From 52616a7a6a550efd75ed56e7e246132453506002 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sat, 21 Jan 2017 14:34:59 +0000
Subject: [PATCH] - improved fqu allows to prove fqu_cpx_trans and its
 derivatives - parked frees_fqus_drops that does not hold anymore

---
 .../basic_2/etc_new/frees/frees_fqus.etc      | 111 +++++++++++
 .../basic_2/rt_transition/cpx_fqus.ma         |  54 +++---
 .../basic_2/s_computation/fqup_drops.ma       |   8 +-
 .../lambdadelta/basic_2/s_computation/fqus.ma |  88 ++++++---
 .../basic_2/s_computation/fqus_drops.ma       |   3 +-
 .../lambdadelta/basic_2/s_transition/fqu.ma   | 181 +++++++++---------
 .../lambdadelta/basic_2/static/frees.ma       |  18 ++
 .../lambdadelta/basic_2/static/frees_fqus.ma  |  93 ---------
 .../lambdadelta/basic_2/web/basic_2_src.tbl   |   2 +-
 .../lambdadelta/ground_2/lib/arith.ma         |   4 +
 10 files changed, 321 insertions(+), 241 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees_fqus.etc
 delete mode 100644 matita/matita/contribs/lambdadelta/basic_2/static/frees_fqus.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees_fqus.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees_fqus.etc
new file mode 100644
index 000000000..618d858ff
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees_fqus.etc
@@ -0,0 +1,111 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/relocation/drops_weight.ma".
+include "basic_2/s_computation/fqup_weight.ma".
+include "basic_2/s_computation/fqus_weight.ma".
+include "basic_2/static/frees.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Properties with star-iterated supclosure *********************************)
+
+(* Note: Restrict fqu_drop on atoms to prove frees_fqus_drops
+         Counterex: ⦃⋆, ⋆.ⓛ#0, ⓛ#1.#0⦄ ⊐ ⦃⋆, ⋆, ⓛ#0.#0⦄ ⊐ ⦃⋆, ⋆.ⓛ#0, #0⦄
+*)
+lemma frees_fqus_drops: ∀G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 →
+                        ∀L2,T2. ⦃G, L1, T1⦄ ⊐* ⦃G, L2, T2⦄ →
+                        ∀I,n. ⬇*[n] L1 ≡ L2.ⓑ{I}T2 →
+                        ∃∃f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ ⫱*[⫯n] f1.
+#G #L1 #T1 @(fqup_wf_ind_eq … G L1 T1) -G -L1 -T1
+#G0 #L0 #T0 #IH #G #L1 * *
+[ #s #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct
+  lapply (frees_inv_sort … Hf1) -Hf1 #Hf1
+  elim (fqus_inv_sort1 … H12) -H12 * [| elim (nat_split n) ]
+  [ -IH #H1 #H2 #H3 destruct
+    lapply (drops_fwd_lw … HL12) -HL12 #HL12
+    elim (lt_le_false … HL12) -HL12 //
+  | -IH #Hn #J #L #V #H12 #H destruct
+    lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct
+    <(fqus_inv_refl_atom3 … H12) -H12
+    /3 width=3 by frees_sort_gen, sle_isid_sn, ex2_intro/
+  | * #m #Hn #J #L #V #H12 #H destruct
+    lapply (drops_inv_drop1 … HL12) -HL12 #HL12
+    elim (IH … H12 … HL12) -IH -H12 -HL12
+    /3 width=3 by frees_sort_gen, fqu_fqup, fqu_drop, lifts_sort, isid_tl, ex2_intro/
+  ]
+| #i #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct
+  elim (fqus_inv_lref1 … H12) -H12 * [|2,3: elim (nat_split n) ]
+  [ -IH #H1 #H2 #H3 destruct
+    lapply (drops_fwd_lw … HL12) -HL12 #HL12
+    elim (lt_le_false … HL12) -HL12 //
+  | -IH #Hn #J #L #V #H12 #H1 #H2 destruct
+    elim (frees_inv_zero_pair … Hf1) -Hf1 #g1 #Hg1 #H destruct
+    lapply (drops_fwd_isid … HL12 ?) -HL12 -H12 // #H destruct
+    /2 width=3 by sle_refl, ex2_intro/
+  | * #m #Hn #J #L #V #H12 #H1 #H2 destruct
+    elim (frees_inv_zero_pair … Hf1) -Hf1 #g1 #Hg1 #H destruct
+    lapply (drops_inv_drop1 … HL12) -HL12 #HL12
+    elim (IH … H12 … HL12) -IH -H12 -HL12
+    /2 width=3 by fqu_fqup, fqu_lref_O, ex2_intro/
+  | -IH #Hn #J #L #V #j #H12 #H1 #H2 destruct
+    elim (frees_inv_lref_pair … Hf1) -Hf1 #g1 #Hg1 #H destruct
+    lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct
+    <(fqus_inv_refl_atom3 … H12) -H12 /2 width=3 by sle_refl, ex2_intro/
+  | * #m #Hn #J #L #V #j #H12 #H1 #H2 destruct
+    elim (frees_inv_lref_pair … Hf1) -Hf1 #g1 #Hg1 #H destruct
+    lapply (drops_inv_drop1 … HL12) -HL12 #HL12
+    elim (IH … H12 … HL12) -IH -H12 -HL12 /2 width=3 by fqu_fqup, ex2_intro/
+  ]
+| #l #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct
+  lapply (frees_inv_gref … Hf1) -Hf1 #Hf1
+  elim (fqus_inv_gref1 … H12) -H12 * [| elim (nat_split n) ]
+  [ -IH #H1 #H2 #H3 destruct
+    lapply (drops_fwd_lw … HL12) -HL12 #HL12
+    elim (lt_le_false … HL12) -HL12 //
+  | -IH #Hn #J #L #V #H12 #H destruct
+    lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct
+    <(fqus_inv_refl_atom3 … H12) -H12
+    /3 width=3 by frees_gref_gen, sle_isid_sn, ex2_intro/
+  | * #m #Hn #J #L #V #H12 #H destruct
+    lapply (drops_inv_drop1 … HL12) -HL12 #HL12
+    elim (IH … H12 … HL12) -IH -H12 -HL12
+    /3 width=3 by frees_gref_gen, fqu_fqup, fqu_drop, lifts_sort, isid_tl, ex2_intro/
+  ]
+| #p #I #V1 #T1 #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct
+  elim (frees_inv_bind … Hf1) -Hf1 #f1V #f1T #Hf1V #Hf1T #Hf1
+  elim (fqus_inv_bind1 … H12) -H12
+  [ -IH -f1V -f1T * #H1 #H2 #H3 destruct
+    lapply (drops_fwd_lw … HL12) -HL12 #HL12
+    elim (lt_le_false … HL12) -HL12 //
+  | #H12 elim (IH … H12 … HL12) -IH -H12 -HL12
+    /4 width=6 by sor_tls, sor_inv_sle_sn_trans, ex2_intro/
+  | #H12 elim (IH … H12 I (⫯n)) -IH -H12 [|*: /2 width=3 by drops_drop, drops_refl/ ] -HL12
+    <tls_xn /4 width=6 by ex2_intro, sor_tls, sor_inv_sle_dx_trans/
+  | * #J #L #V #X1 #H12 #HXT1 #H destruct
+  ]
+| #I #V1 #T1 #HG #HL #HT #f1 #Hf1 #L2 #T2 #H12 #I #n #HL12 destruct
+  elim (frees_inv_flat … Hf1) -Hf1 #f1V #f1T #Hf1V #Hf1T #Hf1
+  elim (fqus_inv_flat1 … H12) -H12
+  [ -IH -f1V -f1T * #H1 #H2 #H3 destruct
+    lapply (drops_fwd_lw … HL12) -HL12 #HL12
+    elim (lt_le_false … HL12) -HL12 //
+  | #H12 elim (IH … H12 … HL12) -IH -H12 -HL12
+    /4 width=6 by sor_tls, sor_inv_sle_sn_trans, ex2_intro/
+  | #H12 elim (IH … H12 … HL12) -IH -H12 -HL12
+    /4 width=6 by ex2_intro, sor_tls, sor_inv_sle_dx_trans/
+  | * #J #L #V #X1 #H12 #HXT1 #H destruct 
+  ]
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.ma
index 41763dfa2..c8aaacf06 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.ma
@@ -14,35 +14,38 @@
 
