From 583c59b229ba770c9694c703b381542ff2e67f4e Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sun, 3 Mar 2013 18:52:14 +0000
Subject: [PATCH] - lambdadelta: we removed focalized reduction from snv
 preservation - matitadep: bug fix in loop detection

---
 .../binaries/matitadep/matitadep.ml           | 29 +++---
 .../basic_2/computation/cprs_cprs.ma          | 27 ++++-
 .../basic_2/computation/dxprs_dxprs.ma        |  2 +-
 .../lambdadelta/basic_2/dynamic/snv.ma        | 15 +++
 .../lambdadelta/basic_2/dynamic/snv_aaa.ma    |  8 +-
 .../lambdadelta/basic_2/dynamic/snv_cpcs.ma   | 98 ++++++++++++++-----
 .../lambdadelta/basic_2/dynamic/snv_ltpr.ma   | 18 ++--
 .../lambdadelta/basic_2/dynamic/snv_ssta.ma   | 31 ++----
 .../{snv_ltpr_ssta.ma => snv_ssta_ltpr.ma}    | 41 ++++----
 .../lambdadelta/basic_2/dynamic/snv_sstas.ma  | 26 +++++
 .../basic_2/equivalence/cpcs_cpcs.ma          | 29 ++++--
 .../basic_2/reducibility/cpr_lift.ma          | 10 +-
 .../lambdadelta/basic_2/unwind/sstas.ma       |  3 +
 .../lambdadelta/basic_2/unwind/sstas_lift.ma  | 10 ++
 14 files changed, 237 insertions(+), 110 deletions(-)
 rename matita/matita/contribs/lambdadelta/basic_2/dynamic/{snv_ltpr_ssta.ma => snv_ssta_ltpr.ma} (82%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_sstas.ma

diff --git a/matita/components/binaries/matitadep/matitadep.ml b/matita/components/binaries/matitadep/matitadep.ml
index fdf5e980f..da82b9fc1 100644
--- a/matita/components/binaries/matitadep/matitadep.ml
+++ b/matita/components/binaries/matitadep/matitadep.ml
@@ -7,6 +7,10 @@ type file = {
 
 let graph = Hashtbl.create 503
 
+let rec purge dname bdeps = match bdeps with
+   | []       -> bdeps
+   | hd :: tl -> if hd = dname then bdeps else purge dname tl 
+
 let add fname =
    if Hashtbl.mem graph fname then () else
    Hashtbl.add graph fname {ddeps = []; rdeps = None}
@@ -18,25 +22,28 @@ let add_ddep fname dname =
 let init fname dname =
    add fname; add dname; add_ddep fname dname 
 
-let rec compute fname file = match file.rdeps with
+let rec compute bdeps fname file = match file.rdeps with
    | Some rdeps -> rdeps
    | None       ->
-      let rdeps = List.fold_left (iter fname) StringSet.empty file.ddeps in
+      let bdeps = fname :: bdeps in
+      let rdeps = List.fold_left (iter fname bdeps) StringSet.empty file.ddeps in
       Hashtbl.replace graph fname {file with rdeps = Some rdeps};
       rdeps
 
-and iter fname rdeps dname =
-   if StringSet.mem dname rdeps then
-      begin Printf.printf "%s: redundant %s\n" fname dname; rdeps end
-   else
+and iter fname bdeps rdeps dname =
+   if StringSet.mem dname rdeps then begin
+      Printf.printf "%s: redundant %s\n" fname dname;
+      rdeps 
+   end else if List.mem dname bdeps then begin
+      let loop = purge dname (List.rev bdeps) in
+      Printf.printf "circular: %s\n" (String.concat " " loop);
+      StringSet.add dname rdeps
+   end else
      let file = Hashtbl.find graph dname in
-     StringSet.add dname (StringSet.union (compute dname file) rdeps)
+     StringSet.add dname (StringSet.union (compute bdeps dname file) rdeps)
 
 let check () = 
-   let iter fname file = 
-      if StringSet.mem fname (compute fname file) then
-         Printf.printf "%s: circular\n" fname 
-   in
+   let iter fname file = ignore (compute [] fname file) in
    Hashtbl.iter iter graph 
 
