From 5d5461cffbec45aebc0db0fe2fcb072592a90efe Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Fri, 17 Jan 2014 19:10:33 +0000
Subject: [PATCH] commit of the "static" component

---
 .../lambdadelta/basic_2/static/aaa.ma         |  16 +--
 .../lambdadelta/basic_2/static/aaa_fqus.ma    |   2 +-
 .../lambdadelta/basic_2/static/aaa_lift.ma    |  53 ++++-----
 .../lambdadelta/basic_2/static/aaa_lifts.ma   |   6 +-
 .../contribs/lambdadelta/basic_2/static/da.ma |  16 +--
 .../lambdadelta/basic_2/static/da_lift.ma     |  66 ++++++------
 .../lambdadelta/basic_2/static/lsuba_ldrop.ma |  40 +++----
 .../lambdadelta/basic_2/static/lsubd.ma       |  56 +++++-----
 .../lambdadelta/basic_2/static/ssta.ma        |  40 +++----
 .../lambdadelta/basic_2/static/ssta_lift.ma   | 101 ++++++++++--------
 10 files changed, 206 insertions(+), 190 deletions(-)

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa.ma
index cc1ed23c2..5e93f55f1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa.ma
@@ -22,7 +22,7 @@ include "basic_2/relocation/ldrop.ma".
 (* activate genv *)
 inductive aaa: relation4 genv lenv term aarity ≝
 | aaa_sort: ∀G,L,k. aaa G L (⋆k) (⓪)
-| aaa_lref: ∀I,G,L,K,V,B,i. ⇩[0, i] L ≡ K. ⓑ{I}V → aaa G K V B → aaa G L (#i) B
+| aaa_lref: ∀I,G,L,K,V,B,i. ⇩[i] L ≡ K. ⓑ{I}V → aaa G K V B → aaa G L (#i) B
 | aaa_abbr: ∀a,G,L,V,T,B,A.
             aaa G L V B → aaa G (L.ⓓV) T A → aaa G L (ⓓ{a}V.T) A
 | aaa_abst: ∀a,G,L,V,T,B,A.
@@ -51,10 +51,10 @@ lemma aaa_inv_sort: ∀G,L,A,k. ⦃G, L⦄ ⊢ ⋆k ⁝ A → A = ⓪.
 /2 width=6 by aaa_inv_sort_aux/ qed-.
 
 fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #i →
-                       ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
+                       ∃∃I,K,V. ⇩[i] L ≡ K.ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
 #G #L #T #A * -G -L -T -A
 [ #G #L #k #i #H destruct
-| #I #G #L #K #V #B #j #HLK #HB #i #H destruct /2 width=5/
+| #I #G #L #K #V #B #j #HLK #HB #i #H destruct /2 width=5 by ex2_3_intro/
 | #a #G #L #V #T #B #A #_ #_ #i #H destruct
 | #a #G #L #V #T #B #A #_ #_ #i #H destruct
 | #G #L #V #T #B #A #_ #_ #i #H destruct
@@ -63,7 +63,7 @@ fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #i →
 qed-.
 
 lemma aaa_inv_lref: ∀G,L,A,i. ⦃G, L⦄ ⊢ #i ⁝ A →
-                    ∃∃I,K,V. ⇩[0, i] L ≡ K. ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
+                    ∃∃I,K,V. ⇩[i] L ≡ K. ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
 /2 width=3 by aaa_inv_lref_aux/ qed-.
 
 fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p. T = §p → ⊥.
@@ -85,7 +85,7 @@ fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀a,W,U. T = ⓓ{
 #G #L #T #A * -G -L -T -A
 [ #G #L #k #a #W #U #H destruct
 | #I #G #L #K #V #B #i #_ #_ #a #W #U #H destruct
-| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=2/
+| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=2 by ex2_intro/
 | #b #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
 | #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
 | #G #L #V #T #A #_ #_ #a #W #U #H destruct
@@ -102,7 +102,7 @@ fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀a,W,U. T = ⓛ{
 [ #G #L #k #a #W #U #H destruct
 | #I #G #L #K #V #B #i #_ #_ #a #W #U #H destruct
 | #b #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=5/
+| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=5 by ex3_2_intro/
 | #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
 | #G #L #V #T #A #_ #_ #a #W #U #H destruct
 ]
@@ -119,7 +119,7 @@ fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U
 | #I #G #L #K #V #B #i #_ #_ #W #U #H destruct
 | #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
 | #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #G #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=3/
+| #G #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=3 by ex2_intro/
 | #G #L #V #T #A #_ #_ #W #U #H destruct
 ]
 qed-.
@@ -136,7 +136,7 @@ fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW.
 | #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
 | #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
 | #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #G #L #V #T #A #HV #HT #W #U #H destruct /2 width=1/
+| #G #L #V #T #A #HV #HT #W #U #H destruct /2 width=1 by conj/
 ]
 qed-.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_fqus.ma
index cc3777262..f9bd61ada 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_fqus.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_fqus.ma
@@ -39,7 +39,7 @@ lemma aaa_fqu_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄
   [ elim (aaa_inv_appl … H)
   | elim (aaa_inv_cast … H)
   ] -H /2 width=2 by ex_intro/
-| /3 width=8 by aaa_inv_lift, ex_intro/
+| /3 width=9 by aaa_inv_lift, ex_intro/
 ]
 qed-.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lift.ma
index ea039002f..f0b5f8a40 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lift.ma
@@ -19,54 +19,55 @@ include "basic_2/static/aaa.ma".
 
