From 5117f4af452934db436f22326f35d90f757bdf8a Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sun, 31 Aug 2014 18:13:55 +0000
Subject: [PATCH] - new stand-alone definition of lstas (sta and old lstas
 parked in etc) - PARTIAL COMMIT: we issue just the components "static" and
 "unfold"

---
 .../lambdadelta/basic_2/etc/sta/da_aaa.ma     |  24 ++
 .../basic_2/{static => etc/sta}/da_sta.ma     |   0
 .../lambdadelta/basic_2/etc/sta/lstas.ma      | 133 ++++++++++
 .../lambdadelta/basic_2/etc/sta/lstas_aaa.ma  |  23 ++
 .../basic_2/{unfold => etc/sta}/lstas_alt.ma  |   0
 .../lambdadelta/basic_2/etc/sta/lstas_da.ma   |  40 +++
 .../lambdadelta/basic_2/etc/sta/lstas_lift.ma |  81 +++++++
 .../basic_2/etc/sta/lstas_lstas.ma            |  51 ++++
 .../basic_2/{static => etc/sta}/sta.ma        |   0
 .../basic_2/{static => etc/sta}/sta_aaa.ma    |   0
 .../basic_2/{static => etc/sta}/sta_lift.ma   |   0
 .../{static => etc/sta}/sta_llpx_sn.ma        |   0
 .../basic_2/{static => etc/sta}/sta_sta.ma    |   0
 .../relations => etc/sta}/statictype_5.ma     |   0
 .../sta}/statictypestaralt_6.ma               |   0
 .../lambdadelta/basic_2/static/da_aaa.ma      |  13 +-
 .../lambdadelta/basic_2/unfold/lstas.ma       | 229 +++++++++++-------
 .../lambdadelta/basic_2/unfold/lstas_aaa.ma   |  39 ++-
 .../lambdadelta/basic_2/unfold/lstas_da.ma    |  79 +++++-
 .../lambdadelta/basic_2/unfold/lstas_lift.ma  | 202 +++++++++++----
 .../lambdadelta/basic_2/unfold/lstas_lstas.ma | 111 +++++++--
 21 files changed, 853 insertions(+), 172 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/sta/da_aaa.ma
 rename matita/matita/contribs/lambdadelta/basic_2/{static => etc/sta}/da_sta.ma (100%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_aaa.ma
 rename matita/matita/contribs/lambdadelta/basic_2/{unfold => etc/sta}/lstas_alt.ma (100%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_da.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_lift.ma
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_lstas.ma
 rename matita/matita/contribs/lambdadelta/basic_2/{static => etc/sta}/sta.ma (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/{static => etc/sta}/sta_aaa.ma (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/{static => etc/sta}/sta_lift.ma (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/{static => etc/sta}/sta_llpx_sn.ma (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/{static => etc/sta}/sta_sta.ma (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/{notation/relations => etc/sta}/statictype_5.ma (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/{notation/relations => etc/sta}/statictypestaralt_6.ma (100%)

diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/sta/da_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/da_aaa.ma
new file mode 100644
index 000000000..dccb10f9b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/da_aaa.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/da_sta.ma".
+include "basic_2/static/sta_aaa.ma".
+
+(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma aaa_da: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃l. ⦃G, L⦄ ⊢ T ▪[h, g] l.
+#h #g #G #L #T #A #H elim (aaa_sta h … H) -A /2 width=2 by sta_da/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/da_sta.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/da_sta.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/static/da_sta.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/sta/da_sta.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas.ma
new file mode 100644
index 000000000..feed03e3f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas.ma
@@ -0,0 +1,133 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/notation/relations/statictypestar_6.ma".
+include "basic_2/static/sta.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+definition lstas: ∀h. genv → lenv → nat → relation term ≝
+                  λh,G,L. lstar … (sta h G L).
+
+interpretation "nat-iterated static type assignment (term)"
+   'StaticTypeStar h G L l T U = (lstas h G L l T U).
+
+(* Basic eliminators ********************************************************)
+
+lemma lstas_ind_sn: ∀h,G,L,U2. ∀R:relation2 nat term.
+                    R 0 U2 → (
+                       ∀l,T,U1. ⦃G, L⦄ ⊢ T •[h] U1 → ⦃G, L⦄ ⊢ U1 •* [h, l] U2 →
+                       R l U1 → R (l+1) T
+                    ) →
+                    ∀l,T. ⦃G, L⦄ ⊢ T •*[h, l] U2 → R l T.
+/3 width=5 by lstar_ind_l/ qed-.
+
+lemma lstas_ind_dx: ∀h,G,L,T. ∀R:relation2 nat term.
+                    R 0 T → (
+                       ∀l,U1,U2. ⦃G, L⦄ ⊢ T •* [h, l] U1 →  ⦃G, L⦄ ⊢ U1 •[h] U2 →
+                       R l U1 → R (l+1) U2
+                    ) →
+                    ∀l,U. ⦃G, L⦄ ⊢ T •*[h, l] U → R l U.
+/3 width=5 by lstar_ind_r/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lstas_inv_O: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •*[h, 0] U → T = U.
+/2 width=4 by lstar_inv_O/ qed-.
+
+lemma lstas_inv_SO: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •*[h, 1] U → ⦃G, L⦄ ⊢ T •[h] U.
+/2 width=1 by lstar_inv_step/ qed-.
+
+lemma lstas_inv_step_sn: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
+                         ∃∃T. ⦃G, L⦄ ⊢ T1 •[h] T & ⦃G, L⦄ ⊢ T •*[h, l] T2.
+/2 width=3 by lstar_inv_S/ qed-.
+
+lemma lstas_inv_step_dx: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
+                         ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, l] T & ⦃G, L⦄ ⊢ T •[h] T2.
+/2 width=3 by lstar_inv_S_dx/ qed-.
+
+lemma lstas_inv_sort1: ∀h,G,L,X,k,l. ⦃G, L⦄ ⊢ ⋆k •*[h, l] X → X = ⋆((next h)^l k).
+#h #G #L #X #k #l #H @(lstas_ind_dx … H) -X -l //
+#l #X #X0 #_ #H #IHX destruct
+lapply (sta_inv_sort1 … H) -H #H destruct
+>iter_SO //
+qed-.
+
+lemma lstas_inv_gref1: ∀h,G,L,X,p,l. ⦃G, L⦄ ⊢ §p •*[h, l+1] X → ⊥.
+#h #G #L #X #p #l #H elim (lstas_inv_step_sn … H) -H
+#U #H #HUX elim (sta_inv_gref1 … H)
+qed-.
+
+lemma lstas_inv_bind1: ∀h,a,I,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, l] X →
+                       ∃∃U. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U & X = ⓑ{a,I}V.U.
+#h #a #I #G #L #V #T #X #l #H @(lstas_ind_dx … H) -X -l /2 width=3 by ex2_intro/
+#l #X #X0 #_ #HX0 * #U #HTU #H destruct
+elim (sta_inv_bind1 … HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/
+qed-.
+
+lemma lstas_inv_appl1: ∀h,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓐV.T •*[h, l] X →
+                       ∃∃U. ⦃G, L⦄ ⊢ T •*[h, l] U & X = ⓐV.U.
+#h #G #L #V #T #X #l #H @(lstas_ind_dx … H) -X -l /2 width=3 by ex2_intro/
+#l #X #X0 #_ #HX0 * #U #HTU #H destruct
+elim (sta_inv_appl1 … HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/
+qed-.
+
+lemma lstas_inv_cast1: ∀h,G,L,W,T,U,l. ⦃G, L⦄ ⊢ ⓝW.T •*[h, l+1] U → ⦃G, L⦄ ⊢ T •*[h, l+1] U.
+#h #G #L #W #T #X #l #H elim (lstas_inv_step_sn … H) -H
+#U #H #HUX lapply (sta_inv_cast1 … H) -H /2 width=3 by lstar_S/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lstas_refl: ∀h,G,L. reflexive … (lstas h G L 0).
+// qed.
+
+lemma sta_lstas: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ T •*[h, 1] U.
+/2 width=1 by lstar_step/ qed.
+
+lemma lstas_step_sn: ∀h,G,L,T1,U1,U2,l. ⦃G, L⦄ ⊢ T1 •[h] U1 → ⦃G, L⦄ ⊢ U1 •*[h, l] U2 →
+                     ⦃G, L⦄ ⊢ T1 •*[h, l+1] U2.
+/2 width=3 by lstar_S/ qed.
+
+lemma lstas_step_dx: ∀h,G,L,T1,T2,U2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 → ⦃G, L⦄ ⊢ T2 •[h] U2 →
+                     ⦃G, L⦄ ⊢ T1 •*[h, l+1] U2.
+/2 width=3 by lstar_dx/ qed.
+
+lemma lstas_split: ∀h,G,L. inv_ltransitive … (lstas h G L).
+/2 width=1 by lstar_inv_ltransitive/ qed-.
+
+lemma lstas_sort: ∀h,G,L,l,k. ⦃G, L⦄ ⊢ ⋆k •*[h, l] ⋆((next h)^l k).
+#h #G #L #l @(nat_ind_plus … l) -l //
+#l #IHl #k >iter_SO /2 width=3 by sta_sort, lstas_step_dx/
+qed.
+
+lemma lstas_bind: ∀h,I,G,L,V,T,U,l. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U →
+                  ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, l] ⓑ{a,I}V.U.
+#h #I #G #L #V #T #U #l #H @(lstas_ind_dx … H) -U -l /3 width=3 by sta_bind, lstar_O, lstas_step_dx/
+qed.
+
+lemma lstas_appl: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U →
+                  ∀V.⦃G, L⦄ ⊢ ⓐV.T •*[h, l] ⓐV.U.
+#h #G #L #T #U #l #H @(lstas_ind_dx … H) -U -l /3 width=3 by sta_appl, lstar_O, lstas_step_dx/
+qed.
+
+lemma lstas_cast: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l+1] U →
+                  ∀W. ⦃G, L⦄ ⊢ ⓝW.T •*[h, l+1] U.
+#h #G #L #T #U #l #H elim (lstas_inv_step_sn … H) -H /3 width=3 by sta_cast, lstas_step_sn/
+qed.
+
+(* Basic_1: removed theorems 7:
+            sty1_abbr sty1_appl sty1_bind sty1_cast2
+            sty1_correct sty1_lift sty1_trans
+*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_aaa.ma
new file mode 100644
index 000000000..7497f44b6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_aaa.ma
@@ -0,0 +1,23 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/sta_aaa.ma".
+include "basic_2/unfold/lstas.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma lstas_aaa_conf: ∀h,G,L,l. Conf3 … (aaa G L) (lstas h G L l).
+/3 width=6 by sta_aaa_conf, lstar_Conf3/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_alt.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_alt.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_alt.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_da.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_da.ma
new file mode 100644
index 000000000..8b11d1dc0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_da.ma
@@ -0,0 +1,40 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/unfold/lstas.ma".
+include "basic_2/static/da_sta.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Properties on degree assignment for terms ********************************)
+
+lemma lstas_da_conf: ∀h,g,G,L,T,U,l1. ⦃G, L⦄ ⊢ T •*[h, l1] U →
+                     ∀l2. ⦃G, L⦄ ⊢ T ▪[h, g] l2 → ⦃G, L⦄ ⊢ U ▪[h, g] l2-l1.
+#h #g #G #L #T #U #l1 #H @(lstas_ind_dx … H) -U -l1 //
+#l1 #U #U0 #_ #HU0 #IHTU #l2 #HT
+<minus_plus /3 width=3 by da_sta_conf/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lstas_inv_refl_pos: ∀h,G,L,T,l. ⦃G, L⦄ ⊢ T •*[h, l+1] T → ⊥.
+#h #G #L #T #l #H elim (lstas_inv_step_sn … H)
+#U #HTU #_ elim (sta_da_ge … (l+1) HTU) -U
+#g #l0 #HT #Hl0 lapply (lstas_da_conf … H … HT) -H
+#H0T lapply (da_mono … HT … H0T) -h -G -L -T
+#H elim (discr_x_minus_xy … H) -H
+[ #H destruct /2 width=3 by le_plus_xSy_O_false/
+| -Hl0 <plus_n_Sm #H destruct 
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_lift.ma
new file mode 100644
index 000000000..947c16216
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_lift.ma
@@ -0,0 +1,81 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/sta_lift.ma".
+include "basic_2/unfold/lstas.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Properties on relocation *************************************************)
+
+lemma lstas_lift: ∀h,G,l. l_liftable (llstar … (sta h G) l).
+/3 width=10 by l_liftable_llstar, sta_lift/ qed.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma lstas_inv_lift1: ∀h,G,l. l_deliftable_sn (llstar … (sta h G) l).
+/3 width=6 by l_deliftable_sn_llstar, sta_inv_lift1/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lstas_inv_lref1: ∀h,G,L,U,i,l. ⦃G, L⦄ ⊢ #i •*[h, l+1] U →
+                       (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, l+1] W &
+                                 ⇧[0, i+1] W ≡ U
+                       ) ∨
+                       (∃∃K,W,V,V0. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •[h] V0 &
+                                    ⦃G, K⦄ ⊢ W •*[h, l] V & ⇧[0, i+1] V ≡ U
+                        ).
+#h #G #L #U #i #l #H elim (lstas_inv_step_sn … H) -H
+#X #H #HXU elim (sta_inv_lref1 … H) -H
+* #K #V #W #HLK #HVW #HWX
+lapply (drop_fwd_drop2 … HLK) #H0LK
+elim (lstas_inv_lift1 … HXU … H0LK … HWX) -H0LK -X
+/4 width=8 by lstas_step_sn, ex4_4_intro, ex3_3_intro, or_introl, or_intror/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lstas_fwd_correct: ∀h,G,L,T1,U1. ⦃G, L⦄ ⊢ T1 •[h] U1 →
+                         ∀T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 →
+                         ∃U2. ⦃G, L⦄ ⊢ T2 •[h] U2.
+#h #G #L #T1 #U1 #HTU1 #T2 #l #H @(lstas_ind_dx … H) -l -T2 /2 width=3 by ex_intro/ -HTU1
+#l #T #T2 #_ #HT2 #_ -T1 -U1 -l
+elim (sta_fwd_correct … HT2) -T /2 width=2 by ex_intro/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lstas_total: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U →
+                   ∀l. ∃U0. ⦃G, L⦄ ⊢ T •*[h, l] U0.
+#h #G #L #T #U #HTU #l @(nat_ind_plus … l) -l /2 width=2 by ex_intro/
+#l * #U0 #HTU0 elim (lstas_fwd_correct … HTU … HTU0) -U
+/3 width=4 by lstas_step_dx, ex_intro/
+qed-.
+
+lemma lstas_ldef: ∀h,G,L,K,V,i. ⇩[i] L ≡ K.ⓓV →
+                  ∀W,l. ⦃G, K⦄ ⊢ V •*[h, l+1] W →
+                  ∀U. ⇧[0, i+1] W ≡ U → ⦃G, L⦄ ⊢ #i •*[h, l+1] U.
+#h #G #L #K #V #i #HLK #W #l #HVW #U #HWU
+lapply (drop_fwd_drop2 … HLK)
+elim (lstas_inv_step_sn … HVW) -HVW #W0
+elim (lift_total W0 0 (i+1)) /3 width=12 by lstas_step_sn, sta_ldef, lstas_lift/
+qed.
+
+lemma lstas_ldec: ∀h,G,L,K,W,i. ⇩[i] L ≡ K.ⓛW → ∀V0. ⦃G, K⦄ ⊢ W •[h] V0 →
+                  ∀V,l. ⦃G, K⦄ ⊢ W •*[h, l] V →
+                  ∀U. ⇧[0, i+1] V ≡ U → ⦃G, L⦄ ⊢ #i •*[h, l+1] U.
+#h #G #L #K #W #i #HLK #V0 #HWV0 #V #l #HWV #U #HVU
+lapply (drop_fwd_drop2 … HLK) #H
+elim (lift_total W 0 (i+1)) /3 width=12 by lstas_step_sn, sta_ldec, lstas_lift/
+qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_lstas.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_lstas.ma
new file mode 100644
index 000000000..58614e3ff
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/lstas_lstas.ma
@@ -0,0 +1,51 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/sta_sta.ma".
+include "basic_2/unfold/lstas_lift.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* Main properties **********************************************************)
+
+theorem lstas_trans: ∀h,G,L. ltransitive … (lstas h G L).
+/2 width=3 by lstar_ltransitive/ qed-.
+
+theorem lstas_mono: ∀h,G,L,l. singlevalued … (lstas h G L l).
+/3 width=7 by sta_mono, lstar_singlevalued/ qed-.
+
+theorem lstas_conf_le: ∀h,G,L,T,U1,l1. ⦃G, L⦄ ⊢ T •*[h, l1] U1 →
+                       ∀U2,l2. l1 ≤ l2 → ⦃G, L⦄ ⊢ T •*[h, l2] U2 →
+                       ⦃G, L⦄ ⊢ U1 •*[h, l2-l1] U2.
+#h #G #L #T #U1 #l1 #HTU1 #U2 #l2 #Hl12
+>(plus_minus_m_m … Hl12) in ⊢ (%→?); -Hl12 >commutative_plus #H
+elim (lstas_split … H) -H #U #HTU
+>(lstas_mono … HTU … HTU1) -T //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lstas_sta_conf_pos: ∀h,G,L,T,U1. ⦃G, L⦄ ⊢ T •[h] U1 →
+                          ∀U2,l. ⦃G, L⦄ ⊢ T •*[h, l+1] U2 → ⦃G, L⦄ ⊢ U1 •*[h, l] U2.
+#h #G #L #T #U1 #HTU1 #U2 #l #HTU2
+lapply (lstas_conf_le … T U1 1 … HTU2) -HTU2 /2 width=1 by sta_lstas/
+qed-.
+
+lemma lstas_strip_pos: ∀h,G,L,T1,U1. ⦃G, L⦄ ⊢ T1 •[h] U1 →
+                       ∀T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
+                       ∃∃U2. ⦃G, L⦄ ⊢ T2 •[h] U2 & ⦃G, L⦄ ⊢ U1 •*[h, l+1] U2.
+#h #G #L #T1 #U1 #HTU1 #T2 #l #HT12
+elim (lstas_fwd_correct … HTU1 … HT12)
+lapply (lstas_sta_conf_pos … HTU1 … HT12) -T1 /3 width=5 by lstas_step_dx, ex2_intro/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/sta.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/static/sta.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/sta_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta_aaa.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/static/sta_aaa.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta_aaa.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/sta_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta_lift.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/static/sta_lift.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta_lift.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/sta_llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta_llpx_sn.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/static/sta_llpx_sn.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta_llpx_sn.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/sta_sta.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta_sta.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/static/sta_sta.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/sta/sta_sta.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/statictype_5.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/statictype_5.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/notation/relations/statictype_5.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/sta/statictype_5.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/statictypestaralt_6.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/sta/statictypestaralt_6.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/notation/relations/statictypestaralt_6.ma
rename to matita/matita/contribs/lambdadelta/basic_2/etc/sta/statictypestaralt_6.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/da_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/static/da_aaa.ma
index dccb10f9b..7e9e4cc88 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/da_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/da_aaa.ma
@@ -12,13 +12,20 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/static/da_sta.ma".
-include "basic_2/static/sta_aaa.ma".
+include "basic_2/static/aaa_lift.ma".
+include "basic_2/static/da.ma".
 
