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proof_cldeque.v
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From iris.algebra Require Import list excl_auth.
From iris.bi Require Import derived_laws_later.
From iris.bi.lib Require Import fractional.
From iris.base_logic.lib Require Import invariants ghost_var.
From smr Require Import mono_list mono_nat atomic.
From smr.lang Require Import proofmode notation.
From iris.prelude Require Import options.
From smr.no_recl Require Import spec_cldeque code_cldeque.
From smr Require Import helpers.
Local Ltac extended_auto :=
eauto;
try rewrite Nat2Z.id;
try rewrite length_replicate;
try rewrite Qp.half_half;
repeat rewrite Loc.add_0; repeat rewrite Loc.add_assoc;
try by (
repeat iNext; repeat iIntros; repeat intros;
try case_decide; try iPureIntro;
try rewrite lookup_lt_is_Some;
try lia; done
).
Local Ltac fr :=
repeat iIntros; repeat iSplit; extended_auto;
repeat iIntros; repeat iExists _;
try iFrame "arr↦"; try iFrame "arr↦1"; try iFrame "arr↦2";
iFrame; eauto.
(** Ghost state for the deque *)
Class cldequeG Σ := CLDequeG {
(* spec *)
#[local] deque_tokG :: inG Σ (excl_authR $ listO valO);
(* info: era, arrptr, C, bot, popping *)
#[local] deque_infoG :: ghost_varG Σ (nat * blk * list val * nat * bool);
(* RA *)
#[local] topbotG :: mono_natG Σ;
#[local] topeltG :: mono_listG val Σ;
#[local] roomG :: mono_listG (gname * blk * nat) Σ;
#[local] museumG :: mono_listG (list val * nat * nat) Σ
}.
Definition cldequeΣ : gFunctors :=
#[ (*invariant *)
GFunctor (excl_authR $ listO valO);
ghost_varΣ (nat * blk * list val * nat * bool);
(* RA *)
mono_natΣ;
mono_listΣ val;
mono_listΣ (gname * blk * nat);
mono_listΣ (list val * nat * nat)
].
Global Instance subG_cldequeΣ {Σ} : subG cldequeΣ Σ → cldequeG Σ.
Proof. solve_inG. Qed.
Section dqst.
Context `{!heapGS Σ, !cldequeG Σ}.
Notation iProp := (iProp Σ).
Definition dqst_gnames : Type := gname*gname*gname*gname.
Definition top_bot_state (t b : nat) : nat :=
2*t + (if bool_decide (t < b) then 1 else 0).
Definition dqst_frag (γdqst : dqst_gnames) (era : nat)
(C : blk) (l : list val) (t b : nat) : iProp :=
let (γ' , γmus) := γdqst in
let (γ'' , γroom) := γ' in
let (γtb, γelt) := γ'' in
∃ (γtbe : gname),
⌜1 ≤ t ≤ b ∧ b < t + length l ∧ length l ≠ 0⌝ ∗
(* top-bot profile *)
( mono_nat_lb_own γtb (top_bot_state t b) ∗
mono_nat_lb_own γtbe (top_bot_state t b)
) ∗
(* top element preservation *)
( ∃ (elts : list val),
mono_list_lb_own γelt elts ∗
⌜t = b ∨ mod_get l t = elts !! t⌝
) ∗
(* museum of past gnames and circles *)
( ∃ (room : list (gname * blk * nat)),
⌜room !! era = Some (γtbe, C, length l)⌝ ∗
mono_list_lb_own γroom room
).
Definition dqst_archived (γdqst : dqst_gnames) (era : nat)
(C : blk) (l : list val) (t b : nat) : iProp :=
let (γ' , γmus) := γdqst in
let (γ'' , γroom) := γ' in
let (γtb, γelt) := γ'' in
∃ (γtbe : gname),
⌜1 ≤ t ≤ b ∧ b < t + length l ∧ length l ≠ 0⌝ ∗
(* top-bot profile *)
( mono_nat_lb_own γtb (top_bot_state t b) ∗
mono_nat_persistent γtbe (top_bot_state t b)
) ∗
(* top element preservation *)
( ∃ (elts : list val),
mono_list_lb_own γelt elts ∗
⌜t = b ∨ mod_get l t = elts !! t⌝
) ∗
(* museum of past gnames and circles *)
( ∃ (room : list (gname * blk * nat))
(museum : list (list val * nat * nat)),
⌜room !! era = Some (γtbe, C, length l)⌝ ∗
⌜museum !! era = Some (l, t, b)⌝ ∗
mono_list_lb_own γroom room ∗
mono_list_lb_own γmus museum
) ∗
(* persistent circle *)
(C +ₗ carr) ↦∗□ l.