 (* Properties on basic relocation *******************************************)
 
-lemma aaa_lift: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀L2,d,e. ⇩[d, e] L2 ≡ L1 →
+lemma aaa_lift: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀L2,s,d,e. ⇩[s, d, e] L2 ≡ L1 →
                 ∀T2. ⇧[d, e] T1 ≡ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
 #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
-[ #G #L1 #k #L2 #d #e #_ #T2 #H
+[ #G #L1 #k #L2 #s #d #e #_ #T2 #H
   >(lift_inv_sort1 … H) -H //
-| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #d #e #HL21 #T2 #H
+| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #s #d #e #HL21 #T2 #H
   elim (lift_inv_lref1 … H) -H * #Hid #H destruct
   [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
     elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
-    /3 width=8/
-  | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 /3 width=8 by aaa_lref/
+    /3 width=9 by aaa_lref/
+  | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1
+    /3 width=9 by aaa_lref, ldrop_inv_gen/
   ]
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #d #e #HL21 #X #H
   elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
-  /4 width=4 by aaa_abbr, ldrop_skip/
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H
+  /4 width=5 by aaa_abbr, ldrop_skip/
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #d #e #HL21 #X #H
   elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
-  /4 width=4 by aaa_abst, ldrop_skip/
-| #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H
+  /4 width=5 by aaa_abst, ldrop_skip/
+| #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #d #e #HL21 #X #H
   elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
-  /3 width=4 by aaa_appl/
-| #G #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #d #e #HL21 #X #H
+  /3 width=5 by aaa_appl/
+| #G #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #s #d #e #HL21 #X #H
   elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
-  /3 width=4 by aaa_cast/
+  /3 width=5 by aaa_cast/
 ]
 qed.
 
-lemma aaa_inv_lift: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀L1,d,e. ⇩[d, e] L2 ≡ L1 →
+lemma aaa_inv_lift: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀L1,s,d,e. ⇩[s, d, e] L2 ≡ L1 →
                     ∀T1. ⇧[d, e] T1 ≡ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
 #G #L2 #T2 #A #H elim H -G -L2 -T2 -A
-[ #G #L2 #k #L1 #d #e #_ #T1 #H
+[ #G #L2 #k #L1 #s #d #e #_ #T1 #H
   >(lift_inv_sort2 … H) -H //
-| #I #G #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #d #e #HL21 #T1 #H
+| #I #G #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #s #d #e #HL21 #T1 #H
   elim (lift_inv_lref2 … H) -H * #Hid #H destruct
-  [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by aaa_lref/
-  | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 /3 width=8 by aaa_lref/
+  [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 /3 width=9 by aaa_lref/
+  | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 /3 width=9 by aaa_lref/
   ]
-| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #d #e #HL21 #X #H
+| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #d #e #HL21 #X #H
   elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
-  /4 width=4 by aaa_abbr, ldrop_skip/
-| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #d #e #HL21 #X #H
+  /4 width=5 by aaa_abbr, ldrop_skip/
+| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #d #e #HL21 #X #H
   elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
-  /4 width=4 by aaa_abst, ldrop_skip/
-| #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #d #e #HL21 #X #H
+  /4 width=5 by aaa_abst, ldrop_skip/
+| #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #d #e #HL21 #X #H
   elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
-  /3 width=4 by aaa_appl/
-| #G #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #d #e #HL21 #X #H
+  /3 width=5 by aaa_appl/
+| #G #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #s #d #e #HL21 #X #H
   elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
-  /3 width=4 by aaa_cast/
+  /3 width=5 by aaa_cast/
 ]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lifts.ma
index 6cd3855c5..b803d42ae 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lifts.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lifts.ma
@@ -19,12 +19,12 @@ include "basic_2/static/aaa_lift.ma".
 
 (* Properties concerning generic relocation *********************************)
 
-lemma aaa_lifts: ∀G,L1,L2,T2,A,des. ⇩*[des] L2 ≡ L1 → ∀T1. ⇧*[des] T1 ≡ T2 →
+lemma aaa_lifts: ∀G,L1,L2,T2,A,s,des. ⇩*[s, des] L2 ≡ L1 → ∀T1. ⇧*[des] T1 ≡ T2 →
                  ⦃G, L1⦄ ⊢ T1 ⁝ A → ⦃G, L2⦄ ⊢ T2 ⁝ A.
-#G #L1 #L2 #T2 #A #des #H elim H -L1 -L2 -des
+#G #L1 #L2 #T2 #A #s #des #H elim H -L1 -L2 -des
 [ #L #T1 #H #HT1
   <(lifts_inv_nil … H) -H //
 | #L1 #L #L2 #des #d #e #_ #HL2 #IHL1 #T1 #H #HT1
-  elim (lifts_inv_cons … H) -H /3 width=9/
+  elim (lifts_inv_cons … H) -H /3 width=10 by aaa_lift/
 ]
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/da.ma b/matita/matita/contribs/lambdadelta/basic_2/static/da.ma
index dc594f68d..ad4d8c182 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/da.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/da.ma
@@ -22,8 +22,8 @@ include "basic_2/static/sd.ma".
 (* activate genv *)
 inductive da (h:sh) (g:sd h): relation4 genv lenv term nat ≝
 | da_sort: ∀G,L,k,l. deg h g k l → da h g G L (⋆k) l
-| da_ldef: ∀G,L,K,V,i,l. ⇩[0, i] L ≡ K.ⓓV → da h g G K V l → da h g G L (#i) l
-| da_ldec: ∀G,L,K,W,i,l. ⇩[0, i] L ≡ K.ⓛW → da h g G K W l → da h g G L (#i) (l+1)
+| da_ldef: ∀G,L,K,V,i,l. ⇩[i] L ≡ K.ⓓV → da h g G K V l → da h g G L (#i) l
+| da_ldec: ∀G,L,K,W,i,l. ⇩[i] L ≡ K.ⓛW → da h g G K W l → da h g G L (#i) (l+1)
 | da_bind: ∀a,I,G,L,V,T,l. da h g G (L.ⓑ{I}V) T l → da h g G L (ⓑ{a,I}V.T) l
 | da_flat: ∀I,G,L,V,T,l. da h g G L T l → da h g G L (ⓕ{I}V.T) l
 .
@@ -48,22 +48,22 @@ lemma da_inv_sort: ∀h,g,G,L,k,l. ⦃G, L⦄ ⊢ ⋆k ▪[h, g] l → deg h g k
 /2 width=5 by da_inv_sort_aux/ qed-.
 
