### vastly simplify the counterexample

parent c6668f89
 ... ... @@ -90,36 +90,22 @@ Module inv. Section inv. Hypothesis inv_open : forall i P Q R, (P ★ Q ⊢ pvs0 (P ★ R)) → (inv i P ★ Q ⊢ pvs1 R). (* We have tokens for a little "three-state STS": [fresh] -> [start n] -> [finish n]. The [auth_*] tokens are in the invariant and assert an exact state. [fresh] also asserts the exact state; it is owned by threads (i.e., there's a token needed to transition to [start].) [started] and [finished] are *lower bounds*. We don't need "auth_finish" because the state will never change again, so [finished] is just as good. *) Context (auth_fresh fresh : iProp). Context (auth_start started finished : name → iProp). Hypothesis fresh_start : forall n, auth_fresh ★ fresh ⊢ pvs0 (auth_start n ★ started n). Hypotheses start_finish : forall n, auth_start n ⊢ pvs0 (finished n). Hypothesis fresh_not_start : forall n, auth_start n ★ fresh ⊢ False. Hypothesis fresh_not_finished : forall n, finished n ★ fresh ⊢ False. Hypothesis started_not_fresh : forall n, auth_fresh ★ started n ⊢ False. Hypothesis finished_not_start : forall n m, auth_start n ★ finished m ⊢ False. Hypothesis started_start_agree : forall n m, auth_start n ★ started m ⊢ n = m. Hypothesis started_finished_agree : forall n m, finished n ★ started m ⊢ n = m. Hypothesis finished_agree : forall n m, finished n ★ finished m ⊢ n = m. Hypothesis started_dup : forall n, started n ⊢ started n ★ started n. Hypothesis finished_dup : forall n, finished n ⊢ finished n ★ finished n. (* We have tokens for a little "two-state STS": [start] -> [finish]. state. [start] also asserts the exact state; it is only ever owned by the invariant. [finished] is duplicable. *) Context (gname : Type). Context (start finished : gname → iProp). Hypothesis sts_alloc : True ⊢ pvs0 (∃ γ, start γ). Hypotheses start_finish : forall γ, start γ ⊢ pvs0 (finished γ). Hypothesis finished_not_start : forall γ, start γ ★ finished γ ⊢ False. Hypothesis finished_dup : forall γ, finished γ ⊢ finished γ ★ finished γ. (* We have that we cannot view shift from the initial state to false (because the initial state is actually achievable). *) Hypothesis soundness : ¬ (auth_fresh ★ fresh ⊢ pvs1 False). Hypothesis soundness : ¬ (True ⊢ pvs1 False). (** Some general lemmas and proof mode compatibility. *) Lemma inv_open' i P R: ... ... @@ -191,144 +177,73 @@ Module inv. Section inv. apply pvs1_mono. by rewrite -HP -(uPred.exist_intro a). Qed. (* "Weak box" -- a weak form of □ for non-persistent assertions. *) Definition wbox P : iProp := ∃ Q, Q ★ □(Q → P) ★ □(Q → Q ★ Q). Lemma wbox_dup P : wbox P ⊢ wbox P ★ wbox P. Proof. iIntros "H". iDestruct "H" as (Q) "(HQ & #HP & #Hdup)". iDestruct ("Hdup" with "HQ") as "[HQ HQ']". iSplitL "HQ"; iExists Q; iSplit; eauto. Qed. Lemma wbox_out P : wbox P ⊢ P. Proof. iIntros "H". iDestruct "H" as (Q) "(HQ & #HP & _)". iApply "HP". done. Qed. (** Now to the actual