From 514f515ecb8765c68720e880460c2457898d74dc Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Thu, 28 Feb 2013 18:14:27 +0000
Subject: [PATCH] - lambdadelta: first recursive part of preservation finally
 proved! - lambda: bugfix in include paths, now they are relative to the root
 - matita.lang: keyword "fact" added (used in lambdadelta)

---
 .../lambdadelta/basic_2/computation/cprs.ma   |   5 +-
 .../lambdadelta/basic_2/computation/fprs.ma   |   3 +
 .../lambdadelta/basic_2/computation/lfprs.ma  |   3 +
 .../lambdadelta/basic_2/computation/ltprs.ma  |   3 +
 .../lambdadelta/basic_2/computation/tprs.ma   |   3 +
 .../lambdadelta/basic_2/dynamic/snv_cpcs.ma   | 123 ++++++++++++
 .../lambdadelta/basic_2/dynamic/snv_cpr.ma    | 113 -----------
 .../basic_2/dynamic/snv_cpr_ssta.ma           | 190 ------------------
 .../lambdadelta/basic_2/dynamic/snv_dxprs.ma  |  69 -------
 .../lambdadelta/basic_2/dynamic/snv_ltpr.ma   |  62 ++++++
 .../basic_2/dynamic/snv_ltpr_ssta.ma          | 121 +++++++++++
 .../lambdadelta/basic_2/equivalence/cpcs.ma   |   3 +
 .../lambdadelta/basic_2/equivalence/fpcs.ma   |   3 +
 .../lambdadelta/basic_2/equivalence/lfpcs.ma  |   5 +-
 .../basic_2/equivalence/lfpcs_lfpcs.ma        |   4 +-
 .../lambdadelta/basic_2/unfold/ldrops.ma      |   3 +-
 .../lambdadelta/basic_2/unfold/tpss.ma        |   3 +
 .../matita/lib/lambda/background/preamble.ma  |   6 +-
 .../paths/alternative_standard_order.ma       |   2 +-
 .../lib/lambda/paths/decomposed_trace.ma      |   2 +-
 .../lib/lambda/paths/dst_computation.ma       |   4 +-
 .../paths/labeled_sequential_computation.ma   |   6 +-
 .../paths/labeled_sequential_reduction.ma     |   4 +-
 .../lambda/paths/labeled_st_computation.ma    |   6 +-
 .../lib/lambda/paths/labeled_st_reduction.ma  |   6 +-
 matita/matita/lib/lambda/paths/path.ma        |   2 +-
 .../matita/lib/lambda/paths/standard_order.ma |   2 +-
 .../lib/lambda/paths/standard_precedence.ma   |   2 +-
 .../matita/lib/lambda/paths/standard_trace.ma |   4 +-
 matita/matita/lib/lambda/paths/trace.ma       |   2 +-
 matita/matita/lib/lambda/subterms/boolean.ma  |   2 +-
 .../matita/lib/lambda/subterms/booleanized.ma |   2 +-
 matita/matita/lib/lambda/subterms/carrier.ma  |   4 +-
 .../subterms/relocating_substitution.ma       |   2 +-
 .../matita/lib/lambda/subterms/relocation.ma  |   2 +-
 matita/matita/lib/lambda/subterms/subterms.ma |   2 +-
 .../terms/labeled_sequential_computation.ma   |   2 +-
 .../matita/lib/lambda/terms/multiplicity.ma   |   2 +-
 .../lib/lambda/terms/parallel_computation.ma  |   2 +-
 .../lib/lambda/terms/parallel_reduction.ma    |   4 +-
 .../lambda/terms/relocating_substitution.ma   |   2 +-
 matita/matita/lib/lambda/terms/relocation.ma  |   2 +-
 .../lambda/terms/sequential_computation.ma    |   2 +-
 .../lib/lambda/terms/sequential_reduction.ma  |   2 +-
 matita/matita/lib/lambda/terms/size.ma        |   2 +-
 matita/matita/lib/lambda/terms/term.ma        |   2 +-
 matita/matita/matita.lang                     |   3 +-
 47 files changed, 382 insertions(+), 421 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
 delete mode 100644 matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpr.ma
 delete mode 100644 matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpr_ssta.ma
 delete mode 100644 matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_dxprs.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr_ssta.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma
index 40608cf5c..e20ce0559 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma
@@ -17,7 +17,6 @@ include "basic_2/computation/tprs.ma".
 
 (* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
 
-(* Basic_1: includes: pr3_pr2 *)
 definition cprs: lenv → relation term ≝
                  λL. TC … (cpr L).
 
@@ -42,6 +41,10 @@ qed-.
 
 (* Basic properties *********************************************************)
 
+(* Basic_1: was: pr3_pr2 *)
+lemma cpr_cprs: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ T1 ➡* T2.
+/2 width=1/ qed.
+
 (* Basic_1: was: pr3_refl *)
 lemma cprs_refl: ∀L,T. L ⊢ T ➡* T.
 /2 width=1/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fprs.ma
index 61c720754..27ea45c33 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fprs.ma
@@ -35,6 +35,9 @@ lemma fprs_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
 
 (* Basic properties *********************************************************)
 
+lemma fpr_fprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ➡* ⦃L2, T2⦄.
+/2 width=1/ qed.
+
 lemma fprs_refl: bi_reflexive … fprs.
 /2 width=1/ qed.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs.ma
index a193f3c0a..c27e7673b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lfprs.ma
@@ -40,6 +40,9 @@ qed-.
 
 (* Basic properties *********************************************************)
 
+lemma lfpr_lfprs: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ➡* ⦃L2⦄.
+/2 width=1/ qed.
+
 lemma lfprs_refl: ∀L. ⦃L⦄ ➡* ⦃L⦄.
 /2 width=1/ qed.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs.ma
index 34ad45039..8e0c32e26 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/ltprs.ma
@@ -44,6 +44,9 @@ qed-.
 lemma ltprs_refl: reflexive … ltprs.
 /2 width=1/ qed.
 
+lemma ltpr_ltprs: ∀L1,L2. L1 ➡ L2 → L1 ➡* L2.
+/2 width=1/ qed.
+
 lemma ltprs_strap1: ∀L1,L,L2. L1 ➡* L → L ➡ L2 → L1 ➡* L2.
 /2 width=3/ qed.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/tprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/tprs.ma
index b094e66c1..22f09eda0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/tprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/tprs.ma
@@ -43,6 +43,9 @@ qed-.
 lemma tprs_refl: reflexive … tprs.
 /2 width=1/ qed.
 