 (* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
 
+include "basic_2/s_computation/fqus_fqup.ma".
+include "basic_2/rt_transition/cpx_drops.ma".
+
 (* Properties on supclosure *************************************************)
 
-lemma fqu_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                     ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 →
-                     ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 
+lemma fqu_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                     ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
+                     ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
 /3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpx_pair_sn, cpx_bind, cpx_flat, ex2_intro/
-[ #I #G #L #V2 #U2 #HVU2
-  elim (lift_total U2 0 1)
-  /4 width=7 by fqu_drop, cpx_delta, drop_pair, drop_drop, ex2_intro/
-| #G #L #K #T1 #U1 #k #HLK1 #HTU1 #T2 #HTU2
-  elim (lift_total T2 0 (k+1))
-  /3 width=11 by cpx_lift, fqu_drop, ex2_intro/
+[ #I #G #L2 #V2 #X2 #HVX2
+  elim (lifts_total X2 (𝐔❴1❵))
+  /3 width=3 by fqu_drop, cpx_delta, ex2_intro/
+| #I #G #L2 #V #T2 #X2 #HTX2 #U2 #HTU2
+  elim (cpx_lifts … HTU2 (Ⓣ) … (L2.ⓑ{I}V) … HTX2)
+  /3 width=3 by fqu_drop, drops_refl, drops_drop, ex2_intro/
 ]
 qed-.
 
-lemma fquq_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
-                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 →
-                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
+lemma fquq_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
+[ #HT12 #U2 #HTU2 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
 ]
 qed-.
 
-lemma fqup_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 →
-                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+lemma fqup_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
 [ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
   /3 width=3 by fqu_fqup, ex2_intro/
 | #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
@@ -51,15 +54,15 @@ lemma fqup_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2,
 ]
 qed-.
 
-lemma fqus_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 →
-                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fqus_inv_gen … H) -H
-[ #HT12 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
+lemma fqus_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H
+[ #HT12 #U2 #HTU2 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
 ]
 qed-.
-
+(*
 lemma fqu_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
@@ -120,3 +123,4 @@ lemma fqus_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2,
 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/
 ]
 qed-.
+*)
\ No newline at end of file
diff --git a/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_drops.ma
index 472b4697f..e9acd8aed 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_drops.ma
@@ -18,7 +18,7 @@ include "basic_2/s_computation/fqup.ma".
 (* PLUS-ITERATED SUPCLOSURE *************************************************)
 
