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(*   Copyright Dominique Larchey-Wendling [*]                 *)
(*             Yannick Forster            [+]                 *)
(*                                                            *)
(*                             [*] Affiliation LORIA -- CNRS  *)
(*                             [+] Affiliation Saarland Univ. *)
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(*      This file is distributed under the terms of the       *)
(*         CeCILL v2 FREE SOFTWARE LICENSE AGREEMENT          *)
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(* Direct reductions from Halting of 1-TM to binary PCP *)
From Undecidability.Shared.Libs.DLW
From Undecidability.TM
From Undecidability.StringRewriting
From Undecidability.PCP
From Undecidability.PCP.Reductions

Lemma PCP_chain_iPCPb : ⎩PCP ⪯ₘ PCPb ⪯ₘ iPCPb⎭.  Proof. split; apply HaltTM_1_chain_iPCPb. Qed.
Lemma PCP_chain_PCPb : ⎩PCP ⪯ₘ PCPb⎭.            Proof. apply PCP_chain_iPCPb. Qed.
Lemma PCPb_chain_iPCPb : ⎩PCPb ⪯ₘiPCPb⎭.         Proof. apply PCP_chain_iPCPb. Qed.