Require Import Definitions.

PCP to CFP


Section PCP_CFP.

  Variable P : SRS.

  Definition Sigma := sym P.
  Notation "#" := (fresh Sigma).

  Definition gamma (A : SRS) := map (fun '(x,y) => (x, rev y)) A.

  Lemma sigma_gamma a A :
    sigma a (gamma A) = tau1 A ++ a ++ rev (tau2 A).
  Proof.
    induction A as [ | (x & y) ]; unfold gamma in *; cbn.
    - reflexivity.
    - rewrite IHA. now simpl_list.
  Qed.

  Definition palin X (A : list X) := A = rev A.

  Lemma main A a :
    ~ a el sym A ->
    tau1 A = tau2 A <-> palin (sigma a (gamma A)).
  Proof.
    rewrite sigma_gamma. unfold palin.
    simpl_list. intuition.
    - now rewrite H0.
    - eapply list_prefix_inv in H0; firstorder using tau1_sym, tau2_sym.
  Qed.

  Lemma gamma_invol A :
    gamma (gamma A) = A.
  Proof.
    induction A as [ | (x,y) ]; cbn.
    - reflexivity.
    - simpl_list. now rewrite <- IHA at 2.
  Qed.

  Lemma gamma_mono A B :
    A <<= gamma B -> gamma A <<= B.
  Proof.
    induction A as [ | (x,y) ]; cbn; intros.
    - firstorder.
    - intros ? [ <- | ].
      + assert ( x/y el gamma B) by firstorder.
        unfold gamma in H0. eapply in_map_iff in H0 as ((x',y') & ? & ?).
        inv H0. now simpl_list.
      + firstorder.
  Qed.

End PCP_CFP.

Theorem PCP_CFP :
  PCP CFP.
Proof.
  exists (fun P => (gamma P, fresh (sym P))).
  intros P. split; intros.
  - destruct H as (A & Hi & He & H).
    exists (gamma A). repeat split.
    + eapply gamma_mono. now rewrite gamma_invol.
    + destruct A; cbn in *; congruence.
    + eapply main. intros F % (sym_mono (P := P)) % fresh_spec; now try eapply F. eauto.
  - destruct H as (B & Hi & He & H).
    exists (gamma B). repeat split.
    + now eapply gamma_mono.
    + destruct B; cbn in *; congruence.
    + eapply main with (a := fresh (sym P)).
      * intros F % (sym_mono (P := P)) % fresh_spec. now eapply F. now eapply gamma_mono.
      * rewrite gamma_invol. eassumption.
Qed.