 (* Properties with generic slicing for local environments *******************)
-(*
+
 lemma fqup_drops_succ: ∀G,K,T,l,L,U. ⬇*[⫯l] L ≡ K → ⬆*[⫯l] T ≡ U →
                        ⦃G, L, U⦄ ⊐+ ⦃G, K, T⦄.
 #G #K #T #l elim l -l
@@ -41,6 +41,6 @@ lemma fqup_drops_strap1: ∀G1,G2,L1,K1,K2,T1,T2,U1,l. ⬇*[l] L1 ≡ K1 → ⬆
 | /3 width=5 by fqup_strap1, fqup_drops_succ/
 ]
 qed-.
-*)
-axiom fqup_lref: ∀I,G,L,K,V,i. ⬇*[i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+ ⦃G, K, V⦄.
-(* /2 width=6 by fqup_drops_strap1/ qed. *)
+
+lemma fqup_lref: ∀I,G,L,K,V,i. ⬇*[i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+ ⦃G, K, V⦄.
+/2 width=6 by fqup_drops_strap1/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus.ma
index d62a80894..8f37fd071 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus.ma
@@ -64,56 +64,84 @@ lemma fqus_inv_fqu_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2
 * #HG #HL #HT #_ destruct //
 qed-.
 
+lemma fqus_inv_sort1: ∀G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐* ⦃G2, L2, T2⦄ →
+                      (∧∧ G1 = G2 & L1 = L2 & ⋆s = T2) ∨
+                      ∃∃J,L,V. ⦃G1, L, ⋆s⦄ ⊐* ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V.
+#G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
+#G #L #T #H elim (fqu_inv_sort1 … H) -H /3 width=5 by ex2_3_intro, or_intror/
+qed-.
+
+lemma fqus_inv_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐* ⦃G2, L2, T2⦄ →
+                      ∨∨ ∧∧ G1 = G2 & L1 = L2 & #i = T2
+                       | ∃∃J,L,V. ⦃G1, L, V⦄ ⊐* ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V & i = 0
+                       | ∃∃J,L,V,j. ⦃G1, L, #j⦄ ⊐* ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V & i = ⫯j.
+#G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/
+#G #L #T #H elim (fqu_inv_lref1 … H) -H * /3 width=7 by or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro/
+qed-.
+
+lemma fqus_inv_gref1: ∀G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐* ⦃G2, L2, T2⦄ →
+                      (∧∧ G1 = G2 & L1 = L2 & §l = T2) ∨
+                      ∃∃J,L,V. ⦃G1, L, §l⦄ ⊐* ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V.
+#G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
+#G #L #T #H elim (fqu_inv_gref1 … H) -H /3 width=5 by ex2_3_intro, or_intror/
+qed-.
+
+lemma fqus_inv_bind1: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                      ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2
+                       | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄
+                       | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐* ⦃G2, L2, T2⦄
+                       | ∃∃J,L,V,T. ⦃G1, L, T⦄ ⊐* ⦃G2, L2, T2⦄ & ⬆*[1] T ≡ ⓑ{p,I}V1.T1 & L1 = L.ⓑ{J}V.
+#p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or4_intro0/
+#G #L #T #H elim (fqu_inv_bind1 … H) -H *
+[3: #J #V ] #H1 #H2 #H3 #H destruct
+/3 width=7 by or4_intro1, or4_intro2, or4_intro3, ex3_4_intro/
+qed-.
+
+lemma fqus_inv_flat1: ∀I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓕ{I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                      ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓕ{I}V1.T1 = T2
+                       | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄
+                       | ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄
+                       | ∃∃J,L,V,T. ⦃G1, L, T⦄ ⊐* ⦃G2, L2, T2⦄ & ⬆*[1] T ≡ ⓕ{I}V1.T1 & L1 = L.ⓑ{J}V.
+#I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or4_intro0/
+#G #L #T #H elim (fqu_inv_flat1 … H) -H *
+[3: #J #V ] #H1 #H2 #H3 #H destruct
+/3 width=7 by or4_intro1, or4_intro2, or4_intro3, ex3_4_intro/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
 lemma fqus_inv_atom1: ∀I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐* ⦃G2, L2, T2⦄ →
                       ∧∧ G1 = G2 & ⋆ = L2 & ⓪{I} = T2.
 #I #G1 #G2 #L2 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /2 width=1 by and3_intro/
 #G #L #T #H elim (fqu_inv_atom1 … H)
 qed-.
 
-lemma fqus_inv_sort1: ∀I,G1,G2,L1,L2,V1,T2,s. ⦃G1, L1.ⓑ{I}V1, ⋆s⦄ ⊐* ⦃G2, L2, T2⦄ →
-                      (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & ⋆s = T2) ∨ ⦃G1, L1, ⋆s⦄ ⊐* ⦃G2, L2, T2⦄.
+lemma fqus_inv_sort1_pair: ∀I,G1,G2,L1,L2,V1,T2,s. ⦃G1, L1.ⓑ{I}V1, ⋆s⦄ ⊐* ⦃G2, L2, T2⦄ →
+                           (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & ⋆s = T2) ∨ ⦃G1, L1, ⋆s⦄ ⊐* ⦃G2, L2, T2⦄.
 #I #G1 #G2 #L1 #L2 #V #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
-#G #L #T #H elim (fqu_inv_sort1 … H) -H
+#G #L #T #H elim (fqu_inv_sort1_pair … H) -H
 #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
 qed-.
 
-lemma fqus_inv_zero1: ∀I,G1,G2,L1,L2,V1,T2. ⦃G1, L1.ⓑ{I}V1, #0⦄ ⊐* ⦃G2, L2, T2⦄ →
-                      (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄.
+lemma fqus_inv_zero1_pair: ∀I,G1,G2,L1,L2,V1,T2. ⦃G1, L1.ⓑ{I}V1, #0⦄ ⊐* ⦃G2, L2, T2⦄ →
+                           (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄.
 #I #G1 #G2 #L1 #L2 #V1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
-#G #L #T #H elim (fqu_inv_zero1 … H) -H
+#G #L #T #H elim (fqu_inv_zero1_pair … H) -H
 #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
 qed-.
 
