. bi_TC … (ypr h g).
+
+interpretation "'big tree' parallel computation (closure)"
+   'BTPRedStar h g L1 T1 L2 T2 = (yprs h g L1 T1 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma yprs_ind: ∀h,g,L1,T1. ∀R:relation2 lenv term. R L1 T1 →
+                (∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≽[g] ⦃L2, T2⦄ → R L T → R L2 T2) →
+                ∀L2,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄ → R L2 T2.
+/3 width=7 by bi_TC_star_ind/ qed-.
+
+lemma yprs_ind_dx: ∀h,g,L2,T2. ∀R:relation2 lenv term. R L2 T2 →
+                   (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → R L T → R L1 T1) →
+                   ∀L1,T1. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄ → R L1 T1.
+/3 width=7 by bi_TC_star_ind_dx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma yprs_refl: ∀h,g. bi_reflexive … (yprs h g).
+/2 width=1/ qed.
+
+lemma ypr_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L2, T2⦄ →
+                h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+/2 width=1/ qed.
+
+lemma yprs_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ →
+                   h ⊢ ⦃L, T⦄ ≽[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+/2 width=4/ qed-.
+
+lemma yprs_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L, T⦄ →
+                   h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+/2 width=4/ qed-.
+
+lemma fw_yprs: ∀h,g,L1,L2,T1,T2. ♯{L2, T2} < ♯{L1, T1} →
+               h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+/3 width=1/ qed.
+
+lemma cprs_yprs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → h ⊢ ⦃L, T1⦄ ≥[g] ⦃L, T2⦄.
+#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4 by ypr_cpr, yprs_strap1/
+qed.
+
+lemma ltprs_yprs: ∀h,g,L1,L2,T. L1 ➡* L2 → h ⊢ ⦃L1, T⦄ ≥[g] ⦃L2, T⦄.
+#h #g #L1 #L2 #T #H @(ltprs_ind … H) -L2 // /3 width=4 by ypr_ltpr, yprs_strap1/
+qed.
+
+lemma sstas_yprs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 →
+                  h ⊢ ⦃L, T1⦄ ≥[g] ⦃L, T2⦄.
+#h #g #L #T1 #T2 #H @(sstas_ind … H) -T2 // /3 width=4 by ypr_ssta, yprs_strap1/
+qed.
+
+lemma ltpr_cprs_yprs: ∀h,g,L1,L2,T1,T2. L1 ➡ L2 → L2 ⊢ T1 ➡* T2 →
+                      h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+/3 width=4 by yprs_strap2, ypr_ltpr, cprs_yprs/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/yprs_yprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/yprs_yprs.ma
new file mode 100644
index 000000000..4fd2d2a60
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/yprs_yprs.ma
@@ -0,0 +1,20 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "basic_2/computation/yprs.ma".
+
+(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
+
+theorem yprs_trans: ∀h,g. bi_transitive … (yprs h g).
+/2 width=4/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
index 7078c9f3e..7f325d41a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
@@ -14,43 +14,28 @@
 
 include "basic_2/static/ssta_ssta.ma".
 include "basic_2/computation/ygt.ma".
+include "basic_2/equivalence/cpcs_ltpr.ma".
 include "basic_2/dynamic/snv_ltpss_dx.ma".
 include "basic_2/dynamic/snv_sstas.ma".
 
 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
 
 (* Inductive premises for the preservation results **************************)
-(*
-definition IH_snv_dxprs: ∀h:sh. sd h → relation2 lenv term ≝
-                         λh,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
-                         ∀L2. L1 ➡ L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g].
-*)
-definition IH_ssta_ltpr_tpr: ∀h:sh. sd h → relation2 lenv term ≝
-                             λh,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
-                             ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                             ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 →
-                             ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & L1 ⊢ U1 ⬌* U2.
 
 definition IH_snv_ltpr_tpr: ∀h:sh. sd h → relation2 lenv term ≝
                             λh,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
                             ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊩ T2 :[g].
 
+definition IH_ssta_ltpr_tpr: ∀h:sh. sd h → relation2 lenv term ≝
+                             λh,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
+                             ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+                             ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 →
+                             ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & L2 ⊢ U1 ⬌* U2.
+
 definition IH_snv_ssta: ∀h:sh. sd h → relation2 lenv term ≝
                         λh,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
                         ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l + 1] U1 → ⦃h, L1⦄ ⊩ U1 :[g].
 
