Commit 10517549 authored by Joseph Tassarotti's avatar Joseph Tassarotti

rvs is (classically) equivalent to a kind of double negation

parent a3ef214f
......@@ -63,6 +63,7 @@ algebra/updates.v
algebra/local_updates.v
algebra/gset.v
algebra/coPset.v
algebra/double_negation.v
program_logic/model.v
program_logic/adequacy.v
program_logic/lifting.v
......
From iris.algebra Require Import upred.
Import upred.
(* In this file we show that the rvs can be thought of a kind of
step-indexed double-negation when our meta-logic is classical *)
(* To define this, we need a way to talk about iterated later modalities: *)
Definition uPred_laterN {M} (n : nat) (P : uPred M) : uPred M :=
Nat.iter n uPred_later P.
Instance: Params (@uPred_laterN) 2.
Notation "▷^ n P" := (uPred_laterN n P)
(at level 20, n at level 9, right associativity,
format "▷^ n P") : uPred_scope.
Definition uPred_nnvs {M} (P: uPred M) : uPred M :=
n, (P - ^n False) - ^n False.
Notation "|=n=> Q" := (uPred_nnvs Q)
(at level 99, Q at level 200, format "|=n=> Q") : uPred_scope.
Notation "P =n=> Q" := (P |=n=> Q)
(at level 99, Q at level 200, only parsing) : C_scope.
Notation "P =n=★ Q" := (P - |=n=> Q)%I
(at level 99, Q at level 200, format "P =n=★ Q") : uPred_scope.
(* Our goal is to prove that:
(1) |=n=> has (nearly) all the properties of the |=r=> modality that are used in Iris
(2) If our meta-logic is classical, then |=n=> and |=r=> are equivalent
*)
Section rvs_nn.
Context {M : ucmraT}.
Implicit Types φ : Prop.
Implicit Types P Q : uPred M.
Implicit Types A : Type.
Implicit Types x : M.
Import uPred.
(* Helper lemmas about iterated later modalities *)
Lemma laterN_big n a x φ: {n} x a n (^a ( φ))%I n x φ.
Proof.
induction 2 as [| ?? IHle].
- induction a; repeat (rewrite //= || uPred.unseal).
intros Hlater. apply IHa; auto using cmra_validN_S.
move:Hlater; repeat (rewrite //= || uPred.unseal).
- intros. apply IHle; auto using cmra_validN_S.
eapply uPred_closed; eauto using cmra_validN_S.
Qed.
Lemma laterN_small n a x φ: {n} x n < a (^a ( φ))%I n x.
Proof.
induction 2.
- induction n as [| n IHn]; [| move: IHn];
repeat (rewrite //= || uPred.unseal).
naive_solver eauto using cmra_validN_S.
- induction n as [| n IHn]; [| move: IHle];
repeat (rewrite //= || uPred.unseal).
red; rewrite //=. intros.
eapply (uPred_closed _ _ (S n)); eauto using cmra_validN_S.
Qed.
(* First we prove that rvs implies nn *)
Lemma rvs_nn P: (|=r=> P) |=n=> P.
Proof.
split. rewrite /uPred_nnvs. repeat uPred.unseal. intros n x ? Hrvs a.
red; rewrite //= => n' yf ??.
edestruct Hrvs as (x'&?&?); eauto.
case (decide (a n')).
- intros Hle Hwand.
exfalso. eapply laterN_big; last (uPred.unseal; eapply (Hwand n' x')); eauto.
* rewrite comm. done.
* rewrite comm. done.
- intros; assert (n' < a). omega.
move: laterN_small. uPred.unseal.
naive_solver.
Qed.
Lemma nn_intro P : P =n=> P.
Proof. apply forall_intro=>?. apply wand_intro_l, wand_elim_l. Qed.
Lemma nn_mono P Q : (P Q) (|=n=> P) =n=> Q.
Proof.
intros HPQ. apply forall_intro=>n.
apply wand_intro_l. rewrite -{1}HPQ.
rewrite /uPred_nnvs (forall_elim n).
apply wand_elim_r.
Qed.
(* Question: is there a clean direct proof of this? *)
(*
Lemma nn_trans P : (|=n=> |=n=> P) =n=> P.
Proof.
apply forall_intro=>n. apply wand_intro_l.
rewrite /uPred_nnvs.
rewrite {1}(nn_intro (P -★ ▷^ n False)).
rewrite /uPred_nnvs. rewrite comm (forall_elim n).
apply wand_elim_r. Qed.
*)
Lemma nn_frame_r P R : (|=n=> P) R =n=> P R.
Proof.
apply forall_intro=>n. apply wand_intro_r.
rewrite (comm _ P) -wand_curry.
rewrite /uPred_nnvs (forall_elim n).
by rewrite -assoc wand_elim_r wand_elim_l.
Qed.
Lemma nn_ownM_updateP x (Φ : M Prop) :
x ~~>: Φ uPred_ownM x =n=> y, Φ y uPred_ownM y.
Proof. intros. rewrite -rvs_nn. by apply rvs_ownM_updateP. Qed.
Lemma except_last_nn P : (|=n=> P) (|=n=> P).
Proof.
rewrite /uPred_except_last. apply or_elim.
- by rewrite -nn_intro -or_intro_l.
- by apply nn_mono, or_intro_r.
Qed.
(* Now we show, nn implies rvs, for which we need a classical axiom: *)
Require Coq.Logic.Classical_Pred_Type.
Lemma nn_rvs P: (|=n=> P) (|=r=> P).
Proof.
rewrite /uPred_nnvs.
split. uPred.unseal; red; rewrite //=.
intros n x ? Hforall k yf Hle ?.
apply Classical_Pred_Type.not_all_not_ex.
intros Hfal.
specialize (Hforall k k).
eapply laterN_big; last (uPred.unseal; red; rewrite //=; eapply Hforall);
eauto.
red; rewrite //= => n' x' ???.
case (decide (n' = k)); intros.
- subst. exfalso. eapply Hfal. rewrite (comm op); eauto.
- assert (n' < k). omega.
move: laterN_small. uPred.unseal. naive_solver.
Qed.
(* Questions:
1) Can one prove an adequacy theorem for the |=n=> modality without axioms?
2) If not, can we prove a weakened form of adequacy, such as :
Lemma adequacy' φ n : (True ⊢ Nat.iter n (λ P, |=n=> ▷ P) (■ φ)) → ¬¬ φ.
3) Do the basic properties of the |=r=> modality (rvs_intro, rvs_mono, rvs_trans, rvs_frame_r,
rvs_ownM_updateP, and adequacy) characterize |=r=>?
*)
End rvs_nn.
\ No newline at end of file
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