+lemma tpr_tprs: ∀T1,T2. T1 ➡ T2 → T2 ➡* T2.
+/2 width=1/ qed.
+
 lemma tprs_strap1: ∀T1,T,T2. T1 ➡* T → T ➡ T2 → T1 ➡* T2.
 /2 width=3/ 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
new file mode 100644
index 000000000..b1430bd59
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
@@ -0,0 +1,123 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_ssta.ma".
+include "basic_2/computation/ygt.ma".
+include "basic_2/equivalence/fpcs_cpcs.ma".
+include "basic_2/dynamic/snv_ltpss_dx.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Inductive premises for the preservation results **************************)
+(*
+definition IH_ssta_cprs: ∀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. L2 ⊢ T1 ➡* T2 →
+                         ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
+
+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].
+
+fact ssta_cpcs_aux: ∀h,g,L,T1,T2. IH_ssta_cprs h g L T1 → IH_ssta_cprs h 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 #L #T1 #T2 #IH1 #IH2 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #H
+elim (cpcs_inv_cprs … H) -H #T #H1 #H2
+elim (IH1 … HT1 … HTU1 … H1) -T1 // #W1 #H1 #HUW1
+elim (IH2 … HT2 … HTU2 … H2) -T2 // #W2 #H2 #HUW2
+elim (ssta_mono … H1 … H2) -T #H1 #H2 destruct
+lapply (fpcs_canc_dx … HUW1 … HUW2) -W2 #HU12
+lapply (fpcs_inv_cpcs … HU12) -HU12 /2 width=1/
+qed-.
+*)
+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⦄ ⬌* ⦃L2, 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_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⦄ ⬌* ⦃L2, U2⦄.
+#h #g #L1 #T1 #IH #HT1 #U1 #l #HTU1 #L2 #HL12 #T2 * #T #HT1T #HTT2
+elim (IH … HTU1 … HL12 … HT1T) // -HL12 -T1 #U #HTU #HU1
+elim (ssta_tpss_conf … HTU … HTT2) -T #U2 #HTU2 #HU2
+lapply (fpcs_fpr_strap1 … HU1 L2 U2 ?) -HU1 /2 width=3/ /3 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].
+#h #g #L1 #T1 #IH #HT1 #L2 #HL12 #T2 * #T #HT1T #HTT2
+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/
+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 
+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
+lapply (fpcs_inv_cpcs … HU2) -HU2 #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
+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_mono … H1 … H2) -h -T #H1 #H2 destruct
+lapply (cpcs_canc_dx … HUW1 … HUW2) -W2 /2 width=1/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpr.ma
deleted file mode 100644
index a4850ff61..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpr.ma
+++ /dev/null
@@ -1,113 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "basic_2/dynamic/snv_ltpss_sn.ma".
-include "basic_2/dynamic/snv_cpr_ssta.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on context-free parallel reduction for closures ***************)
-
-axiom snv_ltpr_tpr_aux: ∀h,g,n. (
-                          ∀L1,T1. ♯{L1, T1} < n →
-                          ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                          ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
-                          ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
-                       ) → (
-                          ∀L1,T1. ♯{L1, T1} < n → ⦃h, L1⦄ ⊩ T1 :[g] →
-                          ∀L2. L1 ➡ L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
-                       ) →
-                       ∀L1,T1. ♯{L1, T1} = n → ⦃h, L1⦄ ⊩ T1 :[g] →
-                       ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊩ T2 :[g].
-(*
-#h #g #n #IH2 #IH1 #L1 * * [||||*]
-[ #k #Hn #H1 #L2 #_ #X #H2 destruct -IH2 -IH1 -L1
-  >(tpr_inv_atom1 … H2) -X //
-| #i #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
-  elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
-  >(tpr_inv_atom1 … H2) -X
-  elim (ltpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
-  elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
-  lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) -HLK1 #HLK1
-  lapply (IH1 … HV1 … HK12 V2 ?) -IH1 -HV1 -HK12 //
-  [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
-  ] -HV12 /2 width=5/
-| #p #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
-  elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #Hn #H1 #L2 #HL12 #X #H2 destruct -IH2
-  elim (snv_inv_bind … H1) -H1 #HV1 #HT1
-  elim (tpr_inv_bind1 … H2) -H2 *
-  [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
-    lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
-    lapply (cpr_intro (L2.ⓑ{I}V2) … T2 0 1 HT10 ?) -HT10 /2 width=1/ -HT02 #HT12
-    lapply (IH1 … HV1 … HL12 V2 ?) -HV1 //
-    [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
-    ] #HV2
-    lapply (IH1 … HT1 (L2.ⓑ{I}V2) … T2 ?) -IH1 -HT1 /3 width=1/
-  | #T2 #HT12 #HXT2 #H1 #H2 destruct
-    lapply (IH1 … HT1 (L2.ⓓV1) … T2 ?) -IH1 -HT1 // /2 width=2/
-    [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
-    ] -HT12 -HL12 #HT2
-    lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
-  ]
-| #V1 #T1 #Hn #H1 #L2 #HL12 #X #H2 destruct
-  elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1
-  elim (tpr_inv_appl1 … H2) -H2 *
-  [ #V2 #T2 #HV12 #HT12 #H destruct
-    lapply (IH1 … HV1 … HL12 V2 ?) 
-    [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
-    | //
-    ] #HV2
-    lapply (IH1 … HT1 … HL12 T2 ?)
-    [ @cprs_dxprs /3 width=1/ (**) (* auto: /4 width=1/ fails *)
-    | //
-    ] #HT2
-    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
-    lapply (fpcs_canc_sn L1 L1 … W10 W1 … HW12) -HW12 /3 width=1/ -W10 #HW12
-    @(snv_appl … HV2 HT2 HVW2)
-*)
-
-fact snv_cpr_aux: ∀h,g,n. (
-                     ∀L1,T1. ♯{L1, T1} < n →
-                     ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                     ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
-                     ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
-                  ) → (
-                     ∀L1,T1. ♯{L1, T1} < n → ⦃h, L1⦄ ⊩ T1 :[g] →
-                     ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
-                  ) →
-                  ∀L1,T1. ♯{L1, T1} = n → ⦃h, L1⦄ ⊩ T1 :[g] →
-                  ∀T2. L1 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T2 :[g].
-#h #g #n #IH2 #IH1 #L1 #T1 #Hn #HT1 #T2 * #T0 #HT10 #HT02
-lapply (snv_ltpr_tpr_aux  … Hn HT1 … HT10) -Hn -HT1 -HT10 [ // | skip | /3 width=6/ | /3 width=5/ ] -n -T1 #HT0
-lapply (snv_tpss_conf … HT0 … HT02) -T0 //
-qed-.
-
-fact snv_lfpr_aux: ∀h,g,n. (
-                      ∀L1,T1. ♯{L1, T1} < n →
-                      ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                      ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
-                      ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
-                   ) → (
-                      ∀L1,T1. ♯{L1, T1} < n → ⦃h, L1⦄ ⊩ T1 :[g] →
-                      ∀L2. ⦃L1⦄ ➡ ⦃L2⦄  → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