 fact da_inv_lref_aux: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l → ∀j. T = #j →
-                      (∃∃K,V. ⇩[0, j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, g] l) ∨
-                      (∃∃K,W,l0. ⇩[0, j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l0 &
+                      (∃∃K,V. ⇩[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, g] l) ∨
+                      (∃∃K,W,l0. ⇩[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l0 &
                                  l = l0 + 1
                        ).
 #h #g #G #L #T #l * -G -L -T -l
 [ #G #L #k #l #_ #j #H destruct
-| #G #L #K #V #i #l #HLK #HV #j #H destruct /3 width=4/
-| #G #L #K #W #i #l #HLK #HW #j #H destruct /3 width=6/
+| #G #L #K #V #i #l #HLK #HV #j #H destruct /3 width=4 by ex2_2_intro, or_introl/
+| #G #L #K #W #i #l #HLK #HW #j #H destruct /3 width=6 by ex3_3_intro, or_intror/
 | #a #I #G #L #V #T #l #_ #j #H destruct
 | #I #G #L #V #T #l #_ #j #H destruct
 ]
 qed-.
 
 lemma da_inv_lref: ∀h,g,G,L,j,l. ⦃G, L⦄ ⊢ #j ▪[h, g] l →
-                   (∃∃K,V. ⇩[0, j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, g] l) ∨
-                   (∃∃K,W,l0. ⇩[0, j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l0 & l = l0+1).
+                   (∃∃K,V. ⇩[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, g] l) ∨
+                   (∃∃K,W,l0. ⇩[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l0 & l = l0+1).
 /2 width=3 by da_inv_lref_aux/ qed-.
 
 fact da_inv_gref_aux: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l → ∀p0. T = §p0 → ⊥.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/da_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/static/da_lift.ma
index c0f13ceb7..78d575b8d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/da_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/da_lift.ma
@@ -20,53 +20,59 @@ include "basic_2/static/da.ma".
 (* Properties on relocation *************************************************)
 
 lemma da_lift: ∀h,g,G,L1,T1,l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l →
-               ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → ∀T2. ⇧[d, e] T1 ≡ T2 →
+               ∀L2,s,d,e. ⇩[s, d, e] L2 ≡ L1 → ∀T2. ⇧[d, e] T1 ≡ T2 →
                ⦃G, L2⦄ ⊢ T2 ▪[h, g] l.
 #h #g #G #L1 #T1 #l #H elim H -G -L1 -T1 -l
-[ #G #L1 #k #l #Hkl #L2 #d #e #_ #X #H
-  >(lift_inv_sort1 … H) -X /2 width=1/
-| #G #L1 #K1 #V1 #i #l #HLK1 #_ #IHV1 #L2 #d #e #HL21 #X #H
+[ #G #L1 #k #l #Hkl #L2 #s #d #e #_ #X #H
+  >(lift_inv_sort1 … H) -X /2 width=1 by da_sort/
+| #G #L1 #K1 #V1 #i #l #HLK1 #_ #IHV1 #L2 #s #d #e #HL21 #X #H
   elim (lift_inv_lref1 … H) * #Hid #H destruct
-  [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2/ #X #HLK2 #H
-    elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
-    /3 width=7/
-  | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=7/
+  [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+    /3 width=9 by da_ldef/
+  | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1
+    /3 width=8 by da_ldef, ldrop_inv_gen/
   ]
-| #G #L1 #K1 #W1 #i #l #HLK1 #_ #IHW1 #L2 #d #e #HL21 #X #H
+| #G #L1 #K1 #W1 #i #l #HLK1 #_ #IHW1 #L2 #s #d #e #HL21 #X #H
   elim (lift_inv_lref1 … H) * #Hid #H destruct
-  [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2/ #X #HLK2 #H
-    elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
-    /3 width=7/
-  | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=7/
+  [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
+    /3 width=8 by da_ldec/
+  | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1
+    /3 width=8 by da_ldec, ldrop_inv_gen/
   ]
-| #a #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #d #e #HL21 #X #H
-  elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct /4 width=4/
-| #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #d #e #HL21 #X #H
-  elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct /3 width=4/
+| #a #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #s #d #e #HL21 #X #H
+  elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct
+  /4 width=5 by da_bind, ldrop_skip/
+| #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #s #d #e #HL21 #X #H
+  elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct
+  /3 width=5 by da_flat/
 ]
 qed.
 