counterexample. We start with a weird for of saved propositions. *) Definition saved (i : name) (P : iProp) : iProp := ∃ F : name → iProp, P = F i ★ started i ★ inv i (auth_fresh ∨ ∃ j, auth_start j ∨ (finished j ★ wbox (F j))). Definition saved (γ : gname) (P : iProp) : iProp := ∃ i, inv i (start γ ∨ (finished γ ★ □P)). Global Instance : forall γ P, PersistentP (saved γ P) := _. Lemma saved_dup i P : saved i P ⊢ saved i P ★ saved i P. Lemma saved_alloc (P : gname → iProp) : True ⊢ pvs1 (∃ γ, saved γ (P γ)). Proof. iIntros "H". iDestruct "H" as (F) "(#? & Hs & #?)". iDestruct (started_dup with "Hs") as "[Hs Hs']". iSplitL "Hs". - iExists F. eauto. - iExists F. eauto. Qed. Lemma saved_alloc (P : name → iProp) : auth_fresh ★ fresh ⊢ pvs1 (∃ i, saved i (P i)). Proof. iIntros "[Haf Hf]". iVs (inv_alloc (auth_fresh ∨ ∃ j, auth_start j ∨ (finished j ★ wbox (P j))) with "[Haf]") as (i) "#Hi". iIntros "". iVs (sts_alloc) as (γ) "Hs". iVs (inv_alloc (start γ ∨ (finished γ ★ □ (P γ))) with "[Hs]") as (i) "#Hi". { iLeft. done. } iExists i. iApply inv_open'. iSplit; first done. iIntros "[Haf|Has]"; last first. { iExFalso. iDestruct "Has" as (j) "[Has | [Haf _]]". - iApply fresh_not_start. iSplitL "Has"; done. - iApply fresh_not_finished. iSplitL "Haf"; done. } iVs ((fresh_start i) with "[Hf Haf]") as "[Has Hs]"; first by iFrame. iDestruct (started_dup with "Hs") as "[Hs Hs']". iApply pvs0_intro. iSplitR "Hs'". - iRight. iExists i. iLeft. done. - iApply pvs1_intro. iExists P. iSplit; first done. by iFrame. iApply pvs1_intro. iExists γ, i. done. Qed. Lemma saved_cast i P Q : saved i P ★ saved i Q ★ wbox P ⊢ pvs1 (wbox Q). Lemma saved_cast γ P Q : saved γ P ★ saved γ Q ★ □ P ⊢ pvs1 (□ Q). Proof. iIntros "(HsP & HsQ & HP)". iDestruct "HsP" as (FP) "(% & HsP & #HiP)". iIntros "(#HsP & #HsQ & #HP)". iDestruct "HsP" as (i) "HiP". iApply (inv_open' i). iSplit; first done. iIntros "[HaP|HaP]". { iExFalso. iApply started_not_fresh. iSplitL "HaP"; done. } (* Can I state a view-shift and immediately run it? *) iAssert (pvs0 (finished i)) with "[HaP HsP]" as "Hf". { iDestruct "HaP" as (j) "[Hs | [Hf _]]". - iApply start_finish. iDestruct (started_start_agree with "[#]") as "%"; first by iSplitL "Hs". subst j. done. - iApply pvs0_intro. iDestruct (started_finished_agree with "[#]") as "%"; first by iSplitL "Hf". subst j. done. } iVs "Hf" as "Hf". iApply pvs0_intro. iDestruct (finished_dup with "Hf") as "[Hf Hf']". iSplitL "Hf' HP". { iRight. iExists i. iRight. subst. iSplitL "Hf'"; done. } iDestruct "HsQ" as (FQ) "(% & HsQ & HiQ)". iApply (inv_open' i). iSplit; first iExact "HiQ". iIntros "[HaQ | HaQ]". { iExFalso. iApply started_not_fresh. iSplitL "HaQ"; done. } iDestruct "HaQ" as (j) "[HaS | [Hf' HQ]]". { iExFalso. iApply finished_not_start. iSplitL "HaS"; done. } iApply pvs0_intro. iDestruct (finished_dup with "Hf'") as "[Hf' Hf'']". iDestruct (wbox_dup with "HQ") as "[HQ HQ']". iSplitL "Hf'' HQ'". { iRight. iExists