-fact ssta_ltpr_cpr_aux: ∀h,g,L1,T1. IH_ssta_ltpr_tpr h g L1 T1 →
-                        ⦃h, L1⦄ ⊩ T1 :[g] →
-                        ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                        ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡ T2 →
-                        ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #L1 #T1 #IH #HT1 #U1 #l #HTU1 #L2 #HL12 #T2 * #T #HT1T #HTT2
-elim (IH … HTU1 … HL12 … HT1T) // -T1 #U #HTU #HU1
-elim (ssta_tpss_conf … HTU … HTT2) -T #U2 #HTU2 #HU2
-lapply (ltpr_cpr_trans … HL12 U U2 ?) -HL12 /2 width=3/ -HU2 #HU2
-lapply (cpcs_cprs_strap1 … HU1 … HU2) -U /2 width=3/
-qed-.
-
 fact snv_ltpr_cpr_aux: ∀h,g,L1,T1. IH_snv_ltpr_tpr h g L1 T1 →
                        ⦃h, L1⦄ ⊩ T1 :[g] →
                        ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L2⦄ ⊩ T2 :[g].
@@ -59,44 +44,56 @@ lapply (IH … HL12 … HT1T) -HL12 // -T1 #HT0
 lapply (snv_tpss_conf … HT0 … HTT2) -T //
 qed-.
 
-fact snv_cprs_aux: ∀h,g,L0,T0.
-                   (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                   ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                   ∀T2. L1 ⊢ T1 ➡* T2 → ⦃h, L1⦄ ⊩ T2 :[g].
-#h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #T2 #H
-@(cprs_ind … H) -T2 // -HT1
-/4 width=6 by snv_ltpr_cpr_aux, ygt_cprs_trans/
+fact ssta_ltpr_cpr_aux: ∀h,g,L1,T1. IH_ssta_ltpr_tpr h g L1 T1 →
+                        ⦃h, L1⦄ ⊩ T1 :[g] →
+                        ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+                        ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡ T2 →
+                        ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & L2 ⊢ U1 ⬌* U2.
+#h #g #L1 #T1 #IH #HT1 #U1 #l #HTU1 #L2 #HL12 #T2 * #T #HT1T #HTT2
+elim (IH … HTU1 … HL12 … HT1T) // -L1 -T1 #U #HTU #HU1
+elim (ssta_tpss_conf … HTU … HTT2) -T #U2 #HTU2 #HU2
+lapply (cpcs_cpr_strap1 … HU1 U2 ?) /2 width=3/
+qed-.
+
+fact snv_ltpr_cprs_aux: ∀h,g,L0,T0.
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+                        ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
+                        ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊩ T2 :[g].
+#h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #L2 #HL12 #T2 #H
+@(cprs_ind … H) -T2 [ /2 width=6 by snv_ltpr_cpr_aux/ ] -HT1
+/5 width=6 by snv_ltpr_cpr_aux, ygt_yprs_trans, ltpr_cprs_yprs/
 qed-.
 
-fact ssta_cprs_aux: ∀h,g,L0,T0.
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                    ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                    ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 → ∀T2. L1 ⊢ T1 ➡* T2 →
-                    ∃∃U2. ⦃h, L1⦄ ⊢ T2 •[g, l] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #l #HTU1 #T2 #H
-@(cprs_ind … H) -T2 [ /2 width=3/ ]
-#T #T2 #HT1T #HTT2 * #U #HTU #HU1 
+fact ssta_ltpr_cprs_aux: ∀h,g,L0,T0.
+                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+                         ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
+                         ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+                         ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 →
+                         ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & L2 ⊢ U1 ⬌* U2.
+#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #l #HTU1 #L2 #HL12 #T2 #H
+@(cprs_ind … H) -T2 [ /2 width=7 by ssta_ltpr_cpr_aux/ ]
+#T #T2 #HT1T #HTT2 * #U #HTU #HU1
 elim (ssta_ltpr_cpr_aux … HTU … HTT2) //
-[2: /3 width=7 by snv_cprs_aux, ygt_cprs_trans/
-|3: /3 width=3 by ygt_cprs_trans/
-] -L0 -T0 -T1 -T #U2 #HTU2 #HU2
+[2: /3 width=9 by snv_ltpr_cprs_aux/
+|3: /5 width=6 by ygt_yprs_trans, ltpr_cprs_yprs/
+] -L0 -L1 -T0 -T1 -T #U2 #HTU2 #HU2
 lapply (cpcs_trans … HU1 … HU2) -U /2 width=3/
 qed-.
 
