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diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_aaa.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_aaa.etc
new file mode 100644
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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_ltpss_dx.ma".
+include "basic_2/static/lsuba_aaa.ma".
+include "basic_2/reducibility/ltpr_ldrop.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties about atomic arity assignment on terms ************************)
+
+lemma aaa_ltpr_tpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀L2. L1 ➡ L2 →
+                         ∀T2. T1 ➡ T2 → L2 ⊢ T2 ⁝ A.
+#L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * *
+[1,2,3:
+  #i #Hn #X #H1 #L2 #HL12 #Y #H2 destruct
+  >(tpr_inv_atom1 … H2) -Y
+|4,5: [ #a ] * #V1 #T1 #Hn #X #H1 #L2 #HL12 #Y #H2 destruct
+]
+[ >(aaa_inv_sort … H1) -X //
+| elim (aaa_inv_lref … H1) #I #K1 #V1 #HLK1 #HA
+  lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
+  elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #Y #H #HLK2
+  elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
+  lapply (IH … HKV1 … HA … HK12 … HV12) -L1 -K1 -V1 /2 width=5/
+| elim (aaa_inv_gref … H1)
+| elim (aaa_inv_abbr … H1) -H1 #B #HB #HA
+  elim (tpr_inv_abbr1 … H2) -H2 *
+  [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct
+    lapply (tps_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+    lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB
+    lapply (IH … HA … (L2.ⓓV2) … HT1) -IH -HA -HT1 /width=5/ -T1 /2 width=1/ -L1 -V1 /3 width=5/
+  | -B #T #HT1 #HXT #H destruct
+    lapply (IH … HA … (L2.ⓓV1) … HT1) /width=5/ -T1 /2 width=1/ -L1 #HA
+    @(aaa_inv_lift … HA … HXT) /2 width=1/
+  ]
+| elim (aaa_inv_abst … H1) -H1 #B #A #HB #HA #H destruct
+  elim (tpr_inv_abst1 … H2) -H2 #V2 #T2 #HV12 #HT12 #H destruct
+  lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB
+  lapply (IH … HA … (L2.ⓛV2) … HT12) -IH -HA -HT12 /width=5/ -T1 /2 width=1/
+| elim (aaa_inv_appl … H1) -H1 #B #HB #HA
+  elim (tpr_inv_appl1 … H2) -H2 *
+  [ #V2 #T2 #HV12 #HT12 #H destruct
+    lapply (IH … HB … HL12 … HV12) -HB -HV12 /width=5/ #HB
+    lapply (IH … HA … HL12 … HT12) -IH -HA -HL12 -HT12 /width=5/ /2 width=3/
+  | #b #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct
+    elim (aaa_inv_abst … HA) -HA #B0 #A0 #HB0 #HA0 #H destruct
+    lapply (IH … HB … HL12 … HV12) -HB -HV12 /width=5/ #HB
+    lapply (IH … HB0  … HL12 W2 ?) -HB0 /width=5/ #HB0
+    lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 // /2 width=1/ -T0 -L1 -V1 /4 width=7/
+  | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct
+    elim (aaa_inv_abbr … HA) -HA #B0 #HW0 #HT0
+    lapply (IH … HW0  … HL12 … HW02) -HW0 /width=5/ #HW2
+    lapply (IH … HB … HL12 … HV10) -HB -HV10 /width=5/ #HV0
+    lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 // /2 width=1/ -V1 -T0 -L1 -W0 #HT2
+    @(aaa_abbr … HW2) -HW2
+    @(aaa_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *)
+  ]
+| elim (aaa_inv_cast … H1) -H1 #HV1 #HT1
+  elim (tpr_inv_cast1 … H2) -H2
+  [ * #V2 #T2 #HV12 #HT12 #H destruct
+    lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2
+    lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/
+  | -HV1 #HT1X
+     lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/
+  ]
+]
+qed.
+
+lemma aaa_ltpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ➡ L2 → L2 ⊢ T ⁝ A.
+/2 width=5/ qed.
+
+lemma aaa_tpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. T1 ➡ T2 → L ⊢ T2 ⁝ A.
+/2 width=5/ qed.