 (* DEGREE ASSIGNMENT FOR TERMS **********************************************)
 
 (* Properties on atomic arity assignment for terms **************************)
 
 lemma aaa_da: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃l. ⦃G, L⦄ ⊢ T ▪[h, g] l.
-#h #g #G #L #T #A #H elim (aaa_sta h … H) -A /2 width=2 by sta_da/
+#h #g #G #L #T #A #H elim H -G -L -T -A
+[ #G #L #k elim (deg_total h g k) /3 width=2 by da_sort, ex_intro/
+| * #G #L #K #V #B #i #HLK #_ * /3 width=5 by da_ldef, da_ldec, ex_intro/
+| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_bind, ex_intro/
+| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_bind, ex_intro/
+| #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_flat, ex_intro/
+| #G #L #W #T #A #_ #_ #_ * /3 width=2 by da_flat, ex_intro/
+]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas.ma
index feed03e3f..2a81f4c06 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas.ma
@@ -13,119 +13,176 @@
 (**************************************************************************)
 
 include "basic_2/notation/relations/statictypestar_6.ma".
-include "basic_2/static/sta.ma".
+include "basic_2/grammar/genv.ma".
+include "basic_2/substitution/drop.ma".
+include "basic_2/static/sh.ma".
 
 (* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
 
-definition lstas: ∀h. genv → lenv → nat → relation term ≝
-                  λh,G,L. lstar … (sta h G L).
+(* activate genv *)
+inductive lstas (h): nat → relation4 genv lenv term term ≝
+| lstas_sort: ∀G,L,l,k. lstas h l G L (⋆k) (⋆((next h)^l k))
+| lstas_ldef: ∀G,L,K,V,W,U,i,l. ⇩[i] L ≡ K.ⓓV → lstas h l G K V W →
+              ⇧[0, i+1] W ≡ U → lstas h l G L (#i) U
+| lstas_zero: ∀G,L,K,W,V,i. ⇩[i] L ≡ K.ⓛW → lstas h 0 G K W V →
+              lstas h 0 G L (#i) (#i)
+| lstas_succ: ∀G,L,K,W,V,U,i,l. ⇩[i] L ≡ K.ⓛW → lstas h l G K W V →
+              ⇧[0, i+1] V ≡ U → lstas h (l+1) G L (#i) U
+| lstas_bind: ∀a,I,G,L,V,T,U,l. lstas h l G (L.ⓑ{I}V) T U →
+              lstas h l G L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
+| lstas_appl: ∀G,L,V,T,U,l. lstas h l G L T U → lstas h l G L (ⓐV.T) (ⓐV.U)
+| lstas_cast: ∀G,L,W,T,U,l. lstas h l G L T U → lstas h l G L (ⓝW.T) U
+.
 
 interpretation "nat-iterated static type assignment (term)"
-   'StaticTypeStar h G L l T U = (lstas h G L l T U).
-
-(* Basic eliminators ********************************************************)
-
-lemma lstas_ind_sn: ∀h,G,L,U2. ∀R:relation2 nat term.
-                    R 0 U2 → (
-                       ∀l,T,U1. ⦃G, L⦄ ⊢ T •[h] U1 → ⦃G, L⦄ ⊢ U1 •* [h, l] U2 →
-                       R l U1 → R (l+1) T
-                    ) →
-                    ∀l,T. ⦃G, L⦄ ⊢ T •*[h, l] U2 → R l T.
-/3 width=5 by lstar_ind_l/ qed-.
-
-lemma lstas_ind_dx: ∀h,G,L,T. ∀R:relation2 nat term.
-                    R 0 T → (
-                       ∀l,U1,U2. ⦃G, L⦄ ⊢ T •* [h, l] U1 →  ⦃G, L⦄ ⊢ U1 •[h] U2 →
-                       R l U1 → R (l+1) U2
-                    ) →
-                    ∀l,U. ⦃G, L⦄ ⊢ T •*[h, l] U → R l U.
-/3 width=5 by lstar_ind_r/ qed-.
+   'StaticTypeStar h G L l T U = (lstas h l G L T U).
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma lstas_inv_O: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •*[h, 0] U → T = U.
-/2 width=4 by lstar_inv_O/ qed-.
-
-lemma lstas_inv_SO: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •*[h, 1] U → ⦃G, L⦄ ⊢ T •[h] U.
-/2 width=1 by lstar_inv_step/ qed-.
-
-lemma lstas_inv_step_sn: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
-                         ∃∃T. ⦃G, L⦄ ⊢ T1 •[h] T & ⦃G, L⦄ ⊢ T •*[h, l] T2.
-/2 width=3 by lstar_inv_S/ qed-.
-
-lemma lstas_inv_step_dx: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
-                         ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, l] T & ⦃G, L⦄ ⊢ T •[h] T2.
-/2 width=3 by lstar_inv_S_dx/ qed-.
+fact lstas_inv_sort1_aux: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ∀k0. T = ⋆k0 →
+                          U = ⋆((next h)^l k0).
+#h #G #L #T #U #l * -G -L -T -U -l
+[ #G #L #l #k #k0 #H destruct //
+| #G #L #K #V #W #U #i #l #_ #_ #_ #k0 #H destruct
+| #G #L #K #W #V #i #_ #_ #k0 #H destruct
+| #G #L #K #W #V #U #i #l #_ #_ #_ #k0 #H destruct
+| #a #I #G #L #V #T #U #l #_ #k0 #H destruct
+| #G #L #V #T #U #l #_ #k0 #H destruct
+| #G #L #W #T #U #l #_ #k0 #H destruct
+qed-.
 
+(* Basic_1: was just: sty0_gen_sort *)
 lemma lstas_inv_sort1: ∀h,G,L,X,k,l. ⦃G, L⦄ ⊢ ⋆k •*[h, l] X → X = ⋆((next h)^l k).
-#h #G #L #X #k #l #H @(lstas_ind_dx … H) -X -l //
-#l #X #X0 #_ #H #IHX destruct
-lapply (sta_inv_sort1 … H) -H #H destruct
->iter_SO //
+/2 width=5 by lstas_inv_sort1_aux/
 qed-.
 
