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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                  *)
(*                                                                        *)
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include "basic_2/unfold/ltpss_sn_alt.ma".
include "basic_2/static/aaa_ltpss_dx.ma".

(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)

(* Properties about sn parallel unfold **************************************)

lemma aaa_ltpss_sn_conf: ∀L1,T,A. L1 ⊢ T ⁝ A →
                         ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 → L2 ⊢ T ⁝ A.
#L1 #T #A #HT #L2 #d #e #HL12
lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12
@(TC_Conf3 … (λL,A. L ⊢ T ⁝ A) … HT ? HL12) /2 width=5/
qed.

lemma aaa_ltpss_sn_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
                              ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
                              ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A.
/3 width=5/ qed.

lemma aaa_ltpss_sn_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
                             ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
                             ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A.
/3 width=5/ qed.