Commit b0bd1855 authored by Robbert Krebbers's avatar Robbert Krebbers

Merge branch 'iris3.0' of gitlab.mpi-sws.org:FP/iris-coq into iris3.0

parents 252130c9 b907771f
......@@ -3,7 +3,7 @@ From iris.proofmode Require Import tactics.
(** This proves that we need the ▷ in a "Saved Proposition" construction with
name-dependend allocation. *)
Section savedprop.
Module savedprop. Section savedprop.
Context (M : ucmraT).
Notation iProp := (uPred M).
Notation "¬ P" := ( (P False))%I : uPred_scope.
......@@ -57,4 +57,194 @@ Section savedprop.
apply (@uPred.adequacy M False 1); simpl.
rewrite -uPred.later_intro. apply rvs_false.
Qed.
End savedprop.
End savedprop. End savedprop.
(** This proves that we need the ▷ when opening invariants. *)
(** We fork in [uPred M] for any M, but the proof would work in any BI. *)
Module inv. Section inv.
Context (M : ucmraT).
Notation iProp := (uPred M).
Implicit Types P : iProp.
(** Assumptions *)
(* We have view shifts (two classes: empty/full mask) *)
Context (pvs0 pvs1 : iProp iProp).
Hypothesis pvs0_intro : forall P, P pvs0 P.
Hypothesis pvs0_mono : forall P Q, (P Q) pvs0 P pvs0 Q.
Hypothesis pvs0_pvs0 : forall P, pvs0 (pvs0 P) pvs0 P.
Hypothesis pvs0_frame_l : forall P Q, P pvs0 Q pvs0 (P Q).
Hypothesis pvs1_mono : forall P Q, (P Q) pvs1 P pvs1 Q.
Hypothesis pvs1_pvs1 : forall P, pvs1 (pvs1 P) pvs1 P.
Hypothesis pvs1_frame_l : forall P Q, P pvs1 Q pvs1 (P Q).
Hypothesis pvs0_pvs1 : forall P, pvs0 P pvs1 P.
(* We have invariants *)
Context (name : Type) (inv : name iProp iProp).
Hypothesis inv_persistent : forall i P, PersistentP (inv i P).
Hypothesis inv_alloc :
forall (P : iProp), P pvs1 ( i, inv i P).
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 "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 assume that we cannot view shift to false. *)
Hypothesis soundness : ¬ (True pvs1 False).
(** Some general lemmas and proof mode compatibility. *)
Lemma inv_open' i P R:
inv i P (P - pvs0 (P pvs1 R)) pvs1 R.
Proof.
iIntros "(#HiP & HP)". iApply pvs1_pvs1. iApply inv_open; last first.
{ iSplit; first done. iExact "HP". }
iIntros "(HP & HPw)". by iApply "HPw".
Qed.
Lemma pvs1_intro P : P pvs1 P.
Proof. rewrite -pvs0_pvs1. apply pvs0_intro. Qed.
Instance pvs0_mono' : Proper (() ==> ()) pvs0.
Proof. intros ?**. by apply pvs0_mono. Qed.
Instance pvs0_proper : Proper (() ==> ()) pvs0.
Proof.
intros P Q Heq.
apply (anti_symm ()); apply pvs0_mono; by rewrite ?Heq -?Heq.
Qed.
Instance pvs1_mono' : Proper (() ==> ()) pvs1.
Proof. intros ?**. by apply pvs1_mono. Qed.
Instance pvs1_proper : Proper (() ==> ()) pvs1.
Proof.
intros P Q Heq.
apply (anti_symm ()); apply pvs1_mono; by rewrite ?Heq -?Heq.
Qed.
Lemma pvs0_frame_r : forall P Q, (pvs0 P Q) pvs0 (P Q).
Proof.
intros. rewrite comm pvs0_frame_l. apply pvs0_mono. by rewrite comm.
Qed.
Lemma pvs1_frame_r : forall P Q, (pvs1 P Q) pvs1 (P Q).
Proof.
intros. rewrite comm pvs1_frame_l. apply pvs1_mono. by rewrite comm.
Qed.
Global Instance elim_pvs0_pvs0 P Q :
ElimVs (pvs0 P) P (pvs0 Q) (pvs0 Q).
Proof.
rewrite /ElimVs. etrans; last eapply pvs0_pvs0.
rewrite pvs0_frame_r. apply pvs0_mono. by rewrite uPred.wand_elim_r.
Qed.
Global Instance elim_pvs1_pvs1 P Q :
ElimVs (pvs1 P) P (pvs1 Q) (pvs1 Q).
Proof.
rewrite /ElimVs. etrans; last eapply pvs1_pvs1.
rewrite pvs1_frame_r. apply pvs1_mono. by rewrite uPred.wand_elim_r.
Qed.
Global Instance elim_pvs0_pvs1 P Q :
ElimVs (pvs0 P) P (pvs1 Q) (pvs1 Q).
Proof.
rewrite /ElimVs. rewrite pvs0_pvs1. apply elim_pvs1_pvs1.
Qed.
Global Instance exists_split_pvs0 {A} P (Φ : A iProp) :
FromExist P Φ FromExist (pvs0 P) (λ a, pvs0 (Φ a)).
Proof.
rewrite /FromExist=>HP. apply uPred.exist_elim=> a.
apply pvs0_mono. by rewrite -HP -(uPred.exist_intro a).
Qed.
Global Instance exists_split_pvs1 {A} P (Φ : A iProp) :
FromExist P Φ FromExist (pvs1 P) (λ a, pvs1 (Φ a)).
Proof.
rewrite /FromExist=>HP. apply uPred.exist_elim=> a.
apply pvs1_mono. by rewrite -HP -(uPred.exist_intro a).
Qed.
(** Now to the actual counterexample. We start with a weird for of saved propositions. *)
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_alloc (P : gname iProp) :
True pvs1 ( γ, saved γ (P γ)).
Proof.
iIntros "". iVs (sts_alloc) as (γ) "Hs".
iVs (inv_alloc (start γ (finished γ (P γ))) with "[Hs]") as (i) "#Hi".
{ iLeft. done. }
iApply pvs1_intro. iExists γ, i. done.
Qed.
Lemma saved_cast γ P Q :
saved γ P saved γ Q P pvs1 ( Q).
Proof.
iIntros "(#HsP & #HsQ & #HP)". iDestruct "HsP" as (i) "HiP".
iApply (inv_open' i). iSplit; first done.
(* Can I state a view-shift and immediately run it? *)
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" := ( (P - pvs1 False))%I : uPred_scope.
Definition A i : iProp := P, ¬P saved i P.
Global Instance : forall i, PersistentP (A i) := _.
Lemma A_alloc :
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 !# #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". iPoseProof (alloc_NA with "Hi") as "HNA".
iExists (A i). iSplit; done.
Qed.
Lemma contradiction : False.
Proof.
apply soundness. iIntros "".
iVs A_alloc as (i) "#H".
iPoseProof (alloc_NA with "H") as "HN".
iApply "HN".
iApply alloc_A. done.
Qed.
End inv. End inv.
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