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Commit 60df6185 authored by Robbert Krebbers's avatar Robbert Krebbers
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Simplify soundness of the base logic. 

Now that we have the plain modality, we can get rid of the basic updates
in the soundness statement.
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theories/base_logic/lib/counter_examples.v
+ 2
− 4
View file @ 60df6185
......@@ -40,13 +40,11 @@ Module savedprop. Section savedprop.
Lemma contradiction : False.
Proof using All.
apply (@soundness M False 1); simpl.
apply (@consistency M); simpl.
iIntros "". iMod A_alloc as (i) "#H".
iPoseProof (saved_NA with "H") as "HN".
iModIntro. iNext.
iApply "HN". iApply saved_A. done.
iApply "HN". by iApply saved_A.
Qed.
End savedprop. End savedprop.
(** This proves that we need the ▷ when opening invariants. *)
......
theories/base_logic/soundness.v
+ 5
− 9
View file @ 60df6185
......@@ -6,18 +6,14 @@ Section adequacy.
Context {M : ucmraT}.
(** Consistency and adequancy statements *)
Lemma soundness φ n : (Nat.iter n (λ P, |==> ▷ P) (@uPred_pure M φ))%I → φ.
Lemma soundness φ n : (▷^n ⌜ φ ⌝ : uPred M)%I → φ.
Proof.
cut (∀ x, ✓{n} x → Nat.iter n (λ P, |==> ▷ P)%I (@uPred_pure M φ) n x → φ).
{ intros help H. eapply (help ∅); eauto using ucmra_unit_validN.
eapply H; try unseal; by eauto using ucmra_unit_validN. }
unseal. induction n as [|n IH]=> x Hx Hupd; auto.
destruct (Hupd (S n) ε) as (x'&?&?); rewrite ?right_id; auto.
eapply IH with x'; eauto using cmra_validN_S, cmra_validN_op_l.
cut ((▷^n ⌜ φ ⌝ : uPred M)%I n ε → φ).
{ intros help H. eapply help, H; eauto using ucmra_unit_validN. by unseal. }
rewrite /uPred_laterN; unseal. induction n as [|n IH]=> H; auto.
Qed.
Corollary consistency_modal n :
¬ (Nat.iter n (λ P, |==> ▷ P) (False : uPred M))%I.
Corollary consistency_modal n : ¬ (▷^n False : uPred M)%I.
Proof. exact (soundness False n). Qed.
Corollary consistency : ¬ (False : uPred M)%I.
......
theories/program_logic/adequacy.v
+ 28
− 30
View file @ 60df6185
......@@ -106,8 +106,11 @@ Proof.
iModIntro; iNext; iMod "H" as ">?". by iApply IH.
Qed.
Instance bupd_iter_mono n : Proper ((⊢) ==> (⊢)) (Nat.iter n (λ P, |==> ▷ P)%I).
Proof. intros P Q HP. induction n; simpl; do 2?f_equiv; auto. Qed.
Lemma bupd_iter_laterN_mono n P Q `{!Plain Q} :
(P ⊢ Q) → Nat.iter n (λ P, |==> ▷ P)%I P ⊢ ▷^n Q.
Proof.
intros HPQ. induction n as [|n IH]=> //=. by rewrite IH bupd_plain.
Qed.
Lemma bupd_iter_frame_l n R Q :
R ∗ Nat.iter n (λ P, |==> ▷ P) Q ⊢ Nat.iter n (λ P, |==> ▷ P) (R ∗ Q).
......@@ -118,44 +121,42 @@ Qed.
Lemma wptp_result n e1 t1 v2 t2 σ1 σ2 φ :
nsteps step n (e1 :: t1, σ1) (of_val v2 :: t2, σ2) →
world σ1 ∗ WP e1 {{ v, ⌜φ v⌝ }} ∗ wptp t1 ⊢
Nat.iter (S (S n)) (λ P, |==> ▷ P) ⌜φ v2⌝.
world σ1 ∗ WP e1 {{ v, ⌜φ v⌝ }} ∗ wptp t1 ⊢ ▷^(S (S n)) ⌜φ v2⌝.
Proof.
intros. rewrite wptp_steps //.
rewrite (Nat_iter_S_r (S n)). apply bupd_iter_mono.
intros. rewrite wptp_steps // laterN_later. apply: bupd_iter_laterN_mono.
iDestruct 1 as (e2 t2' ?) "((Hw & HE & _) & H & _)"; simplify_eq.
iDestruct (wp_value_inv with "H") as "H". rewrite fupd_eq /fupd_def.
iMod ("H" with "[Hw HE]") as ">(_ & _ & $)"; iFrame; auto.
Qed.
Lemma wp_safe e σ Φ :
world σ ∗ WP e {{ Φ }} ==∗ ▷ ⌜is_Some (to_val e) ∨ reducible e σ⌝.
world σ -∗ WP e {{ Φ }} ==∗ ▷ ⌜is_Some (to_val e) ∨ reducible e σ⌝.
Proof.
rewrite wp_unfold /wp_pre. iIntros "[(Hw&HE&Hσ) H]".
rewrite wp_unfold /wp_pre. iIntros "(Hw&HE&Hσ) H".
destruct (to_val e) as [v|] eqn:?; [eauto 10|].
rewrite fupd_eq. iMod ("H" with "Hσ [-]") as ">(?&?&%&?)"; first by iFrame.
eauto 10.
rewrite fupd_eq. iMod ("H" with "Hσ [-]") as ">(?&?&%&?)"; eauto 10 with iFrame.
Qed.
