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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/substitution/fpa_fpa.ma".
include "basic_2/multiple/fpas.ma".

(* MULTIPLE AJUSTMENT *******************************************************)

(* Advanced properties ******************************************************)

lemma fpas_strip: ∀L0,L1,L2,T0,T1,T2,s. ⦃L0, T0⦄ ⇳[s] ⦃L1, T1⦄ → ⦃L0, T0⦄ ⇳*[s] ⦃L2, T2⦄ →
                  ∃∃L,T. ⦃L1, T1⦄ ⇳*[s] ⦃L, T⦄ & ⦃L2, T2⦄ ⇳[s] ⦃L, T⦄.
/3 width=4 by fpa_conf, bi_TC_strip/ qed-.

(* Main properties **********************************************************)

theorem fpas_conf: ∀s. bi_confluent … (fpas s).
/3 width=4 by fpa_conf, bi_TC_confluent/ qed-.

theorem fpas_trans: ∀s. bi_transitive … (fpas s).
/2 width=4 by bi_TC_transitive/ qed-.

theorem fpas_canc_sn: ∀L,L1,L2,T,T1,T2,s.
                      ⦃L, T⦄ ⇳*[s] ⦃L1, T1⦄→ ⦃L, T⦄ ⇳*[s] ⦃L2, T2⦄ → ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
/3 width=4 by fpas_trans, fpas_sym/ qed-.

theorem fpas_canc_dx: ∀L1,L2,L,T1,T2,T,s.
                      ⦃L1, T1⦄ ⇳*[s] ⦃L, T⦄ → ⦃L2, T2⦄ ⇳*[s] ⦃L, T⦄ → ⦃L1, T1⦄ ⇳*[s] ⦃L2, T2⦄.
/3 width=4 by fpas_trans, fpas_sym/ qed-.