invariants.v 4.8 KB
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From stdpp Require Export namespaces.
From iris.proofmode Require Import tactics.
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From iris.algebra Require Import gmap.
From iris.base_logic.lib Require Export fancy_updates.
From iris.base_logic.lib Require Import wsat.
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Set Default Proof Using "Type".
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Import uPred.
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(** Derived forms and lemmas about them. *)
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Definition inv_def `{!invG Σ} (N : namespace) (P : iProp Σ) : iProp Σ :=
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  ( i P', i  (N:coPset)    (P'  P)  ownI i P')%I.
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Definition inv_aux : seal (@inv_def). by eexists. Qed.
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Definition inv {Σ i} := inv_aux.(unseal) Σ i.
Definition inv_eq : @inv = @inv_def := inv_aux.(seal_eq).
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Instance: Params (@inv) 3 := {}.
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Typeclasses Opaque inv.
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Section inv.
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Context `{!invG Σ}.
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Implicit Types i : positive.
Implicit Types N : namespace.
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Implicit Types P Q R : iProp Σ.
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Global Instance inv_contractive N : Contractive (inv N).
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Proof. rewrite inv_eq. solve_contractive. Qed.
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Global Instance inv_ne N : NonExpansive (inv N).
Proof. apply contractive_ne, _. Qed.

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Global Instance inv_proper N : Proper (() ==> ()) (inv N).
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Proof. apply ne_proper, _. Qed.

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Global Instance inv_persistent N P : Persistent (inv N P).
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Proof. rewrite inv_eq /inv; apply _. Qed.
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Lemma inv_iff N P Q :   (P  Q) - inv N P - inv N Q.
Proof.
  iIntros "#HPQ". rewrite inv_eq. iDestruct 1 as (i P') "(?&#HP&?)".
  iExists i, P'. iFrame. iNext; iAlways; iSplit.
  - iIntros "HP'". iApply "HPQ". by iApply "HP".
  - iIntros "HQ". iApply "HP". by iApply "HPQ".
Qed.

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Lemma fresh_inv_name (E : gset positive) N :  i, i  E  i  (N:coPset).
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Proof.
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  exists (coPpick ( N  gset_to_coPset E)).
  rewrite -elem_of_gset_to_coPset (comm and) -elem_of_difference.
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  apply coPpick_elem_of=> Hfin.
  eapply nclose_infinite, (difference_finite_inv _ _), Hfin.
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  apply gset_to_coPset_finite.
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Qed.

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Lemma inv_alloc N E P :  P ={E}= inv N P.
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Proof.
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  rewrite inv_eq /inv_def uPred_fupd_eq. iIntros "HP [Hw $]".
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  iMod (ownI_alloc ( (N : coPset)) P with "[$HP $Hw]")
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    as (i ?) "[$ ?]"; auto using fresh_inv_name.
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  do 2 iModIntro. iExists i, P. rewrite -(iff_refl True%I). auto.
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Qed.
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Lemma inv_alloc_open N E P :
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  N  E  (|={E, E∖↑N}=> inv N P  (P ={E∖↑N, E}= True))%I.
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Proof.
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  rewrite inv_eq /inv_def uPred_fupd_eq. iIntros (Sub) "[Hw HE]".
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  iMod (ownI_alloc_open ( (N : coPset)) P with "Hw")
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    as (i ?) "(Hw & #Hi & HD)"; auto using fresh_inv_name.
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  iAssert (ownE {[i]}  ownE ( N  {[i]})  ownE (E   N))%I
    with "[HE]" as "(HEi & HEN\i & HE\N)".
  { rewrite -?ownE_op; [|set_solver..].
    rewrite assoc_L -!union_difference_L //. set_solver. }
  do 2 iModIntro. iFrame "HE\N". iSplitL "Hw HEi"; first by iApply "Hw".
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  iSplitL "Hi".
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  { iExists i, P. rewrite -(iff_refl True%I). auto. }
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  iIntros "HP [Hw HE\N]".
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  iDestruct (ownI_close with "[$Hw $Hi $HP $HD]") as "[$ HEi]".
  do 2 iModIntro. iSplitL; [|done].
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  iCombine "HEi HEN\i HE\N" as "HEN".
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  rewrite -?ownE_op; [|set_solver..].
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  rewrite assoc_L -!union_difference_L //; set_solver.
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Qed.

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Lemma inv_open E N P :
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  N  E  inv N P ={E,E∖↑N}=  P  ( P ={E∖↑N,E}= True).
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Proof.
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  rewrite inv_eq /inv_def uPred_fupd_eq /uPred_fupd_def.
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  iDestruct 1 as (i P') "(Hi & #HP' & #HiP)".
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  iDestruct "Hi" as % ?%elem_of_subseteq_singleton.
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  rewrite {1 4}(union_difference_L ( N) E) // ownE_op; last set_solver.
  rewrite {1 5}(union_difference_L {[ i ]} ( N)) // ownE_op; last set_solver.
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  iIntros "(Hw & [HE $] & $) !> !>".
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  iDestruct (ownI_open i with "[$Hw $HE $HiP]") as "($ & HP & HD)".
  iDestruct ("HP'" with "HP") as "$".
  iIntros "HP [Hw $] !> !>". iApply (ownI_close _ P'). iFrame "HD Hw HiP".
  iApply "HP'". iFrame.
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Qed.

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Lemma inv_open_strong E N P :
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  N  E  inv N P ={E,E∖↑N}=  P   E',  P ={E',N  E'}= True.
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Proof.
  iIntros (?) "Hinv".
  iPoseProof (inv_open ( N) N P 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 "[$ H]"; iModIntro.
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  iIntros (E') "HP".
  iPoseProof (fupd_mask_frame_r _ _ E' with "(H HP)") as "H"; first set_solver.
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  by rewrite left_id_L.
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Qed.

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Global Instance into_inv_inv N P : IntoInv (inv N P) N := {}.
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Global Instance into_acc_inv E N P :
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  IntoAcc (X:=unit) (inv N P) 
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          (N  E) True (fupd E (E∖↑N)) (fupd (E∖↑N) E)
          (λ _,  P)%I (λ _,  P)%I (λ _, None)%I.
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Proof.
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  rewrite /IntoAcc /accessor exist_unit.
  iIntros (?) "#Hinv _". iApply inv_open; done.
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Qed.

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Lemma inv_open_timeless E N P `{!Timeless P} :
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  N  E  inv N P ={E,E∖↑N}= P  (P ={E∖↑N,E}= True).
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Proof.
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  iIntros (?) "Hinv". iMod (inv_open with "Hinv") as "[>HP Hclose]"; auto.
  iIntros "!> {$HP} HP". iApply "Hclose"; auto.
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Qed.
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End inv.