-                   ) →
-                   ∀L1,T1. ♯{L1, T1} = n → ⦃h, L1⦄ ⊩ T1 :[g] →
-                   ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃h, L2⦄ ⊩ T1 :[g].
-#h #g #n #IH2 #IH1 #L1 #T1 #Hn #HT1 #L2 * #L #HL1 #HL2
-lapply (snv_ltpr_tpr_aux  … Hn HT1 … HL1 ??) -Hn -HT1 -HL1 [ // | skip | /3 width=6/ | /3 width=5/ ] -n -L1 #HT1
-lapply (snv_ltpss_sn_conf … HL2 … HT1) -L //
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpr_ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpr_ssta.ma
deleted file mode 100644
index 9aa991939..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpr_ssta.ma
+++ /dev/null
@@ -1,190 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "basic_2/static/ssta_ltpss_sn.ma".
-include "basic_2/computation/dxprs_lift.ma".
-include "basic_2/equivalence/lsubse_ssta.ma".
-include "basic_2/equivalence/fpcs_cpcs.ma".
-include "basic_2/equivalence/lfpcs_fpcs.ma".
-include "basic_2/dynamic/snv_ssta.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on context-sensitive parallel reduction for terms *************)
-
-fact ssta_ltpr_tpr_aux: ∀h,g,n. (
-                           ∀L,T2. ♯{L,T2} < n →
-                           ∀T1. L ⊢ T1 ⬌* 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 ⊢ U1 ⬌* U2 ∧ l1 = l2
-                        ) → (
-                           ∀L,T. ♯{L,T} < n → ⦃h, L⦄ ⊩ T :[g] →
-                           ∀U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U → ⦃h, L⦄ ⊩ U :[g]
-                        ) → (
-                           ∀L1,T1. ♯{L1,T1} < n →
-                           ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                           ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] → 
-                           ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
-                        ) →
-                        ∀L1,T1. ♯{L1,T1} = n →
-                        ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                        ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
-                        ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
-#h #g #n #IH3 #IH2 #IH1 #L1 * * [|||| *]
-[ #k #_ #Y #l #H1 #L2 #HL12 #X #H2 #_ -IH3 -IH2 -IH1
-  elim (ssta_inv_sort1 … H1) -H1 #Hkl #H destruct
-  >(tpr_inv_atom1 … H2) -X /4 width=6/
-| #i #Hn #U1 #l #H1 #L2 #HL12 #X #H2 #H3 destruct -IH3 -IH2
-  elim (ssta_inv_lref1 … H1) -H1 * #K1
-  >(tpr_inv_atom1 … H2) -X
-  elim (snv_inv_lref … H3) -H3 #I0 #K0 #V0 #H #HV1
-  [ #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_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
-    elim (IH1 … HVW1 K2 … HV12) -IH1 -HVW1 -HV12 // -HV1 -HKV1 #W2 #HVW2 #HW12
-    lapply (ldrop_fwd_ldrop2 … HLK1) -V1 #H1
-    lapply (ldrop_fwd_ldrop2 … HLK2) #H2
-    elim (lift_total W2 0 (i+1)) #U2 #HWU2
-    lapply (fpcs_lift … HW12 … H1 H2 … HWU1 … HWU2) -H1 -H2 -W1 [ /3 width=1/ ] /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
-    elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 #X #H #HLK2
-    elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
-    elim (IH1 … HVW1 K2 … HV12) -IH1 -HVW1 // -HV1 -HK12 -HKV1 #W2 #HVW2 #_ -W1
-    elim (lift_total V2 0 (i+1)) #U2 #HVU2
-    lapply (tpr_lift … HV12 … HVU1 … HVU2) -V1 /4 width=6/
-  ]
-| #p #Hn #U1 #l #H1 -IH3 -IH1
-  elim (ssta_inv_gref1 … H1)
-| #a #I #V1 #T1 #Hn #Y #l #H1 #L2 #HL12 #X #H2 #H3 destruct -IH3 -IH2
-  elim (ssta_inv_bind1 … H1) -H1 #U1 #HTU1 #H destruct
-  elim (snv_inv_bind … H3) -H3 #_ #HT1
-  elim (tpr_inv_bind1 … H2) -H2 *
-  [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
-    elim (IH1 … HTU1 (L2.ⓑ{I}V2) … HT10) -IH1 -HTU1 -HT10 // -T1 /3 width=1/ -HL12 #U0 #HTU0 #HU10
-    lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 [ /2 width=1/ ] #HT02
-    elim (ssta_tps_conf … HTU0 … HT02) -T0 #U2 #HTU2 #HU02
-    lapply (cpr_intro … U0 … HU02) -HU02 // #HU02
-    lapply (fpcs_fpr_strap1 … HU10 (L2.ⓑ{I}V2) U2 ?) [ /2 width=1/ ] -U0 #HU12
-    lapply (fpcs_fwd_shift … HU12 a) -HU12 /3 width=3/
-  | #T2 #HT12 #HT2 #H1 #H2 destruct
-    elim (IH1 … HTU1 (L2.ⓓV1) … HT12) -IH1 -HTU1 -HT12 // -T1 [2: /3 width=1/ ] -HL12 #U2 #HTU2 #HU12
-    lapply (fpcs_fwd_shift … HU12 true) -HU12 #HU12
-    elim (ssta_inv_lift1 … HTU2 … HT2) -T2 [3: /2 width=1/ |2: skip ] #U #HXU #HU2
-    lapply (fpcs_fpr_strap1 … HU12 L2 U ?) -HU12 [ /3 width=3/ ] -U2 /2 width=3/
-  ]
-| #V1 #T1 #Hn #Y #l #H1 #L2 #HL12 #X #H2 #H3 destruct
-  elim (ssta_inv_appl1 … H1) -H1 #U1 #HTU1 #H destruct
-  elim (snv_inv_appl … H3) -H3 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #HVW1 #HW10 #HTU10
-  elim (tpr_inv_appl1 … H2) -H2 *
-  [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2
-    elim (IH1 … HTU1 … HL12 … HT12 HT1) -IH1 -HTU1 -HL12 -HT12 -HT1 // /3 width=5/
-  | #b #V2 #W #T2 #T20 #HV12 #HT20 #H1 #H2 destruct
-    elim (snv_inv_bind … HT1) -HT1 #HW #HT2
-    elim (ssta_inv_bind1 … HTU1) -HTU1 #U2 #HTU2 #H destruct
-    elim (dxprs_inv_abst1 … HTU10) -HTU10 #W0 #U0 #HW0 #_ #H destruct
-    lapply (cprs_div … HW10 … HW0) -W0 #HW1
-    elim (ssta_fwd_correct … HVW1) <minus_plus_m_m #X1 #HWX1
-    elim (snv_ssta … HW) #V #l1 #HWV
-    elim (IH3 … HW1 … HW … HWX1 … HWV) -IH3 -HWX1 // [2: /2 width=5/ ] -IH2 #_ #H destruct -X1
-    elim (IH1 … HVW1 … HL12 … HV12) -HVW1 // -HV1 #W2 #HVW2 #HW12
-    elim (IH1 … HWV … HL12 W) -HWV // -HW #V0 #HWV0 #_
-    elim (IH1 … HTU2 (L2.ⓛW) … HT20 HT2) -IH1 -HTU2 -HT20 -HT2 // [2: /2 width=1/ ] #U20 #HTU20 #HU20
-    lapply (lfpr_inv_fpr L1 L2 … W) [ /2 width=1/ ] -HL12 #HL12
-    elim (lsubse_ssta_trans … HTU20 (L2.ⓓV2) ?) -HTU20
-    [ #U #HTU20 #HU20 -HWV0 -HL12 -W1 -W2
-      @(ex2_intro … (ⓓ{b}V2.U)) [ /2 width=1/ ] -h -l -l1 -V -V0 -T2 -T20 -U0
-      @(fpcs_fprs_strap2 ? L1 … (ⓓ{b}V2.U2)) [ /4 width=1/ ] -V1
-      /4 width=4 by fpcs_fwd_shift, fpcs_canc_dx, cpcs_fpcs, fpcs_fwd_abst13/
-    | -b -l -V -V1 -T2 -T20 -U0 -U2 -U20
-      /6 width=6 by lsubse_abbr, fpcs_inv_cpcs, fpcs_canc_sn, fpcs_fprs_strap1, cpcs_fpcs, bi_inj/
-    ]
-  | #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 HT0) -IH1 -HTU0 -HT02 -HT0 // -T0 [2: /2 width=1/ ] -HL12 -HW02 #U2 #HTU2 #HU02
-    lapply (fpcs_fwd_shift … HU02 b) -HU02 #HU02
-    lapply (fpcs_flat_dx_tpr … HU02 … HV10 Appl) -HV10 -HU02 #HU02
-    lapply (fpcs_fpr_strap1 … HU02 L2 (ⓓ{b}W2.ⓐV2.U2) ?) -HU02 [ @ltpr_tpr_fpr // /2 width=3/ ] -V0 /4 width=3/ 
-  ]
-| #U0 #T1 #Hn #U1 #l #H1 #L2 #HL12 #X #H2 #H3 destruct -IH3 -IH2
-  lapply (ssta_inv_cast1 … H1) -H1 #HTU1
-  elim (snv_inv_cast … H3) -H3 #T0 #l0 #_ #HT1 #HT10 #_
-  elim (ssta_mono … HT10 … HTU1) -HT10 #H1 #H2 destruct
-  elim (tpr_inv_cast1 … H2) -H2
-  [ * #U2 #T2 #_ #HT12 #H destruct
-    elim (IH1 … HTU1 … HL12 … HT12 HT1) -IH1 -HTU1 -HL12 -HT12 -HT1 // -T1 -U0 /3 width=3/
-  | #HT1X