 (* Inversion lemmas on relocation *******************************************)
 
 lemma da_inv_lift: ∀h,g,G,L2,T2,l. ⦃G, L2⦄ ⊢ T2 ▪[h, g] l →
-                   ∀L1,d,e. ⇩[d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 →
+                   ∀L1,s,d,e. ⇩[s, d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 →
                    ⦃G, L1⦄ ⊢ T1 ▪[h, g] l.
 #h #g #G #L2 #T2 #l #H elim H -G -L2 -T2 -l
-[ #G #L2 #k #l #Hkl #L1 #d #e #_ #X #H
-  >(lift_inv_sort2 … H) -X /2 width=1/
-| #G #L2 #K2 #V2 #i #l #HLK2 #HV2 #IHV2 #L1 #d #e #HL21 #X #H
+[ #G #L2 #k #l #Hkl #L1 #s #d #e #_ #X #H
+  >(lift_inv_sort2 … H) -X /2 width=1 by da_sort/
+| #G #L2 #K2 #V2 #i #l #HLK2 #HV2 #IHV2 #L1 #s #d #e #HL21 #X #H
   elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HV2 | -IHV2 ]
-  [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 // /3 width=7/
-  | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // /2 width=4/
+  [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldef/
+  | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldef/
   ]
-| #G #L2 #K2 #W2 #i #l #HLK2 #HW2 #IHW2 #L1 #d #e #HL21 #X #H
+| #G #L2 #K2 #W2 #i #l #HLK2 #HW2 #IHW2 #L1 #s #d #e #HL21 #X #H
   elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HW2 | -IHW2 ]
-  [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 // /3 width=7/
-  | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // /2 width=4/
+  [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldec/
+  | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldec/
   ]
-| #a #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #d #e #HL21 #X #H
-  elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct /4 width=4/
-| #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #d #e #HL21 #X #H
-  elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct /3 width=4/
+| #a #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #s #d #e #HL21 #X #H
+  elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+  /4 width=5 by da_bind, ldrop_skip/
+| #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #s #d #e #HL21 #X #H
+  elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+  /3 width=5 by da_flat/
 ]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_ldrop.ma
index 05afc6ebd..43de659bf 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_ldrop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba_ldrop.ma
@@ -19,45 +19,45 @@ include "basic_2/static/lsuba.ma".
 (* Properties concerning basic local environment slicing ********************)
 
 (* Note: the constant 0 cannot be generalized *)
-lemma lsuba_ldrop_O1_conf: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀K1,e. ⇩[0, e] L1 ≡ K1 →
-                           ∃∃K2. G ⊢ K1 ⁝⊑ K2 & ⇩[0, e] L2 ≡ K2.
+lemma lsuba_ldrop_O1_conf: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀K1,s,e. ⇩[s, 0, e] L1 ≡ K1 →
+                           ∃∃K2. G ⊢ K1 ⁝⊑ K2 & ⇩[s, 0, e] L2 ≡ K2.
 #G #L1 #L2 #H elim H -L1 -L2
 [ /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #e #H
   elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
   [ destruct
-    elim (IHL12 L1 0) -IHL12 // #X #HL12 #H
-    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
-  | elim (IHL12 … HLK1) -L1 /3 width=3/
+    elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, ldrop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
   ]
-| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #e #H
+| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #s #e #H
   elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
   [ destruct
-    elim (IHL12 L1 0) -IHL12 // #X #HL12 #H
-    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
-  | elim (IHL12 … HLK1) -L1 /3 width=3/
+    elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_abbr, ldrop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
   ]
 ]
 qed-.
 
 (* Note: the constant 0 cannot be generalized *)
-lemma lsuba_ldrop_O1_trans: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
-                            ∃∃K1. G ⊢ K1 ⁝⊑ K2 & ⇩[0, e] L1 ≡ K1.
+lemma lsuba_ldrop_O1_trans: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀K2,s,e. ⇩[s, 0, e] L2 ≡ K2 →
+                            ∃∃K1. G ⊢ K1 ⁝⊑ K2 & ⇩[s, 0, e] L1 ≡ K1.
 #G #L1 #L2 #H elim H -L1 -L2
 [ /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #e #H
   elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
   [ destruct
-    elim (IHL12 L2 0) -IHL12 // #X #HL12 #H
-    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
-  | elim (IHL12 … HLK2) -L2 /3 width=3/
+    elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, ldrop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
   ]
-| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #e #H
+| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #s #e #H
   elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
   [ destruct
-    elim (IHL12 L2 0) -IHL12 // #X #HL12 #H
-    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
-  | elim (IHL12 … HLK2) -L2 /3 width=3/
+    elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_abbr, ldrop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
   ]
 ]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsubd.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsubd.ma
index 0644771eb..32ca7ad3d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lsubd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsubd.ma
@@ -33,7 +33,7 @@ interpretation
 (* Basic_forward lemmas *****************************************************)
 
 lemma lsubd_fwd_lsubr: ∀h,g,G,L1,L2. G ⊢ L1 ▪⊑[h, g] L2 → L1 ⊑ L2.
-#h #g #G #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+#h #g #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_bind, lsubr_abst/
 qed-.
 
 (* Basic inversion lemmas ***************************************************)
@@ -57,8 +57,8 @@ fact lsubd_inv_pair1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ▪⊑[h, g] L2 →
                                       I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
 #h #g #G #L1 #L2 * -L1 -L2
 [ #J #K1 #X #H destruct
-| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3/
-| #L1 #L2 #W #V #l #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9/
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #l #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by ex6_4_intro, or_intror/
 ]
 qed-.
 