-lemma lstas_inv_gref1: ∀h,G,L,X,p,l. ⦃G, L⦄ ⊢ §p •*[h, l+1] X → ⊥.
-#h #G #L #X #p #l #H elim (lstas_inv_step_sn … H) -H
-#U #H #HUX elim (sta_inv_gref1 … H)
+fact lstas_inv_lref1_aux: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ∀j. T = #j → ∨∨
+                          (∃∃K,V,W. ⇩[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, l] W &
+                                    ⇧[0, j+1] W ≡ U
+                          ) |
+                          (∃∃K,W,V. ⇩[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V & 
+                                    U = #j & l = 0
+                          ) |
+                          (∃∃K,W,V,l0. ⇩[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, l0] V &
+                                       ⇧[0, j+1] V ≡ U & l = l0+1
+                          ).
+#h #G #L #T #U #l * -G -L -T -U -l
+[ #G #L #l #k #j #H destruct
+| #G #L #K #V #W #U #i #l #HLK #HVW #HWU #j #H destruct /3 width=6 by or3_intro0, ex3_3_intro/
+| #G #L #K #W #V #i #HLK #HWV #j #H destruct /3 width=5 by or3_intro1, ex4_3_intro/
+| #G #L #K #W #V #U #i #l #HLK #HWV #HWU #j #H destruct /3 width=8 by or3_intro2, ex4_4_intro/ 
+| #a #I #G #L #V #T #U #l #_ #j #H destruct
+| #G #L #V #T #U #l #_ #j #H destruct
+| #G #L #W #T #U #l #_ #j #H destruct
+]
 qed-.
 
-lemma lstas_inv_bind1: ∀h,a,I,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, l] X →
-                       ∃∃U. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U & X = ⓑ{a,I}V.U.
-#h #a #I #G #L #V #T #X #l #H @(lstas_ind_dx … H) -X -l /2 width=3 by ex2_intro/
-#l #X #X0 #_ #HX0 * #U #HTU #H destruct
-elim (sta_inv_bind1 … HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/
+lemma lstas_inv_lref1: ∀h,G,L,X,i,l. ⦃G, L⦄ ⊢ #i •*[h, l] X → ∨∨
+                       (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, l] W &
+                                 ⇧[0, i+1] W ≡ X
+                       ) |
+                       (∃∃K,W,V. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V & 
+                                 X = #i & l = 0
+                       ) |                      
+                       (∃∃K,W,V,l0. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, l0] V &
+                                    ⇧[0, i+1] V ≡ X & l = l0+1
+                       ).
+/2 width=3 by lstas_inv_lref1_aux/
 qed-.
 
-lemma lstas_inv_appl1: ∀h,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓐV.T •*[h, l] X →
-                       ∃∃U. ⦃G, L⦄ ⊢ T •*[h, l] U & X = ⓐV.U.
-#h #G #L #V #T #X #l #H @(lstas_ind_dx … H) -X -l /2 width=3 by ex2_intro/
-#l #X #X0 #_ #HX0 * #U #HTU #H destruct
-elim (sta_inv_appl1 … HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/
+lemma lstas_inv_lref1_O: ∀h,G,L,X,i. ⦃G, L⦄ ⊢ #i •*[h, 0] X →
+                         (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, 0] W &
+                                   ⇧[0, i+1] W ≡ X
+                         ) ∨
+                         (∃∃K,W,V. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V & 
+                                   X = #i
+                         ).
+#h #G #L #X #i #H elim (lstas_inv_lref1 … H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/
+#K #W #V #l #_ #_ #_ <plus_n_Sm #H destruct
 qed-.
 
-lemma lstas_inv_cast1: ∀h,G,L,W,T,U,l. ⦃G, L⦄ ⊢ ⓝW.T •*[h, l+1] U → ⦃G, L⦄ ⊢ T •*[h, l+1] U.
-#h #G #L #W #T #X #l #H elim (lstas_inv_step_sn … H) -H
-#U #H #HUX lapply (sta_inv_cast1 … H) -H /2 width=3 by lstar_S/
+(* Basic_1: was just: sty0_gen_lref *)
+lemma lstas_inv_lref1_S: ∀h,G,L,X,i,l. ⦃G, L⦄ ⊢ #i •*[h, l+1] X →
+                         (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, l+1] W &
+                                   ⇧[0, i+1] W ≡ X
+                         ) ∨                      
+                         (∃∃K,W,V. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, l] V &
+                                   ⇧[0, i+1] V ≡ X
+                         ).
+#h #G #L #X #i #l #H elim (lstas_inv_lref1 … H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/
+#K #W #V #_ #_ #_ <plus_n_Sm #H destruct
 qed-.
 
-(* Basic properties *********************************************************)
-
-lemma lstas_refl: ∀h,G,L. reflexive … (lstas h G L 0).
-// qed.
-
-lemma sta_lstas: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ T •*[h, 1] U.
-/2 width=1 by lstar_step/ qed.
+fact lstas_inv_gref1_aux: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ∀p0. T = §p0 → ⊥.
+#h #G #L #T #U #l * -G -L -T -U -l
+[ #G #L #l #k #p0 #H destruct
+| #G #L #K #V #W #U #i #l #_ #_ #_ #p0 #H destruct
+| #G #L #K #W #V #i #_ #_ #p0 #H destruct
+| #G #L #K #W #V #U #i #l #_ #_ #_ #p0 #H destruct
+| #a #I #G #L #V #T #U #l #_ #p0 #H destruct
+| #G #L #V #T #U #l #_ #p0 #H destruct
+| #G #L #W #T #U #l #_ #p0 #H destruct
+qed-.
 
-lemma lstas_step_sn: ∀h,G,L,T1,U1,U2,l. ⦃G, L⦄ ⊢ T1 •[h] U1 → ⦃G, L⦄ ⊢ U1 •*[h, l] U2 →
-                     ⦃G, L⦄ ⊢ T1 •*[h, l+1] U2.
-/2 width=3 by lstar_S/ qed.
+lemma lstas_inv_gref1: ∀h,G,L,X,p,l. ⦃G, L⦄ ⊢ §p •*[h, l] X → ⊥.
+/2 width=9 by lstas_inv_gref1_aux/
+qed-.
 
-lemma lstas_step_dx: ∀h,G,L,T1,T2,U2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 → ⦃G, L⦄ ⊢ T2 •[h] U2 →
-                     ⦃G, L⦄ ⊢ T1 •*[h, l+1] U2.
-/2 width=3 by lstar_dx/ qed.
+fact lstas_inv_bind1_aux: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ∀b,J,X,Y. T = ⓑ{b,J}Y.X →
+                          ∃∃Z. ⦃G, L.ⓑ{J}Y⦄ ⊢ X •*[h, l] Z & U = ⓑ{b,J}Y.Z.
+#h #G #L #T #U #l * -G -L -T -U -l
+[ #G #L #l #k #b #J #X #Y #H destruct
+| #G #L #K #V #W #U #i #l #_ #_ #_ #b #J #X #Y #H destruct
+| #G #L #K #W #V #i #_ #_ #b #J #X #Y #H destruct
+| #G #L #K #W #V #U #i #l #_ #_ #_ #b #J #X #Y #H destruct
+| #a #I #G #L #V #T #U #l #HTU #b #J #X #Y #H destruct /2 width=3 by ex2_intro/
+| #G #L #V #T #U #l #_ #b #J #X #Y #H destruct
+| #G #L #W #T #U #l #_ #b #J #X #Y #H destruct
+]
+qed-.
 
-lemma lstas_split: ∀h,G,L. inv_ltransitive … (lstas h G L).
-/2 width=1 by lstar_inv_ltransitive/ qed-.
+(* Basic_1: was just: sty0_gen_bind *)
+lemma lstas_inv_bind1: ∀h,a,I,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, l] X →
+                       ∃∃U. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U & X = ⓑ{a,I}V.U.
+/2 width=3 by lstas_inv_bind1_aux/
+qed-.
 
-lemma lstas_sort: ∀h,G,L,l,k. ⦃G, L⦄ ⊢ ⋆k •*[h, l] ⋆((next h)^l k).
-#h #G #L #l @(nat_ind_plus … l) -l //
-#l #IHl #k >iter_SO /2 width=3 by sta_sort, lstas_step_dx/
-qed.
+fact lstas_inv_appl1_aux: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ∀X,Y. T = ⓐY.X →
+                          ∃∃Z. ⦃G, L⦄ ⊢ X •*[h, l] Z & U = ⓐY.Z.
+#h #G #L #T #U #l * -G -L -T -U -l
+[ #G #L #l #k #X #Y #H destruct
+| #G #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
+| #G #L #K #W #V #i #_ #_ #X #Y #H destruct
+| #G #L #K #W #V #U #i #l #_ #_ #_ #X #Y #H destruct
+| #a #I #G #L #V #T #U #l #_ #X #Y #H destruct
+| #G #L #V #T #U #l #HTU #X #Y #H destruct /2 width=3 by ex2_intro/
+| #G #L #W #T #U #l #_ #X #Y #H destruct
+]
+qed-.
 