Definition dqst_auth (γdqst : dqst_gnames) (era : nat)
(C : blk) (l : list val) (t b : nat) : iProp :=
let (γ' , γmus) := γdqst in
let (γ'' , γroom) := γ' in
let (γtb, γelt) := γ'' in
∃ (γtbe : gname),
⌜1 ≤ t ≤ b ∧ b < t + length l ∧ length l ≠ 0⌝ ∗
(* top-bot profile *)
(mono_nat_auth_own γtb 1 (top_bot_state t b) ∗
mono_nat_auth_own γtbe 1 (top_bot_state t b)
) ∗
(* top element preservation *)
(∃ (elts : list val),
mono_list_auth_own γelt 1 elts ∗
⌜if (bool_decide (t = b))
then length elts = t
else (length elts = S t ∧ mod_get l t = elts !! t)⌝
) ∗
(* museum of past gnames and circles *)
( ∃ (proom : list (gname * blk * nat))
(museum : list (list val * nat * nat)),
⌜length proom = era ∧ length museum = era⌝ ∗
mono_list_auth_own γroom 1 (proom ++ [(γtbe, C, length l)]) ∗
mono_list_auth_own γmus 1 museum ∗
[∗ list] i ↦ gcl ; ltbi ∈ proom ; museum, (
dqst_archived γdqst i (gcl.1.2) (ltbi.1.1) (ltbi.1.2) (ltbi.2)
)
).
(* Timeless & Persistent *)
Ltac desγ H :=
destruct H as (((γtb, γelt), γroom), γmuseum).
Global Instance dqst_frag_timeless γdqst era C l t b :
Timeless (dqst_frag γdqst era C l t b).
Proof. desγ γdqst. apply _. Qed.
Global Instance dqst_frag_persistent γdqst era C l t b :
Persistent (dqst_frag γdqst era C l t b).
Proof. desγ γdqst. apply _. Qed.
Global Instance dqst_archived_timeless γdqst era C l t b :
Timeless (dqst_archived γdqst era C l t b).
Proof. desγ γdqst. apply _. Qed.
Global Instance dqst_archived_persistent γdqst era C l t b :
Persistent (dqst_archived γdqst era C l t b).
Proof. desγ γdqst. apply _. Qed.
Global Instance dqst_auth_timeless γdqst era C l t b :
Timeless (dqst_auth γdqst era C l t b).
Proof. desγ γdqst. apply _. Qed.
Lemma top_bot_state_le t1 b1 t2 b2 :
top_bot_state t1 b1 ≤ top_bot_state t2 b2 ↔
t1 ≤ t2 ∧ (t1 = t2 ∧ t1 < b1 → t2 < b2).
Proof. unfold top_bot_state. do 2 case_bool_decide; lia. Qed.
Lemma dqst_auth_alloc C l :
length l ≠ 0 →
⊢ |==> ∃ (γdqst : dqst_gnames),
dqst_auth γdqst 0 C l 1 1.
Proof.
intros Hl. unfold dqst_auth.
iMod (mono_nat_own_alloc 2) as (γtb) "[tb _]".
iMod (mono_nat_own_alloc 2) as (γtbe) "[tbe _]".
iMod (mono_list_own_alloc ([NONEV])) as (γelt) "[topelt _]".
iMod (mono_list_own_alloc ([(γtbe, C, length l)])) as (γroom) "[room _]".
iMod (mono_list_own_alloc ([] : list (list val * nat * nat))) as (γmus) "[museum _]".
iExists (γtb, γelt, γroom, γmus).
iModIntro. fr. fr.
iSplit; fr.
Qed.
Lemma dqst_frag_agree γdqst era C1 l1 t1 b1 C2 l2 t2 b2 :
dqst_frag γdqst era C1 l1 t1 b1 -∗
dqst_frag γdqst era C2 l2 t2 b2 -∗
⌜C1 = C2 ∧ length l1 = length l2⌝.