-    elim (IH1 … HTU1 … HL12 … HT1X HT1) -IH1 -HTU1 -HL12 -HT1X -HT1 // -U0 -T1 /2 width=3/
-  ]
-]
-qed-.
-
-fact ssta_cpr_aux: ∀h,g,n. (
-                      ∀L,T2. ♯{L,T2} < n →
-                      ∀T1. L ⊢ T1 ⬌* 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 ⊢ U1 ⬌* U2 ∧ l1 = l2
-                   ) → (
-                      ∀L,T. ♯{L,T} < n → ⦃h, L⦄ ⊩ T :[g] →
-                      ∀U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U → ⦃h, L⦄ ⊩ U :[g]
-                   ) → (
-                      ∀L1,T1. ♯{L1,T1} < n →
-                      ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                      ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
-                      ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
-                   ) →
-                   ∀L1,T1. ♯{L1,T1} = n →
-                   ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                   ∀T2. L1 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
-                   ∃∃U2. ⦃h, L1⦄ ⊢ T2 •[g, l] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #n #IH3 #IH2 #IH1 #L1 #T1 #Hn #U1 #l #HTU1 #T2 * #T0 #HT10 #HT02 #HT1
-elim (ssta_ltpr_tpr_aux … IH3 IH2 … Hn … HTU1  … HT10 HT1)
--T1 -IH3 -IH2 [3: // |2: skip |4: /3 width=5/ ] -n #U0 #HTU0 #HU10
-lapply (fpcs_inv_cpcs … HU10) -HU10 #HU10
-elim (ssta_tpss_conf … HTU0 … HT02) -T0 #U2 #HTU2 #HU02
-lapply (cpcs_cpr_strap1 … HU10 U2 ?) -HU10 /2 width=3/
-qed-.
-
-fact ssta_lfpr_aux: ∀h,g,n. (
-                       ∀L,T2. ♯{L,T2} < n →
-                       ∀T1. L ⊢ T1 ⬌* 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 ⊢ U1 ⬌* U2 ∧ l1 = l2
-                    ) → (
-                       ∀L,T. ♯{L,T} < n → ⦃h, L⦄ ⊩ T :[g] →
-                       ∀U,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U → ⦃h, L⦄ ⊩ U :[g]
-                    ) → (
-                       ∀L1,T1. ♯{L1,T1} < n →
-                       ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                       ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
-                       ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
-                    ) →
-                    ∀L1,T1. ♯{L1,T1} = n →
-                    ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                    ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                    ∃∃U2. ⦃h, L2⦄ ⊢ T1 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄.
-#h #g #n #IH3 #IH2 #IH1 #L1 #T1 #Hn #U1 #l #HTU1 #L2 * #L #HL1 #HL2 #HT1
-elim (ssta_ltpr_tpr_aux … IH3 IH2 … Hn … HTU1 … HL1 … HT1) -HTU1 -HL1 -HT1
--IH3 -IH2 [3: // |2: skip |4: /3 width=5/ ] -n #U0 #HTU0 #HU10
-elim (ssta_ltpss_sn_conf … HTU0 … HL2) -HTU0 #U2 #HTU2 #HU02
-lapply (fpcs_fpr_strap1 … HU10 L U2 ?) -HU10 /3 width=3/ -HU02 #HU12
-lapply (fpcs_fpr_strap1 … HU12 L2 U2 ?) /2 width=3/ -h -L1 -T1 -U1 -U0 -l /3 width=4/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_dxprs.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_dxprs.ma
deleted file mode 100644
index 39525c45e..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_dxprs.ma
+++ /dev/null
@@ -1,69 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "basic_2/computation/ygt.ma".
-include "basic_2/dynamic/snv_cpr.ma".
-
-(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-
-(* Properties on context-free parallel computation for closures *************)
-
-fact ssta_cprs_aux: ∀h,g,L0,T0. (
-                       ∀L,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T2⦄ →
-                       ∀T1. L ⊢ T1 ⬌* 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 ⊢ U1 ⬌* U2 ∧ l1 = l2
-                    ) → (
-                       ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                       ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
-                    ) → (
-                       ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ →
-                       ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 → ⦃h, L1⦄ ⊩ T1 :[g] →
-                       ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. L2 ⊢ T1 ➡* T2 →
-                       ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
-                   ) →
-                   ∀L1,T1,T2. L1 ⊢ T1 ➡* T2 → L0 = L1 → T0 = T1 →
-                   ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 → ⦃h, L1⦄ ⊩ T1 :[g] →
-                   ∃∃U2. ⦃h, L1⦄ ⊢ T2 •[g, l] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #T2 #H
-@(cprs_ind_dx … H) -T1 [ /2 width=3/ ]
-#T1 #T #HT1T #HTT2 #IH #H1 #H2 #U1 #l #HTU1 #HT1 destruct
-elim (term_eq_dec T1 T) #H destruct [ /2 width=1/ ] -IH
-elim (ssta_cpr_aux … HTU1 … HT1T HT1) -HTU1
-[2: // |3: skip |4,5,6: /3 width=9 by inj, dxprs_strap2, fw_ygt/ ] -IH3 #U #HTU #HU1
-lapply (snv_cpr_aux … HT1 … HT1T) -HT1
-[ // | skip |3,4: /3 width=6 by inj, fw_ygt/ ] -IH2 #HT
-elim (IH1 … HTU HT … HTT2) -IH1 -HTU -HT -HTT2 // [2: /3 width=1/ ] -T #U2 #HTU2 #HU2
-lapply (fpcs_inv_cpcs … HU2) -HU2 #HU2
-lapply (cpcs_trans … HU1 … HU2) -U /2 width=3/
-qed-.
-
-fact snv_cprs_aux: ∀h,g,L0,T0. (
-                      ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ →
-                      ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
-                      ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. L2 ⊢ T1 ➡* T2 → ⦃h, L1⦄ ⊩ T1 :[g] →
-                      ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & ⦃L1, U1⦄ ⬌* ⦃L2, U2⦄
-                   ) → (
-                      ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
-                      ∀L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L2⦄ ⊩ T2 :[g]
-                   ) →
-                   ∀L1,T1. L0 = L1 → T0 = T1 → ⦃h, L1⦄ ⊩ T1 :[g] →
-                   ∀T2. L1 ⊢ T1 ➡* T2 → ⦃h, L1⦄ ⊩ T2 :[g].
-#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H1 #H2 #HT1 #T2 #H
-@(cprs_ind … H) -T2 // -HT1
-#T #T2 #HT1T #HTT2 #HT destruct
-lapply (snv_cpr_aux … HT … HTT2) -HT -HTT2 [1,5: // |2: skip ]
-/4 width=6 by cprs_ygt_trans, inj, fw_ygt/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr.ma
new file mode 100644
index 000000000..4bcfa9889
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr.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 "basic_2/dynamic/snv_ltpr_ssta.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-free parallel reduction for local environments *****)
+
+fact snv_ltpr_tpr_aux: ∀h,g,L0,T0.
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
+                       ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_ltpr_tpr h g L1 T1.
+#h #g #L0 #T0 #IH2 #IH1 #L1 * * [||||*]
+[ #k #HL0 #HT0 #H1 #L2 #_ #X #H2 destruct -IH2 -IH1 -L1
+  >(tpr_inv_atom1 … H2) -X //
+| #i #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH2
+  elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