 let rec read ich = 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma
index 8f94b0fa7..3326b8b46 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma
@@ -21,8 +21,8 @@ include "basic_2/computation/cprs_lfpr.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma cprs_abst_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2.
-                    L.ⓛV ⊢ T1 ➡* T2 → ∀a. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2.
+lemma cprs_abst_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡* T2 →
+                    ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
 #L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a @(cprs_ind … HT12) -T2
 [ /3 width=2/
 | /3 width=6 by cprs_strap1, cpr_abst/ (**) (* /3 width=6/ is too slow *)
@@ -96,9 +96,9 @@ lemma cprs_flat: ∀I,L,T1,T2. L ⊢ T1 ➡* T2 → ∀V1,V2. L ⊢ V1 ➡* V2 
 @(cprs_trans … IHV1) -IHV1 /2 width=1/
 qed.
 
-lemma cprs_abst: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀V,T1,T2.
-                 L.ⓛV ⊢ T1 ➡* T2 → ∀a. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2.
-#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a @(cprs_ind … HV12) -V2
+lemma cprs_abst: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡* T2 →
+                 ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
+#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a #I @(cprs_ind … HV12) -V2
 [ lapply (cprs_lsubs_trans … HT12 (L.ⓛV1) ?) -HT12 /2 width=2/
 | #V0 #V2 #_ #HV02 #IHV01
   @(cprs_trans … IHV01) -V1 /2 width=2/
@@ -112,6 +112,11 @@ lemma cprs_abbr1: ∀L,V1,T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → ∀V2. L ⊢ V1 ➡
 @(cprs_trans … IHV1) -IHV1 /2 width=1/
 qed.
 
+lemma cprs_bind1: ∀I,L,V1,T1,T2. L. ⓑ{I}V1 ⊢ T1 ➡* T2 → ∀V2. L ⊢ V1 ➡* V2 →
+                  ∀a. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
+* /2 width=1/ /2 width=2/
+qed. 
+
 lemma cprs_abbr2_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡* T2 →
                      ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2.
 #L #V1 #V2 #HV12 #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1
@@ -130,6 +135,11 @@ lapply (IHV1 T1 T1 ? a) -IHV1 // #HV1
 @(cprs_trans … HV1) -HV1 /2 width=1/
 qed.
 
+lemma cprs_bind2: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀I,T1,T2. L. ⓑ{I}V2 ⊢ T1 ➡* T2 → 
+                  ∀a. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
+#L #V1 #V2 #HV12 * /2 width=1/ /2 width=2/
+qed. 
+
 lemma cprs_beta_dx: ∀L,V1,V2,W,T1,T2.
                     L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡* T2 →
                     ∀a.L ⊢ ⓐV1.ⓛ{a}W.T1 ➡* ⓓ{a}V2.T2.
@@ -141,6 +151,13 @@ lemma cprs_beta_dx: ∀L,V1,V2,W,T1,T2.
 ]
 qed.
 
+lemma ltpr_cprs_trans: ∀L1,L2. L1 ➡ L2 →
+                       ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
+#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT2
+@(cprs_trans … IHT2) /2 width=3/
+qed.
+
 (* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
 lemma lcpr_cprs_trans: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ →
                        ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/dxprs_dxprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/dxprs_dxprs.ma
index 772cbd94a..c0aff4efb 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/dxprs_dxprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/dxprs_dxprs.ma
@@ -21,7 +21,7 @@ include "basic_2/computation/dxprs.ma".
 (* Advanced properties ******************************************************)
 