-fact ssta_cpcs_aux: ∀h,g,L0,T0.
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                    ∀L,T1,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T1⦄ → h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T2⦄ →
-                    ⦃h, L⦄ ⊩ T1 :[g] → ⦃h, L⦄ ⊩ T2 :[g] →
-                    ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g, l1] U1 →
-                    ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g, l2] U2 →
-                    L ⊢ T1 ⬌* T2 →
-                    l1 = l2 ∧ L ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L #T1 #T2 #HLT01 #HLT02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #H
+fact ssta_ltpr_cpcs_aux: ∀h,g,L0,T0.
+                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+                         ∀L1,L2,T1,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → h ⊢ ⦃L0, T0⦄ >[g] ⦃L2, T2⦄ →
+                         ⦃h, L1⦄ ⊩ T1 :[g] → ⦃h, L2⦄ ⊩ T2 :[g] →
+                         ∀U1,l1. ⦃h, L1⦄ ⊢ T1 •[g, l1] U1 →
+                         ∀U2,l2. ⦃h, L2⦄ ⊢ T2 •[g, l2] U2 →
+                         L1 ➡ L2 → L2 ⊢ T1 ⬌* T2 →
+                         l1 = l2 ∧ L2 ⊢ U1 ⬌* U2.
+#h #g #L0 #T0 #IH2 #IH1 #L1 #L2 #T1 #T2 #HLT01 #HLT02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #HL12 #H
 elim (cpcs_inv_cprs … H) -H #T #H1 #H2
-elim (ssta_cprs_aux … HLT01 HT1 … HTU1 … H1) -T1 /2 width=1/ #W1 #H1 #HUW1
-elim (ssta_cprs_aux … HLT02 HT2 … HTU2 … H2) -T2 /2 width=1/ #W2 #H2 #HUW2 -L0 -T0
+elim (ssta_ltpr_cprs_aux … HLT01 HT1 … HTU1 … H1) -T1 /2 width=1/ #W1 #H1 #HUW1
+elim (ssta_ltpr_cprs_aux … HLT02 HT2 … HTU2 … H2) -T2 /2 width=1/ #W2 #H2 #HUW2 -L1 -L0 -T0
 elim (ssta_mono … H1 … H2) -h -T #H1 #H2 destruct
 lapply (cpcs_canc_dx … HUW1 … HUW2) -W2 /2 width=1/
 qed-.
@@ -106,55 +103,43 @@ fact snv_sstas_aux: ∀h,g,L0,T0.
                     ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
                     ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ⦃h, L1⦄ ⊩ U1 :[g].
 #h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #U1 #H
-@(sstas_ind … H) -U1 // -HT1 /3 width=5 by ygt_sstas_trans/
+@(sstas_ind … H) -U1 // -HT1 /4 width=5 by ygt_yprs_trans, sstas_yprs/
 qed-.
 
-fact sstas_cprs_aux: ∀h,g,L0,T0.
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                     ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                     ∀T2. L1 ⊢ T1 ➡* T2 → ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
-                     ∃∃U2. ⦃h, L1⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #T2 #HT12 #U1 #H
+fact sstas_ltpr_cprs_aux: ∀h,g,L0,T0.
+                          (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                          (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+                          (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+                          ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
+                          ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 → ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+                          ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ⬌* U2.
+#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #L2 #HL12 #T2 #HT12 #U1 #H
 @(sstas_ind … H) -U1 [ /3 width=3/ ]
 #U1 #W1 #l1 #HTU1 #HUW1 * #U2 #HTU2 #HU12
-lapply (snv_cprs_aux … IH2 … HT1 … HT12) // #HT2
+lapply (snv_ltpr_cprs_aux … IH2 … HT1 … HT12) // #HT2
 elim (snv_sstas_fwd_correct … HTU2) // #W2 #l2 #HUW2
-elim (ssta_cpcs_aux … IH2 IH1 … HUW1 … HUW2 … HU12) -IH2 -IH1 -HUW1 -HU12
-[2,3: /3 width=7 by snv_sstas_aux, ygt_cprs_trans/
-|4,5: /3 width=3 by ygt_sstas_trans, ygt_cprs_trans/
+elim (ssta_ltpr_cpcs_aux … IH2 IH1 … HUW1 … HUW2 … HU12) -IH2 -IH1 -HUW1 -HU12 //
+[2: /4 width=8 by snv_sstas_aux, ygt_yprs_trans, ltpr_cprs_yprs/
+|3: /3 width=7 by snv_sstas_aux, ygt_yprs_trans, cprs_yprs/
+|4: /4 width=5 by ygt_yprs_trans, ltpr_cprs_yprs, sstas_yprs/
+|5: /3 width=4 by ygt_yprs_trans, cprs_yprs, sstas_yprs/
 ] -L0 -T0 -T1 -HT2 #H #HW12 destruct /3 width=4/
 qed-.
 
