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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_aaa.ma
new file mode 100644
index 000000000..6f4b3b2a0
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/aaa_lift.ma".
+include "basic_2/static/lsuba_aaa.ma".
+include "basic_2/reduction/lpx_drop.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+(* Note: lemma 500 *)
+lemma cpx_lpx_aaa_conf: ∀h,o,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
+                        ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, o] T2 →
+                        ∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+#h #o #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
+[ #o #L1 #s #X #H
+  elim (cpx_inv_sort1 … H) -H // * //
+| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
+  elim (cpx_inv_lref1 … H) -H
+  [ #H destruct
+    elim (lpx_drop_conf … HLK1 … HL12) -L1 #X #H #HLK2
+    elim (lpx_inv_pair1 … H) -H
+    #K2 #V2 #HK12 #HV12 #H destruct /3 width=6 by aaa_lref/
+  | * #J #Y #Z #V2 #H #HV12 #HV2
+    lapply (drop_mono … H … HLK1) -H #H destruct
+    elim (lpx_drop_conf … HLK1 … HL12) -L1 #Z #H #HLK2
+    elim (lpx_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
+    /3 width=8 by aaa_lift, drop_fwd_drop2/
+  ]
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+  elim (cpx_inv_abbr1 … H) -H *
+  [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=2 by lpx_pair, aaa_abbr/
+  | #T2 #HT12 #HT2 #H destruct -IHV1
+    /4 width=8 by lpx_pair, aaa_inv_lift, drop_drop/
+  ]
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+  elim (cpx_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+  /4 width=1 by lpx_pair, aaa_abst/
+| #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+  elim (cpx_inv_appl1 … H) -H *
+  [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=3 by aaa_appl/
+  | #b #V2 #W1 #W2 #U1 #U2 #HV12 #HW12 #HU12 #H1 #H2 destruct
+    lapply (IHV1 … HV12 … HL12) -IHV1 -HV12 #HV2
+    lapply (IHT1 (ⓛ{b}W2.U2) … HL12) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
+    elim (aaa_inv_abst … H) -H #B0 #A0 #HW1 #HU2 #H destruct
+    /5 width=6 by lsuba_aaa_trans, lsuba_beta, aaa_abbr, aaa_cast/
+  | #b #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
+    lapply (aaa_lift G L2 … B … (L2.ⓓW2) … HV2) -HV2 /2 width=2 by drop_drop/ #HV2
+    lapply (IHT1 (ⓓ{b}W2.U2) … HL12) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
+    elim (aaa_inv_abbr … H) -H /3 width=3 by aaa_abbr, aaa_appl/
+  ]
+| #G #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+  elim (cpx_inv_cast1 … H) -H
+  [ * #V2 #T2 #HV12 #HT12 #H destruct /3 width=1 by aaa_cast/
+  | -IHV1 /2 width=1 by/
+  | -IHT1 /2 width=1 by/
+  ]
+]
+qed-.
+
+lemma cpx_aaa_conf: ∀h,o,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
+/2 width=7 by cpx_lpx_aaa_conf/ qed-.
+
+lemma lpx_aaa_conf: ∀h,o,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+/2 width=7 by cpx_lpx_aaa_conf/ qed-.
+
+lemma cpr_aaa_conf: ∀G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
+/3 width=5 by cpx_aaa_conf, cpr_cpx/ qed-.
+
+lemma lpr_aaa_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+/3 width=5 by lpx_aaa_conf, lpr_lpx/ qed-.