@@ -87,8 +87,8 @@ fact lsubd_inv_pair2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ▪⊑[h, g] L2 →
                                     G ⊢ K1 ▪⊑[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
 #h #g #G #L1 #L2 * -L1 -L2
 [ #J #K2 #U #H destruct
-| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3/
-| #L1 #L2 #W #V #l #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7/
+| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #l #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by ex5_3_intro, or_intror/
 ]
 qed-.
 
@@ -101,51 +101,51 @@ lemma lsubd_inv_pair2: ∀h,g,I,G,L1,K2,W. G ⊢ L1 ▪⊑[h, g] K2.ⓑ{I}W →
 (* Basic properties *********************************************************)
 
 lemma lsubd_refl: ∀h,g,G,L. G ⊢ L ▪⊑[h, g] L.
-#h #g #G #L elim L -L // /2 width=1/
+#h #g #G #L elim L -L /2 width=1 by lsubd_pair/
 qed.
 
 (* Note: the constant 0 cannot be generalized *)
 lemma lsubd_ldrop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ▪⊑[h, g] L2 →
-                           ∀K1,e. ⇩[0, e] L1 ≡ K1 →
-                           ∃∃K2. G ⊢ K1 ▪⊑[h, g] K2 & ⇩[0, e] L2 ≡ K2.
+                           ∀K1,s,e. ⇩[s, 0, e] L1 ≡ K1 →
+                           ∃∃K2. G ⊢ K1 ▪⊑[h, g] K2 & ⇩[s, 0, e] L2 ≡ K2.
 #h #g #G #L1 #L2 #H elim H -L1 -L2
-[ /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #e #H
   elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
   [ destruct
-    elim (IHL12 L1 0) -IHL12 // #X #HL12 #H
-    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
-  | elim (IHL12 … HLK1) -L1 /3 width=3/
+    elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, ldrop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
   ]
-| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K1 #e #H
+| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K1 #s #e #H
   elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
   [ destruct
-    elim (IHL12 L1 0) -IHL12 // #X #HL12 #H
-    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
-  | elim (IHL12 … HLK1) -L1 /3 width=3/
+    elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_abbr, ldrop_pair, ex2_intro/
+  | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
   ]
 ]
 qed-.
 
 (* Note: the constant 0 cannot be generalized *)
 lemma lsubd_ldrop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ▪⊑[h, g] L2 →
-                            ∀K2,e. ⇩[0, e] L2 ≡ K2 →
-                            ∃∃K1. G ⊢ K1 ▪⊑[h, g] K2 & ⇩[0, e] L1 ≡ K1.
+                            ∀K2,s,e. ⇩[s, 0, e] L2 ≡ K2 →
+                            ∃∃K1. G ⊢ K1 ▪⊑[h, g] K2 & ⇩[s, 0, e] L1 ≡ K1.
 #h #g #G #L1 #L2 #H elim H -L1 -L2
-[ /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #e #H
   elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
   [ destruct
-    elim (IHL12 L2 0) -IHL12 // #X #HL12 #H
-    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
-  | elim (IHL12 … HLK2) -L2 /3 width=3/
+    elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, ldrop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
   ]
-| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K2 #e #H
+| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K2 #s #e #H
   elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
   [ destruct
-    elim (IHL12 L2 0) -IHL12 // #X #HL12 #H
-    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
-  | elim (IHL12 … HLK2) -L2 /3 width=3/
+    elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+    <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_abbr, ldrop_pair, ex2_intro/
+  | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
   ]
 ]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ssta.ma
index eede2acc8..f09d03b86 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/ssta.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/ssta.ma
@@ -20,9 +20,9 @@ include "basic_2/static/da.ma".
 (* activate genv *)
 inductive ssta (h) (g): relation4 genv lenv term term ≝
 | ssta_sort: ∀G,L,k. ssta h g G L (⋆k) (⋆(next h k))
-| ssta_ldef: ∀G,L,K,V,U,W,i. ⇩[0, i] L ≡ K.ⓓV → ssta h g G K V W →
+| ssta_ldef: ∀G,L,K,V,U,W,i. ⇩[i] L ≡ K.ⓓV → ssta h g G K V W →
              ⇧[0, i + 1] W ≡ U → ssta h g G L (#i) U
-| ssta_ldec: ∀G,L,K,W,U,l,i. ⇩[0, i] L ≡ K.ⓛW → ⦃G, K⦄ ⊢ W ▪[h, g] l →
+| ssta_ldec: ∀G,L,K,W,U,l,i. ⇩[i] L ≡ K.ⓛW → ⦃G, K⦄ ⊢ W ▪[h, g] l →
              ⇧[0, i + 1] W ≡ U → ssta h g G L (#i) U
 | ssta_bind: ∀a,I,G,L,V,T,U. ssta h g G (L.ⓑ{I}V) T U →
              ssta h g G L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
@@ -51,16 +51,16 @@ lemma ssta_inv_sort1: ∀h,g,G,L,U,k. ⦃G, L⦄ ⊢ ⋆k •[h, g] U → U = 
 /2 width=6 by ssta_inv_sort1_aux/ qed-.
 