j. iRight. by iSplitR "HQ'". } iPoseProof (finished_agree with "[#]") as "H". { iFrame "Hf Hf'". done. } iDestruct "H" as %<-. iApply pvs1_intro. subst Q. done. iIntros "HaP". iAssert (pvs0 (finished γ)) with "[HaP]" as "Hf". { iDestruct "HaP" as "[Hs | [Hf _]]". - by iApply start_finish. - by iApply pvs0_intro. } iVs "Hf" as "Hf". iDestruct (finished_dup with "Hf") as "[Hf Hf']". iApply pvs0_intro. iSplitL "Hf'"; first by eauto. (* Step 2: Open the Q-invariant. *) iClear "HiP". clear i. iDestruct "HsQ" as (i) "HiQ". iApply (inv_open' i). iSplit; first done. iIntros "[HaQ | [_ #HQ]]". { iExFalso. iApply finished_not_start. iSplitL "HaQ"; done. } iApply pvs0_intro. iSplitL "Hf". { iRight. by iSplitL "Hf". } by iApply pvs1_intro. Qed. (** And now we tie a bad knot. *) Notation "¬ P" := (wbox (P -★ pvs1 False))%I : uPred_scope. Notation "¬ P" := (□ (P -★ pvs1 False))%I : uPred_scope. Definition A i : iProp := ∃ P, ¬P ★ saved i P. Lemma A_dup i : A i ⊢ A i ★ A i. Proof. iIntros "HA". iDestruct "HA" as (P) "[HNP HsP]". iDestruct (wbox_dup with "HNP") as "[HNP HNP']". iDestruct (saved_dup with "HsP") as "[HsP HsP']". iSplitL "HNP HsP"; iExists P. - by iSplitL "HNP". - by iSplitL "HNP'". Qed. Lemma A_wbox i : A i ⊢ wbox (A i). Proof. iIntros "H". iExists (A i). iSplitL "H"; first done. iSplit; first by iIntros "!# ?". iIntros "!# HA". by iApply A_dup. Qed. Global Instance : forall i, PersistentP (A i) := _. Lemma A_alloc : auth_fresh ★ fresh ⊢ pvs1 (∃ i, saved i (A i)). True ⊢ pvs1 (∃ i, saved i (A i)). Proof. by apply saved_alloc. Qed. Lemma alloc_NA i : saved i (A i) ⊢ (¬A i). Proof. iIntros "Hi". iExists (saved i (A i)). iSplitL "Hi"; first done. iSplit; last by (iIntros "!# ?"; iApply saved_dup). iIntros "!# Hi HAi". iDestruct (A_dup with "HAi") as "[HAi HAi']". iDestruct "HAi'" as (P) "[HNP Hi']". iVs ((saved_cast i) with "[Hi Hi' HAi]") as "HP". { iSplitL "Hi"; first done. iSplitL "Hi'"; first done. by iApply A_wbox. } iPoseProof (wbox_out with "HNP") as "HNP". iApply "HNP". iApply wbox_out. done. iIntros "#Hi !# #HA". iPoseProof "HA" as "HA'". iDestruct "HA'" as (P) "#[HNP Hi']". iVs ((saved_cast i) with "[]") as "HP". { iSplit; first iExact "Hi". iSplit; first iExact "Hi'". done. } by iApply "HNP". Qed. Lemma alloc_A i : saved i (A i) ⊢ A i. Proof. iIntros "Hi". iDestruct (saved_dup with "Hi") as "[Hi Hi']". iPoseProof (alloc_NA with "Hi") as "HNA". iExists (A i). iSplitL "HNA"; done. iIntros "#Hi". iPoseProof (alloc_NA with "Hi") as "HNA". iExists (A i). iSplit; done. Qed. Lemma contradiction : False. Proof. apply soundness. iIntros "H". iVs (A_alloc with "H") as "H". iDestruct "H" as (i) "H". iDestruct (saved_dup with "H") as "[H H']". apply soundness. iIntros "". iVs A_alloc as (i) "#H". iPoseProof (alloc_NA with "H") as "HN". iPoseProof (wbox_out with "HN") as "HN". iApply "HN". iApply alloc_A. done. Qed. ... ...
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