Proof.
desγ γdqst.
iIntros "F1 F2".
iDestruct "F1" as (γtbe1) "(%Hlt1 & tb1 & elt1 & muse1)".
iDestruct "muse1" as (room1) "[%Hroom1 Lb1]".
iDestruct "F2" as (γtbe2) "(%Hlt2 & tb2 & elt2 & muse2)".
iDestruct "muse2" as (room2) "[%Hroom2 Lb2]".
iDestruct (mono_list_lb_valid with "Lb1 Lb2") as "[%Pref|%Pref]".
- eapply prefix_lookup_Some in Hroom1; eauto.
rewrite Hroom2 in Hroom1. by injection Hroom1.
- eapply prefix_lookup_Some in Hroom2; eauto.
rewrite Hroom2 in Hroom1. by injection Hroom1.
Qed.
Lemma dqst_get_frag γdqst era C l t b :
dqst_auth γdqst era C l t b -∗
dqst_frag γdqst era C l t b.
Proof with extended_auto.
desγ γdqst.
iIntros "Auth".
iDestruct "Auth" as (γtbeO) "(%HltO & [tbO tbeO] & eltO & museO)".
iDestruct "eltO" as (elts) "[Elt %Heltslen]".
iDestruct "museO" as (room museum) "museO".
iDestruct "museO" as "([%Hroomlen %Hmuslen] & museO)".
iDestruct "museO" as "(Room & Museum & Archives)".
iDestruct (mono_nat_lb_own_get with "tbO") as "#lb".
iDestruct (mono_nat_lb_own_get with "tbeO") as "#lbe".
iDestruct (mono_list_lb_own_get with "Elt") as "#eltlb".
iDestruct (mono_list_lb_own_get with "Room") as "#rlb".
fr. fr.
- fr. case_bool_decide; [iLeft|iRight]... destruct Heltslen...
- fr. rewrite lookup_app_r... replace (era - length room) with 0...
Qed.
Lemma dqst_get_archived γdqst era1 C1 l1 t1 b1
era2 C2 l2 t2 b2 :
(* era1 is later than era2 *)
era1 ≠ era2 →
dqst_auth γdqst era1 C1 l1 t1 b1 -∗
dqst_frag γdqst era2 C2 l2 t2 b2 -∗
∃ l' t' b', dqst_archived γdqst era2 C2 l' t' b'.
Proof with extended_auto.
desγ γdqst.
iIntros (Hneq) "Auth F".
iDestruct "Auth" as (γtbeO) "(%HltO & tbO & eltO & museO)".
iDestruct "museO" as (room museum) "museO".
iDestruct "museO" as "([%Hroomlen %Hmuslen] & museO)".
iDestruct "museO" as "(Room & Museum & Archives)".
iDestruct "F" as (γtbe) "(%Hlt & tb & elt & muse)".
iDestruct "muse" as (room') "[%Hroom'2 Lb']".
iDestruct (mono_list_auth_lb_valid with "Room Lb'") as "%Pref".
destruct Pref as [_ Pref].
eapply prefix_lookup_Some in Hroom'2; eauto.
assert (era2 < era1) as Hera21.
{ apply lookup_lt_Some in Hroom'2.
rewrite length_app Hroomlen in Hroom'2. simpl in Hroom'2... }
assert (is_Some (museum !! era2)) as [ltbera2 Hltbera2].
{ rewrite lookup_lt_is_Some... }
rewrite lookup_app_l in Hroom'2...
iDestruct (big_sepL2_lookup with "Archives") as "Arch2"...
Qed.
Lemma dqst_get_lb γdqst era1 C1 l1 t1 b1
era2 C2 l2 t2 b2 :
(* era1 is later than era2 *)
dqst_auth γdqst era1 C1 l1 t1 b1 -∗
dqst_frag γdqst era2 C2 l2 t2 b2 -∗
⌜t2 ≤ t1 ∧ (
(t2 = t1 ∧ t2 < b2) →
(t1 < b1 ∧ mod_get l2 t2 = mod_get l1 t1)
)⌝.