-lemma fqus_inv_lref1: ∀I,G1,G2,L1,L2,V1,T2,i. ⦃G1, L1.ⓑ{I}V1, #⫯i⦄ ⊐* ⦃G2, L2, T2⦄ →
-                      (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #(⫯i) = T2) ∨ ⦃G1, L1, #i⦄ ⊐* ⦃G2, L2, T2⦄.
+lemma fqus_inv_lref1_pair: ∀I,G1,G2,L1,L2,V1,T2,i. ⦃G1, L1.ⓑ{I}V1, #⫯i⦄ ⊐* ⦃G2, L2, T2⦄ →
+                           (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #(⫯i) = T2) ∨ ⦃G1, L1, #i⦄ ⊐* ⦃G2, L2, T2⦄.
 #I #G1 #G2 #L1 #L2 #V #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
-#G #L #T #H elim (fqu_inv_lref1 … H) -H
+#G #L #T #H elim (fqu_inv_lref1_pair … H) -H
 #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
 qed-.
 
-lemma fqus_inv_gref1: ∀I,G1,G2,L1,L2,V1,T2,l. ⦃G1, L1.ⓑ{I}V1, §l⦄ ⊐* ⦃G2, L2, T2⦄ →
-                      (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & §l = T2) ∨ ⦃G1, L1, §l⦄ ⊐* ⦃G2, L2, T2⦄.
+lemma fqus_inv_gref1_pair: ∀I,G1,G2,L1,L2,V1,T2,l. ⦃G1, L1.ⓑ{I}V1, §l⦄ ⊐* ⦃G2, L2, T2⦄ →
+                           (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & §l = T2) ∨ ⦃G1, L1, §l⦄ ⊐* ⦃G2, L2, T2⦄.
 #I #G1 #G2 #L1 #L2 #V #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
-#G #L #T #H elim (fqu_inv_gref1 … H) -H
+#G #L #T #H elim (fqu_inv_gref1_pair … H) -H
 #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
 qed-.
 
-lemma fqus_inv_bind1: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                      ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2
-                       | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄
-                       | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
-#p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/
-#G #L #T #H elim (fqu_inv_bind1 … H) -H *
-#H1 #H2 #H3 #H destruct /2 width=1 by or3_intro1, or3_intro2/
-qed-.
-
-lemma fqus_inv_flat1: ∀I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓕ{I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                      ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓕ{I}V1.T1 = T2
-                       | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄
-                       | ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
-#I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/
-#G #L #T #H elim (fqu_inv_flat1 … H) -H *
-#H1 #H2 #H3 #H destruct /2 width=1 by or3_intro1, or3_intro2/
-qed-.
-
 (* Basic_2A1: removed theorems 1: fqus_drop *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_drops.ma
index 217e82894..8850c1f54 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_drops.ma
@@ -18,10 +18,9 @@ include "basic_2/s_computation/fqus_fqup.ma".
 (* STAR-ITERATED SUPCLOSURE *************************************************)
 
 (* Properties with generic slicing for local environments *******************)
-(*
+
 lemma fqus_drops: ∀G,L,K,T,U,l. ⬇*[l] L ≡ K → ⬆*[l] T ≡ U →
                   ⦃G, L, U⦄ ⊐* ⦃G, K, T⦄.
 #G #L #K #T #U * /3 width=3 by fqup_drops_succ, fqup_fqus/
 #HLK #HTU <(lifts_fwd_isid … HTU) -U // <(drops_fwd_isid … HLK) -K //
 qed.
-*)
\ No newline at end of file
diff --git a/matita/matita/contribs/lambdadelta/basic_2/s_transition/fqu.ma b/matita/matita/contribs/lambdadelta/basic_2/s_transition/fqu.ma
index 7ab53a700..043d01598 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/s_transition/fqu.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_transition/fqu.ma
@@ -20,14 +20,16 @@ include "basic_2/relocation/lifts.ma".
 (* SUPCLOSURE ***************************************************************)
 
 (* activate genv *)
-(* Note: frees_total requires fqu_drop for all atoms *)
+(* Note: frees_total requires fqu_drop for all atoms
+         fqu_cpx_trans requires fqu_drop for all terms
+         frees_fqus_drops requires fqu_drop restricted on atoms
+*)
 inductive fqu: tri_relation genv lenv term ≝
 | fqu_lref_O : ∀I,G,L,V. fqu G (L.ⓑ{I}V) (#0) G L V
 | fqu_pair_sn: ∀I,G,L,V,T. fqu G L (②{I}V.T) G L V
 | fqu_bind_dx: ∀p,I,G,L,V,T. fqu G L (ⓑ{p,I}V.T) G (L.ⓑ{I}V) T
 | fqu_flat_dx: ∀I,G,L,V,T. fqu G L (ⓕ{I}V.T) G L T
-| fqu_drop   : ∀I,I1,I2,G,L,V. ⬆*[1] ⓪{I2} ≡ ⓪{I1} →
-               fqu G (L.ⓑ{I}V) (⓪{I1}) G L (⓪{I2})
+| fqu_drop   : ∀I,G,L,V,T,U. ⬆*[1] T ≡ U → fqu G (L.ⓑ{I}V) U G L T
 .
 
 interpretation
@@ -36,129 +38,136 @@ interpretation
 
 (* Basic properties *********************************************************)
 
