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From stdpp Require Export namespaces.
From iris.proofmode Require Import tactics.
From iris.algebra Require Import gmap.
From iris.base_logic.lib Require Export fancy_updates.
From iris.base_logic.lib Require Import wsat.
Set Default Proof Using "Type".
(** Semantic Invariants *)
Definition inv_def `{!invG Σ} (N : namespace) (P : iProp Σ) : iProp Σ :=
(□ ∀ E, ⌜↑N ⊆ E⌝ → |={E,E ∖ ↑N}=> ▷ P ∗ (▷ P ={E ∖ ↑N,E}=∗ True))%I.
Definition inv_aux : seal (@inv_def). by eexists. Qed.
Definition inv {Σ i} := inv_aux.(unseal) Σ i.
Definition inv_eq : @inv = @inv_def := inv_aux.(seal_eq).
Instance: Params (@inv) 3 := {}.
Typeclasses Opaque inv.
(** * Invariants *)
Section inv.
Context `{!invG Σ}.
Implicit Types i : positive.
Implicit Types N : namespace.
Implicit Types E : coPset.
Implicit Types P Q R : iProp Σ.
(** ** Internal model of invariants *)
Definition own_inv (N : namespace) (P : iProp Σ) : iProp Σ :=
(∃ i P', ⌜i ∈ (↑N:coPset)⌝ ∧ ▷ □ (P' ↔ P) ∧ ownI i P')%I.
Lemma own_inv_open E N P :
↑N ⊆ E → own_inv N P ={E,E∖↑N}=∗ ▷ P ∗ (▷ P ={E∖↑N,E}=∗ True).
iDestruct 1 as (i P') "(Hi & #HP' & #HiP)".
iDestruct "Hi" as % ?%elem_of_subseteq_singleton.
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.
iIntros "(Hw & [HE $] & $) !> !>".
iDestruct (ownI_open i with "[$Hw $HE $HiP]") as "($ & HP & HD)".
iDestruct ("HP'" with "HP") as "$".
iIntros "HP [Hw $] !> !>". iApply (ownI_close _ P'). iFrame "HD Hw HiP".
iApply "HP'". iFrame.
Qed.
Lemma fresh_inv_name (E : gset positive) N : ∃ i, i ∉ E ∧ i ∈ (↑N:coPset).
Proof.
exists (coPpick (↑ N ∖ gset_to_coPset E)).
rewrite -elem_of_gset_to_coPset (comm and) -elem_of_difference.
apply coPpick_elem_of=> Hfin.
eapply nclose_infinite, (difference_finite_inv _ _), Hfin.
apply gset_to_coPset_finite.
Qed.
Lemma own_inv_alloc N E P : ▷ P ={E}=∗ own_inv N P.
iMod (ownI_alloc (.∈ (↑N : coPset)) P with "[$HP $Hw]")
as (i ?) "[$ ?]"; auto using fresh_inv_name.
do 2 iModIntro. iExists i, P. rewrite -(iff_refl True%I). auto.
Qed.
Lemma own_inv_alloc_open N E P :
↑N ⊆ E → (|={E, E∖↑N}=> own_inv N P ∗ (▷P ={E∖↑N, E}=∗ True))%I.
iMod (ownI_alloc_open (.∈ (↑N : coPset)) P with "Hw")
as (i ?) "(Hw & #Hi & HD)"; auto using fresh_inv_name.
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".
iSplitL "Hi".
{ iExists i, P. rewrite -(iff_refl True%I). auto. }
iIntros "HP [Hw HE\N]".
iDestruct (ownI_close with "[$Hw $Hi $HP $HD]") as "[$ HEi]".
do 2 iModIntro. iSplitL; [|done].
iCombine "HEi HEN\i HE\N" as "HEN".
rewrite -?ownE_op; [|set_solver..].
rewrite assoc_L -!union_difference_L //; set_solver.
Qed.
Lemma own_inv_to_inv M P: own_inv M P -∗ inv M P.
Proof.
iIntros "#I". rewrite inv_eq. iIntros (E H).
iPoseProof (own_inv_open with "I") as "H"; eauto.
Qed.
(** ** Public API of invariants *)
Global Instance inv_contractive N : Contractive (inv N).