-fact dxprs_cprs_aux: ∀h,g,L0,T0.
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                     ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                     ∀T2. L1 ⊢ T1 ➡* T2 → ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
-                     ∃∃U2. ⦃h, L1⦄ ⊢ T2 •*➡*[g] U2 & L1 ⊢ U1 ➡* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #T2 #HT12 #U1 * #W1 #HTW1 #HWU1
-elim (sstas_cprs_aux … IH3 IH2 IH1 … H01 … HT12 … HTW1) // -L0 -T0 -T1 #W2 #HTW2 #HW12
-lapply (cpcs_cprs_conf … HWU1 … HW12) -W1 #H
+fact dxprs_ltpr_cprs_aux: ∀h,g,L0,T0.
+                          (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                          (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+                          (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+                          ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
+                          ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
+                          ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 →
+                          ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 & L2 ⊢ U1 ➡* U2.
+#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 * #W1 #HTW1 #HWU1 #L2 #HL12 #T2 #HT12
+elim (sstas_ltpr_cprs_aux … IH3 IH2 IH1 … H01 … HT12 … HTW1) // -L0 -T0 -T1 #W2 #HTW2 #HW12
+lapply (ltpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
+lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
 elim (cpcs_inv_cprs … H) -H /3 width=3/
 qed-.
-
-fact sstas_ltpr_aux: ∀h,g,L0,T0.
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                     ∀L1,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T⦄ → ⦃h, L1⦄ ⊩ T :[g] →
-                     ∀L2. L1 ➡ L2 → ∀U1. ⦃h, L1⦄ ⊢ T •*[g] U1 →
-                     ∃∃U2. ⦃h, L2⦄ ⊢ T •*[g] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L1 #T #H01 #HT #L2 #HL12 #U1 #H
-@(sstas_ind … H) -U1 [ /4 width=3/ ]
-#W1 #U1 #l1 #HTW1 #HWU1 * #W2 #HTW2 #HW12
-lapply (snv_sstas_aux … IH2 … HTW1) // -IH2 -HT #HW1
-
-(*
-elim (IH1 … HWU1 … HL12 W1) // -IH1 -HW1 -HWU1 -HL12
-[2: /2 width=3 by ygt_sstas_trans/ ] #U2 
-*)
 (*
 fact sstas_dxprs_aux: ∀h,g,L0,T0.
                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
@@ -164,4 +149,4 @@ fact sstas_dxprs_aux: ∀h,g,L0,T0.
                       ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ∀T2. ⦃h, L1⦄ ⊢ T1 •*➡*[g] T2 →
                       ∃∃U2. ⦃h, L1⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U1 ⬌* U2.
 #h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #HTU1 #T2 * #T #HT1T #HTT2
-*)
\ No newline at end of file
+*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr.ma
index 04f7dcb4c..5b4df6be5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr.ma
@@ -12,6 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/computation/dxprs_dxprs.ma".
 include "basic_2/dynamic/snv_cpcs.ma".
 