-lemma fqu_lref_S: ∀I,G,L,V,i. ⦃G, L.ⓑ{I}V, #(⫯i)⦄ ⊐ ⦃G, L, #(i)⦄.
+lemma fqu_lref_S: ∀I,G,L,V,i. ⦃G, L.ⓑ{I}V, #⫯i⦄ ⊐ ⦃G, L, #i⦄.
 /2 width=1 by fqu_drop/ qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact fqu_inv_atom1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                        ∀I. L1 = ⋆ → T1 = ⓪{I} → ⊥.
-#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #H destruct
-| #I #G #L #V #T #J #_ #H destruct
-| #p #I #G #L #V #T #J #_ #H destruct
-| #I #G #L #V #T #J  #_ #H destruct
-| #I #I1 #I2 #G #L #V #_ #J #H destruct
-]
-qed-.
-
-lemma fqu_inv_atom1: ∀I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐ ⦃G2, L2, T2⦄ → ⊥.
-/2 width=10 by fqu_inv_atom1_aux/ qed-.
-
 fact fqu_inv_sort1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                        ∀I,K,V,s. L1 = K.ⓑ{I}V → T1 = ⋆s →
-                        ∧∧ G1 = G2 & L2 = K & T2 = ⋆s.
-#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #K #W #s #_ #H destruct
-| #I #G #L #V #T #J #K #W #s #_ #H destruct
-| #p #I #G #L #V #T #J #K #W #s #_ #H destruct
-| #I #G #L #V #T #J #K #W #s #_ #H destruct
-| #I #I1 #I2 #G #L #V #HI12 #J #K #W #s #H1 #H2 destruct
-  lapply (lifts_inv_sort2 … HI12) -HI12 /2 width=1 by and3_intro/
-]
-qed-.
-
-lemma fqu_inv_sort1: ∀I,G1,G2,K,L2,V,T2,s. ⦃G1, K.ⓑ{I}V, ⋆s⦄ ⊐ ⦃G2, L2, T2⦄ →
-                     ∧∧ G1 = G2 & L2 = K & T2 = ⋆s.
-/2 width=7 by fqu_inv_sort1_aux/ qed-.
-
-fact fqu_inv_zero1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                        ∀I,K,V. L1 = K.ⓑ{I}V → T1 = #0 →
-                        ∧∧ G1 = G2 & L2 = K & T2 = V.
+                        ∀s. T1 = ⋆s →
+                        ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = ⋆s.
 #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #K #W #H1 #H2 destruct /2 width=1 by and3_intro/
-| #I #G #L #V #T #J #K #W #_ #H destruct
-| #p #I #G #L #V #T #J #K #W #_ #H destruct
-| #I #G #L #V #T #J #K #W #_ #H destruct
-| #I #I1 #I2 #G #L #V #HI12 #J #K #W #H1 #H2 destruct
-  elim (lifts_inv_lref2_uni_lt … HI12) -HI12 //
+[ #I #G #L #T #s #H destruct
+| #I #G #L #V #T #s #H destruct
+| #p #I #G #L #V #T #s #H destruct
+| #I #G #L #V #T #s #H destruct
+| #I #G #L #V #T #U #HI12 #s #H destruct
+  lapply (lifts_inv_sort2 … HI12) -HI12 /2 width=3 by ex3_2_intro/
 ]
 qed-.
 
-lemma fqu_inv_zero1: ∀I,G1,G2,K,L2,V,T2. ⦃G1, K.ⓑ{I}V, #0⦄ ⊐ ⦃G2, L2, T2⦄ →
-                     ∧∧ G1 = G2 & L2 = K & T2 = V.
-/2 width=9 by fqu_inv_zero1_aux/ qed-.
+lemma fqu_inv_sort1: ∀G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐ ⦃G2, L2, T2⦄ →
+                     ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = ⋆s.
+/2 width=3 by fqu_inv_sort1_aux/ qed-.
 
 fact fqu_inv_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                        ∀I,K,V,i. L1 = K.ⓑ{I}V → T1 = #(⫯i) →
-                        ∧∧ G1 = G2 & L2 = K & T2 = #i.
+                        ∀i. T1 = #i →
+                        (∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨
+                        ∃∃J,V,j. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = #j & i = ⫯j.
 #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #K #W #i #_ #H destruct
-| #I #G #L #V #T #J #K #W #i #_ #H destruct
-| #p #I #G #L #V #T #J #K #W #i #_ #H destruct
-| #I #G #L #V #T #J #K #W #i #_ #H destruct
-| #I #I1 #I2 #G #L #V #HI12 #J #K #W #i #H1 #H2 destruct
-  lapply (lifts_inv_lref2_uni_ge … HI12) -HI12 /2 width=1 by and3_intro/
+[ #I #G #L #T #i #H destruct /3 width=4 by ex4_2_intro, or_introl/
+| #I #G #L #V #T #i #H destruct
+| #p #I #G #L #V #T #i #H destruct
+| #I #G #L #V #T #i #H destruct
+| #I #G #L #V #T #U #HI12 #i #H destruct
+  elim (lifts_inv_lref2_uni … HI12) -HI12 /3 width=3 by ex4_3_intro, or_intror/
 ]
 qed-.
 
