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diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/aaa_lpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/aaa_lpss.etc
new file mode 100644
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+++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/lsubr/aaa_lpss.etc
@@ -0,0 +1,54 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/substitution/lpss_ldrop.ma".
+include "basic_2/static/aaa_lift.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties about sn parallel substitution ********************************)
+
+(* Note: lemma 500 *)
+lemma aaa_cpss_lpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀T2. L1 ⊢ T1 ▶* T2 →
+                          ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T2 ⁝ A.
+#L1 #T1 #A #H elim H -L1 -T1 -A
+[ #L1 #k #X #H
+  >(cpss_inv_sort1 … H) -H //
+| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
+  elim (cpss_inv_lref1 … H) -H
+  [ #H destruct
+    elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
+    elim (lpss_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/
+  | * #Y #Z #V2 #H #HV12 #HV2
+    lapply (ldrop_mono … H … HLK1) -H #H destruct
+    elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #Z #H #HLK2
+    elim (lpss_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
+    lapply (ldrop_fwd_ldrop2 … HLK2) -V0 /3 width=7/
+  ]
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+  elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=2/
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+  elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=1/
+| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+  elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=3/
+| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+  elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=1/
+]
+qed-.
+
+lemma aaa_cpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ▶* T2 → L ⊢ T2 ⁝ A.
+/2 width=5 by aaa_cpss_lpss_conf/ qed-.
+
+lemma aaa_lpss_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T ⁝ A.
+/2 width=5 by aaa_cpss_lpss_conf/ qed-.