-lemma lstas_bind: ∀h,I,G,L,V,T,U,l. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U →
-                  ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, l] ⓑ{a,I}V.U.
-#h #I #G #L #V #T #U #l #H @(lstas_ind_dx … H) -U -l /3 width=3 by sta_bind, lstar_O, lstas_step_dx/
-qed.
+(* Basic_1: was just: sty0_gen_appl *)
+lemma lstas_inv_appl1: ∀h,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓐV.T •*[h, l] X →
+                       ∃∃U. ⦃G, L⦄ ⊢ T •*[h, l] U & X = ⓐV.U.
+/2 width=3 by lstas_inv_appl1_aux/
+qed-.
 
-lemma lstas_appl: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U →
-                  ∀V.⦃G, L⦄ ⊢ ⓐV.T •*[h, l] ⓐV.U.
-#h #G #L #T #U #l #H @(lstas_ind_dx … H) -U -l /3 width=3 by sta_appl, lstar_O, lstas_step_dx/
-qed.
+fact lstas_inv_cast1_aux: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ∀X,Y. T = ⓝY.X →
+                          ⦃G, L⦄ ⊢ X •*[h, l] U.
+#h #G #L #T #U #l * -G -L -T -U -l
+[ #G #L #l #k #X #Y #H destruct
+| #G #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
+| #G #L #K #W #V #i #_ #_ #X #Y #H destruct
+| #G #L #K #W #V #U #i #l #_ #_ #_ #X #Y #H destruct
+| #a #I #G #L #V #T #U #l #_ #X #Y #H destruct
+| #G #L #V #T #U #l #_ #X #Y #H destruct
+| #G #L #W #T #U #l #HTU #X #Y #H destruct //
+]
+qed-.
 
-lemma lstas_cast: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l+1] U →
-                  ∀W. ⦃G, L⦄ ⊢ ⓝW.T •*[h, l+1] U.
-#h #G #L #T #U #l #H elim (lstas_inv_step_sn … H) -H /3 width=3 by sta_cast, lstas_step_sn/
-qed.
+(* Basic_1: was just: sty0_gen_cast *)
+lemma lstas_inv_cast1: ∀h,G,L,W,T,U,l. ⦃G, L⦄ ⊢ ⓝW.T •*[h, l] U → ⦃G, L⦄ ⊢ T •*[h, l] U.
+/2 width=4 by lstas_inv_cast1_aux/
+qed-.
 
 (* Basic_1: removed theorems 7:
             sty1_abbr sty1_appl sty1_bind sty1_cast2
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_aaa.ma
index 7497f44b6..434c6e991 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_aaa.ma
@@ -12,12 +12,43 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/static/sta_aaa.ma".
-include "basic_2/unfold/lstas.ma".
+include "basic_2/static/aaa_lift.ma".
+include "basic_2/unfold/lstas_lstas.ma".
 
 (* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
 
 (* Properties on atomic arity assignment for terms **************************)
 
-lemma lstas_aaa_conf: ∀h,G,L,l. Conf3 … (aaa G L) (lstas h G L l).
-/3 width=6 by sta_aaa_conf, lstar_Conf3/ qed-.
+lemma aaa_lstas: ∀h,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀l.
+                 ∃∃U. ⦃G, L⦄ ⊢ T •*[h, l] U & ⦃G, L⦄ ⊢ U ⁝ A.
+#h #G #L #T #A #H elim H -G -L -T -A
+[ /2 width=3 by ex2_intro/
+| * #G #L #K #V #B #i #HLK #HV #IHV #l
+  [ elim (IHV l) -IHV #W
+    elim (lift_total W 0 (i+1))
+    lapply (drop_fwd_drop2 … HLK)
+    /3 width=10 by lstas_ldef, aaa_lift, ex2_intro/
+  | @(nat_ind_plus … l) -l
+    [ elim (IHV 0) -IHV /3 width=7 by lstas_zero, aaa_lref, ex2_intro/
+    | #l #_ elim (IHV l) -IHV #W
+      elim (lift_total W 0 (i+1))
+      lapply (drop_fwd_drop2 … HLK)
+      /3 width=10 by lstas_succ, aaa_lift, ex2_intro/
+    ]
+  ]
+| #a #G #L #V #T #B #A #HV #_ #_ #IHT #l elim (IHT l) -IHT
+  /3 width=7 by lstas_bind, aaa_abbr, ex2_intro/
+| #a #G #L #V #T #B #A #HV #_ #_ #IHT #l elim (IHT l) -IHT
+  /3 width=6 by lstas_bind, aaa_abst, ex2_intro/
+| #G #L #V #T #B #A #HV #_ #_ #IHT #l elim (IHT l) -IHT
+  /3 width=6 by lstas_appl, aaa_appl, ex2_intro/
+| #G #L #W #T #A #HW #_ #_ #IHT #l elim (IHT l) -IHT
+  /3 width=3 by lstas_cast, aaa_cast, ex2_intro/
+]
+qed-.
+
+lemma lstas_aaa_conf: ∀h,G,L,l. Conf3 … (aaa G L) (lstas h l G L).
+#h #G #L #l #A #T #HT #U #HTU
+elim (aaa_lstas h … HT l) -HT #X #HTX
+lapply (lstas_mono … HTX … HTU) -T //
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_da.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_da.ma
index 8b11d1dc0..8d1feca9f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_da.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_da.ma
@@ -12,29 +12,84 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/unfold/lstas.ma".
-include "basic_2/static/da_sta.ma".
+include "basic_2/static/da_da.ma".
+include "basic_2/unfold/lstas_lstas.ma".
 
 (* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
 
 (* Properties on degree assignment for terms ********************************)
 
