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diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_delift.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_delift.ma
index 58697b9c1..810083108 100644
--- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_delift.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_delift.ma
@@ -12,18 +12,70 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/unfold/delift_lift.ma".
 include "basic_2/unfold/delift_delift.ma".
 include "basic_2/computation/cprs_delift.ma".
 include "basic_2/equivalence/cpcs_cpcs.ma".
 
 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
 
+(* Advanced inversion lemmas ************************************************)
+(* does not holds
+axiom cprs_inv_appl1_cpcs: ∀L,V1,T1,U2. L ⊢ ⓐV1. T1 ➡* U2 → (
+                           ∃∃V2,T2.    L ⊢ V1 ➡* V2 & L ⊢ T1 ➡* T2 &
+                                       L ⊢ U2 ➡* ⓐV2. T2
+                           ) ∨
+                           ∃∃a,V2,W,T. L ⊢ V1 ➡* V2 &
+                                       L ⊢ T1 ➡* ⓛ{a}W. T & L ⊢ ⓓ{a}V2. T ⬌* U2.
+#L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
+#U #U2 #_ #HU2 * *
+[ #V0 #T0 #HV10 #HT10 #HUT0
+  elim (cprs_strip … HUT0 … HU2) -U #U #H #HU2
+  elim (cpr_inv_appl1 … H) -H *
+  [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
+  | #b #V2 #W2 #T #T2 #HV02 #HT2 #H1 #H2 destruct
+    lapply (cprs_strap1 … HV10 HV02) -V0 #HV12
+    lapply (cprs_div ? (ⓓ{b}V2.T) ? ? ? HU2) -HU2 /2 width=1/ /3 width=7/
+  | #b #V #V2 #W0 #W2 #T #T2 #HV0 #HW02 #HT2 #HV2 #H1 #H2 destruct
+    lapply (cprs_strap1 … HV10 HV0) -V0 #HV1
+    lapply (cprs_trans … HT10 (ⓓ{b}W2.T2) ?) -HT10 /2 width=1/ -W0 -T #HT1
+    elim (sfr_delift (L.ⓓW2) (ⓐV2.T2) 0 1 ? ?) // #X #H1
+    lapply (cprs_zeta_delift … H1) #H2
+    lapply (cprs_trans … HU2 … H2) -HU2 -H2 #HU2T3
+    elim (delift_inv_flat1 … H1) -H1 #V3 #T3 #HV23 #HT23 #H destruct
+    lapply (delift_inv_lift1_eq … HV23 … HV2) -V2 [ /2 width=1/ | skip ] #H destruct
+    lapply (cprs_zeta_delift … HT23) -HT23 #H
+    lapply (cprs_trans … HT1 … H) -W2 -T2 /3 width=5/
+  ]
+| /4 width=8/
+]
+qed-.
+*)
+(* maybe holds
+axiom cprs_inv_appl_abst: ∀L,V,T,W,U. L ⊢ ⓐV.T ➡* ⓛW.U →
+                          ∃∃W0,T0,V1,T1. L ⊢ T ➡* ⓛW0.T0 &
+                                         L ⊢ ⓓV.T0 ➡* ⓛV1.T1 &
+                                         L ⊢ ⓛW.U ➡* ⓛV1.T1.
+#L #V #T #W #U #H
+elim (cprs_inv_appl1_cpcs … H) -H *
+[ #V0 #T0 #HV0 #HT0 #H
+  elim (cprs_inv_abst1 Abst W … H) -H #W0 #U0 #_ #_ #H destruct
+| #V0 #W0 #T0 #HV0 #HT0 #H
+  elim (cpcs_inv_abst2 … H) -H #V1 #T1 #H1 #H2
+  lapply (cprs_trans … (ⓓV.T0) … H1) -H1 /2 width=1/ -V0 /2 width=7/
+]
+qed-.
+*)
 (* Properties on inverse basic term relocation ******************************)
 
+lemma cpcs_zeta_delift_comm: ∀L,V,T1,T2. L.ⓓV ⊢ ▼*[O, 1] T1 ≡ T2 →
+                             L ⊢ T2 ⬌* +ⓓV.T1.
+/3 width=1/ qed.
+
 (* Basic_1: was only: pc3_gen_cabbr *)
 lemma thin_cpcs_delift_mono: ∀L,U1,U2. L ⊢ U1 ⬌* U2 →
-                             ∀K,d,e. L [d, e] ≡ K → ∀T1. L ⊢ U1 [d, e] ≡ T1 →
-                             ∀T2. L ⊢ U2 [d, e] ≡ T2 → K ⊢ T1 ⬌* T2.
+                             ∀K,d,e. ▼*[d, e] L ≡ K → ∀T1. L ⊢ ▼*[d, e] U1 ≡ T1 →
+                             ∀T2. L ⊢ ▼*[d, e] U2 ≡ T2 → K ⊢ T1 ⬌* T2.
 #L #U1 #U2 #H #K #d #e #HLK #T1 #HTU1 #T2 #HTU2
 elim (cpcs_inv_cprs … H) -H #U #HU1 #HU2
 elim (thin_cprs_delift_conf … HU1 … HLK … HTU1) -U1 #T #HT1 #HUT