+  >(tpr_inv_atom1 … H2) -X
+  elim (ltpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
+  elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+  lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) -HLK1 #HLK1
+  lapply (IH1 … HK12 … HV12) -IH1 -HV12 -HK12 // -HV1 [ /2 width=1/ ] -HLK1 /2 width=5/
+| #p #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH2 -IH1
+  elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH2
+  elim (snv_inv_bind … H1) -H1 #HV1 #HT1
+  elim (tpr_inv_bind1 … H2) -H2 *
+  [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
+    lapply (tps_lsubs_trans … HT02 (L2.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
+    lapply (IH1 … HL12 … HV12) // [ /2 width=1/ ] #HV2
+    lapply (snv_ltpr_cpr_aux … HT1  (L2.ⓑ{I}V2) … T2 ?) -HT1
+    [ /3 width=5 by cpr_intro, tps_tpss/ | /2 width=1/ | /3 width=1/ ] -IH1 -T0 /2 width=1/
+  | #T2 #HT12 #HXT2 #H1 #H2 destruct
+    lapply (IH1 … HT1 (L2.ⓓV1) … HT12) -IH1 -HT1 -HT12 [1,2: /2 width=1/ ] -HL12 #HT2
+    lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
+  ]
+| #V1 #T1 #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct
+  elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1
+  elim (tpr_inv_appl1 … H2) -H2 *
+  [ #V2 #T2 #HV12 #HT12 #H destruct
+    lapply (IH1 … HV1 … HL12 … HV12) [ /2 width=1/ ] #HV2
+    lapply (IH1 … HT1 … HL12 … HT12) [ /2 width=1/ ] #HT2
+    elim (IH2 … HVW1 … HL12 … HV12) -IH2 -HVW1 -HV12 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
+    lapply (fpcs_canc_sn L1 L1 … W10 W1 … HW12) -HW12 /3 width=1/ -W10 #HW12
+    @(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
+*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr_ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr_ssta.ma
new file mode 100644
index 000000000..cabeceb25
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr_ssta.ma
@@ -0,0 +1,121 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_ltpss_sn.ma".
+include "basic_2/equivalence/lsubse_ssta.ma".
+include "basic_2/equivalence/lfpcs_fpcs.ma".
+include "basic_2/dynamic/snv_ssta.ma".
+include "basic_2/dynamic/snv_cpcs.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Properties on context-free parallel reduction for local environments *****)
+
+fact ssta_ltpr_tpr_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. L0 = L1 → T0 = T1 → IH_ssta_ltpr_tpr h g L1 T1.
+#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 * * [|||| *]
+[ #k #_ #_ #_ #X2 #l #H2 #L2 #HL12 #X3 #H3 -IH3 -IH2 -IH1
+  elim (ssta_inv_sort1 … H2) -H2 #Hkl #H destruct
+  >(tpr_inv_atom1 … H3) -X3 /4 width=6/
+| #i #HL0 #HT0 #H1 #X2 #l #H2 #L2 #HL12 #X3 #H3 destruct -IH3 -IH2
+  elim (snv_inv_lref … H1) -H1 #I0 #K0 #V0 #H #HV1
+  elim (ssta_inv_lref1 … H2) -H2 * #K1
+  >(tpr_inv_atom1 … H3) -X3
+  [ #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_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
+    lapply (ldrop_fwd_ldrop2 … HLK2) #H2
+    elim (lift_total W2 0 (i+1)) #U2 #HWU2
+    lapply (fpcs_lift … HW12 … H1 H2 … HWU1 … HWU2) -H1 -H2 -W1 [ /3 width=1/ ] /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
+    elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 #X #H #HLK2
+    elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+    elim (IH1 … HVW1 … HK12 … HV12) -IH1 -HVW1 -HK12 // [2: /2 width=1/ ] -HV1 -HKV1 #W2 #HVW2 #_ -W1
+    elim (lift_total V2 0 (i+1)) #U2 #HVU2
+    lapply (tpr_lift … HV12 … HVU1 … HVU2) -V1 /4 width=6/
+  ]
+| #p #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
+  elim (snv_inv_gref … H1)
+| #a #I #V1 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #L2 #HL12 #X3 #H3 destruct -IH3 -IH2
+  elim (snv_inv_bind … H1) -H1 #_ #HT1
+  elim (ssta_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
+  elim (tpr_inv_bind1 … H3) -H3 *
+  [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
+    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 -HV12 #U2 #HTU2 #HU12
+    lapply (fpcs_fwd_shift … HU12 a) -HU12 /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 (fpcs_fwd_shift … HU12 true) -HU12 #HU12
+    elim (ssta_inv_lift1 … HTU2 … HT2) -T2 [3: /2 width=1/ |2: skip ] #U #HXU #HU2
+    lapply (fpcs_fpr_strap1 … HU12 L2 U ?) -HU12 [ /3 width=3/ ] -U2 /2 width=3/
+  ]
+| #V1 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #L2 #HL12 #X3 #H3 destruct
+  elim (snv_inv_appl … H1) -H1 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #HVW1 #HW10 #HTU10
+  elim (ssta_inv_appl1 … H2) -H2 #U1 #HTU1 #H destruct
+  elim (tpr_inv_appl1 … H3) -H3 *
+  [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2
+    elim (IH1 … HTU1 … HL12 … HT12) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -T1 /3 width=5/
+  | #b #V2 #W #T2 #T20 #HV12 #HT20 #H1 #H2 destruct
+    elim (snv_inv_bind … HT1) -HT1 #HW #HT2
+    elim (ssta_inv_bind1 … HTU1) -HTU1 #U2 #HTU2 #H destruct
+    elim (dxprs_inv_abst1 … HTU10) -HTU10 #W0 #U0 #HW0 #_ #H destruct
+    lapply (cprs_div … HW10 … HW0) -W0 #HW1W
+    elim (ssta_fwd_correct … HVW1) <minus_plus_m_m #X1 #HWX1
+    elim (snv_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 (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 (lfpr_inv_fpr L1 L2 … W) [ /2 width=1/ ] -HL12 #HL12
+    elim (lsubse_ssta_trans … HTU20 (L2.ⓓV2) ?) -HTU20
+    [ #U #HTU20 #HU20 -HWV0 -HL12 -W1 -W2
+      @(ex2_intro … (ⓓ{b}V2.U)) [ /2 width=1/ ] -h -l -l1 -V -V0 -T2 -T20 -U0
+      @(fpcs_fprs_strap2 ? L1 … (ⓓ{b}V2.U2)) [ /4 width=1/ ] -V1
+      /4 width=4 by fpcs_fwd_shift, fpcs_canc_dx, cpcs_fpcs, fpcs_fwd_abst13/
+    | -b -l -V -V1 -T2 -T20 -U0 -U2 -U20
+      /6 width=6 by lsubse_abbr, fpcs_inv_cpcs, fpcs_canc_sn, fpcs_fprs_strap1, cpcs_fpcs, bi_inj/
+    ]
+  | #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 -HW02 #U2 #HTU2 #HU02
+    lapply (fpcs_fwd_shift … HU02 b) -HU02 #HU02
+    lapply (fpcs_flat_dx_tpr … HU02 … HV10 Appl) -HV10 -HU02 #HU02
+    lapply (fpcs_fpr_strap1 … HU02 L2 (ⓓ{b}W2.ⓐV2.U2) ?) -HU02 [ @ltpr_tpr_fpr // /2 width=3/ ] -V0 /4 width=3/ 
+  ]
+| #U0 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #L2 #HL12 #X3 #H3 destruct -IH3 -IH2
+  elim (snv_inv_cast … H1) -H1 #T0 #l0 #_ #HT1 #HT10 #_
+  lapply (ssta_inv_cast1 … H2) -H2 #HTU1
+  elim (ssta_mono … HT10 … HTU1) -HT10 #H1 #H2 destruct
+  elim (tpr_inv_cast1 … H3) -H3
+  [ * #U2 #T2 #_ #HT12 #H destruct
+    elim (IH1 … HTU1 … HL12 … HT12) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -T1 -U0 /3 width=3/
+  | #HT1X3
+    elim (IH1 … HTU1 … HL12 … HT1X3) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -U0 -T1 /2 width=3/
+  ]
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs.ma
index a979b7bae..8b2259dac 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs.ma
@@ -47,6 +47,9 @@ lemma cpcs_refl: ∀L. reflexive … (cpcs L).
 lemma cpcs_sym: ∀L. symmetric … (cpcs L).
 /3 width=1/ qed.
 
