From b03f7081d56a98c88cd9a6f367b34c24b97485fd Mon Sep 17 00:00:00 2001
From: Ralf Jung <jung@mpi-sws.org>
Date: Thu, 6 Feb 2020 14:46:24 +0100
Subject: [PATCH] another variant of the linear-inv counterexample, based on an
 accessor that gives back the token earlier

---
 theories/bi/lib/counterexamples.v | 70 +++++++++++++++++++++++++++++--
 1 file changed, 67 insertions(+), 3 deletions(-)

diff --git a/theories/bi/lib/counterexamples.v b/theories/bi/lib/counterexamples.v
index 1747b2cca..3c4467d8b 100644
--- a/theories/bi/lib/counterexamples.v
+++ b/theories/bi/lib/counterexamples.v
@@ -217,8 +217,10 @@ Module inv. Section inv.
 End inv. End inv.
 
 (** This proves that if we have linear impredicative invariants, we can still
-drop arbitrary resources (i.e., we can "defeat" linearity). *)
-Module linear. Section linear.
+drop arbitrary resources (i.e., we can "defeat" linearity).
+Variant 1: a strong invariant creation lemma that lets us create invariants
+in the "open" state. *)
+Module linear1. Section linear1.
   Context {PROP: sbi}.
   Implicit Types P : PROP.
 
@@ -265,5 +267,67 @@ Module linear. Section linear.
     iMod (cinv_alloc_open INV) as (γ) "(Hinv & Htok & Hclose)".
     iApply "Hclose". iNext. iExists γ, _. iFrame.
   Qed.
+End linear1. End linear1.
 
-End linear. End linear.
+(** This proves that if we have linear impredicative invariants, we can still
+drop arbitrary resources (i.e., we can "defeat" linearity).
+Variant 2: the invariant can depend on the chose gname, and we also have
+an accessor that gives back the invariant token immediately,
+not just after the invariant got closed again. *)
+Module linear2. Section linear2.
+  Context {PROP: sbi}.
+  Implicit Types P : PROP.
+
+  (** Assumptions. *)
+  (** We have the mask-changing update modality (two classes: empty/full mask) *)
+  Inductive mask := M0 | M1.
+  Context (fupd : mask → mask → PROP → PROP).
+  Arguments fupd _ _ _%I.
+  Hypothesis fupd_intro : ∀ E P, P ⊢ fupd E E P.
+  Hypothesis fupd_mono : ∀ E1 E2 P Q, (P ⊢ Q) → fupd E1 E2 P ⊢ fupd E1 E2 Q.
+  Hypothesis fupd_fupd : ∀ E1 E2 E3 P, fupd E1 E2 (fupd E2 E3 P) ⊢ fupd E1 E3 P.
+  Hypothesis fupd_frame_l : ∀ E1 E2 P Q, P ∗ fupd E1 E2 Q ⊢ fupd E1 E2 (P ∗ Q).
+
+  (** We have cancellable invariants. (Really they would have fractions at
+  [cinv_own] but we do not need that. They would also have a name matching
+  the [mask] type, but we do not need that either.) *)
+  Context (gname : Type) (cinv : gname → PROP → PROP) (cinv_own : gname → PROP).
+  Hypothesis cinv_alloc : ∀ E,
+    fupd E E (∃ γ, cinv_own γ ∗ (∀ P, ▷ P -∗ fupd E E (cinv γ P)))%I.
+  Hypothesis cinv_access : ∀ P γ,
+    cinv γ P -∗ cinv_own γ -∗ fupd M1 M0 (▷ P ∗ cinv_own γ ∗ (▷ P -∗ fupd M0 M1 emp)).
+  Hypothesis cinv_own_excl : ∀ E γ,
+    ▷ cinv_own γ -∗ ▷ cinv_own γ -∗ fupd E E False.
+
+  (** Some general lemmas and proof mode compatibility. *)
+  Instance fupd_mono' E1 E2 : Proper ((⊢) ==> (⊢)) (fupd E1 E2).
+  Proof. intros P Q ?. by apply fupd_mono. Qed.
+  Instance fupd_proper E1 E2 : Proper ((⊣⊢) ==> (⊣⊢)) (fupd E1 E2).
+  Proof.
+    intros P Q; rewrite !bi.equiv_spec=> -[??]; split; by apply fupd_mono.
+  Qed.
+
+  Lemma fupd_frame_r E1 E2 P Q : fupd E1 E2 P ∗ Q ⊢ fupd E1 E2 (P ∗ Q).
+  Proof. by rewrite comm fupd_frame_l comm. Qed.
+
+  Global Instance elim_fupd_fupd p E1 E2 E3 P Q :
+    ElimModal True p false (fupd E1 E2 P) P (fupd E1 E3 Q) (fupd E2 E3 Q).
+  Proof.
+    by rewrite /ElimModal bi.intuitionistically_if_elim
+      fupd_frame_r bi.wand_elim_r fupd_fupd.
+  Qed.
+
+  (** Counterexample: now we can make any resource disappear. *)
+  Lemma leak P : P -∗ fupd M1 M1 emp.
+  Proof.
+    iIntros "HP".
+    iMod cinv_alloc as (γ) "(Htok & Hmkinv)".
+    set (INV := (P ∨ cinv_own γ ∗ P)%I).
+    iMod ("Hmkinv" $! INV with "[HP]") as "Hinv".
+    { iLeft. done. }
+    iMod (cinv_access with "Hinv Htok") as "([HP|Hinv] & Htok & Hclose)"; last first.
+    { iDestruct "Hinv" as "(Htok' & ?)".
+      iMod (cinv_own_excl with "Htok Htok'") as "[]". }
+    iApply "Hclose". iNext. iRight. iFrame.
+  Qed.
+End linear2. End linear2.
-- 
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