-lemma lstas_da_conf: ∀h,g,G,L,T,U,l1. ⦃G, L⦄ ⊢ T •*[h, l1] U →
-                     ∀l2. ⦃G, L⦄ ⊢ T ▪[h, g] l2 → ⦃G, L⦄ ⊢ U ▪[h, g] l2-l1.
-#h #g #G #L #T #U #l1 #H @(lstas_ind_dx … H) -U -l1 //
-#l1 #U #U0 #_ #HU0 #IHTU #l2 #HT
-<minus_plus /3 width=3 by da_sta_conf/
+lemma da_lstas: ∀h,g,G,L,T,l1. ⦃G, L⦄ ⊢ T ▪[h, g] l1 → ∀l2.
+                ∃∃U. ⦃G, L⦄ ⊢ T •*[h, l2] U & ⦃G, L⦄ ⊢ U ▪[h, g] l1-l2.
+#h #g #G #L #T #l1 #H elim H -G -L -T -l1
+[ /4 width=3 by da_sort, deg_iter, ex2_intro/
+| #G #L #K #V #i #l1 #HLK #_ #IHV #l2
+  elim (IHV l2) -IHV #W
+  elim (lift_total W 0 (i+1))
+  lapply (drop_fwd_drop2 … HLK)
+  /3 width=10 by lstas_ldef, da_lift, ex2_intro/
+| #G #L #K #W #i #l1 #HLK #HW #IHW #l2 @(nat_ind_plus … l2) -l2
+  [ elim (IHW 0) -IHW /3 width=6 by lstas_zero, da_ldec, ex2_intro/
+  | #l #_ elim (IHW l) -IHW #V
+    elim (lift_total V 0 (i+1))
+    lapply (drop_fwd_drop2 … HLK)
+    /3 width=10 by lstas_succ, da_lift, ex2_intro/
+  ]
+| #a #I #G #L #V #T #l1 #_ #IHT #l2 elim (IHT … l2) -IHT
+  /3 width=6 by lstas_bind, da_bind, ex2_intro/
+| * #G #L #V #T #l1 #_ #IHT #l2 elim (IHT … l2) -IHT
+  /3 width=5 by lstas_appl, lstas_cast, da_flat, ex2_intro/
+]
+qed-.
+
+lemma lstas_da_conf: ∀h,g,G,L,T,U,l2. ⦃G, L⦄ ⊢ T •*[h, l2] U →
+                     ∀l1. ⦃G, L⦄ ⊢ T ▪[h, g] l1 → ⦃G, L⦄ ⊢ U ▪[h, g] l1-l2.
+#h #g #G #L #T #U #l2 #HTU #l1 #HT
+elim (da_lstas … HT l2) -HT #X #HTX
+lapply (lstas_mono … HTX … HTU) -T //
+qed-.
+
+(* inversion lemmas on degree assignment for terms **************************)
+
+lemma lstas_inv_da: ∀h,g,G,L,T,U,l2. ⦃G, L⦄ ⊢ T •*[h, l2] U →
+                    ∃∃l1. ⦃G, L⦄ ⊢ T ▪[h, g] l1 & ⦃G, L⦄ ⊢ U ▪[h, g] l1-l2.
+#h #g #G #L #T #U #l2 #H elim H -G -L -T -U -l2
+[ #G #L #l2 #k elim (deg_total h g k) /4 width=3 by da_sort, deg_iter, ex2_intro/
+| #G #L #K #V #W #U #i #l2 #HLK #_ #HWU *
+  lapply (drop_fwd_drop2 … HLK) /3 width=10 by da_lift, da_ldef, ex2_intro/
+| #G #L #K #W #V #i #HLK #_ * /3 width=6 by da_ldec, ex2_intro/
+| #G #L #K #W #V #U #i #l2 #HLK #_ #HVU *
+  lapply (drop_fwd_drop2 … HLK) /3 width=10 by da_lift, da_ldec, ex2_intro/
+| #a #I #G #L #V #T #U #l2 #_ * /3 width=3 by da_bind, ex2_intro/
+| #G #L #V #T #U #l2 #_ * /3 width=3 by da_flat, ex2_intro/
+| #G #L #W #T #U #l2 #_ * /3 width=3 by da_flat, ex2_intro/
+]
+qed-.
+
+lemma lstas_inv_da_ge: ∀h,G,L,T,U,l2,l. ⦃G, L⦄ ⊢ T •*[h, l2] U →
+                       ∃∃g,l1. ⦃G, L⦄ ⊢ T ▪[h, g] l1 & ⦃G, L⦄ ⊢ U ▪[h, g] l1-l2 & l ≤ l1.
+#h #G #L #T #U #l2 #l #H elim H -G -L -T -U -l2
+[ /4 width=5 by da_sort, deg_iter, ex3_2_intro/
+| #G #L #K #V #W #U #i #l2 #HLK #_ #HWU *
+  lapply (drop_fwd_drop2 … HLK) /3 width=10 by da_lift, da_ldef, ex3_2_intro/
+| #G #L #K #W #V #i #HLK #_ * 
+  #g #l1 #HW #HV #Hl1 /4 width=6 by da_ldec, lt_to_le, le_S_S, ex3_2_intro/
+| #G #L #K #W #V #U #i #l2 #HLK #_ #HVU *
+  lapply (drop_fwd_drop2 … HLK)
+  /4 width=10 by da_lift, da_ldec, lt_to_le, le_S_S, ex3_2_intro/
+| #a #I #G #L #V #T #U #l2 #_ * /3 width=5 by da_bind, ex3_2_intro/
+| #G #L #V #T #U #l2 #_ * /3 width=5 by da_flat, ex3_2_intro/
+| #G #L #W #T #U #l2 #_ * /3 width=5 by da_flat, ex3_2_intro/
+]
 qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
 lemma lstas_inv_refl_pos: ∀h,G,L,T,l. ⦃G, L⦄ ⊢ T •*[h, l+1] T → ⊥.
-#h #G #L #T #l #H elim (lstas_inv_step_sn … H)
-#U #HTU #_ elim (sta_da_ge … (l+1) HTU) -U
-#g #l0 #HT #Hl0 lapply (lstas_da_conf … H … HT) -H
-#H0T lapply (da_mono … HT … H0T) -h -G -L -T
+#h #G #L #T #l2 #H elim (lstas_inv_da_ge … (l2+1) H) -H
+#g #l1 #HT1 #HT12 #Hl21 lapply (da_mono … HT1 … HT12) -h -G -L -T
 #H elim (discr_x_minus_xy … H) -H
 [ #H destruct /2 width=3 by le_plus_xSy_O_false/
-| -Hl0 <plus_n_Sm #H destruct 
+| -l1 <plus_n_Sm #H destruct 
 ]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lift.ma
index 947c16216..9239d34df 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lift.ma
@@ -12,70 +12,176 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/static/sta_lift.ma".
+include "basic_2/substitution/drop_drop.ma".
 include "basic_2/unfold/lstas.ma".
 
 (* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
 
 (* Properties on relocation *************************************************)
 
-lemma lstas_lift: ∀h,G,l. l_liftable (llstar … (sta h G) l).
-/3 width=10 by l_liftable_llstar, sta_lift/ qed.
+(* Basic_1: was just: sty0_lift *)
+lemma lstas_lift: ∀h,G,l. l_liftable (lstas h G l).
+#h #G #l #L1 #T1 #U1 #H elim H -G -L1 -T1 -U1 -l
+[ #G #L1 #l #k #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
+  >(lift_inv_sort1 … H1) -X1
+  >(lift_inv_sort1 … H2) -X2 //
+| #G #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
+  elim (lift_inv_lref1 … H) * #Hid #H destruct
+  [ elim (lift_trans_ge … HW1 … HWU2) -W // #W2 #HW12 #HWU2
+    elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+    /3 width=9 by lstas_ldef/
+  | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
+    lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by lstas_ldef, drop_inv_gen/
+  ]
+| #G #L1 #K1 #V1 #W1 #i #HLK1 #_ #IHVW1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
+  >(lift_mono … HWU2 … H) -U2
+  elim (lift_inv_lref1 … H) * #Hid #H destruct
+  [ elim (lift_total W1 (d-i-1) e) #W2 #HW12
+    elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+    /3 width=10 by lstas_zero/
+  | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
+    /3 width=10 by lstas_zero, drop_inv_gen/
+  ]
+| #G #L1 #K1 #W1 #V1 #W #i #l #HLK1 #_ #HW1 #IHWV1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
+  elim (lift_inv_lref1 … H) * #Hid #H destruct
+  [ elim (lift_trans_ge … HW1 … HWU2) -W // <minus_plus #W #HW1 #HWU2
+    elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
+    /3 width=9 by lstas_succ/
+  | lapply (lift_trans_be … HW1 … HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
+    lapply (drop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=9 by lstas_succ, drop_inv_gen/
+  ]
+| #a #I #G #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
+  elim (lift_inv_bind1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
+  elim (lift_inv_bind1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
+  lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by lstas_bind, drop_skip/
+| #G #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
+  elim (lift_inv_flat1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
+  elim (lift_inv_flat1 … H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
+  lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by lstas_appl/
+| #G #L1 #W1 #T1 #U1 #l #_ #IHTU1 #L2 #s #d #e #HL21 #X #H #U2 #HU12
+  elim (lift_inv_flat1 … H) -H #W2 #T2 #_ #HT12 #H destruct /3 width=6 by lstas_cast/
+]
+qed.
 
 (* Inversion lemmas on relocation *******************************************)
 
-lemma lstas_inv_lift1: ∀h,G,l. l_deliftable_sn (llstar … (sta h G) l).
-/3 width=6 by l_deliftable_sn_llstar, sta_inv_lift1/ qed-.
+(* Note: apparently this was missing in basic_1 *)
+lemma lstas_inv_lift1: ∀h,G,l. l_deliftable_sn (lstas h G l).
+#h #G #l #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2 -l
+[ #G #L2 #l #k #L1 #s #d #e #_ #X #H
+  >(lift_inv_sort2 … H) -X /2 width=3 by lstas_sort, lift_sort, ex2_intro/
+| #G #L2 #K2 #V2 #W2 #W #i #l #HLK2 #HVW2 #HW2 #IHVW2 #L1 #s #d #e #HL21 #X #H
+  elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ]
+  [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12
+    elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HW12 #HVW1
+    elim (lift_trans_le … HW12 … HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by lstas_ldef, ex2_intro/
+  | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
+    elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
+    elim (lift_split … HW2 d (i-e+1)) -HW2 /2 width=1 by le_S_S, le_S/
+    #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by lstas_ldef, le_S, ex2_intro/
+  ]
+| #G #L2 #K2 #W2 #V2 #i #HLK2 #HWV2 #IHWV2 #L1 #s #d #e #HL21 #X #H
+  elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HWV2 | -IHWV2 ]
+  [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
+    elim (IHWV2 … HK21 … HW12) -K2
+    /3 width=5 by lstas_zero, lift_lref_lt, ex2_intro/
+  | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2
+    /3 width=5 by lstas_zero, lift_lref_ge_minus, ex2_intro/
+  ]
+| #G #L2 #K2 #W2 #V2 #W #i #l #HLK2 #HWV2 #HW2 #IHWV2 #L1 #s #d #e #HL21 #X #H
+  elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HWV2 | -IHWV2 ]
+  [ elim (drop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
+    elim (IHWV2 … HK21 … HW12) -K2 #V1 #HV12 #HWV1
+    elim (lift_trans_le … HV12 … HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by lstas_succ, ex2_intro/
+  | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
+    elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
+    elim (lift_split … HW2 d (i-e+1)) -HW2 /2 width=1 by le_S_S, le_S/
+    #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by lstas_succ, le_S, ex2_intro/
+  ]
+| #a #I #G #L2 #V2 #T2 #U2 #l #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
+  elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+  elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12 /3 width=5 by lstas_bind, drop_skip, lift_bind, ex2_intro/
+| #G #L2 #V2 #T2 #U2 #l #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
+  elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+  elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=5 by lstas_appl, lift_flat, ex2_intro/
+| #G #L2 #W2 #T2 #U2 #l #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
+  elim (lift_inv_flat2 … H) -H #W1 #T1 #_ #HT12 #H destruct
+  elim (IHTU2 … HL21 … HT12) -L2 -HT12 /3 width=3 by lstas_cast, ex2_intro/
+]
+qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma lstas_inv_lref1: ∀h,G,L,U,i,l. ⦃G, L⦄ ⊢ #i •*[h, l+1] U →
-                       (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, l+1] W &
-                                 ⇧[0, i+1] W ≡ U
-                       ) ∨
-                       (∃∃K,W,V,V0. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •[h] V0 &
-                                    ⦃G, K⦄ ⊢ W •*[h, l] V & ⇧[0, i+1] V ≡ U
-                        ).
-#h #G #L #U #i #l #H elim (lstas_inv_step_sn … H) -H
-#X #H #HXU elim (sta_inv_lref1 … H) -H
-* #K #V #W #HLK #HVW #HWX
-lapply (drop_fwd_drop2 … HLK) #H0LK
-elim (lstas_inv_lift1 … HXU … H0LK … HWX) -H0LK -X
-/4 width=8 by lstas_step_sn, ex4_4_intro, ex3_3_intro, or_introl, or_intror/
+lemma zero_eq_plus: ∀x,y. 0 = x + y → 0 = x ∧ 0 = y.
+* /2 width=1 by conj/ #x #y normalize #H destruct
 qed-.
 