+lemma cpc_cpcs: ∀L,T1,T2. L ⊢ T1 ⬌ T2 → L ⊢ T2 ⬌* T2.
+/2 width=1/ qed.
+
 lemma cpcs_strap1: ∀L,T1,T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → L ⊢ T1 ⬌* T2.
 /2 width=3/ qed.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs.ma
index c0e02359b..45be26781 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/fpcs.ma
@@ -41,6 +41,9 @@ lemma fpcs_refl: bi_reflexive … fpcs.
 lemma fpcs_sym: bi_symmetric … fpcs.
 /3 width=1/ qed.
 
+lemma fpc_fpcs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
+/2 width=1/ qed.
+
 lemma fpcs_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
 /2 width=4/ qed.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs.ma
index fded17bd1..5ad9180ac 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs.ma
@@ -41,9 +41,12 @@ qed-.
 lemma lfpcs_refl: reflexive … lfpcs.
 /2 width=1/ qed.
 
-lemma lfprs_sym: symmetric … lfpcs.
+lemma lfpcs_sym: symmetric … lfpcs.
 /3 width=1/ qed.
 
+lemma lfpc_lfpcs: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
+/2 width=1/ qed.
+
 lemma lfpcs_strap1: ∀L1,L,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L⦄ ⬌ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
 /2 width=3/ qed.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_lfpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_lfpcs.ma
index 434068e4f..1e9521736 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_lfpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs_lfpcs.ma
@@ -44,7 +44,7 @@ theorem lfpcs_trans: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L
 /2 width=3/ qed.
 
 theorem lfpcs_canc_sn: ∀L,L1,L2. ⦃L⦄ ⬌* ⦃L1⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/3 width=3 by lfpcs_trans, lfprs_sym/ qed.
+/3 width=3 by lfpcs_trans, lfpcs_sym/ qed.
 
 theorem lfpcs_canc_dx: ∀L,L1,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L2⦄ ⬌* ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
-/3 width=3 by lfpcs_trans, lfprs_sym/ qed.
+/3 width=3 by lfpcs_trans, lfpcs_sym/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops.ma
index b899bd273..9af21a0f1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/ldrops.ma
@@ -86,5 +86,4 @@ lemma ldrops_skip: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → ∀V1,V2. ⇧*[des] V2
 ].
 qed.
 
-(* Basic_1: removed theorems 1: drop1_getl_trans
-*)
+(* Basic_1: removed theorems 1: drop1_getl_trans *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss.ma
index cc5be0eca..faafb36d1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/tpss.ma
@@ -40,6 +40,9 @@ qed-.
 
 (* Basic properties *********************************************************)
 
+lemma tps_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
+/2 width=1/ qed.
+
 lemma tpss_strap1: ∀L,T1,T,T2,d,e.
                    L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2. 
 /2 width=3/ qed.
diff --git a/matita/matita/lib/lambda/background/preamble.ma b/matita/matita/lib/lambda/background/preamble.ma
index 6455bfcf4..a4400abce 100644
--- a/matita/matita/lib/lambda/background/preamble.ma
+++ b/matita/matita/lib/lambda/background/preamble.ma
@@ -16,9 +16,9 @@ include "basics/star.ma".
 include "basics/lists/lstar.ma".
 include "arithmetics/exp.ma".
 
-include "background/xoa_notation.ma".
-include "background/xoa.ma".
-include "background/notation.ma".
+include "lambda/background/xoa_notation.ma".
+include "lambda/background/xoa.ma".
+include "lambda/background/notation.ma".
 
 (* logic *)
 
diff --git a/matita/matita/lib/lambda/paths/alternative_standard_order.ma b/matita/matita/lib/lambda/paths/alternative_standard_order.ma
index e8eca4f11..523cba51b 100644
--- a/matita/matita/lib/lambda/paths/alternative_standard_order.ma
+++ b/matita/matita/lib/lambda/paths/alternative_standard_order.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "paths/standard_precedence.ma".
+include "lambda/paths/standard_precedence.ma".
 
 (* ALTERNATIVE STANDARD ORDER ***********************************************)
 
diff --git a/matita/matita/lib/lambda/paths/decomposed_trace.ma b/matita/matita/lib/lambda/paths/decomposed_trace.ma
index 2241fab10..9c391c1a9 100644
--- a/matita/matita/lib/lambda/paths/decomposed_trace.ma
+++ b/matita/matita/lib/lambda/paths/decomposed_trace.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "paths/trace.ma".
+include "lambda/paths/trace.ma".
 
 (* DECOMPOSED TRACE *********************************************************)
 
diff --git a/matita/matita/lib/lambda/paths/dst_computation.ma b/matita/matita/lib/lambda/paths/dst_computation.ma
index a532e7b15..85874e21c 100644
--- a/matita/matita/lib/lambda/paths/dst_computation.ma
+++ b/matita/matita/lib/lambda/paths/dst_computation.ma
@@ -12,8 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "paths/standard_trace.ma".
-include "paths/labeled_sequential_computation.ma".
+include "lambda/paths/standard_trace.ma".
+include "lambda/paths/labeled_sequential_computation.ma".
 