 lemma dxprs_cprs_trans: ∀h,g,L,T1,T,T2.
-                   ⦃h, L⦄ ⊢ T1 •*➡*[g] T → L ⊢ T ➡* T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
+                        ⦃h, L⦄ ⊢ T1 •*➡*[g] T → L ⊢ T ➡* T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
 #h #g #L #T1 #T #T2 * #T0 #HT10 #HT0 #HT2
 lapply (cprs_trans … HT0 … HT2) -T /2 width=3/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma
index e39a5764a..982d7edaf 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma
@@ -111,3 +111,18 @@ lemma snv_inv_cast: ∀h,g,L,W,T. ⦃h, L⦄ ⊩ ⓝW.T :[g] →
                     ∃∃U,l. ⦃h, L⦄ ⊩ W :[g] & ⦃h, L⦄ ⊩ T :[g] &
                            ⦃h, L⦄ ⊢ T •[g, l + 1] U & L ⊢ U ⬌* W.
 /2 width=3/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma snv_fwd_ssta: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → ∃∃U,l. ⦃h, L⦄ ⊢ T •[g, l] U.
+#h #g #L #T #H elim H -L -T
+[ #L #k elim (deg_total h g k) /3 width=3/
+| * #L #K #V #i #HLK #_ * #W #l0 #HVW
+  [ elim (lift_total W 0 (i+1)) /3 width=8/
+  | elim (lift_total V 0 (i+1)) /3 width=8/
+  ]
+| #a #I #L #V #T #_ #_ #_ * /3 width=3/
+| #a #L #V #W #W1 #T0 #T1 #l #_ #_ #_ #_ #_ #_ * /3 width=3/
+| #L #W #T #U #l #_ #_ #HTU #_ #_ #_ /3 width=3/ (**) (* auto fails without the last #_ *) 
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma
index 010a1ad7d..f492c019b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma
@@ -19,9 +19,9 @@ include "basic_2/dynamic/snv.ma".
 
 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
 
-(* Properties on atomic arity assignment for terms **************************)
+(* Forward lemmas on atomic arity assignment for terms **********************)
 
-lemma snv_aaa: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → ∃A. L ⊢ T ⁝ A.
+lemma snv_fwd_aaa: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → ∃A. L ⊢ T ⁝ A.
 #h #g #L #T #H elim H -L -T
 [ /2 width=2/
 | #I #L #K #V #i #HLK #_ * /3 width=6/
@@ -36,6 +36,6 @@ lemma snv_aaa: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → ∃A. L ⊢ T ⁝ A.
 ]
 qed-.
 
-lemma snv_csn: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → L ⊢ ⬊* T.
-#h #g #L #T #H elim (snv_aaa … H) -H /2 width=2/
+lemma snv_fwd_csn: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → L ⊢ ⬊* T.
+#h #g #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2/
 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 b1430bd59..7078c9f3e 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 +14,22 @@
 
 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".
+include "basic_2/dynamic/snv_sstas.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⦄.
+                             ∃∃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] →
@@ -64,11 +43,12 @@ 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⦄.
+                        ∃∃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) // -HL12 -T1 #U #HTU #HU1
+elim (IH … HTU1 … HL12 … HT1T) // -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/
+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 →
@@ -101,7 +81,6 @@ 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-.
 
@@ -121,3 +100,68 @@ elim (ssta_cprs_aux … HLT02 HT2 … HTU2 … H2) -T2 /2 width=1/ #W2 #H2 #HUW2
 elim (ssta_mono … H1 … H2) -h -T #H1 #H2 destruct
 lapply (cpcs_canc_dx … HUW1 … HUW2) -W2 /2 width=1/
 qed-.
+
+fact snv_sstas_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⦄ → ⦃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/
+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
+@(sstas_ind … H) -U1 [ /3 width=3/ ]
+#U1 #W1 #l1 #HTU1 #HUW1 * #U2 #HTU2 #HU12
+lapply (snv_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/
+] -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
+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) →
+                      (∀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 → ∀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 4bcfa9889..04f7dcb4c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/dynamic/snv_ltpr_ssta.ma".
+include "basic_2/dynamic/snv_cpcs.ma".
 
 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
 
@@ -20,21 +20,22 @@ include "basic_2/dynamic/snv_ltpr_ssta.ma".
 