 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
@@ -54,11 +55,10 @@ fact snv_ltpr_tpr_aux: ∀h,g,L0,T0.
     lapply (IH1 … HV1 … HL12 … HV12) [ /2 width=1/ ] #HV2
     lapply (IH1 … HT1 … HL12 … HT12) [ /2 width=1/ ] #HT2
     elim (IH3 … HVW1 … HL12 … HV12) -HVW1 -HV12 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
-    lapply (cpcs_cprs_conf … HW10 … HW12) -W10 #HW12
-    elim (dxprs_cprs_aux … IH2 IH1 IH3 … T2 … HTU1) // [2: /3 width=1/ |3: /2 width=1/ ] -IH2 -IH1 -IH3 -HT1 -HT12 -HTU1 #U2 #HTU2 #HU12
-    @(snv_appl … HV2 HT2 HVW2)
-(*
-    lapply (IH1 … HT1 … HTU1) -IH1 // #H
-    elim (snv_inv_bind … H) -H #HW1 #HU1
-    elim (IH2 … HVW1 … HL12 … HV12 HV1) -IH2 -HVW1 -HV12 -HV1 // #W2 #HVW2 #HW12
-*)
+    elim (dxprs_ltpr_cprs_aux … IH2 IH1 IH3 … HTU1 … HL12 T2) // [2: /3 width=1/ |3: /2 width=1/ ] -IH2 -IH1 -IH3 -HT1 -HT12 -HTU1 #X #HTU2 #H
+    elim (cprs_inv_abst1 Abst W1 … H) -H #W20 #U2 #HW120 #_ #H destruct
+    lapply (ltpr_cprs_conf … HL12 … HW10) -L1 #HW10
+    lapply (cpcs_cprs_strap1 … HW10 … HW120) -W1 #HW120
+    lapply (cpcs_canc_sn … HW12 HW120) -W10 #HW20
+    elim (cpcs_inv_cprs … HW20) -HW20 #W0 #HW20 #HW200
+    lapply (dxprs_cprs_trans … (ⓛ{a}W0.U2) HTU2 ?) [ /2 width=1/ ] -HW200 /2 width=8/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta_ltpr.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta_ltpr.ma
index 5ee41883b..a82640426 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta_ltpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta_ltpr.ma
@@ -12,7 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/equivalence/cpcs_ltpr.ma".
 include "basic_2/equivalence/lsubse_ssta.ma".
 include "basic_2/dynamic/snv_cpcs.ma".
 