 (* DECOMPOSED STANDARD COMPUTATION ***********************************************)
 
diff --git a/matita/matita/lib/lambda/paths/labeled_sequential_computation.ma b/matita/matita/lib/lambda/paths/labeled_sequential_computation.ma
index 57a88baa8..db94f26ba 100644
--- a/matita/matita/lib/lambda/paths/labeled_sequential_computation.ma
+++ b/matita/matita/lib/lambda/paths/labeled_sequential_computation.ma
@@ -12,9 +12,9 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "terms/labeled_sequential_computation.ma".
-include "paths/trace.ma".
-include "paths/labeled_sequential_reduction.ma".
+include "lambda/terms/labeled_sequential_computation.ma".
+include "lambda/paths/trace.ma".
+include "lambda/paths/labeled_sequential_reduction.ma".
 
 (* PATH-LABELED SEQUENTIAL COMPUTATION (MULTISTEP) *******************************)
 
diff --git a/matita/matita/lib/lambda/paths/labeled_sequential_reduction.ma b/matita/matita/lib/lambda/paths/labeled_sequential_reduction.ma
index 8864a708e..8e9d693b6 100644
--- a/matita/matita/lib/lambda/paths/labeled_sequential_reduction.ma
+++ b/matita/matita/lib/lambda/paths/labeled_sequential_reduction.ma
@@ -12,8 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "paths/path.ma".
-include "terms/sequential_reduction.ma".
+include "lambda/paths/path.ma".
+include "lambda/terms/sequential_reduction.ma".
 
 (* PATH-LABELED SEQUENTIAL REDUCTION (SINGLE STEP) **************************)
 
diff --git a/matita/matita/lib/lambda/paths/labeled_st_computation.ma b/matita/matita/lib/lambda/paths/labeled_st_computation.ma
index 29dd72769..a9c19fbb1 100644
--- a/matita/matita/lib/lambda/paths/labeled_st_computation.ma
+++ b/matita/matita/lib/lambda/paths/labeled_st_computation.ma
@@ -12,9 +12,9 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "paths/standard_trace.ma".
-include "paths/labeled_sequential_computation.ma".
-include "paths/labeled_st_reduction.ma".
+include "lambda/paths/standard_trace.ma".
+include "lambda/paths/labeled_sequential_computation.ma".
+include "lambda/paths/labeled_st_reduction.ma".
 
 (* PATH-LABELED STANDARD COMPUTATION (MULTISTEP) ****************************)
 
diff --git a/matita/matita/lib/lambda/paths/labeled_st_reduction.ma b/matita/matita/lib/lambda/paths/labeled_st_reduction.ma
index 2f93b35af..7e8c1578c 100644
--- a/matita/matita/lib/lambda/paths/labeled_st_reduction.ma
+++ b/matita/matita/lib/lambda/paths/labeled_st_reduction.ma
@@ -12,9 +12,9 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "subterms/booleanized.ma".
-include "paths/labeled_sequential_reduction.ma".
-include "paths/standard_order.ma".
+include "lambda/subterms/booleanized.ma".
+include "lambda/paths/labeled_sequential_reduction.ma".
+include "lambda/paths/standard_order.ma".
 
 (* PATH-LABELED STANDARD REDUCTION ON SUBTERMS (SINGLE STEP) ****************)
 
diff --git a/matita/matita/lib/lambda/paths/path.ma b/matita/matita/lib/lambda/paths/path.ma
index 743ec4c34..2c2aaab70 100644
--- a/matita/matita/lib/lambda/paths/path.ma
+++ b/matita/matita/lib/lambda/paths/path.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "terms/term.ma".
+include "lambda/terms/term.ma".
 
 (* PATH *********************************************************************)
 
diff --git a/matita/matita/lib/lambda/paths/standard_order.ma b/matita/matita/lib/lambda/paths/standard_order.ma
index 75a3ca97f..c9af9bf52 100644
--- a/matita/matita/lib/lambda/paths/standard_order.ma
+++ b/matita/matita/lib/lambda/paths/standard_order.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "paths/standard_precedence.ma".
+include "lambda/paths/standard_precedence.ma".
 
 (* STANDARD ORDER ************************************************************)
 
diff --git a/matita/matita/lib/lambda/paths/standard_precedence.ma b/matita/matita/lib/lambda/paths/standard_precedence.ma
index a4ff9e9fb..cd61cb223 100644
--- a/matita/matita/lib/lambda/paths/standard_precedence.ma
+++ b/matita/matita/lib/lambda/paths/standard_precedence.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "paths/path.ma".
+include "lambda/paths/path.ma".
 
 (* STANDARD PRECEDENCE ******************************************************)
 
diff --git a/matita/matita/lib/lambda/paths/standard_trace.ma b/matita/matita/lib/lambda/paths/standard_trace.ma
index f9549d983..c76069686 100644
--- a/matita/matita/lib/lambda/paths/standard_trace.ma
+++ b/matita/matita/lib/lambda/paths/standard_trace.ma
@@ -12,8 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "paths/trace.ma".
-include "paths/standard_order.ma".
+include "lambda/paths/trace.ma".
+include "lambda/paths/standard_order.ma".
 
 (* STANDARD TRACE ***********************************************************)
 
diff --git a/matita/matita/lib/lambda/paths/trace.ma b/matita/matita/lib/lambda/paths/trace.ma
index fb042bdf6..108790c92 100644
--- a/matita/matita/lib/lambda/paths/trace.ma
+++ b/matita/matita/lib/lambda/paths/trace.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "paths/path.ma".
+include "lambda/paths/path.ma".
 
 (* TRACE ********************************************************************)
 
diff --git a/matita/matita/lib/lambda/subterms/boolean.ma b/matita/matita/lib/lambda/subterms/boolean.ma
index f49b23b29..66447a3bb 100644
--- a/matita/matita/lib/lambda/subterms/boolean.ma
+++ b/matita/matita/lib/lambda/subterms/boolean.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "subterms/carrier.ma".
+include "lambda/subterms/carrier.ma".
 
 (* BOOLEAN (EMPTY OR FULL) SUBSET *******************************************)
 
diff --git a/matita/matita/lib/lambda/subterms/booleanized.ma b/matita/matita/lib/lambda/subterms/booleanized.ma
index b3258d224..d247ee73f 100644
--- a/matita/matita/lib/lambda/subterms/booleanized.ma
+++ b/matita/matita/lib/lambda/subterms/booleanized.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "subterms/boolean.ma".
+include "lambda/subterms/boolean.ma".
 
 (* BOOLEANIZED SUBSET (EMPTY OR FULL) ***************************************)
 
diff --git a/matita/matita/lib/lambda/subterms/carrier.ma b/matita/matita/lib/lambda/subterms/carrier.ma
index 8936792e2..a15e11180 100644
--- a/matita/matita/lib/lambda/subterms/carrier.ma
+++ b/matita/matita/lib/lambda/subterms/carrier.ma
@@ -12,8 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "terms/relocating_substitution.ma".
-include "subterms/relocating_substitution.ma".
+include "lambda/terms/relocating_substitution.ma".
+include "lambda/subterms/relocating_substitution.ma".
 