 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_ssta 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
+#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 * * [||||*]
+[ #k #HL0 #HT0 #H1 #L2 #_ #X #H2 destruct -IH3 -IH2 -IH1 -L1
   >(tpr_inv_atom1 … H2) -X //
-| #i #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH2
+| #i #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH3 -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
+| #p #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH3 -IH2 -IH1
   elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH2
+| #a #I #V1 #T1 #HL0 #HT0 #H1 #L2 #HL12 #X #H2 destruct -IH3 -IH2
   elim (snv_inv_bind … H1) -H1 #HV1 #HT1
   elim (tpr_inv_bind1 … H2) -H2 *
   [ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct
@@ -52,8 +53,9 @@ fact snv_ltpr_tpr_aux: ∀h,g,L0,T0.
   [ #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
+    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
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta.ma
index ea04940d8..2bee82488 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta.ma
@@ -12,33 +12,20 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/dynamic/snv.ma".
+include "basic_2/dynamic/snv_cpcs.ma".
 
 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
 
 (* Properties on stratified static type assignment for terms ****************)
 
-lemma snv_ssta: ∀h,g,L,T. ⦃h, L⦄ ⊩ T :[g] → ∃∃U,l. ⦃h, L⦄ ⊢ T •[g, l] U.
-#h #g #L #T #H elim H -L -T
-[ #L #k elim (deg_total h g k) /3 width=3/
-| * #L #K #V #i #HLK #_ * #W #l0 #HVW
-  [ elim (lift_total W 0 (i+1)) /3 width=8/
-  | elim (lift_total V 0 (i+1)) /3 width=8/
-  ]
-| #a #I #L #V #T #_ #_ #_ * /3 width=3/
-| #a #L #V #W #W1 #T0 #T1 #l #_ #_ #_ #_ #_ #_ * /3 width=3/
-| #L #W #T #U #l #_ #_ #HTU #_ #_ #_ /3 width=3/ (**) (* auto fails without the last #_ *) 
-]
-qed-.
-
-fact snv_ssta_conf_aux: ∀h,g,L,T. (
-                           ∀L0,T0. ⦃h, L0⦄ ⊩ T0 :[g] →
-                           ∀U0,l. ⦃h, L0⦄ ⊢ T0 •[g, l + 1] U0 →
-                           ♯{L0, T0} < ♯{L, T} → ⦃h, L0⦄ ⊩ U0 :[g]
-                        ) →
-                        ∀L0,T0. ⦃h, L0⦄ ⊩ T0 :[g] →
-                        ∀U0,l. ⦃h, L0⦄ ⊢ T0 •[g, l + 1] U0 →
-                        L0 = L → T0 = T → ⦃h, L0⦄ ⊩ U0 :[g].
+fact snv_ssta_aux: ∀h,g,L,T. (
+                      ∀L0,T0. ⦃h, L0⦄ ⊩ T0 :[g] →
+                      ∀U0,l. ⦃h, L0⦄ ⊢ T0 •[g, l + 1] U0 →
+                      ♯{L0, T0} < ♯{L, T} → ⦃h, L0⦄ ⊩ U0 :[g]
+                   ) →
+                   ∀L0,T0. ⦃h, L0⦄ ⊩ T0 :[g] →
+                   ∀U0,l. ⦃h, L0⦄ ⊢ T0 •[g, l + 1] U0 →
+                   L0 = L → T0 = T → ⦃h, L0⦄ ⊩ U0 :[g].
 #h #g #L #T #IH1 #L0 #T0 * -L0 -T0
 [
 |
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr_ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta_ltpr.ma
similarity index 82%
rename from matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr_ssta.ma
rename to matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta_ltpr.ma
index cabeceb25..5ee41883b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ltpr_ssta.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta_ltpr.ma
@@ -12,10 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/static/ssta_ltpss_sn.ma".
+include "basic_2/equivalence/cpcs_ltpr.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 *************************************)
@@ -42,9 +40,8 @@ fact ssta_ltpr_tpr_aux: ∀h,g,L0,T0.
     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/
+    lapply (cpcs_lift … H1 … HWU1 … HWU2 HW12) -H1 -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
@@ -63,49 +60,51 @@ fact ssta_ltpr_tpr_aux: ∀h,g,L0,T0.
   [ #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/
+    [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/
   | #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
+    lapply (cpcs_bind1 … HU12 … V1 … 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/
+    lapply (cpcs_cpr_strap1 … HU12 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/
+    elim (IH1 … HTU1 … HL12 … HT12) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -T1 #U2 #HTU2 #HU12
+    lapply (cpcs_flat … V1 V2 … HU12) -HU12 [ /2 width=1/ ] -HV12 /3 width=3/
   | #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
+    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 (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
+    [ #U #HTU20 #HUU20 -HWV0 -W1 -W2
+      lapply (ltpr_cpcs_trans (L1.ⓓV2) … HUU20) -HUU20 [ /2 width=1/ ] -HL12 #HUU20
       @(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/
+      @(cpcs_cprs_strap2 … (ⓓ{b}V2.U2)) [ /4 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 #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/ 
+    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_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/ 
   ]
 | #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 #_
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_sstas.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_sstas.ma
new file mode 100644
index 000000000..34dc2e9a2
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_sstas.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_lift.ma".
+include "basic_2/dynamic/snv.ma".
+
+(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
+
+(* Forward_lemmas on iterated stratified static type assignment for terms ***)
+
+lemma snv_sstas_fwd_correct: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊩ T1 :[g] → ⦃h, L⦄ ⊢ T1 •* [g] T2 →
+                             ∃∃U2,l. ⦃h, L⦄ ⊢ T2 •[g, l] U2.
+#h #g #L #T1 #T2 #HT1 #HT12
+elim (snv_fwd_ssta … HT1) -HT1 /2 width=5 by sstas_fwd_correct/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
index ac9de9300..35dcf5fc3 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
@@ -90,17 +90,24 @@ qed-.
 