-lemma fqu_inv_lref1: ∀I,G1,G2,K,L2,V,T2,i. ⦃G1, K.ⓑ{I}V, #(⫯i)⦄ ⊐ ⦃G2, L2, T2⦄ →
-                     ∧∧ G1 = G2 & L2 = K & T2 = #i.
-/2 width=9 by fqu_inv_lref1_aux/ qed-.
+lemma fqu_inv_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐ ⦃G2, L2, T2⦄ →
+                     (∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨
+                     ∃∃J,V,j. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = #j & i = ⫯j.
+/2 width=3 by fqu_inv_lref1_aux/ qed-.
 
 fact fqu_inv_gref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                        ∀I,K,V,l. L1 = K.ⓑ{I}V → T1 = §l →
-                        ∧∧ G1 = G2 & L2 = K & T2 = §l.
+                        ∀l. T1 = §l →
+                        ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = §l.
 #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #K #W #l #_ #H destruct
-| #I #G #L #V #T #J #K #W #l #_ #H destruct
-| #p #I #G #L #V #T #J #K #W #l #_ #H destruct
-| #I #G #L #V #T #J #K #W #l #_ #H destruct
-| #I #I1 #I2 #G #L #V #HI12 #J #K #W #l #H1 #H2 destruct
-  lapply (lifts_inv_gref2 … HI12) -HI12 /2 width=1 by and3_intro/
+[ #I #G #L #T #l #H destruct
+| #I #G #L #V #T #l #H destruct
+| #p #I #G #L #V #T #l #H destruct
+| #I #G #L #V #T #s #H destruct
+| #I #G #L #V #T #U #HI12 #l #H destruct
+  lapply (lifts_inv_gref2 … HI12) -HI12 /2 width=3 by ex3_2_intro/
 ]
 qed-.
 
-lemma fqu_inv_gref1: ∀I,G1,G2,K,L2,V,T2,l. ⦃G1, K.ⓑ{I}V, §l⦄ ⊐ ⦃G2, L2, T2⦄ →
-                     ∧∧ G1 = G2 & L2 = K & T2 = §l.
-/2 width=7 by fqu_inv_gref1_aux/ qed-.
+lemma fqu_inv_gref1: ∀G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐ ⦃G2, L2, T2⦄ →
+                     ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = §l.
+/2 width=3 by fqu_inv_gref1_aux/ qed-.
 
 fact fqu_inv_bind1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
                         ∀p,I,V1,U1. T1 = ⓑ{p,I}V1.U1 →
-                        (∧∧ G1 = G2 & L1 = L2 & V1 = T2) ∨
-                        (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2).
+                        ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+                         | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
+                         | ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & ⬆*[1] T2 ≡ ⓑ{p,I}V1.U1.
 #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #q #J #W #U #H destruct
-| #I #G #L #V #T #q #J #W #U #H destruct /3 width=1 by and3_intro, or_introl/
-| #p #I #G #L #V #T #q #J #W #U #H destruct /3 width=1 by and3_intro, or_intror/
-| #I #G #L #V #T #q #J #W #U #H destruct
-| #I #I1 #I2 #G #L #V #_ #q #J #W #U #H destruct
+[ #I #G #L #T #q #J #V0 #U0 #H destruct
+| #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro0/
+| #p #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro1/
+| #I #G #L #V #T #q #J #V0 #U0 #H destruct
+| #I #G #L #V #T #U #HTU #q #J #V0 #U0 #H destruct /3 width=3 by or3_intro2, ex3_2_intro/
 ]
 qed-.
 
 lemma fqu_inv_bind1: ∀p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                     (∧∧ G1 = G2 & L1 = L2 & V1 = T2) ∨
-                     (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2).
+                     ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+                      | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
+                      | ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & ⬆*[1] T2 ≡ ⓑ{p,I}V1.U1.
 /2 width=4 by fqu_inv_bind1_aux/ qed-.
 
 fact fqu_inv_flat1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
                         ∀I,V1,U1. T1 = ⓕ{I}V1.U1 →
-                        (∧∧ G1 = G2 & L1 = L2 & V1 = T2) ∨
-                        (∧∧ G1 = G2 & L1 = L2 & U1 = T2).
+                        ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+                         | ∧∧ G1 = G2 & L1 = L2 & U1 = T2
+                         | ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & ⬆*[1] T2 ≡ ⓕ{I}V1.U1.
 #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #W #U #H destruct
-| #I #G #L #V #T #J #W #U #H destruct /3 width=1 by and3_intro, or_introl/
-| #p #I #G #L #V #T #J #W #U #H destruct
-| #I #G #L #V #T #J #W #U #H destruct /3 width=1 by and3_intro, or_intror/
-| #I #I1 #I2 #G #L #V #_ #J #W #U #H destruct
+[ #I #G #L #T #J #V0 #U0 #H destruct
+| #I #G #L #V #T #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro0/
+| #p #I #G #L #V #T #J #V0 #U0 #H destruct
+| #I #G #L #V #T #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro1/
+| #I #G #L #V #T #U #HTU #J #V0 #U0 #H destruct /3 width=3 by or3_intro2, ex3_2_intro/
 ]
 qed-.
 
 lemma fqu_inv_flat1: ∀I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓕ{I}V1.U1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                     (∧∧ G1 = G2 & L1 = L2 & V1 = T2) ∨
-                     (∧∧ G1 = G2 & L1 = L2 & U1 = T2).
+                     ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+                      | ∧∧ G1 = G2 & L1 = L2 & U1 = T2
+                      | ∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & ⬆*[1] T2 ≡ ⓕ{I}V1.U1.
 /2 width=4 by fqu_inv_flat1_aux/ qed-.
 