 (* CARRIER (UNDERLYING TERM) ************************************************)
 
diff --git a/matita/matita/lib/lambda/subterms/relocating_substitution.ma b/matita/matita/lib/lambda/subterms/relocating_substitution.ma
index af5efe023..6d7fad314 100644
--- a/matita/matita/lib/lambda/subterms/relocating_substitution.ma
+++ b/matita/matita/lib/lambda/subterms/relocating_substitution.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "subterms/relocation.ma".
+include "lambda/subterms/relocation.ma".
 
 (* RELOCATING SUBSTITUTION **************************************************)
 
diff --git a/matita/matita/lib/lambda/subterms/relocation.ma b/matita/matita/lib/lambda/subterms/relocation.ma
index 0943d3fc2..99c8ded5a 100644
--- a/matita/matita/lib/lambda/subterms/relocation.ma
+++ b/matita/matita/lib/lambda/subterms/relocation.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "subterms/subterms.ma".
+include "lambda/subterms/subterms.ma".
 
 (* RELOCATION FOR SUBTERMS **************************************************)
 
diff --git a/matita/matita/lib/lambda/subterms/subterms.ma b/matita/matita/lib/lambda/subterms/subterms.ma
index 6c75a3f4c..c4ec383b5 100644
--- a/matita/matita/lib/lambda/subterms/subterms.ma
+++ b/matita/matita/lib/lambda/subterms/subterms.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "background/preamble.ma".
+include "lambda/background/preamble.ma".
 
 (* SUBSETS OF SUBTERMS ******************************************************)
 
diff --git a/matita/matita/lib/lambda/terms/labeled_sequential_computation.ma b/matita/matita/lib/lambda/terms/labeled_sequential_computation.ma
index 2287c6b81..03874d59d 100644
--- a/matita/matita/lib/lambda/terms/labeled_sequential_computation.ma
+++ b/matita/matita/lib/lambda/terms/labeled_sequential_computation.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "terms/sequential_computation.ma".
+include "lambda/terms/sequential_computation.ma".
 
 (* ABSTRACT LABELED SEQUENTIAL COMPUTATION (MULTISTEP) **********************)
 
diff --git a/matita/matita/lib/lambda/terms/multiplicity.ma b/matita/matita/lib/lambda/terms/multiplicity.ma
index 4f41b75db..737ce634b 100644
--- a/matita/matita/lib/lambda/terms/multiplicity.ma
+++ b/matita/matita/lib/lambda/terms/multiplicity.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "terms/relocating_substitution.ma".
+include "lambda/terms/relocating_substitution.ma".
 
 (* MULTIPLICITY *************************************************************)
 
diff --git a/matita/matita/lib/lambda/terms/parallel_computation.ma b/matita/matita/lib/lambda/terms/parallel_computation.ma
index 3b048d133..cd27b3c30 100644
--- a/matita/matita/lib/lambda/terms/parallel_computation.ma
+++ b/matita/matita/lib/lambda/terms/parallel_computation.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "terms/parallel_reduction.ma".
+include "lambda/terms/parallel_reduction.ma".
 
 (* PARALLEL COMPUTATION (MULTISTEP) *****************************************)
 
diff --git a/matita/matita/lib/lambda/terms/parallel_reduction.ma b/matita/matita/lib/lambda/terms/parallel_reduction.ma
index b54c4bc57..111d794b0 100644
--- a/matita/matita/lib/lambda/terms/parallel_reduction.ma
+++ b/matita/matita/lib/lambda/terms/parallel_reduction.ma
@@ -12,8 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "terms/size.ma".
-include "terms/sequential_reduction.ma".
+include "lambda/terms/size.ma".
+include "lambda/terms/sequential_reduction.ma".
 
 (* PARALLEL REDUCTION (SINGLE STEP) *****************************************)
 
diff --git a/matita/matita/lib/lambda/terms/relocating_substitution.ma b/matita/matita/lib/lambda/terms/relocating_substitution.ma
index 4f813396f..826c458d7 100644
--- a/matita/matita/lib/lambda/terms/relocating_substitution.ma
+++ b/matita/matita/lib/lambda/terms/relocating_substitution.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "terms/relocation.ma".
+include "lambda/terms/relocation.ma".
 
 (* RELOCATING SUBSTITUTION **************************************************)
 
diff --git a/matita/matita/lib/lambda/terms/relocation.ma b/matita/matita/lib/lambda/terms/relocation.ma
index 15c7fde8c..3d42e8a5c 100644
--- a/matita/matita/lib/lambda/terms/relocation.ma
+++ b/matita/matita/lib/lambda/terms/relocation.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "terms/term.ma".
+include "lambda/terms/term.ma".
 
 (* RELOCATION ***************************************************************)
 
diff --git a/matita/matita/lib/lambda/terms/sequential_computation.ma b/matita/matita/lib/lambda/terms/sequential_computation.ma
index b248e6027..717a7602a 100644
--- a/matita/matita/lib/lambda/terms/sequential_computation.ma
+++ b/matita/matita/lib/lambda/terms/sequential_computation.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "terms/parallel_computation.ma".
+include "lambda/terms/parallel_computation.ma".
 
 (* SEQUENTIAL COMPUTATION (MULTISTEP) ***************************************)
 
diff --git a/matita/matita/lib/lambda/terms/sequential_reduction.ma b/matita/matita/lib/lambda/terms/sequential_reduction.ma
index becaaac78..f1a6bfaf8 100644
--- a/matita/matita/lib/lambda/terms/sequential_reduction.ma
+++ b/matita/matita/lib/lambda/terms/sequential_reduction.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "terms/multiplicity.ma".
+include "lambda/terms/multiplicity.ma".
 
 (* SEQUENTIAL REDUCTION (SINGLE STEP) ***************************************)
 
diff --git a/matita/matita/lib/lambda/terms/size.ma b/matita/matita/lib/lambda/terms/size.ma
index 1fd9a060e..3396dc415 100644
--- a/matita/matita/lib/lambda/terms/size.ma
+++ b/matita/matita/lib/lambda/terms/size.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "terms/relocation.ma".
+include "lambda/terms/relocation.ma".
 
 (* SIZE *********************************************************************)
 
diff --git a/matita/matita/lib/lambda/terms/term.ma b/matita/matita/lib/lambda/terms/term.ma
index c831236a3..9db0fffa8 100644
--- a/matita/matita/lib/lambda/terms/term.ma
+++ b/matita/matita/lib/lambda/terms/term.ma
@@ -14,7 +14,7 @@
 
 (* Initial invocation: - Patience on us to gain peace and perfection! - *)
 
-include "background/preamble.ma".
+include "lambda/background/preamble.ma".
 
 (* TERM STRUCTURE ***********************************************************)
 
diff --git a/matita/matita/matita.lang b/matita/matita/matita.lang
index be110b9ba..29831639b 100644
--- a/matita/matita/matita.lang
+++ b/matita/matita/matita.lang
@@ -82,7 +82,8 @@
          <keyword>inductive</keyword>
          <keyword>coinductive</keyword>
          <keyword>let</keyword>
-         <keyword>lemma</keyword>
+         <keyword>fact</keyword>
+	 <keyword>lemma</keyword>
          <keyword>remark</keyword>
          <keyword>axiom</keyword>
 
-- 
2.39.2