@@ -36,12 +35,12 @@ fact ssta_ltpr_tpr_aux: ∀h,g,L0,T0.
   [ #V1 #W1 #HLK1 #HVW1 #HWU1
     lapply (ldrop_mono … H … HLK1) -H #H destruct
     lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
-    elim (ltpr_ldrop_conf … HLK1 … HL12) #X #H #HLK2
+    elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
     elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
-    elim (IH1 … HVW1 … HK12 … HV12) -IH1 -HVW1 -HV12 // [2: /2 width=1/ ] -HV1 -HKV1 #W2 #HVW2 #HW12
-    lapply (ldrop_fwd_ldrop2 … HLK1) -V1 #H1
+    elim (IH1 … HVW1 … HK12 … HV12) -IH1 -HVW1 -HV12 // [2: /2 width=1/ ] -V1 #W2 #HVW2 #HW12
+    lapply (ldrop_fwd_ldrop2 … HLK2) #H2
     elim (lift_total W2 0 (i+1)) #U2 #HWU2
-    lapply (cpcs_lift … H1 … HWU1 … HWU2 HW12) -H1 -W1 /3 width=6/
+    lapply (cpcs_lift … H2 … HWU1 … HWU2 HW12) -H2 -W1 /3 width=6/
   | #V1 #W1 #l0 #HLK1 #HVW1 #HVU1 #H destruct
     lapply (ldrop_mono … H … HLK1) -H #H destruct
     lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
@@ -61,7 +60,7 @@ fact ssta_ltpr_tpr_aux: ∀h,g,L0,T0.
     lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 [ /2 width=1/ ] #HT02
     elim (ssta_ltpr_cpr_aux … HT1 … HTU1 (L2.ⓑ{I}V2) … T2) -HT1 -HTU1
     [2: /3 width=5 by cpr_intro, tps_tpss/ |3: /2 width=1/ |4: /3 width=1/ ] -IH1 -T0 -HL12 #U2 #HTU2 #HU12
-    lapply (cpcs_bind1 … HU12 … V2 … a) -HU12 [ /2 width=1/ ] -HV12 /3 width=3/
+    lapply (cpcs_bind2 … V1 … HU12 … a) -HU12 [ /2 width=1/ ] -HV12 /3 width=3/
   | #T2 #HT12 #HT2 #H1 #H2 destruct
     elim (IH1 … HTU1 (L2.ⓓV1) … T2) -IH1 -HTU1 // [2,3: /2 width=1/ ] -T1 -HL12 #U2 #HTU2 #HU12
     lapply (cpcs_bind1 … HU12 … V1 … true) // -HU12 #HU12
@@ -83,26 +82,25 @@ fact ssta_ltpr_tpr_aux: ∀h,g,L0,T0.
     elim (ssta_fwd_correct … HVW1) <minus_plus_m_m #X1 #HWX1
     elim (snv_fwd_ssta … HW) #V #l1 #HWV
     lapply (IH3 … HVW1) -IH3 // [ /2 width=1/ ] #HW1
-    elim (ssta_cpcs_aux … IH2 IH1 … HWX1 … HWV …) -IH2 -HWX1 //
-    [2: /2 width=1/ |3: /4 width=4 by fw_ygt, ysc_ssta, bi_step/ ] #H #_ destruct -X1
+    elim (ssta_ltpr_cpcs_aux … IH2 IH1 … HWX1 … HWV …) -IH2 -HWX1 //
+    [2: /2 width=1/ |3: /3 width=4 by ygt_strap1, fw_ygt, ypr_ssta/ ] #H #_ destruct -X1
     elim (IH1 … HVW1 … HL12 … HV12) -HVW1 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
     elim (IH1 … HWV … HL12 W) -HWV // -HW [2: /2 width=1/ ] #V0 #HWV0 #_
     elim (IH1 … HTU2 (L2.ⓛW) … HT20) -IH1 -HTU2 -HT20 // [2,3: /2 width=1/ ] -HT2 #U20 #HTU20 #HU20
+    lapply (ltpr_cpcs_conf … HL12 … HW1W) -L1 #HW1W
+    lapply (cpcs_canc_sn … HW12 HW1W) -W1 #HW2
     elim (lsubse_ssta_trans … HTU20 (L2.ⓓV2) ?) -HTU20
-    [ #U #HTU20 #HUU20 -HWV0 -W1 -W2
-      lapply (ltpr_cpcs_trans (L1.ⓓV2) … HUU20) -HUU20 [ /2 width=1/ ] -HL12 #HUU20
+    [ #U #HTU20 #HUU20 -HWV0 -W2
       @(ex2_intro … (ⓓ{b}V2.U)) [ /2 width=1/ ] -h -l -l1 -V -V0 -T2 -T20 -U0
-      @(cpcs_cprs_strap2 … (ⓓ{b}V2.U2)) [ /4 width=1/ ] -V1
+      @(cpcs_cprs_strap2 … (ⓓ{b}V2.U2)) [ /3 width=1/ ] -V1
       @(cpcs_canc_dx … (ⓓ{b}V2.U20)) /2 width=2/
-    | -b -l -V -V1 -T2 -T20 -U0 -U2 -U20 -HW1
-      lapply (cpcs_canc_sn … HW12 … HW1W) -HW1W -HW12 #HW2
-      lapply (ltpr_cpcs_conf … HL12 … HW2) -L1 /2 width=6/
+    | -b -l -V -V1 -T2 -T20 -U0 -U2 -U20 /2 width=6/
     ]
   | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH3 -IH2
     elim (ssta_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
     elim (snv_inv_bind … HT1) -HT1 #_ #HT0
     elim (IH1 … HTU0 (L2.ⓓW2) … HT02) -IH1 -HTU0 // [2,3: /2 width=1/ ] -T0 -HL12 #U2 #HTU2 #HU02
-    lapply (cpcs_bind1 … HU02  … W0 … b) -HU02 // #HU02
+    lapply (cpcs_bind2 … W0 … HU02 b) -HU02 [ /2 width=1/ ] #HU02
     lapply (cpcs_flat … V1 V0 … HU02 Appl) [ /2 width=1/ ] -HV10 -HU02 #HU02
     lapply (cpcs_cpr_strap1 … HU02 (ⓓ{b}W2.ⓐV2.U2) ?) -HU02 [ /3 width=3/ ] -V0 -HW02 /4 width=3/ 
   ]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation.ma b/matita/matita/contribs/lambdadelta/basic_2/notation.ma
index d2008a3f8..7b34b1e5b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation.ma
@@ -340,9 +340,13 @@ notation "hvbox( ⦃ term 46 L1 ⦄ ➡ ➡ break ⦃ term 46 L2 ⦄ )"
    non associative with precedence 45
    for @{ 'FocalizedPRedAlt $L1 $L2 }.
 
+notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ≽ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'BTPRed $h $g $L1 $T1 $L2 $T2 }.
+
 notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ≻ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
    non associative with precedence 45
-   for @{ 'BTSuccessor $h $g $L1 $T1 $L2 $T2 }.
+   for @{ 'BTPRedProper $h $g $L1 $T1 $L2 $T2 }.
 
 (* Computation **************************************************************)
 
@@ -406,9 +410,13 @@ notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ • * ⬊ * break [ ter
    non associative with precedence 45
    for @{ 'DecomposedXSN $h $g $L $T }.
 
+notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄  ≥ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'BTPRedStar $h $g $L1 $T1 $L2 $T2 }.
+
 notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ > break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
    non associative with precedence 45
-   for @{ 'BTGreaterThan $h $g $L1 $T1 $L2 $T2 }.
+   for @{ 'BTPRedStarProper $h $g $L1 $T1 $L2 $T2 }.
 
 (* Conversion ***************************************************************)
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/ypr.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ypr.ma
new file mode 100644
index 000000000..4b4122549
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ypr.ma
@@ -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 "basic_2/static/ssta.ma".
+include "basic_2/reducibility/ltpr.ma".
+include "basic_2/reducibility/cpr.ma".
+
+(* "BIG TREE" PARALLEL REDUCTION FOR CLOSURES *******************************)
+
+inductive ypr (h) (g) (L1) (T1): relation2 lenv term ≝
+| ypr_fw  : ∀L2,T2. ♯{L2, T2} < ♯{L1, T1} → ypr h g L1 T1 L2 T2
+| ypr_ltpr: ∀L2. L1 ➡ L2 → ypr h g L1 T1 L2 T1
+| ypr_cpr : ∀T2. L1 ⊢ T1 ➡ T2 → ypr h g L1 T1 L1 T2
+| ypr_ssta: ∀T2,l. ⦃h, L1⦄ ⊢ T1 •[g, l + 1] T2 → ypr h g L1 T1 L1 T2
+.
+
+interpretation
+   "'big tree' parallel reduction (closure)"
+   'BTPRed h g L1 T1 L2 T2 = (ypr h g L1 T1 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma ypr_refl: ∀h,g. bi_reflexive … (ypr h g).
+/2 width=1/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/ysc.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ysc.ma
index 8762efb02..e0aa31061 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reducibility/ysc.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/ysc.ma
@@ -12,10 +12,9 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/static/ssta.ma".
-include "basic_2/reducibility/cpr.ma".
+include "basic_2/reducibility/ypr.ma".
 
-(* "BIG TREE" SUCCESSOR ON CLOSURES *****************************************)
+(* "BIG TREE" PROPER PARALLEL REDUCTION FOR CLOSURES ************************)
 
 inductive ysc (h) (g) (L1) (T1): relation2 lenv term ≝
 | ysc_fw  : ∀L2,T2. ♯{L2, T2} < ♯{L1, T1} → ysc h g L1 T1 L2 T2
@@ -24,5 +23,20 @@ inductive ysc (h) (g) (L1) (T1): relation2 lenv term ≝
 .
 
 interpretation
-   "'big tree' successor (closure)"
-   'BTSuccessor h g L1 T1 L2 T2 = (ysc h g L1 T1 L2 T2).
+   "'big tree' proper parallel reduction (closure)"
+   'BTPRedProper h g L1 T1 L2 T2 = (ysc h g L1 T1 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma ysc_ypr: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ →
+               h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L2, T2⦄.
+#h #g #L1 #L2 #T1 #T2 * -L2 -T2 /2 width=1/ /2 width=2/
+qed.
+
+(* Inversion lemmas on "big tree" parallel reduction for closures ***********)
+
+lemma ypr_inv_ysc: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L2, T2⦄ →
+                   h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ ∨ (L1 ➡ L2 ∧ T1 = T2).
+#h #g #L1 #L2 #T1 #T2 * -L2 -T2 /3 width=1/ /3 width=2/
+#T2 #HT12 elim (term_eq_dec T1 T2) #H destruct /3 width=1/ /4 width=1/
+qed-.
-- 
2.39.2