Library iris.base_logic.lib.cancelable_invariants
From iris.algebra Require Export frac.
From iris.bi.lib Require Import fractional.
From iris.proofmode Require Import proofmode.
From iris.base_logic.lib Require Export invariants.
From iris.prelude Require Import options.
Import uPred.
Class cinvG Σ := { cinv_inG : inG Σ fracR }.
Local Existing Instance cinv_inG.
Definition cinvΣ : gFunctors := #[GFunctor fracR].
Global Instance subG_cinvΣ {Σ} : subG cinvΣ Σ → cinvG Σ.
Proof. solve_inG. Qed.
Section defs.
Context `{!invGS_gen hlc Σ, !cinvG Σ}.
Definition cinv_own (γ : gname) (p : frac) : iProp Σ := own γ p.
Definition cinv (N : namespace) (γ : gname) (P : iProp Σ) : iProp Σ :=
inv N (P ∨ cinv_own γ 1).
End defs.
Global Instance: Params (@cinv) 5 := {}.
Section proofs.
Context `{!invGS_gen hlc Σ, !cinvG Σ}.
Global Instance cinv_own_timeless γ p : Timeless (cinv_own γ p).
Proof. rewrite /cinv_own; apply _. Qed.
Global Instance cinv_contractive N γ : Contractive (cinv N γ).
Proof. solve_contractive. Qed.
Global Instance cinv_ne N γ : NonExpansive (cinv N γ).
Proof. exact: contractive_ne. Qed.
Global Instance cinv_proper N γ : Proper ((≡) ==> (≡)) (cinv N γ).
Proof. exact: ne_proper. Qed.
Global Instance cinv_persistent N γ P : Persistent (cinv N γ P).
Proof. rewrite /cinv; apply _. Qed.
Global Instance cinv_own_fractional γ : Fractional (cinv_own γ).
Proof. intros ??. by rewrite /cinv_own -own_op. Qed.
Global Instance cinv_own_as_fractional γ q :
AsFractional (cinv_own γ q) (cinv_own γ) q.
Proof. split; [done|]. apply _. Qed.
Lemma cinv_own_valid γ q1 q2 : cinv_own γ q1 -∗ cinv_own γ q2 -∗ ⌜q1 + q2 ≤ 1⌝%Qp.
Proof. rewrite -frac_valid -uPred.discrete_valid. apply (own_valid_2 γ q1 q2). Qed.
Lemma cinv_own_1_l γ q : cinv_own γ 1 -∗ cinv_own γ q -∗ False.
Proof.
iIntros "H1 H2".
iDestruct (cinv_own_valid with "H1 H2") as %[]%(exclusive_l 1%Qp).
Qed.
Lemma cinv_iff N γ P Q : cinv N γ P -∗ ▷ □ (P ↔ Q) -∗ cinv N γ Q.
Proof.
iIntros "HI #HPQ". iApply (inv_iff with "HI"). iIntros "!> !>".
iSplit; iIntros "[?|$]"; iLeft; by iApply "HPQ".
Qed.
From iris.bi.lib Require Import fractional.
From iris.proofmode Require Import proofmode.
From iris.base_logic.lib Require Export invariants.
From iris.prelude Require Import options.
Import uPred.
Class cinvG Σ := { cinv_inG : inG Σ fracR }.
Local Existing Instance cinv_inG.
Definition cinvΣ : gFunctors := #[GFunctor fracR].
Global Instance subG_cinvΣ {Σ} : subG cinvΣ Σ → cinvG Σ.
Proof. solve_inG. Qed.
Section defs.
Context `{!invGS_gen hlc Σ, !cinvG Σ}.
Definition cinv_own (γ : gname) (p : frac) : iProp Σ := own γ p.
Definition cinv (N : namespace) (γ : gname) (P : iProp Σ) : iProp Σ :=
inv N (P ∨ cinv_own γ 1).
End defs.
Global Instance: Params (@cinv) 5 := {}.
Section proofs.
Context `{!invGS_gen hlc Σ, !cinvG Σ}.
Global Instance cinv_own_timeless γ p : Timeless (cinv_own γ p).
Proof. rewrite /cinv_own; apply _. Qed.
Global Instance cinv_contractive N γ : Contractive (cinv N γ).
Proof. solve_contractive. Qed.
Global Instance cinv_ne N γ : NonExpansive (cinv N γ).