Proof. rewrite inv_eq. solve_contractive. Qed.
Global Instance inv_ne N : NonExpansive (inv N).
Proof. apply contractive_ne, _. Qed.
Global Instance inv_proper N : Proper (equiv ==> equiv) (inv N).
Global Instance inv_persistent N P : Persistent (inv N P).
Proof. rewrite inv_eq. apply _. Qed.
Lemma inv_acc N P Q:
inv N P -∗ ▷ □ (P -∗ Q ∗ (Q -∗ P)) -∗ inv N Q.
Proof.
rewrite inv_eq. iIntros "#HI #Acc !>" (E H).
iMod ("HI" $! E H) as "[HP Hclose]".
iDestruct ("Acc" with "HP") as "[$ HQP]".
iIntros "!> HQ". iApply "Hclose". iApply "HQP". done.
Qed.
Lemma inv_iff N P Q : ▷ □ (P ↔ Q) -∗ inv N P -∗ inv N Q.
Proof.
iIntros "#HPQ #HI". iApply (inv_acc with "HI").
iIntros "!> !# HP". iSplitL "HP".
Qed.
Lemma inv_alloc N E P : ▷ P ={E}=∗ inv N P.
Proof.
iIntros "HP". iApply own_inv_to_inv.
iApply (own_inv_alloc N E with "HP").
Qed.
Lemma inv_alloc_open N E P :
↑N ⊆ E → (|={E, E∖↑N}=> inv N P ∗ (▷P ={E∖↑N, E}=∗ True))%I.
Proof.
iIntros (?). iMod own_inv_alloc_open as "[HI $]"; first done.
iApply own_inv_to_inv. done.
Qed.
Lemma inv_open E N P :
↑N ⊆ E → inv N P ={E,E∖↑N}=∗ ▷ P ∗ (▷ P ={E∖↑N,E}=∗ True).
Proof.
rewrite inv_eq /inv_def; iIntros (?) "#HI". by iApply "HI".
Global Instance into_inv_inv N P : IntoInv (inv N P) N := {}.
Global Instance into_acc_inv N P E:
IntoAcc (X := unit) (inv N P)
(↑N ⊆ E) True (fupd E (E ∖ ↑N)) (fupd (E ∖ ↑N) E)
(λ _ : (), (▷ P)%I) (λ _ : (), (▷ P)%I) (λ _ : (), None).
Proof.
rewrite inv_eq /IntoAcc /accessor bi.exist_unit.
iIntros (?) "#Hinv _". iApply "Hinv"; done.
Qed.
(** ** Derived properties *)
Lemma inv_open_strong E N P :
↑N ⊆ E → inv N P ={E,E∖↑N}=∗ ▷ P ∗ ∀ E', ▷ P ={E',↑N ∪ E'}=∗ True.
Proof.
iIntros (?) "Hinv".
iPoseProof (inv_open (↑ N) N P with "Hinv") as "H"; first done.
rewrite difference_diag_L.
iPoseProof (fupd_mask_frame_r _ _ (E ∖ ↑ N) with "H") as "H"; first set_solver.
rewrite left_id_L -union_difference_L //. iMod "H" as "[$ H]"; iModIntro.
iIntros (E') "HP".
iPoseProof (fupd_mask_frame_r _ _ E' with "(H HP)") as "H"; first set_solver.
by rewrite left_id_L.
Qed.
Lemma inv_open_timeless E N P `{!Timeless P} :
↑N ⊆ E → inv N P ={E,E∖↑N}=∗ P ∗ (P ={E∖↑N,E}=∗ True).
Proof.
iIntros (?) "Hinv". iMod (inv_open with "Hinv") as "[>HP Hclose]"; auto.
iIntros "!> {$HP} HP". iApply "Hclose"; auto.
Qed.
iIntros "#HI". iApply inv_acc; eauto.
iIntros "!> !# [$ $] $".
Lemma inv_sep N P Q : inv N (P ∗ Q) -∗ inv N P ∗ inv N Q.
iPoseProof (inv_sep_l with "H") as "$".
iPoseProof (inv_sep_r with "H") as "$".

Ralf Jung
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