 (* Advanced properties ******************************************************)
 
-lemma cpr_cprs_conf: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2.
+lemma ltpr_cpcs_trans: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2.
+#L1 #L2 #HL12 #T1 #T2 #H
+elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2
+lapply (ltpr_cprs_trans … HL12 … HT1) -HT1
+lapply (ltpr_cprs_trans … HL12 … HT2) -L2 /2 width=3/
+qed-.
+
+lemma cpr_cprs_conf_cpcs: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2.
 #L #T #T1 #T2 #HT1 #HT2
 elim (cprs_strip … HT1 … HT2) /2 width=3 by cpr_cprs_div/
 qed-.
 
-lemma cprs_cpr_conf: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T2 ⬌* T1.
+lemma cprs_cpr_conf_cpcs: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T2 ⬌* T1.
 #L #T #T1 #T2 #HT1 #HT2
 elim (cprs_strip … HT1 … HT2) /2 width=3 by cprs_cpr_div/
 qed-.
 
-lemma cprs_conf: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡* T2 → L ⊢ T1 ⬌* T2.
+lemma cprs_conf_cpcs: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡* T2 → L ⊢ T1 ⬌* T2.
 #L #T #T1 #T2 #HT1 #HT2
 elim (cprs_conf … HT1 … HT2) /2 width=3/
 qed-.
@@ -117,9 +124,9 @@ lemma cpcs_flat_dx_tpr_rev: ∀L,V1,V2. V2 ➡ V1 → ∀T1,T2. L ⊢ T1 ⬌* T2
                             ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
 /3 width=1/ qed.
 