+(* Advanced inversion lemmas ************************************************)
+
+lemma fqu_inv_atom1: ∀I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐ ⦃G2, L2, T2⦄ → ⊥.
+* #x #G1 #G2 #L2 #T2 #H
+[ elim (fqu_inv_sort1 … H) | elim (fqu_inv_lref1 … H) * | elim (fqu_inv_gref1 … H) ] -H
+#I #V [3: #i ] #_ #H destruct
+qed-.
+
+lemma fqu_inv_sort1_pair: ∀I,G1,G2,K,L2,V,T2,s. ⦃G1, K.ⓑ{I}V, ⋆s⦄ ⊐ ⦃G2, L2, T2⦄ →
+                          ∧∧ G1 = G2 & L2 = K & T2 = ⋆s.
+#I #G1 #G2 #K #L2 #V #T2 #s #H elim (fqu_inv_sort1 … H) -H
+#Z #X #H1 #H2 #H3 destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma fqu_inv_zero1_pair: ∀I,G1,G2,K,L2,V,T2. ⦃G1, K.ⓑ{I}V, #0⦄ ⊐ ⦃G2, L2, T2⦄ →
+                          ∧∧ G1 = G2 & L2 = K & T2 = V.
+#I #G1 #G2 #K #L2 #V #T2 #H elim (fqu_inv_lref1 … H) -H *
+#Z #X [2: #x ] #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma fqu_inv_lref1_pair: ∀I,G1,G2,K,L2,V,T2,i. ⦃G1, K.ⓑ{I}V, #(⫯i)⦄ ⊐ ⦃G2, L2, T2⦄ →
+                          ∧∧ G1 = G2 & L2 = K & T2 = #i.
+#I #G1 #G2 #K #L2 #V #T2 #i #H elim (fqu_inv_lref1 … H) -H *
+#Z #X [2: #x ] #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma fqu_inv_gref1_pair: ∀I,G1,G2,K,L2,V,T2,l. ⦃G1, K.ⓑ{I}V, §l⦄ ⊐ ⦃G2, L2, T2⦄ →
+                          ∧∧ G1 = G2 & L2 = K & T2 = §l.
+#I #G1 #G2 #K #L2 #V #T2 #l #H elim (fqu_inv_gref1 … H) -H
+#Z #X #H1 #H2 #H3 destruct /2 width=1 by and3_intro/
+qed-.
+
 (* Basic_2A1: removed theorems 3:
               fqu_drop fqu_drop_lt fqu_lref_S_lt
 *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/frees.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees.ma
index 3699063ce..5911dddfe 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/frees.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/frees.ma
@@ -146,6 +146,24 @@ lemma frees_inv_flat: ∀f,I,L,V,T. L ⊢ 𝐅*⦃ⓕ{I}V.T⦄ ≡ f →
                       ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≡ f1 & L ⊢ 𝐅*⦃T⦄ ≡ f2 & f1 ⋓ f2 ≡ f.
 /2 width=4 by frees_inv_flat_aux/ qed-.
 
+(* Advanced inversion lemmas ***********************************************)
+
+lemma frees_inv_zero_pair: ∀f,I,K,V. K.ⓑ{I}V ⊢ 𝐅*⦃#0⦄ ≡ f →
+                           ∃∃g. K ⊢ 𝐅*⦃V⦄ ≡ g & f = ⫯g.
+#f #I #K #V #H elim (frees_inv_zero … H) -H *
+[ #H destruct
+| #g #Z #Y #X #Hg #H1 #H2 destruct /3 width=3 by ex2_intro/  
+]
+qed-.
+
+lemma frees_inv_lref_pair: ∀f,I,K,V,i. K.ⓑ{I}V ⊢ 𝐅*⦃#(⫯i)⦄ ≡ f →
+                           ∃∃g. K ⊢ 𝐅*⦃#i⦄ ≡ g & f = ↑g.
+#f #I #K #V #i #H elim (frees_inv_lref … H) -H *
+[ #H destruct
+| #g #Z #Y #X #Hg #H1 #H2 destruct /3 width=3 by ex2_intro/  
+]
+qed-.
+
 (* Basic forward lemmas ****************************************************)
 