 fact ssta_inv_lref1_aux: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U → ∀j. T = #j →
-                         (∃∃K,V,W. ⇩[0, j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h, g] W &
+                         (∃∃K,V,W. ⇩[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h, g] W &
                                    ⇧[0, j + 1] W ≡ U
                          ) ∨
-                         (∃∃K,W,l. ⇩[0, j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l &
+                         (∃∃K,W,l. ⇩[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l &
                                    ⇧[0, j + 1] W ≡ U
                          ).
 #h #g #G #L #T #U * -G -L -T -U
 [ #G #L #k #j #H destruct
-| #G #L #K #V #U #W #i #HLK #HVW #HWU #j #H destruct /3 width=6/
-| #G #L #K #W #U #l #i #HLK #HWl #HWU #j #H destruct /3 width=6/
+| #G #L #K #V #U #W #i #HLK #HVW #HWU #j #H destruct /3 width=6 by ex3_3_intro, or_introl/
+| #G #L #K #W #U #l #i #HLK #HWl #HWU #j #H destruct /3 width=6 by ex3_3_intro, or_intror/
 | #a #I #G #L #V #T #U #_ #j #H destruct
 | #G #L #V #T #U #_ #j #H destruct
 | #G #L #W #T #U #_ #j #H destruct
@@ -68,10 +68,10 @@ fact ssta_inv_lref1_aux: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U → ∀j.
 qed-.
 
 lemma ssta_inv_lref1: ∀h,g,G,L,U,i. ⦃G, L⦄ ⊢ #i •[h, g] U →
-                      (∃∃K,V,W. ⇩[0, i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h, g] W &
+                      (∃∃K,V,W. ⇩[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h, g] W &
                                 ⇧[0, i + 1] W ≡ U
                       ) ∨
-                      (∃∃K,W,l. ⇩[0, i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l &
+                      (∃∃K,W,l. ⇩[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l &
                                 ⇧[0, i + 1] W ≡ U
                       ).
 /2 width=3 by ssta_inv_lref1_aux/ qed-.
@@ -97,7 +97,7 @@ fact ssta_inv_bind1_aux: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U →
 [ #G #L #k #b #J #X #Y #H destruct
 | #G #L #K #V #U #W #i #_ #_ #_ #b #J #X #Y #H destruct
 | #G #L #K #W #U #l #i #_ #_ #_ #b #J #X #Y #H destruct
-| #a #I #G #L #V #T #U #HTU #b #J #X #Y #H destruct /2 width=3/
+| #a #I #G #L #V #T #U #HTU #b #J #X #Y #H destruct /2 width=3 by ex2_intro/
 | #G #L #V #T #U #_ #b #J #X #Y #H destruct
 | #G #L #W #T #U #_ #b #J #X #Y #H destruct
 ]
@@ -114,7 +114,7 @@ fact ssta_inv_appl1_aux: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U → ∀X,Y
 | #G #L #K #V #U #W #i #_ #_ #_ #X #Y #H destruct
 | #G #L #K #W #U #l #i #_ #_ #_ #X #Y #H destruct
 | #a #I #G #L #V #T #U #_ #X #Y #H destruct
-| #G #L #V #T #U #HTU #X #Y #H destruct /2 width=3/
+| #G #L #V #T #U #HTU #X #Y #H destruct /2 width=3 by ex2_intro/
 | #G #L #W #T #U #_ #X #Y #H destruct
 ]
 qed-.
@@ -143,12 +143,12 @@ lemma ssta_inv_cast1: ∀h,g,G,L,X,Y,U. ⦃G, L⦄ ⊢ ⓝY.X •[h, g] U → 
 lemma ssta_inv_da: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U →
                    ∃l. ⦃G, L⦄ ⊢ T ▪[h, g] l.
 #h #g #G #L #T #U #H elim H -G -L -T -U
-[ #G #L #k elim (deg_total h g k) /3 width=2/
-| #G #L #K #V #U #W #i #HLK #_ #_ * /3 width=5/
-| #G #L #K #W #U #l #i #HLK #HWl #_ /3 width=5/
-| #a #I #G #L #V #T #U #_ * /3 width=2/
-| #G #L #V #T #U #_ * /3 width=2/
-| #G #L #W #T #U #_ * /3 width=2/
+[ #G #L #k elim (deg_total h g k) /3 width=2 by da_sort, ex_intro/
+| #G #L #K #V #U #W #i #HLK #_ #_ * /3 width=5 by da_ldef, ex_intro/
+| #G #L #K #W #U #l #i #HLK #HWl #_ /3 width=5 by da_ldec, ex_intro/
+| #a #I #G #L #V #T #U #_ * /3 width=2 by da_bind, ex_intro/
+| #G #L #V #T #U #_ * /3 width=2 by da_flat, ex_intro/
+| #G #L #W #T #U #_ * /3 width=2 by da_flat, ex_intro/
 ]
 qed-.
 
@@ -159,11 +159,11 @@ lemma da_ssta: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l →
 #h #g #G #L #T #l #H elim H -G -L -T -l
 [ /2 width=2/
 | #G #L #K #V #i #l #HLK #_ * #W #HVW
-  elim (lift_total W 0 (i+1)) /3 width=7/
+  elim (lift_total W 0 (i+1)) /3 width=7 by ssta_ldef, ex_intro/
 | #G #L #K #W #i #l #HLK #HW #_
-  elim (lift_total W 0 (i+1)) /3 width=7/
-| #a #I #G #L #V #T #l #_ * /3 width=2/
-| * #G #L #V #T #l #_ * /3 width=2/
+  elim (lift_total W 0 (i+1)) /3 width=7 by ssta_ldec, ex_intro/
+| #a #I #G #L #V #T #l #_ * /3 width=2 by ssta_bind, ex_intro/
+| * #G #L #V #T #l #_ * /3 width=2 by ssta_appl, ssta_cast, ex_intro/
 ]
 qed-.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ssta_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ssta_lift.ma
index ff975d25a..843bc3a1e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/ssta_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/ssta_lift.ma
@@ -21,38 +21,40 @@ include "basic_2/static/ssta.ma".
 