-(* Advanced forward lemmas **************************************************)
-
-lemma lstas_fwd_correct: ∀h,G,L,T1,U1. ⦃G, L⦄ ⊢ T1 •[h] U1 →
-                         ∀T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 →
-                         ∃U2. ⦃G, L⦄ ⊢ T2 •[h] U2.
-#h #G #L #T1 #U1 #HTU1 #T2 #l #H @(lstas_ind_dx … H) -l -T2 /2 width=3 by ex_intro/ -HTU1
-#l #T #T2 #_ #HT2 #_ -T1 -U1 -l
-elim (sta_fwd_correct … HT2) -T /2 width=2 by ex_intro/
+lemma lstas_split_aux: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 → ∀l1,l2. l = l1 + l2 →
+                       ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, l1] T & ⦃G, L⦄ ⊢ T •*[h, l2] T2.
+#h #G #L #T1 #T2 #l #H elim H -G -L -T1 -T2 -l
+[ #G #L #l #k #l1 #l2 #H destruct
+  >commutative_plus >iter_plus /2 width=3 by lstas_sort, ex2_intro/
+| #G #L #K #V1 #V2 #U2 #i #l #HLK #_ #VU2 #IHV12 #l1 #l2 #H destruct
+  elim (IHV12 l1 l2) -IHV12 // #V
+  elim (lift_total V 0 (i+1))
+  lapply (drop_fwd_drop2 … HLK)
+  /3 width=12 by lstas_lift, lstas_ldef, ex2_intro/
+| #G #L #K #W1 #W2 #i #HLK #HW12 #_ #l1 #l2 #H
+  elim (zero_eq_plus … H) -H #H1 #H2 destruct
+  /3 width=5 by lstas_zero, ex2_intro/
+| #G #L #K #W1 #W2 #U2 #i #l #HLK #HW12 #HWU2 #IHW12 #l1 @(nat_ind_plus … l1) -l1
+  [ #l2 normalize #H destruct
+    elim (IHW12 0 l) -IHW12 //
+    lapply (drop_fwd_drop2 … HLK)
+    /3 width=8 by lstas_succ, lstas_zero, ex2_intro/
+  | #l1 #_ #l2 <plus_plus_comm_23 #H lapply (injective_plus_l … H) -H #H
+    elim (IHW12 … H) -l #W
+    elim (lift_total W 0 (i+1))
+    lapply (drop_fwd_drop2 … HLK)
+    /3 width=12 by lstas_lift, lstas_succ, ex2_intro/
+  ]
+| #a #I #G #L #V #T #U #l #_ #IHTU #l1 #l2 #H
+  elim (IHTU … H) -l /3 width=3 by lstas_bind, ex2_intro/
+| #G #L #V #T #U #l #_ #IHTU #l1 #l2 #H
+  elim (IHTU … H) -l /3 width=3 by lstas_appl, ex2_intro/
+| #G #L #W #T #U #l #_ #IHTU #l1 #l2 #H
+  elim (IHTU … H) -l /3 width=3 by lstas_cast, ex2_intro/
+]
 qed-.
 
+lemma lstas_split: ∀h,G,L,T1,T2,l1,l2. ⦃G, L⦄ ⊢ T1 •*[h, l1 + l2] T2 →
+                   ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, l1] T & ⦃G, L⦄ ⊢ T •*[h, l2] T2.
+/2 width=3 by lstas_split_aux/ qed-.
+
 (* Advanced properties ******************************************************)
 
-lemma lstas_total: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U →
-                   ∀l. ∃U0. ⦃G, L⦄ ⊢ T •*[h, l] U0.
-#h #G #L #T #U #HTU #l @(nat_ind_plus … l) -l /2 width=2 by ex_intro/
-#l * #U0 #HTU0 elim (lstas_fwd_correct … HTU … HTU0) -U
-/3 width=4 by lstas_step_dx, ex_intro/
+lemma lstas_lstas: ∀h,G,L,T,T1,l1. ⦃G, L⦄ ⊢ T •*[h, l1] T1 →
+                   ∀l2. ∃T2. ⦃G, L⦄ ⊢ T •*[h, l2] T2.
+#h #G #L #T #T1 #l1 #H elim H -G -L -T -T1 -l1
+[ /2 width=2 by lstas_sort, ex_intro/
+| #G #L #K #V #V1 #U1 #i #l1 #HLK #_ #HVU1 #IHV1 #l2
+  elim (IHV1 l2) -IHV1 #V2
+  elim (lift_total V2 0 (i+1))
+  /3 width=7 by ex_intro, lstas_ldef/
+| #G #L #K #W #W1 #i #HLK #HW1 #IHW1 #l2
+  @(nat_ind_plus … l2) -l2 /3 width=5 by lstas_zero, ex_intro/
+  #l2 #_ elim (IHW1 l2) -IHW1 #W2
+  elim (lift_total W2 0 (i+1))
+  /3 width=7 by lstas_succ, ex_intro/
+| #G #L #K #W #W1 #U1 #i #l #HLK #_ #_ #IHW1 #l2
+  @(nat_ind_plus … l2) -l2
+  [ elim (IHW1 0) -IHW1 /3 width=5 by lstas_zero, ex_intro/
+  | #l2 #_ elim (IHW1 l2) -IHW1
+    #W2 elim (lift_total W2 0 (i+1)) /3 width=7 by ex_intro, lstas_succ/
+  ]
+| #a #I #G #L #V #T #U #l #_ #IHTU #l2
+  elim (IHTU l2) -IHTU /3 width=2 by lstas_bind, ex_intro/
+| #G #L #V #T #U #l #_ #IHTU #l2
+  elim (IHTU l2) -IHTU /3 width=2 by lstas_appl, ex_intro/
+| #G #L #W #T #U #l #_ #IHTU #l2
+  elim (IHTU l2) -IHTU /3 width=2 by lstas_cast, ex_intro/
+]
 qed-.
-
-lemma lstas_ldef: ∀h,G,L,K,V,i. ⇩[i] L ≡ K.ⓓV →
-                  ∀W,l. ⦃G, K⦄ ⊢ V •*[h, l+1] W →
-                  ∀U. ⇧[0, i+1] W ≡ U → ⦃G, L⦄ ⊢ #i •*[h, l+1] U.
-#h #G #L #K #V #i #HLK #W #l #HVW #U #HWU
-lapply (drop_fwd_drop2 … HLK)
-elim (lstas_inv_step_sn … HVW) -HVW #W0
-elim (lift_total W0 0 (i+1)) /3 width=12 by lstas_step_sn, sta_ldef, lstas_lift/
-qed.
-
-lemma lstas_ldec: ∀h,G,L,K,W,i. ⇩[i] L ≡ K.ⓛW → ∀V0. ⦃G, K⦄ ⊢ W •[h] V0 →
-                  ∀V,l. ⦃G, K⦄ ⊢ W •*[h, l] V →
-                  ∀U. ⇧[0, i+1] V ≡ U → ⦃G, L⦄ ⊢ #i •*[h, l+1] U.
-#h #G #L #K #W #i #HLK #V0 #HWV0 #V #l #HWV #U #HVU
-lapply (drop_fwd_drop2 … HLK) #H
-elim (lift_total W 0 (i+1)) /3 width=12 by lstas_step_sn, sta_ldec, lstas_lift/
-qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lstas.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lstas.ma
index 58614e3ff..1b5843657 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lstas.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lstas.ma
@@ -12,18 +12,97 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/static/sta_sta.ma".
 include "basic_2/unfold/lstas_lift.ma".
 
