Commit ac38f1dd authored by Ralf Jung's avatar Ralf Jung

use Iris' semantic invariants

parent f4315521
......@@ -84,7 +84,6 @@ theories/logrel/F_mu_ref_conc/examples/fact.v
theories/logrel_heaplang/ltyping.v
theories/logrel_heaplang/ltyping_safety.v
theories/logrel_heaplang/lib/symbol_adt.v
theories/logrel_heaplang/lib/invariants.v
theories/logrel_heaplang/lib/arrays.v
theories/logrel_heaplang/lib/vectors.v
......
From iris.algebra Require Import auth agree.
From iris.base_logic Require Import invariants.
From iris.heap_lang Require Export lifting metatheory.
From iris.heap_lang Require Import notation proofmode.
From iris_examples.logrel_heaplang.lib Require Import invariants.
From iris_examples.logrel_heaplang Require Import ltyping ltyping_safety.
(** Semantic typing "branded types" for unchecked indexing into a fixed-size
array. *)
(* utility stuff *)
Ltac bool_decide H :=
match goal with
......
(* TODO: merge this into iris *)
From iris.base_logic.lib Require Export fancy_updates.
From stdpp Require Export namespaces.
From iris.base_logic.lib Require Import wsat.
From iris.algebra Require Import gmap.
From iris.proofmode Require Import tactics.
Set Default Proof Using "Type".
Import uPred.
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.
(** * Invariants *)
Section inv.
Context `{!invG Σ}.
(** Internal backing store of invariants *)
Definition internal_inv_def (N : namespace) (P : iProp Σ) : iProp Σ :=
( i P', i (N:coPset) (P' P) ownI i P')%I.
Definition internal_inv_aux : seal (@internal_inv_def). by eexists. Qed.
Definition internal_inv := internal_inv_aux.(unseal).
Definition internal_inv_eq : @internal_inv = @internal_inv_def := internal_inv_aux.(seal_eq).
Typeclasses Opaque internal_inv.
Global Instance internal_inv_persistent N P : Persistent (internal_inv N P).
Proof. rewrite internal_inv_eq /internal_inv; apply _. Qed.
Lemma internal_inv_open E N P :
N E internal_inv N P ={E,E∖↑N}= P ( P ={E∖↑N,E}= True).
Proof.
rewrite internal_inv_eq /internal_inv_def uPred_fupd_eq /uPred_fupd_def.
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 internal_inv_alloc N E P : P ={E}= internal_inv N P.
Proof.
rewrite internal_inv_eq /internal_inv_def uPred_fupd_eq. iIntros "HP [Hw $]".
iMod (ownI_alloc ( (N : coPset)) P with "[$HP $Hw]")
as (i ?) "[$ ?]"; auto using fresh_inv_name.
do 2 iModIntro. iExists i, P. rewrite -(iff_refl True%I). auto.
Qed.
Lemma internal_inv_alloc_open N E P :
N E (|={E, E∖↑N}=> internal_inv N P (P ={E∖↑N, E}= True))%I.
Proof.
rewrite internal_inv_eq /internal_inv_def uPred_fupd_eq. iIntros (Sub) "[Hw HE]".
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.
(** Invariants API *)
Definition inv_def (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 := inv_aux.(unseal).
Definition inv_eq : @inv = @inv_def := inv_aux.(seal_eq).
Typeclasses Opaque inv.
(** Properties about 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).
Proof. apply ne_proper, _. Qed.
Global Instance inv_persistent M P: Persistent (inv M P).
Proof. rewrite inv_eq. typeclasses eauto. Qed.
Lemma inv_acc N P Q:
inv N P - (P - Q (Q - P)) - inv N Q.
Proof.
iIntros "#I #Acc". rewrite inv_eq.
iModIntro. iIntros (E H). iDestruct ("I" $! E H) as "#I'".
iApply fupd_wand_r. iFrame "I'".
iIntros "(P & Hclose)". iSpecialize ("Acc" with "P").
iDestruct "Acc" as "[Q CB]". iFrame.
iIntros "Q". iApply "Hclose". now iApply "CB".
Qed.
Lemma inv_iff N P Q : (P Q) - inv N P - inv N Q.
Proof.
iIntros "#HPQ #I". iApply (inv_acc with "I").
iNext. iIntros "!# P". iSplitL "P".
- by iApply "HPQ".
- iIntros "Q". by iApply "HPQ".
Qed.
Lemma inv_to_inv M P: internal_inv M P - inv M P.
Proof.
iIntros "#I". rewrite inv_eq. iIntros (E H).
iPoseProof (internal_inv_open with "I") as "H"; eauto.
Qed.
Lemma inv_alloc N E P : P ={E}= inv N P.
Proof.
iIntros "P". iPoseProof (internal_inv_alloc N E with "P") as "I".
iApply fupd_mono; last eauto.
iApply inv_to_inv.
Qed.
Lemma inv_alloc_open N E P :
N E (|={E, E∖↑N}=> inv N P (P ={E∖↑N, E}= True))%I.
Proof.
iIntros (H). iPoseProof (internal_inv_alloc_open _ _ _ H) as "H".
iApply fupd_mono; last eauto.
iIntros "[I H]"; iFrame; by iApply inv_to_inv.
Qed.
Lemma inv_open E N P :
N E inv N P ={E,E∖↑N}= P ( P ={E∖↑N,E}= True).
Proof.
rewrite inv_eq /inv_def; iIntros (H) "#I". by iApply "I".
Qed.
Lemma inv_open_strong E N P :
N E inv N P ={E,E∖↑N}= P E', P ={E',N E'}= True.
Proof.
iIntros (?) "Hinv". iPoseProof (inv_open ( N) N P with "Hinv") as "H"; first done.
rewrite difference_diag_L.
iPoseProof (fupd_mask_frame_r _ _ (E N) with "H") as "H"; first set_solver.
rewrite left_id_L -union_difference_L //. iMod "H" as "[$ 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.
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.
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 "!> {$HP} HP". iApply "Hclose"; auto.
Qed.
(* Weakening of semantic invariants *)
Lemma inv_proj_l N P Q: inv N (P Q) - inv N P.
Proof.
iIntros "#I". iApply inv_acc; eauto.
iNext. iIntros "!# [$ Q] P"; iFrame.
Qed.
Lemma inv_proj_r N P Q: inv N (P Q) - inv N Q.
Proof.
rewrite (bi.sep_comm P Q). eapply inv_proj_l.
Qed.
Lemma inv_split N P Q: inv N (P Q) - inv N P inv N Q.
Proof.
iIntros "#H".
iPoseProof (inv_proj_l with "H") as "$".
iPoseProof (inv_proj_r with "H") as "$".
Qed.
End inv.
From iris.algebra Require Import gmap auth agree.
From iris.heap_lang Require Export lifting metatheory.
From iris.heap_lang Require Import notation proofmode.
From iris.base_logic Require Import invariants.
From iris_examples.logrel_heaplang Require Import ltyping ltyping_safety.
From iris.algebra Require Import gmap auth agree.
From iris_examples.logrel_heaplang.lib Require Import invariants arrays.
From iris_examples.logrel_heaplang.lib Require Import arrays.
(** Semantic typing "branded types" for unchecked indexing into a growable
vector. *)
Set Default Proof Using "Type".
......
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