 lemma ssta_lift: ∀h,g,G. l_liftable (ssta h g G).
 #h #g #G #L1 #T1 #U1 #H elim H -G -L1 -T1 -U1
-[ #G #L1 #k #L2 #d #e #HL21 #X1 #H1 #X2 #H2
+[ #G #L1 #k #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
   >(lift_inv_sort1 … H1) -X1
   >(lift_inv_sort1 … H2) -X2 //
-| #G #L1 #K1 #V1 #U1 #W1 #i #HLK1 #_ #HWU1 #IHVW1 #L2 #d #e #HL21 #X #H #U2 #HU12
+| #G #L1 #K1 #V1 #U1 #W1 #i #HLK1 #_ #HWU1 #IHVW1 #L2 #s #d #e #HL21 #X #H #U2 #HU12
   elim (lift_inv_lref1 … H) * #Hid #H destruct
   [ elim (lift_trans_ge … HWU1 … HU12) -U1 // #W2 #HW12 #HWU2
-    elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2/ #X #HLK2 #H
-    elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
-    /3 width=8/
-  | lapply (lift_trans_be … HWU1 … HU12 ? ?) -U1 // /2 width=1/ #HW1U2
-    lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
+    elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+    /3 width=9 by ssta_ldef/
+  | lapply (lift_trans_be … HWU1 … HU12 ? ?) -U1 /2 width=1 by le_S/ #HW1U2
+    lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid
+    /3 width=9 by ssta_ldef, ldrop_inv_gen/
   ]
-| #G #L1 #K1 #W1 #U1 #l #i #HLK1 #HW1l #HWU1 #L2 #d #e #HL21 #X #H #U2 #HU12
+| #G #L1 #K1 #W1 #U1 #l #i #HLK1 #HW1l #HWU1 #L2 #s #d #e #HL21 #X #H #U2 #HU12
   elim (lift_inv_lref1 … H) * #Hid #H destruct
   [ elim (lift_trans_ge … HWU1 … HU12) -U1 // <minus_plus #W2 #HW12 #HWU2
-    elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2/ #X #HLK2 #H
-    elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K2 #W #HK21 #HW1 #H destruct
+    elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+    elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #W #HK21 #HW1 #H destruct
     lapply (lift_mono … HW1 … HW12) -HW1 #H destruct
-    /3 width=10 by da_lift, ssta_ldec/
-  | lapply (lift_trans_be … HWU1 … HU12 ? ?) -U1 // /2 width=1/ #HW1U2
-    lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
+    /3 width=11 by da_lift, ssta_ldec/
+  | lapply (lift_trans_be … HWU1 … HU12 ? ?) -U1 /2 width=1 by le_S/ #HW1U2
+    lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid
+    /3 width=8 by ssta_ldec, ldrop_inv_gen/
   ]
-| #a #I #G #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
+| #a #I #G #L1 #V1 #T1 #U1 #_ #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=5/
-| #G #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL21 #X1 #H1 #X2 #H2
+  lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by ssta_bind, ldrop_skip/
+| #G #L1 #V1 #T1 #U1 #_ #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=5/
-| #G #L1 #W1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL21 #X #H #U2 #HU12
-  elim (lift_inv_flat1 … H) -H #W2 #T2 #_ #HT12 #H destruct /3 width=5/
+  lapply (lift_mono … H1 … HV12) -H1 #H destruct /4 width=6 by ssta_appl/
+| #G #L1 #W1 #T1 #U1 #_ #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 ssta_cast/
 ]
 qed.
 
@@ -60,41 +62,48 @@ qed.
 
