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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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(* A Coq computable reduction from n-registers MM termination
    to 2-registers MMA termination. Beware that the semantics
    of MMA is a bit different than the semantics of MM: 

       the DEC instruction jumps when not zero instead of when zero 

    Compare mm_sss  (ILL/Mm/mm_defs.v)
    and     mma_sss (MM/mma_defs.v)                   

    The reduction goes via regular FRACTRAN termination *)

Require Import Undecidability.Synthetic.Undecidability.

From Undecidability.Shared.Libs.DLW Require Import Utils.utils Vec.pos Vec.vec.
From Undecidability.Shared.Libs.DLW.Code Require Import sss.

From Undecidability.MinskyMachines Require Import MM FRACTRAN_to_MMA2.
From Undecidability.FRACTRAN Require Import FRACTRAN MM_FRACTRAN.

Set Implicit Arguments.

Corollary MM_MMA2_HALTING : MM_HALTING ⪯ MMA2_HALTING.
Proof.
  eapply reduces_transitive. exact MM_FRACTRAN_REG_HALTING.
  exact FRACTRAN_REG_MMA2_HALTING.
Qed.

Check MM_MMA2_HALTING.

From Undecidability.MinskyMachines Require Import mm_defs mma_defs fractran_mma.
From Undecidability.FRACTRAN Require Import fractran_utils prime_seq mm_fractran.

(* This is somewhat for direct proof that does not involve
    testing for (0,_) in the intermediate Fractran program *)

Local Notation "P /MM/ s ↓" := (sss_terminates (@mm_sss _) P s) (at level 70, no associativity).
Local Notation "P /MMA/ s ↓" := (sss_terminates (@mma_sss 2) P s) (at level 70, no associativity).

Theorem mm_mma2 n (P : list (mm_instr (pos n))) :
               { Q : list (mm_instr (pos 2))
               & { f : vec nat n -> vec nat 2
                 | forall v, (1,P) /MM/ (1,v) ↓ <-> (1,Q) /MMA/ (1,f v) ↓ } }.
Proof.
  destruct (mm_fractran_not_zero P) as (l & H1 & H2).
  exists (fractran_mma l), (fun v => ps 1 * exp 1 v ## 0 ## vec_nil).
  intros v; rewrite H2.
  apply fractran_mma_reduction; trivial.
  revert H1; apply Forall_impl; tauto.
Qed.