cancelable_invariants.v 3.87 KB
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From iris.base_logic.lib Require Export invariants.
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From iris.bi.lib Require Import fractional.
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From iris.algebra Require Export frac.
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From iris.proofmode Require Import tactics.
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Set Default Proof Using "Type".
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Import uPred.

Class cinvG Σ := cinv_inG :> inG Σ fracR.
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Definition cinvΣ : gFunctors := #[GFunctor fracR].

Instance subG_cinvΣ {Σ} : subG cinvΣ Σ  cinvG Σ.
Proof. solve_inG. Qed.
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Section defs.
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  Context `{invG Σ, cinvG Σ}.
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  Definition cinv_own (γ : gname) (p : frac) : iProp Σ := own γ p.

  Definition cinv (N : namespace) (γ : gname) (P : iProp Σ) : iProp Σ :=
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    ( P',   (P  P')  inv N (P'  cinv_own γ 1%Qp))%I.
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End defs.

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Instance: Params (@cinv) 5.
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Section proofs.
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  Context `{invG Σ, cinvG Σ}.
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  Global Instance cinv_own_timeless γ p : Timeless (cinv_own γ p).
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  Proof. rewrite /cinv_own; apply _. Qed.

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  Global Instance cinv_contractive N γ : Contractive (cinv N γ).
  Proof. solve_contractive. Qed.
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  Global Instance cinv_ne N γ : NonExpansive (cinv N γ).
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  Proof. exact: contractive_ne. Qed.
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  Global Instance cinv_proper N γ : Proper (() ==> ()) (cinv N γ).
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  Proof. exact: ne_proper. Qed.
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  Global Instance cinv_persistent N γ P : Persistent (cinv N γ P).
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  Proof. rewrite /cinv; apply _. Qed.

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  Global Instance cinv_own_fractionnal γ : Fractional (cinv_own γ).
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  Proof. intros ??. by rewrite /cinv_own -own_op. Qed.
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  Global Instance cinv_own_as_fractionnal γ q :
    AsFractional (cinv_own γ q) (cinv_own γ) q.
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  Proof. split. done. apply _. Qed.
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  Lemma cinv_own_valid γ q1 q2 : cinv_own γ q1 - cinv_own γ q2 -  (q1 + q2)%Qp.
  Proof. apply (own_valid_2 γ q1 q2). Qed.
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  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.
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  Lemma cinv_iff N γ P P' :
      (P  P') - cinv N γ P - cinv N γ P'.
  Proof.
    iIntros "#HP' Hinv". iDestruct "Hinv" as (P'') "[#HP'' Hinv]".
    iExists _. iFrame "Hinv". iNext. iAlways. iSplit.
    - iIntros "?". iApply "HP''". iApply "HP'". done.
    - iIntros "?". iApply "HP'". iApply "HP''". done.
  Qed.

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  Lemma cinv_alloc E N P :  P ={E}=  γ, cinv N γ P  cinv_own γ 1.
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  Proof.
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    iIntros "HP".
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    iMod (own_alloc 1%Qp) as (γ) "H1"; first done.
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    iMod (inv_alloc N _ (P  own γ 1%Qp)%I with "[HP]"); first by eauto.
    iExists _. iFrame. iExists _. iFrame. iIntros "!> !# !>". iSplit; by iIntros "?".
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  Qed.

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  Lemma cinv_cancel E N γ P : N  E  cinv N γ P - cinv_own γ 1 ={E}=  P.
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  Proof.
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    iIntros (?) "#Hinv Hγ". iDestruct "Hinv" as (P') "[#HP' Hinv]".
    iInv N as "[HP|>Hγ']" "Hclose".
    - iMod ("Hclose" with "[Hγ]") as "_"; first by eauto. iModIntro. iNext.
      iApply "HP'". done.
    - iDestruct (cinv_own_1_l with "Hγ Hγ'") as %[].
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  Qed.

  Lemma cinv_open E N γ p P :
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    N  E 
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    cinv N γ P - cinv_own γ p ={E,E∖↑N}=  P  cinv_own γ p  ( P ={E∖↑N,E}= True).
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  Proof.
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    iIntros (?) "#Hinv Hγ". iDestruct "Hinv" as (P') "[#HP' Hinv]".
    iInv N as "[HP | >Hγ']" "Hclose".
    - iIntros "!> {$Hγ}". iSplitL "HP".
      + iNext. iApply "HP'". done.
      + iIntros "HP". iApply "Hclose". iLeft. iNext. by iApply "HP'".
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    - iDestruct (cinv_own_1_l with "Hγ' Hγ") as %[].
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  Qed.
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  Global Instance into_inv_cinv N γ P : IntoInv (cinv N γ P) N.
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  Global Instance elim_inv_cinv p γ E N P Q Q' :
    ( R, ElimModal True (|={E,E∖↑N}=> R) R Q Q') 
    ElimInv (N  E) (cinv N γ P) (cinv_own γ p)
      ( P  cinv_own γ p) ( P ={E∖↑N,E}= True) Q Q'.
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  Proof.
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    rewrite /ElimInv /ElimModal. iIntros (Helim ?) "(#H1&Hown&H2)".
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    iApply (Helim with "[- $H2]"); first done.
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    iMod (cinv_open E N γ p P with "[#] [Hown]") as "(HP&Hown&Hclose)"; auto. 
    by iFrame.
  Qed.
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End proofs.
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Typeclasses Opaque cinv_own cinv.