 (* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
 
 (* Main properties **********************************************************)
 
-theorem lstas_trans: ∀h,G,L. ltransitive … (lstas h G L).
-/2 width=3 by lstar_ltransitive/ qed-.
+theorem lstas_trans: ∀h,G,L,T1,T,l1. ⦃G, L⦄ ⊢ T1 •*[h, l1] T →
+                     ∀T2,l2. ⦃G, L⦄ ⊢ T •*[h, l2] T2 → ⦃G, L⦄ ⊢ T1 •*[h, l1+l2] T2.
+#h #G #L #T1 #T #l1 #H elim H -G -L -T1 -T -l1
+[ #G #L #l1 #k #X #l2 #H >(lstas_inv_sort1 … H) -X
+  <iter_plus /2 width=1 by lstas_sort/
+| #G #L #K #V1 #V #U #i #l1 #HLK #_ #HVU #IHV1 #U2 #l2 #HU2
+  lapply (drop_fwd_drop2 … HLK) #H0
+  elim (lstas_inv_lift1 … HU2 … H0 … HVU)
+  /3 width=6 by lstas_ldef/
+| //
+| #G #L #K #W1 #W #U #i #l1 #HLK #_ #HWU #IHW1 #U2 #l2 #HU2
+  lapply (drop_fwd_drop2 … HLK) #H0
+  elim (lstas_inv_lift1 … HU2 … H0 … HWU)
+  /3 width=6 by lstas_succ/
+| #a #I #G #L #V #T1 #T #l1 #_ #IHT1 #X #l2 #H
+  elim (lstas_inv_bind1 … H) -H #T2 #HT2 #H destruct
+  /3 width=1 by lstas_bind/
+| #G #L #V #T1 #T #l1 #_ #IHT1 #X #l2 #H
+  elim (lstas_inv_appl1 … H) -H #T2 #HT2 #H destruct
+  /3 width=1 by lstas_appl/
+| /3 width=1 by lstas_cast/
+]
+qed-.
+
+(* Note: apparently this was missing in basic_1 *)
+theorem lstas_mono: ∀h,G,L,l. singlevalued … (lstas h l G L).
+#h #G #L #l #T #T1 #H elim H -G -L -T -T1 -l
+[ #G #L #l #k #X #H >(lstas_inv_sort1 … H) -X //
+| #G #L #K #V #V1 #U1 #i #l #HLK #_ #HVU1 #IHV1 #X #H
+  elim (lstas_inv_lref1 … H) -H *
+  #K0 #V0 #W0 [3: #l0 ] #HLK0
+  lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct
+  #HVW0 #HX lapply (IHV1 … HVW0) -IHV1 -HVW0 #H destruct
+  /2 width=5 by lift_mono/
+| #G #L #K #W #W1 #i #HLK #_ #_ #X #H
+  elim (lstas_inv_lref1_O … H) -H *
+  #K0 #V0 #W0 #HLK0
+  lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct //
+| #G #L #K #W #W1 #U1 #i #l #HLK #_ #HWU1 #IHWV #X #H
+  elim (lstas_inv_lref1_S … H) -H * #K0 #W0 #V0 #HLK0
+  lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct
+  #HW0 #HX lapply (IHWV … HW0) -IHWV -HW0 #H destruct
+  /2 width=5 by lift_mono/
+| #a #I #G #L #V #T #U1 #l #_ #IHTU1 #X #H
+  elim (lstas_inv_bind1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
+| #G #L #V #T #U1 #l #_ #IHTU1 #X #H
+  elim (lstas_inv_appl1 … H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
+| #G #L #W #T #U1 #l #_ #IHTU1 #U2 #H
+  lapply (lstas_inv_cast1 … H) -H /2 width=1 by/
+]
+qed-.
 
-theorem lstas_mono: ∀h,G,L,l. singlevalued … (lstas h G L l).
-/3 width=7 by sta_mono, lstar_singlevalued/ qed-.
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was just: sty0_correct *)
+lemma lstas_correct: ∀h,G,L,T1,T,l1. ⦃G, L⦄ ⊢ T1 •*[h, l1] T →
+                     ∀l2. ∃T2. ⦃G, L⦄ ⊢ T •*[h, l2] T2.
+#h #G #L #T1 #T #l1 #H elim H -G -L -T1 -T -l1
+[ /2 width=2 by lstas_sort, ex_intro/
+| #G #L #K #V1 #V #U #i #l #HLK #_ #HVU #IHV1 #l2
+  elim (IHV1 l2) -IHV1 #V2
+  elim (lift_total V2 0 (i+1))
+  lapply (drop_fwd_drop2 … HLK) -HLK
+  /3 width=11 by ex_intro, lstas_lift/
+| #G #L #K #W1 #W #i #HLK #HW1 #IHW1 #l2
+  @(nat_ind_plus … l2) -l2 /3 width=5 by lstas_zero, ex_intro/
+  #l2 #_ elim (IHW1 l2) -IHW1 #W2 #HW2
+  lapply (lstas_trans … HW1 … HW2) -W
+  elim (lift_total W2 0 (i+1))
+  /3 width=7 by lstas_succ, ex_intro/
+| #G #L #K #W1 #W #U #i #l #HLK #_ #HWU #IHW1 #l2
+  elim (IHW1 l2) -IHW1 #W2
+  elim (lift_total W2 0 (i+1))
+  lapply (drop_fwd_drop2 … HLK) -HLK
+  /3 width=11 by ex_intro, lstas_lift/
+| #a #I #G #L #V #T #U #l #_ #IHTU #l2
+  elim (IHTU l2) -IHTU /3 width=2 by lstas_bind, ex_intro/
+| #G #L #V #T #U #l #_ #IHTU #l2
+  elim (IHTU l2) -IHTU /3 width=2 by lstas_appl, ex_intro/
+| #G #L #W #T #U #l #_ #IHTU #l2
+  elim (IHTU l2) -IHTU /2 width=2 by ex_intro/
+]
+qed-.
+
+(* more main properties *****************************************************)
 
 theorem lstas_conf_le: ∀h,G,L,T,U1,l1. ⦃G, L⦄ ⊢ T •*[h, l1] U1 →
                        ∀U2,l2. l1 ≤ l2 → ⦃G, L⦄ ⊢ T •*[h, l2] U2 →
@@ -34,18 +113,12 @@ elim (lstas_split … H) -H #U #HTU
 >(lstas_mono … HTU … HTU1) -T //
 qed-.
 
-(* Advanced properties ******************************************************)
-
-lemma lstas_sta_conf_pos: ∀h,G,L,T,U1. ⦃G, L⦄ ⊢ T •[h] U1 →
-                          ∀U2,l. ⦃G, L⦄ ⊢ T •*[h, l+1] U2 → ⦃G, L⦄ ⊢ U1 •*[h, l] U2.
-#h #G #L #T #U1 #HTU1 #U2 #l #HTU2
-lapply (lstas_conf_le … T U1 1 … HTU2) -HTU2 /2 width=1 by sta_lstas/
-qed-.
-
-lemma lstas_strip_pos: ∀h,G,L,T1,U1. ⦃G, L⦄ ⊢ T1 •[h] U1 →
-                       ∀T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
-                       ∃∃U2. ⦃G, L⦄ ⊢ T2 •[h] U2 & ⦃G, L⦄ ⊢ U1 •*[h, l+1] U2.
-#h #G #L #T1 #U1 #HTU1 #T2 #l #HT12
-elim (lstas_fwd_correct … HTU1 … HT12)
-lapply (lstas_sta_conf_pos … HTU1 … HT12) -T1 /3 width=5 by lstas_step_dx, ex2_intro/
+theorem lstas_conf: ∀h,G,L,T0,T1,l1. ⦃G, L⦄ ⊢ T0 •*[h, l1] T1 →
+                    ∀T2,l2. ⦃G, L⦄ ⊢ T0 •*[h, l2] T2 →
+                    ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, l2] T & ⦃G, L⦄ ⊢ T2 •*[h, l1] T.
+#h #G #L #T0 #T1 #l1 #HT01 #T2 #l2 #HT02
+elim (lstas_lstas … HT01 (l1+l2)) #T #HT0
+lapply (lstas_conf_le … HT01 … HT0) // -HT01 <minus_plus_m_m_commutative
+lapply (lstas_conf_le … HT02 … HT0) // -T0 <minus_plus_m_m
+/2 width=3 by ex2_intro/
 qed-.
-- 
2.39.2