-lemma cpcs_abst: ∀a,L,V1,V2. L ⊢ V1 ⬌* V2 →
-                 ∀V,T1,T2. L.ⓛV ⊢ T1 ⬌* T2 → L ⊢ ⓛ{a}V1. T1 ⬌* ⓛ{a}V2. T2.
-#a #L #V1 #V2 #HV12 #V #T1 #T2 #HT12
+lemma cpcs_abst: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ⬌* T2 →
+                 ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a #I
 elim (cpcs_inv_cprs … HV12) -HV12
 elim (cpcs_inv_cprs … HT12) -HT12
 /3 width=6 by cprs_div, cprs_abst/ (**) (* /3 width=6/ is a bit slow *)
@@ -202,3 +209,13 @@ lemma cpcs_abbr2: ∀a,L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓓV2 ⊢ T1 
 #a #L #V1 #V2 #HV12 #T1 #T2 #HT12
 @(cpcs_trans … (ⓓ{a}V2.T1)) /2 width=1/
 qed.
+
+lemma cpcs_bind1: ∀I,L,V1,T1,T2. L.ⓑ{I}V1 ⊢ T1 ⬌* T2 → ∀V2. L ⊢ V1 ⬌* V2 →
+                  ∀a. L ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+* /2 width=1/ /2 width=2/
+qed.
+
+lemma cpcs_bind2: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀I,T1,T2. L.ⓑ{I}V2 ⊢ T1 ⬌* T2 → 
+                  ∀a. L ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+#L #V1 #V2 #HV12 * /2 width=1/ /2 width=2/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_lift.ma
index bcef7bb74..b59a3e6ed 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reducibility/cpr_lift.ma
@@ -28,12 +28,12 @@ lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
 @ex2_intro [2: // | skip | @tpss_subst /width=6/ ] (**) (* /3 width=6/ is too slow *)
 qed.
 
-lemma cpr_abst: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2.
-                L.ⓛV ⊢ T1 ➡ T2 → ∀a. L ⊢ ⓛ{a}V1. T1 ➡ ⓛ{a}V2. T2.
-#L #V1 #V2 * #V0 #HV10 #HV02 #V #T1 #T2 * #T0 #HT10 #HT02 #a
+lemma cpr_abst: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡ T2 →
+                          ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡ ⓑ{a,I}V2. T2.
+#L #V1 #V2 * #V0 #HV10 #HV02 #V #T1 #T2 * #T0 #HT10 #HT02 #a #I
 lapply (tpss_inv_S2 … HT02 L V ?) -HT02 // #HT02
-lapply (tpss_lsubs_trans … HT02 (L.ⓛV2) ?) -HT02 /2 width=1/ #HT02
-@(ex2_intro … (ⓛ{a}V0.T0)) /2 width=1/ (* explicit constructors *)
+lapply (tpss_lsubs_trans … HT02 (L.ⓑ{I}V2) ?) -HT02 /2 width=1/ #HT02
+@(ex2_intro … (ⓑ{a,I}V0.T0)) /2 width=1/ (* explicit constructors *)
 qed.
 
 lemma cpr_beta: ∀a,L,V1,V2,W,T1,T2.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unwind/sstas.ma b/matita/matita/contribs/lambdadelta/basic_2/unwind/sstas.ma
index 93e63853f..93fe0a7e0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unwind/sstas.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unwind/sstas.ma
@@ -47,6 +47,9 @@ qed-.
 
 (* Basic properties *********************************************************)
 
+lemma sstas_refl: ∀h,g,L. reflexive … (sstas h g L).
+// qed.
+
 lemma ssta_sstas: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l+1] U → ⦃h, L⦄ ⊢ T •*[g] U.
 /3 width=2 by R_to_star, ex_intro/ qed. (**) (* auto fails without trace *)
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unwind/sstas_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/unwind/sstas_lift.ma
index ea4b303ef..e8ba22729 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unwind/sstas_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unwind/sstas_lift.ma
@@ -17,6 +17,16 @@ include "basic_2/unwind/sstas.ma".
 
 (* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
 
+(* Advanced forward lemmas **************************************************)
+
+lemma sstas_fwd_correct: ∀h,g,L,T1,U1,l1. ⦃h, L⦄ ⊢ T1 •[g, l1] U1 →
+                         ∀T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 →
+                         ∃∃U2,l2. ⦃h, L⦄ ⊢ T2 •[g, l2] U2.
+#h #g #L #T1 #U1 #l1 #HTU1 #T2 #H @(sstas_ind … H) -T2 [ /2 width=3/ ] -HTU1
+#T #T2 #l #_ #HT2 * #U #l0 #_ -l0
+elim (ssta_fwd_correct … HT2) -T /2 width=3/
+qed-.
+
 (* Properties on relocation *************************************************)
 
 lemma sstas_lift: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
-- 
2.39.2