Proof. exact: contractive_ne. Qed.
Global Instance cinv_proper N γ : Proper ((≡) ==> (≡)) (cinv N γ).
Proof. exact: ne_proper. Qed.
Global Instance cinv_persistent N γ P : Persistent (cinv N γ P).
Proof. rewrite /cinv; apply _. Qed.
Global Instance cinv_own_fractional γ : Fractional (cinv_own γ).
Proof. intros ??. by rewrite /cinv_own -own_op. Qed.
Global Instance cinv_own_as_fractional γ q :
AsFractional (cinv_own γ q) (cinv_own γ) q.
Proof. split; [done|]. apply _. Qed.
Lemma cinv_own_valid γ q1 q2 : cinv_own γ q1 -∗ cinv_own γ q2 -∗ ⌜q1 + q2 ≤ 1⌝%Qp.
Proof. rewrite -frac_valid -uPred.discrete_valid. apply (own_valid_2 γ q1 q2). Qed.
Lemma cinv_own_1_l γ q : cinv_own γ 1 -∗ cinv_own γ q -∗ False.
Proof.
iIntros "H1 H2".
iDestruct (cinv_own_valid with "H1 H2") as %[]%(exclusive_l 1%Qp).
Qed.
Lemma cinv_iff N γ P Q : cinv N γ P -∗ ▷ □ (P ↔ Q) -∗ cinv N γ Q.
Proof.
iIntros "HI #HPQ". iApply (inv_iff with "HI"). iIntros "!> !>".
iSplit; iIntros "[?|$]"; iLeft; by iApply "HPQ".
Qed.
Lemma cinv_alloc_strong (I : gname → Prop) E N :
pred_infinite I →
⊢ |={E}=> ∃ γ, ⌜ I γ ⌝ ∗ cinv_own γ 1 ∗ ∀ P, ▷ P ={E}=∗ cinv N γ P.
Proof.
iIntros (?). iMod (own_alloc_strong 1%Qp I) as (γ) "[Hfresh Hγ]"; [done|done|].
iExists γ. iIntros "!> {$Hγ $Hfresh}" (P) "HP".
iMod (inv_alloc N _ (P ∨ cinv_own γ 1) with "[HP]"); eauto.
Qed.
pred_infinite I →
⊢ |={E}=> ∃ γ, ⌜ I γ ⌝ ∗ cinv_own γ 1 ∗ ∀ P, ▷ P ={E}=∗ cinv N γ P.
Proof.
iIntros (?). iMod (own_alloc_strong 1%Qp I) as (γ) "[Hfresh Hγ]"; [done|done|].
iExists γ. iIntros "!> {$Hγ $Hfresh}" (P) "HP".
iMod (inv_alloc N _ (P ∨ cinv_own γ 1) with "[HP]"); eauto.
Qed.
The "open" variants create the invariant in the open state, and delay
having to prove P.
These do not imply the other variants because of the extra assumption ↑N ⊆ E.
Lemma cinv_alloc_strong_open (I : gname → Prop) E N :
pred_infinite I →
↑N ⊆ E →
⊢ |={E}=> ∃ γ, ⌜ I γ ⌝ ∗ cinv_own γ 1 ∗ ∀ P,
|={E,E∖↑N}=> cinv N γ P ∗ (▷ P ={E∖↑N,E}=∗ True).
Proof.
iIntros (??). iMod (own_alloc_strong 1%Qp I) as (γ) "[Hfresh Hγ]"; [done|done|].
iExists γ. iIntros "!> {$Hγ $Hfresh}" (P).
iMod (inv_alloc_open N _ (P ∨ cinv_own γ 1)) as "[Hinv Hclose]"; first by eauto.
iIntros "!>". iFrame. iIntros "HP". iApply "Hclose". iLeft. done.
Qed.
Lemma cinv_alloc_cofinite (G : gset gname) E N :
⊢ |={E}=> ∃ γ, ⌜ γ ∉ G ⌝ ∗ cinv_own γ 1 ∗ ∀ P, ▷ P ={E}=∗ cinv N γ P.
Proof.
apply cinv_alloc_strong. apply (pred_infinite_set (C:=gset gname))=> E'.
∃ (fresh (G ∪ E')). apply not_elem_of_union, is_fresh.
Qed.
Lemma cinv_alloc E N P : ▷ P ={E}=∗ ∃ γ, cinv N γ P ∗ cinv_own γ 1.
