Commit c6668f89 authored by Ralf Jung's avatar Ralf Jung

counterexample no longer needs duplicable ghost state

parent 230444d4
......@@ -114,8 +114,8 @@ Module inv. Section inv.
Hypothesis finished_agree :
forall n m, finished n finished m n = m.
Hypothesis started_persistent : forall n, PersistentP (started n).
Hypothesis finished_persistent : forall n, PersistentP (finished n).
Hypothesis started_dup : forall n, started n started n started n.
Hypothesis finished_dup : forall n, finished n finished n finished n.
(* We have that we cannot view shift from the initial state to false
(because the initial state is actually achievable). *)
......@@ -191,60 +191,110 @@ 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 F j)).
inv i (auth_fresh j, auth_start j (finished j wbox (F j))).
Lemma saved_dup i P :
saved i P saved i P saved i P.
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 P j)) with "[Haf]") as (i) "#Hi".
iIntros "[Haf Hf]". iVs (inv_alloc (auth_fresh j, auth_start j (finished j wbox (P j))) with "[Haf]") 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.
iApply pvs0_intro. iSplitL.
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 P. iSplit; first done. by iFrame.
Qed.
Lemma saved_cast i P Q :
saved i P saved i Q P pvs1 (Q).
saved i P saved i Q wbox P pvs1 (wbox Q).
Proof.
iIntros "(#HsP & #HsQ & #HP)". iDestruct "HsP" as (FP) "(% & HsP & HiP)".
iIntros "(HsP & HsQ & HP)". iDestruct "HsP" as (FP) "(% & HsP & #HiP)".
iApply (inv_open' i). iSplit; first done.
iIntros "[HaP|HaP]".
{ iExFalso. iApply started_not_fresh. iSplit; done. }
{ iExFalso. iApply started_not_fresh. iSplitL "HaP"; done. }
(* Can I state a view-shift and immediately run it? *)
iAssert (pvs0 (finished i)) with "[HaP]" as "Hf".
iAssert (pvs0 (finished i)) with "[HaP HsP]" as "Hf".
{ iDestruct "HaP" as (j) "[Hs | [Hf _]]".
- iApply start_finish. (* FIXME: iPoseProof as "%" calls the assertion "%" instead of moving to the Coq context. *)
iPoseProof (started_start_agree with "[#]") as "H"; first by iSplit.
iDestruct "H" as %<-. done.
- iApply pvs0_intro. iPoseProof (started_finished_agree with "[#]") as "H"; first by iSplit.
iDestruct "H" as %<-. done. }
iVs "Hf" as "#Hf". iApply pvs0_intro. iSplitL.
{ iRight. iExists i. iRight. subst. eauto. }
- 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. iSplit; done. }
iDestruct "HaQ" as (j) "[HaS | #[Hf' HQ]]".
{ iExFalso. iApply finished_not_start. eauto. }
iApply pvs0_intro. iSplitL.
{ iRight. iExists j. eauto. }
{ 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.
Qed.
(** And now we tie a bad knot. *)
Notation "¬ P" := ( (P pvs1 False))%I : uPred_scope.
Notation "¬ P" := (wbox (P - pvs1 False))%I : uPred_scope.
Definition A i : iProp := P, ¬P saved i P.
Instance : forall i, PersistentP (A i) := _.
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.
Lemma A_alloc :
auth_fresh fresh pvs1 ( i, saved i (A i)).
......@@ -253,28 +303,33 @@ Module inv. Section inv.
Lemma alloc_NA i :
saved i (A i) (¬A i).
Proof.
iIntros "#Hi !# #HAi". iPoseProof "HAi" as "HAi'".
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 "[]") as "HP".
{ iSplit; first iExact "Hi". iSplit; first iExact "Hi'". done. }
iDestruct "HP" as "#HP". by iApply "HNP".
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.
Qed.
Lemma alloc_A i :
saved i (A i) A i.
Proof.
iIntros "#Hi". iPoseProof (alloc_NA with "[]") as "HNA"; first done.
(* Patterns in iPoseProof don't seem to work; adding a "#" here also does the wrong thing.
Or maybe iPoseProof is the wrong tactic -- but then which is the right one? *)
iDestruct "HNA" as "#HNA". iExists (A i).
iSplit; done.
iIntros "Hi". iDestruct (saved_dup with "Hi") as "[Hi Hi']".
iPoseProof (alloc_NA with "Hi") as "HNA".
iExists (A i). iSplitL "HNA"; done.
Qed.
Lemma contradiction : False.
Proof.
apply soundness. iIntros "H".
iVs (A_alloc with "H") as "H". iDestruct "H" as (i) "#H".
iPoseProof (alloc_NA with "H") as "HN". iApply "HN". (* FIXME: "iApply alloc_NA" does not work. *)
iVs (A_alloc with "H") as "H". iDestruct "H" as (i) "H".
iDestruct (saved_dup with "H") as "[H 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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