 lemma ssta_inv_lift1: ∀h,g,G. l_deliftable_sn (ssta h g G).
 #h #g #G #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2
-[ #G #L2 #k #L1 #d #e #_ #X #H
-  >(lift_inv_sort2 … H) -X /2 width=3/
-| #G #L2 #K2 #V2 #U2 #W2 #i #HLK2 #HVW2 #HWU2 #IHVW2 #L1 #d #e #HL21 #X #H
+[ #G #L2 #k #L1 #s #d #e #_ #X #H
+  >(lift_inv_sort2 … H) -X /2 width=3 by ssta_sort, lift_sort, ex2_intro/
+| #G #L2 #K2 #V2 #U2 #W2 #i #HLK2 #HVW2 #HWU2 #IHVW2 #L1 #s #d #e #HL21 #X #H
   elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HVW2 | -IHVW2 ]
   [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12
     elim (IHVW2 … HK21 … HV12) -K2 -V2 #W1 #HW12 #HVW1
-    elim (lift_trans_le … HW12 … HWU2) -W2 // >minus_plus <plus_minus_m_m // -Hid /3 width=8/
+    elim (lift_trans_le … HW12 … HWU2) -W2 // >minus_plus <plus_minus_m_m
+    /3 width=8 by ssta_ldef, ex2_intro/
   | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
     elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
-    elim (lift_split … HWU2 d (i-e+1)) -HWU2 // [3: /2 width=1/ ]
-    [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /2 width=1/ /3 width=8/
+    elim (lift_split … HWU2 d (i-e+1)) -HWU2 // [3: /2 width=1 by le_S/ ]
+    [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m
+      /3 width=8 by ssta_ldef, le_S, ex2_intro/
     | <le_plus_minus_comm //
     ]
   ]
-| #G #L2 #K2 #W2 #U2 #l #i #HLK2 #HW2l #HWU2 #L1 #d #e #HL21 #X #H
+| #G #L2 #K2 #W2 #U2 #l #i #HLK2 #HW2l #HWU2 #L1 #s #d #e #HL21 #X #H
   elim (lift_inv_lref2 … H) * #Hid #H destruct
   [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
     lapply (da_inv_lift … HW2l … HK21 … HW12) -K2
-    elim (lift_trans_le … HW12 … HWU2) -W2 // >minus_plus <plus_minus_m_m // -Hid /3 width=8/
+    elim (lift_trans_le … HW12 … HWU2) -W2 // >minus_plus <plus_minus_m_m
+    /3 width=8 by ssta_ldec, ex2_intro/
   | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // #HL1K2
     elim (le_inv_plus_l … Hid) -Hid #Hdie #ei
-    elim (lift_split … HWU2 d (i-e+1)) -HWU2 // [3: /2 width=1/ ]
-    [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /2 width=1/ /3 width=8/
+    elim (lift_split … HWU2 d (i-e+1)) -HWU2 // [3: /2 width=1 by le_S/ ]
+    [ #W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m
+      /3 width=8 by ssta_ldec, le_S, ex2_intro/
     | <le_plus_minus_comm //
     ]
   ]
-| #a #I #G #L2 #V2 #T2 #U2 #_ #IHTU2 #L1 #d #e #HL21 #X #H
+| #a #I #G #L2 #V2 #T2 #U2 #_ #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 /2 width=1/ -HL21 /3 width=5/
-| #G #L2 #V2 #T2 #U2 #_ #IHTU2 #L1 #d #e #HL21 #X #H
+  elim (IHTU2 (L1.ⓑ{I}V1) … HT12) -IHTU2 -HT12
+  /3 width=5 by ssta_bind, ldrop_skip, lift_bind, ex2_intro/
+| #G #L2 #V2 #T2 #U2 #_ #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/
-| #G #L2 #W2 #T2 #U2 #_ #IHTU2 #L1 #d #e #HL21 #X #H
+  elim (IHTU2 … HL21 … HT12) -L2 -HT12
+  /3 width=5 by ssta_appl, lift_flat, ex2_intro/
+| #G #L2 #W2 #T2 #U2 #_ #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/
+  elim (IHTU2 … HL21 … HT12) -L2 -HT12
+ /3 width=3 by ssta_cast, ex2_intro/
 ]
 qed-.
 
@@ -104,21 +113,21 @@ lemma ssta_da_conf: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U →
                     ∀l. ⦃G, L⦄ ⊢ T ▪[h, g] l → ⦃G, L⦄ ⊢ U ▪[h, g] l-1.
 #h #g #G #L #T #U #H elim H -G -L -T -U
 [ #G #L #k #l #H
-  lapply (da_inv_sort … H) -H /3 width=1/
+  lapply (da_inv_sort … H) -H /3 width=1 by da_sort, deg_next/
 | #G #L #K #V #U #W #i #HLK #_ #HWU #IHVW #l #H
   elim (da_inv_lref … H) -H * #K0 #V0 [| #l0] #HLK0 #HV0
   lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct
-  lapply (ldrop_fwd_drop2 … HLK) -HLK /3 width=7/
+  lapply (ldrop_fwd_drop2 … HLK) -HLK /3 width=8 by da_lift/
 | #G #L #K #W #U #l0 #i #HLK #_ #HWU #l #H -l0
   elim (da_inv_lref … H) -H * #K0 #V0 [| #l1] #HLK0 #HV0 [| #H0 ]
   lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct
-  lapply (ldrop_fwd_drop2 … HLK) -HLK /3 width=7/
+  lapply (ldrop_fwd_drop2 … HLK) -HLK /3 width=8 by da_lift/
 | #a #I #G #L #V #T #U #_ #IHTU #l #H
-  lapply (da_inv_bind … H) -H /3 width=1/
+  lapply (da_inv_bind … H) -H /3 width=1 by da_bind/
 | #G #L #V #T #U #_ #IHTU #l #H
-  lapply (da_inv_flat … H) -H /3 width=1/
+  lapply (da_inv_flat … H) -H /3 width=1 by da_flat/
 | #G #L #W #T #U #_ #IHTU #l #H
-  lapply (da_inv_flat … H) -H /2 width=1/
+  lapply (da_inv_flat … H) -H /2 width=1 by /
 ]
 qed-.
 
@@ -130,13 +139,13 @@ lemma ssta_fwd_correct: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U →
 [ /2 width=2/
 | #G #L #K #V #U #W #i #HLK #_ #HWU * #T #HWT
   lapply (ldrop_fwd_drop2 … HLK) -HLK #HLK
-  elim (lift_total T 0 (i+1)) /3 width=10/
+  elim (lift_total T 0 (i+1)) /3 width=11 by ssta_lift, ex_intro/
 | #G #L #K #W #U #l #i #HLK #HWl #HWU
   elim (da_ssta … HWl) -HWl #T #HWT
   lapply (ldrop_fwd_drop2 … HLK) -HLK #HLK
-  elim (lift_total T 0 (i+1)) /3 width=10/
-| #a #I #G #L #V #T #U #_ * /3 width=2/
-| #G #L #V #T #U #_ * #T0 #HUT0 /3 width=2/
-| #G #L #W #T #U #_ * /2 width=2/
+  elim (lift_total T 0 (i+1)) /3 width=11 by ssta_lift, ex_intro/
+| #a #I #G #L #V #T #U #_ * /3 width=2 by ssta_bind, ex_intro/
+| #G #L #V #T #U #_ * #T0 #HUT0 /3 width=2 by ssta_appl, ex_intro/
+| #G #L #W #T #U #_ * /2 width=2 by ex_intro/
 ]
 qed-.
-- 
2.39.2