Proof.
iIntros "HP". iMod (cinv_alloc_cofinite ∅ E N) as (γ _) "[Hγ Halloc]".
iExists γ. iFrame "Hγ". by iApply "Halloc".
Qed.
Lemma cinv_alloc_open E N P :
↑N ⊆ E → ⊢ |={E,E∖↑N}=> ∃ γ, cinv N γ P ∗ cinv_own γ 1 ∗ (▷ P ={E∖↑N,E}=∗ True).
Proof.
iIntros (?). iMod (cinv_alloc_strong_open (λ _, True)) as (γ) "(_ & Htok & Hmake)"; [|done|].
{ apply pred_infinite_True. }
iMod ("Hmake" $! P) as "[Hinv Hclose]". iIntros "!>". iExists γ. iFrame.
Qed.
Lemma cinv_acc_strong E N γ p P :
↑N ⊆ E →
cinv N γ P -∗ (cinv_own γ p ={E,E∖↑N}=∗
▷ P ∗ cinv_own γ p ∗ (∀ E' : coPset, ▷ P ∨ cinv_own γ 1 ={E',↑N ∪ E'}=∗ True)).
Proof.
iIntros (?) "Hinv Hown".
iPoseProof (inv_acc (↑ N) N with "Hinv") as "H"; first done.
rewrite difference_diag_L.
iPoseProof (fupd_mask_frame_r _ _ (E ∖ ↑ N) with "H") as "H"; first set_solver.
rewrite left_id_L -union_difference_L //. iMod "H" as "[[$ | >Hown'] H]".
- iIntros "{$Hown} !>" (E') "HP".
iPoseProof (fupd_mask_frame_r _ _ E' with "(H [HP])") as "H"; first set_solver.
{ iDestruct "HP" as "[?|?]"; eauto. }
by rewrite left_id_L.
- iDestruct (cinv_own_1_l with "Hown' Hown") as %[].
Qed.
Lemma cinv_acc E N γ p P :
↑N ⊆ E →
cinv N γ P -∗ cinv_own γ p ={E,E∖↑N}=∗ ▷ P ∗ cinv_own γ p ∗ (▷ P ={E∖↑N,E}=∗ True).
Proof.
iIntros (?) "#Hinv Hγ".
iMod (cinv_acc_strong with "Hinv Hγ") as "($ & $ & H)"; first done.
iIntros "!> HP".
rewrite {2}(union_difference_L (↑N) E)=> //.
iApply "H". by iLeft.
Qed.
Lemma cinv_cancel E N γ P : ↑N ⊆ E → cinv N γ P -∗ cinv_own γ 1 ={E}=∗ ▷ P.
Proof.
iIntros (?) "#Hinv Hγ".
iMod (cinv_acc_strong with "Hinv Hγ") as "($ & Hγ & H)"; first done.
rewrite {2}(union_difference_L (↑N) E)=> //.
iApply "H". by iRight.
Qed.
Global Instance into_inv_cinv N γ P : IntoInv (cinv N γ P) N := {}.
Global Instance into_acc_cinv E N γ P p :
IntoAcc (X:=unit) (cinv N γ P)
(↑N ⊆ E) (cinv_own γ p) (fupd E (E∖↑N)) (fupd (E∖↑N) E)
(λ _, ▷ P ∗ cinv_own γ p)%I (λ _, ▷ P)%I (λ _, None)%I.
Proof.
rewrite /IntoAcc /accessor. iIntros (?) "#Hinv Hown".
rewrite exist_unit -assoc.
iApply (cinv_acc with "Hinv"); done.
Qed.
End proofs.
Global Typeclasses Opaque cinv_own cinv.
pred_infinite I →
↑N ⊆ E →
⊢ |={E}=> ∃ γ, ⌜ I γ ⌝ ∗ cinv_own γ 1 ∗ ∀ P,
|={E,E∖↑N}=> cinv N γ P ∗ (▷ P ={E∖↑N,E}=∗ True).
Proof.
iIntros (??). iMod (own_alloc_strong 1%Qp I) as (γ) "[Hfresh Hγ]"; [done|done|].
iExists γ. iIntros "!> {$Hγ $Hfresh}" (P).
iMod (inv_alloc_open N _ (P ∨ cinv_own γ 1)) as "[Hinv Hclose]"; first by eauto.
iIntros "!>". iFrame. iIntros "HP". iApply "Hclose". iLeft. done.
