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(*   Copyright Dominique Larchey-Wendling *                 *)
(*                                                            *)
(*                             * Affiliation LORIA -- CNRS  *)
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(*      This file is distributed under the terms of the       *)
(*         CeCILL v2 FREE SOFTWARE LICENSE AGREEMENT          *)
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Require Import Undecidability.Synthetic.Undecidability.

From Undecidability.PCP Require Import PCP PCPb_iff_iPCPb.
From Undecidability.StackMachines Require Import BSM iPCPb_to_BSM_HALTING.
From Undecidability.MinskyMachines Require Import MM BSM_MM.

Theorem PCPb_MM_HALTS_ON_ZERO : PCPb ⪯ MM_HALTS_ON_ZERO.
Proof.
  eapply reduces_transitive. exact PCPb_iff_iPCPb.reductionLR.
  eapply reduces_transitive. exact iPCPb_to_BSM_HALTING.
  apply BSM_MM_HALTS_ON_ZERO.
Qed.

Theorem PCPb_MM_HALTING : PCPb ⪯ MM_HALTING.
Proof.
  eapply reduces_transitive. exact PCPb_iff_iPCPb.reductionLR.
  eapply reduces_transitive. exact iPCPb_to_BSM_HALTING.
  apply BSM_MM_HALTING.
Qed.