Commit c371a4a5 authored by Joseph Tassarotti's avatar Joseph Tassarotti

Direct (double negated) adequacy proof for nnvs modality

parent 54bd5cd8
......@@ -322,17 +322,59 @@ Proof.
unseal. rewrite right_id in Hvs *; naive_solver.
Qed.
(* Nevertheless, we can prove a weaker form of adequacy (which is equvialent to adequacy
under classical axioms) directly without passing through the proofs for rvs: *)
Lemma adequacy_helper1 P n k x:
{S n + k} x ¬¬ (Nat.iter (S n) (λ P, |=n=> P)%I P (S n + k) x)
¬¬ ( x', {n + k} (x') Nat.iter n (λ P, |=n=> P)%I P (n + k) (x')).
Proof.
revert k P x. induction n.
- rewrite /uPred_nnvs. unseal=> k P x Hx Hf1 Hf2.
eapply Hf1. intros Hf3.
eapply (laterN_big (S k) (S k)); eauto.
specialize (Hf3 (S k) (S k) ). rewrite right_id in Hf3 *. unseal.
intros Hf3. eapply Hf3; eauto.
intros ??? Hx'. rewrite left_id in Hx' *=> Hx'.
unseal.
assert (n' < S k n' = S k) as [|] by omega.
* intros. move:(laterN_small n' (S k) x' False). rewrite //=. unseal. intros Hsmall.
eapply Hsmall; eauto.
* subst. intros. exfalso. eapply Hf2. exists x'. split; eauto using cmra_validN_S.
- intros k P x Hx. rewrite ?Nat_iter_S_r.
replace (S (S n) + k) with (S n + (S k)) by omega.
replace (S n + k) with (n + (S k)) by omega.
intros. eapply IHn. replace (S n + S k) with (S (S n) + k) by omega. eauto.
rewrite ?Nat_iter_S_r. eauto.
Qed.
(* Questions:
1) Can we prove a weakened form of adequacy for nnvs directly, such as :
Lemma adequacy_helper2 P n k x:
{S n + k} x ¬¬ (Nat.iter (S n) (λ P, |=n=> P)%I P (S n + k) x)
¬¬ ( x', {k} (x') Nat.iter 0 (λ P, |=n=> P)%I P k (x')).
Proof.
revert x. induction n.
- specialize (adequacy_helper1 P 0). rewrite //=. naive_solver.
- intros ?? Hfal%adequacy_helper1; eauto using cmra_validN_S.
intros Hfal'. eapply Hfal. intros (x''&?&?).
eapply IHn; eauto.
Qed.
Lemma adequacy' φ n : (True ⊢ Nat.iter n (λ P, |=n=> ▷ P) (■ φ)) → ¬¬ φ.
Lemma adequacy φ n : (True Nat.iter n (λ P, |=n=> P) ( φ)) ¬¬ φ.
Proof.
cut ( x, {S n} x Nat.iter n (λ P, |=n=> P)%I ( φ)%I (S n) x ¬¬φ).
{ intros help H. eapply (help ); eauto using ucmra_unit_validN.
eapply H; try unseal; eauto using ucmra_unit_validN. red; rewrite //=. }
destruct n.
- rewrite //=; unseal; auto.
- intros ??? Hfal.
eapply (adequacy_helper2 _ n 1); (replace (S n + 1) with (S (S n)) by omega); eauto.
unseal. intros (x'&?&Hphi). simpl in *.
eapply Hfal. auto.
Qed.
One idea may be to prove a limited adequacy theorem for each
nnvs_k and use the limiting argument we did for transitivity.
(* Open question:
2) Do the basic properties of the |=r=> modality (rvs_intro, rvs_mono, rvs_trans, rvs_frame_r,
Do the basic properties of the |=r=> modality (rvs_intro, rvs_mono, rvs_trans, rvs_frame_r,
rvs_ownM_updateP, and adequacy) uniquely characterize |=r=>?
*)
*)
End rvs_nnvs.
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