Qed.
Lemma cinv_alloc_cofinite (G : gset gname) E N :
⊢ |={E}=> ∃ γ, ⌜ γ ∉ G ⌝ ∗ cinv_own γ 1 ∗ ∀ P, ▷ P ={E}=∗ cinv N γ P.
Proof.
apply cinv_alloc_strong. apply (pred_infinite_set (C:=gset gname))=> E'.
∃ (fresh (G ∪ E')). apply not_elem_of_union, is_fresh.
Qed.
Lemma cinv_alloc E N P : ▷ P ={E}=∗ ∃ γ, cinv N γ P ∗ cinv_own γ 1.
Proof.
iIntros "HP". iMod (cinv_alloc_cofinite ∅ E N) as (γ _) "[Hγ Halloc]".
iExists γ. iFrame "Hγ". by iApply "Halloc".
Qed.
Lemma cinv_alloc_open E N P :
↑N ⊆ E → ⊢ |={E,E∖↑N}=> ∃ γ, cinv N γ P ∗ cinv_own γ 1 ∗ (▷ P ={E∖↑N,E}=∗ True).
Proof.
iIntros (?). iMod (cinv_alloc_strong_open (λ _, True)) as (γ) "(_ & Htok & Hmake)"; [|done|].
{ apply pred_infinite_True. }
iMod ("Hmake" $! P) as "[Hinv Hclose]". iIntros "!>". iExists γ. iFrame.
Qed.
Lemma cinv_acc_strong E N γ p P :
↑N ⊆ E →
cinv N γ P -∗ (cinv_own γ p ={E,E∖↑N}=∗
▷ P ∗ cinv_own γ p ∗ (∀ E' : coPset, ▷ P ∨ cinv_own γ 1 ={E',↑N ∪ E'}=∗ True)).
Proof.
iIntros (?) "Hinv Hown".
iPoseProof (inv_acc (↑ N) N with "Hinv") as "H"; first done.
rewrite difference_diag_L.
iPoseProof (fupd_mask_frame_r _ _ (E ∖ ↑ N) with "H") as "H"; first set_solver.
rewrite left_id_L -union_difference_L //. iMod "H" as "[[$ | >Hown'] H]".
- iIntros "{$Hown} !>" (E') "HP".
iPoseProof (fupd_mask_frame_r _ _ E' with "(H [HP])") as "H"; first set_solver.
{ iDestruct "HP" as "[?|?]"; eauto. }
by rewrite left_id_L.
- iDestruct (cinv_own_1_l with "Hown' Hown") as %[].
Qed.
Lemma cinv_acc E N γ p P :
↑N ⊆ E →
cinv N γ P -∗ cinv_own γ p ={E,E∖↑N}=∗ ▷ P ∗ cinv_own γ p ∗ (▷ P ={E∖↑N,E}=∗ True).
Proof.
iIntros (?) "#Hinv Hγ".
iMod (cinv_acc_strong with "Hinv Hγ") as "($ & $ & H)"; first done.
iIntros "!> HP".
rewrite {2}(union_difference_L (↑N) E)=> //.
iApply "H". by iLeft.
Qed.
Lemma cinv_cancel E N γ P : ↑N ⊆ E → cinv N γ P -∗ cinv_own γ 1 ={E}=∗ ▷ P.
Proof.
iIntros (?) "#Hinv Hγ".
iMod (cinv_acc_strong with "Hinv Hγ") as "($ & Hγ & H)"; first done.
rewrite {2}(union_difference_L (↑N) E)=> //.
iApply "H". by iRight.
Qed.
Global Instance into_inv_cinv N γ P : IntoInv (cinv N γ P) N := {}.
Global Instance into_acc_cinv E N γ P p :
IntoAcc (X:=unit) (cinv N γ P)
(↑N ⊆ E) (cinv_own γ p) (fupd E (E∖↑N)) (fupd (E∖↑N) E)
(λ _, ▷ P ∗ cinv_own γ p)%I (λ _, ▷ P)%I (λ _, None)%I.
Proof.
rewrite /IntoAcc /accessor. iIntros (?) "#Hinv Hown".
rewrite exist_unit -assoc.
iApply (cinv_acc with "Hinv"); done.
Qed.
End proofs.
Global Typeclasses Opaque cinv_own cinv.