 lemma frees_fwd_isfin: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≡ f → 𝐅⦃f⦄.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/frees_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees_fqus.ma
deleted file mode 100644
index e80fc704c..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/static/frees_fqus.ma
+++ /dev/null
@@ -1,93 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "basic_2/relocation/drops_weight.ma".
-include "basic_2/s_computation/fqus_weight.ma".
-include "basic_2/static/frees.ma".
-
-(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
-
-(* Properties with star-iterated supclosure *********************************)
-
-lemma frees_fqus_drops: ∀f1,G,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 →
-                        ∀L2,T2. ⦃G, L1, T1⦄ ⊐* ⦃G, L2, T2⦄ →
-                        ∀I,n. ⬇*[n] L1 ≡ L2.ⓑ{I}T2 →
-                        ∃∃f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ ⫱*[⫯n] f1.
-#f1 #G #L1 #T1 #H elim H -f1 -L1 -T1
-[ #f1 #J #Hf1 #L2 #T2 #H12 #I #n #HL12
-  elim (fqus_inv_atom1 … H12) -H12 #H1 #H2 #H3 destruct
-  lapply (drops_fwd_lw … HL12) -HL12 #HL12
-  elim (lt_le_false … HL12) -HL12 //
-| #f1 #J #L1 #V1 #s #Hf1 #IH #L2 #T2 #H12
-  elim (fqus_inv_sort1 … H12) -H12 [ * | #H12 #I * ]
-  [ -IH -Hf1 #H1 #H2 #H3 #I #n #HL12 destruct
-    lapply (drops_fwd_lw … HL12) -HL12 #HL12
-    elim (lt_le_false … HL12) -HL12 //
-  | -IH #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 //
-    #H destruct <(fqus_inv_refl_atom3 … H12) -H12 /2 width=3 by sle_refl, ex2_intro/
-  | -Hf1 #I #HL12 lapply (drops_inv_drop1 … HL12) -HL12
-    #HL12 elim (IH … H12 … HL12) -IH -H12 -HL12 /3 width=3 by ex2_intro/
-  ]
-| #f1 #J #L1 #V1 #Hf1 #IH #L2 #T2 #H12
-  elim (fqus_inv_zero1 … H12) -H12 [ * | #H12 #I * ]
-  [ -IH -Hf1 #H1 #H2 #H3 #I #n #HL12 destruct
-    lapply (drops_fwd_lw … HL12) -HL12 #HL12
-    elim (lt_le_false … HL12) -HL12 //
-  | -IH -H12 #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 //
-     #H destruct /3 width=3 by sle_refl, ex2_intro/
-  | -Hf1 #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
-    #HL12 elim (IH … H12 … HL12) -IH -H12 -HL12 /3 width=3 by ex2_intro/
-  ]
-| #f1 #J #L1 #V1 #i #Hf1 #IH #L2 #T2 #H12
-  elim (fqus_inv_lref1 … H12) -H12 [ * | #H12 #I * ]
-  [ -IH -Hf1 #H1 #H2 #H3 #I #n #HL12 destruct
-    lapply (drops_fwd_lw … HL12) -HL12 #HL12
-    elim (lt_le_false … HL12) -HL12 //
-  | -IH #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 //
-    #H destruct <(fqus_inv_refl_atom3 … H12) -H12 /2 width=3 by sle_refl, ex2_intro/
-  | -Hf1 #I #HL12 lapply (drops_inv_drop1 … HL12) -HL12
-    #HL12 elim (IH … H12 … HL12) -IH -H12 -HL12 /3 width=3 by ex2_intro/
-  ]
-| #f1 #J #L1 #V1 #l #Hf1 #IH #L2 #T2 #H12
-  elim (fqus_inv_gref1 … H12) -H12 [ * | #H12 #I * ]
-  [ -IH -Hf1 #H1 #H2 #H3 #I #n #HL12 destruct
-    lapply (drops_fwd_lw … HL12) -HL12 #HL12
-    elim (lt_le_false … HL12) -HL12 //
-  | -IH #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 //
-    #H destruct <(fqus_inv_refl_atom3 … H12) -H12 /2 width=3 by sle_refl, ex2_intro/
-  | -Hf1 #I #HL12 lapply (drops_inv_drop1 … HL12) -HL12
-    #HL12 elim (IH … H12 … HL12) -IH -H12 -HL12 /3 width=3 by ex2_intro/
-  ]
-| #f1V #f1T #f1 #p #J #L1 #V #T #_ #_ #Hf1 #IHV #IHT #L2 #T2 #H12 #I #n #HL12
-  elim (fqus_inv_bind1 … H12) -H12 [ * |*: #H12 ]
-  [ -IHV -IHT -Hf1 #H1 #H2 #H3 destruct
-    lapply (drops_fwd_lw … HL12) -HL12 #HL12
-    elim (lt_le_false … HL12) -HL12 //
-  | -IHT elim (IHV … H12 … HL12) -IHV -H12 -HL12
-    /4 width=6 by sor_tls, sor_inv_sle_sn_trans, ex2_intro/
-  | -IHV elim (IHT … H12 I (⫯n)) -IHT -H12 /2 width=1 by drops_drop/ -HL12
-    <tls_xn /4 width=6 by ex2_intro, sor_tls, sor_inv_sle_dx_trans/
-  ]
-| #f1V #f1T #f1 #J #L1 #V #T #_ #_ #Hf1 #IHV #IHT #L2 #T2 #H12 #I #n #HL12
-  elim (fqus_inv_flat1 … H12) -H12 [ * |*: #H12 ]
-  [ -IHV -IHT -Hf1 #H1 #H2 #H3 destruct
-    lapply (drops_fwd_lw … HL12) -HL12 #HL12
-    elim (lt_le_false … HL12) -HL12 //
-  | -IHT elim (IHV … H12 … HL12) -IHV -H12 -HL12
-    /4 width=6 by sor_tls, sor_inv_sle_sn_trans, ex2_intro/
-  | -IHV elim (IHT … H12 … HL12) -IHT -H12 -HL12
-    /4 width=6 by ex2_intro, sor_tls, sor_inv_sle_dx_trans/
-  ]
-]
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
index 78203c527..172b43b1d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
+++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
@@ -191,7 +191,7 @@ table {
           }
         ]        
         [ { "context-sensitive free variables" * } {
-             [ "frees ( ? ⊢ 𝐅*⦃?⦄ ≡ ? )" "frees_weight" + "frees_lreq" + "frees_drops" + "frees_fqup" + "frees_fqus" + "frees_frees" * ]
+             [ "frees ( ? ⊢ 𝐅*⦃?⦄ ≡ ? )" "frees_weight" + "frees_lreq" + "frees_drops" + "frees_fqup" + "frees_frees" * ]
           }
         ]
         [ { "restricted ref. for local env." * } {
diff --git a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
index dad1f9382..2c6ad2a56 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
@@ -172,6 +172,10 @@ qed.
 
 (* Inversion & forward lemmas ***********************************************)
 
+lemma nat_split: ∀x. x = 0 ∨ ∃y. ⫯y = x.
+* /3 width=2 by ex_intro, or_introl, or_intror/
+qed-.
+
 lemma max_inv_O3: ∀x,y. (x ∨ y) = 0 → 0 = x ∧ 0 = y.
 /4 width=2 by le_maxr, le_maxl, le_n_O_to_eq, conj/
 qed-.
-- 
2.39.2