Lemma wptp_safe n e1 e2 t1 t2 σ1 σ2 Φ :
nsteps step n (e1 :: t1, σ1) (t2, σ2) → e2 ∈ t2 →
world σ1 ∗ WP e1 {{ Φ }} ∗ wptp t1 ⊢
Nat.iter (S (S n)) (λ P, |==> ▷ P) ⌜is_Some (to_val e2) ∨ reducible e2 σ2⌝.
▷^(S (S n)) ⌜is_Some (to_val e2) ∨ reducible e2 σ2⌝.
Proof.
intros ? He2. rewrite wptp_steps //; rewrite (Nat_iter_S_r (S n)). apply bupd_iter_mono.
intros ? He2. rewrite wptp_steps // laterN_later. apply: bupd_iter_laterN_mono.
iDestruct 1 as (e2' t2' ?) "(Hw & H & Htp)"; simplify_eq.
apply elem_of_cons in He2 as [<-|?]; first (iApply wp_safe; by iFrame "Hw H").
iApply wp_safe. iFrame "Hw". by iApply (big_sepL_elem_of with "Htp").
apply elem_of_cons in He2 as [<-|?].
- iMod (wp_safe with "Hw H") as "$".
- iMod (wp_safe with "Hw [Htp]") as "$". by iApply (big_sepL_elem_of with "Htp").
Qed.
Lemma wptp_invariance n e1 e2 t1 t2 σ1 σ2 φ Φ :
nsteps step n (e1 :: t1, σ1) (t2, σ2) →
(state_interp σ2 ={⊤,∅}=∗ ⌜φ⌝) ∗ world σ1 ∗ WP e1 {{ Φ }} ∗ wptp t1
⊢ Nat.iter (S (S n)) (λ P, |==> ▷ P) ⌜φ⌝.
⊢ ▷^(S (S n)) ⌜φ⌝.
Proof.
intros ?. rewrite wptp_steps //.
rewrite (Nat_iter_S_r (S n)) !bupd_iter_frame_l. apply bupd_iter_mono.
iIntros "[Hback H]". iDestruct "H" as (e2' t2' ->) "[(Hw&HE&Hσ) _]".
intros ?. rewrite wptp_steps // bupd_iter_frame_l laterN_later.
apply: bupd_iter_laterN_mono.
iIntros "[Hback H]"; iDestruct "H" as (e2' t2' ->) "[(Hw&HE&Hσ) _]".
rewrite fupd_eq.
iMod ("Hback" with "Hσ [$Hw $HE]") as "> (_ & _ & $)"; auto.
Qed.
......@@ -170,19 +171,17 @@ Theorem wp_adequacy Σ Λ `{invPreG Σ} e σ φ :
Proof.
intros Hwp; split.
- intros t2 σ2 v2 [n ?]%rtc_nsteps.
eapply (soundness (M:=iResUR Σ) _ (S (S (S n)))); iIntros "".
rewrite Nat_iter_S. iMod wsat_alloc as (Hinv) "[Hw HE]".
eapply (soundness (M:=iResUR Σ) _ (S (S n))).
iMod wsat_alloc as (Hinv) "[Hw HE]".
rewrite fupd_eq in Hwp; iMod (Hwp with "[$Hw $HE]") as ">(Hw & HE & Hwp)".
iDestruct "Hwp" as (Istate) "[HI Hwp]".
iModIntro. iNext. iApply (@wptp_result _ _ (IrisG _ _ Hinv Istate)); eauto.
by iFrame.
iApply (@wptp_result _ _ (IrisG _ _ Hinv Istate)); eauto with iFrame.
- intros t2 σ2 e2 [n ?]%rtc_nsteps ?.
eapply (soundness (M:=iResUR Σ) _ (S (S (S n)))); iIntros "".
rewrite Nat_iter_S. iMod wsat_alloc as (Hinv) "[Hw HE]".
eapply (soundness (M:=iResUR Σ) _ (S (S n))).
iMod wsat_alloc as (Hinv) "[Hw HE]".
rewrite fupd_eq in Hwp; iMod (Hwp with "[$Hw $HE]") as ">(Hw & HE & Hwp)".
iDestruct "Hwp" as (Istate) "[HI Hwp]".
iModIntro. iNext. iApply (@wptp_safe _ _ (IrisG _ _ Hinv Istate)); eauto.
by iFrame.
iApply (@wptp_safe _ _ (IrisG _ _ Hinv Istate)); eauto with iFrame.
Qed.
Theorem wp_invariance Σ Λ `{invPreG Σ} e σ1 t2 σ2 φ :
......@@ -194,10 +193,9 @@ Theorem wp_invariance Σ Λ `{invPreG Σ} e σ1 t2 σ2 φ :
φ.
Proof.
intros Hwp [n ?]%rtc_nsteps.
eapply (soundness (M:=iResUR Σ) _ (S (S (S n)))); iIntros "".
rewrite Nat_iter_S. iMod wsat_alloc as (Hinv) "[Hw HE]".
eapply (soundness (M:=iResUR Σ) _ (S (S n))).
iMod wsat_alloc as (Hinv) "[Hw HE]".
rewrite {1}fupd_eq in Hwp; iMod (Hwp with "[$Hw $HE]") as ">(Hw & HE & Hwp)".
iDestruct "Hwp" as (Istate) "(HIstate & Hwp & Hclose)".
iModIntro. iNext. iApply (@wptp_invariance _ _ (IrisG _ _ Hinv Istate)); eauto.
by iFrame.
iApply (@wptp_invariance _ _ (IrisG _ _ Hinv Istate)); eauto with iFrame.
Qed.
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