Proof with extended_auto.
desγ γdqst.
iIntros "Auth F".
iDestruct "Auth" as (γtbeO) "(%HltO & [tbO tbeO] & eltO & museO)".
iDestruct "F" as (γtbe) "(%Hlt & [tb tbe] & elt & muse)".
iDestruct (mono_nat_lb_own_valid with "tbO tb") as "[_ %Htb]".
apply top_bot_state_le in Htb as [Ht21 Htb21]. fr.
iIntros ([H1 Ht1b2]). subst t2. assert (t1 < b1) as Htb1... fr.
iDestruct "elt" as (elts') "[lb %Helts]"...
destruct Helts as [NO|Helts]...
iDestruct "eltO" as (elts) "[Elts %Heltslen]".
iDestruct (mono_list_auth_lb_lookup t1 with "Elts lb") as "%Heqg"...
{ rewrite -lookup_lt_is_Some -Helts. apply mod_get_is_Some... }
iDestruct (mono_list_auth_lb_valid with "Elts lb") as "[_ %Pref]".
case_bool_decide... destruct Heltslen as [_ Hget].
rewrite Hget Helts...
Qed.
Lemma dqst_archived_get_array γdqst era C l t b :
dqst_archived γdqst era C l t b -∗
(C +ₗ carr) ↦∗□ l.
Proof.
desγ γdqst.
iIntros "Arc".
by iDestruct "Arc" as (γtbeA) "(_&_&_&_& pers)".
Qed.
Lemma dqst_archived_get_frag γdqst era C l t b :
dqst_archived γdqst era C l t b -∗
dqst_frag γdqst era C l t b.
Proof.
desγ γdqst.
iIntros "Arc".
iDestruct "Arc" as (γtbeA) "(%Hlt & [tb tbe] & elt & muse & pers)".
iDestruct "muse" as (room museum) "(%Hroom & Hmuseum & Room & Museum)".
fr. fr. by iApply mono_nat_persistent_lb_own_get.
Qed.
Lemma dqst_archived_get_lb γdqst era C l1 t1 b1 l2 t2 b2 :
dqst_archived γdqst era C l1 t1 b1 -∗
dqst_frag γdqst era C l2 t2 b2 -∗
⌜t2 ≤ t1 ∧ (
(t2 = t1 ∧ t2 < b2) →
(t1 < b1 ∧ mod_get l2 t2 = mod_get l1 t1)
)⌝.
Proof with extended_auto.
desγ γdqst.
iIntros "Arc F".
iDestruct "Arc" as (γtbeA) "(%Hlt1 & [tb1 tbe1] & elt1 & muse1 & pers1)".
iDestruct "muse1" as (room1 museum1) "muse1".
iDestruct "muse1" as "(%Hroom1 & %Hmuse1 & Lbroom1 & Lbmuse1)".
iDestruct "F" as (γtbe) "(%Hlt2 & [tb2 tbe2] & elt2 & muse2)".
iDestruct "muse2" as (room2) "[%Hroom2 Lbroom2]".
iDestruct (mono_list_lb_lookup era with "Lbroom1 Lbroom2") as "%Hr12".
{ apply lookup_lt_Some in Hroom1... }
{ apply lookup_lt_Some in Hroom2... }
rewrite Hroom1 Hroom2 in Hr12. injection Hr12 as [= <-].
iDestruct (mono_nat_persistent_lb_own_valid with "tbe1 tbe2") as "%Htb2".
apply top_bot_state_le in Htb2 as [Hlt21 Htb2].
fr. iIntros ([<- Hlt]). clear Hlt21.
assert (t2 < b1) as Hlt21...
iSplit...
iDestruct "elt1" as (elts1) "[lb1 [%NO|%Hget1]]"...
iDestruct "elt2" as (elts2) "[lb2 [%NO|%Hget2]]"...
rewrite Hget1 Hget2.
iApply (mono_list_lb_lookup with "lb2 lb1").
- rewrite -lookup_lt_is_Some -Hget2. apply mod_get_is_Some...
- rewrite -lookup_lt_is_Some -Hget1. apply mod_get_is_Some...
Qed.
Lemma dqst_auth_write_bot v γdqst era C l t b :
dqst_auth γdqst era C l t b -∗
dqst_auth γdqst era C (mod_set l b v) t b.
Proof.
desγ γdqst.
iIntros "Auth".
iDestruct "Auth" as (γtbeO) "(%HltO & tbO & eltO & museO)".
iDestruct "eltO" as (elts) "[Elts %Heltslen]".
unfold dqst_auth. rewrite mod_set_length. fr.
fr. case_bool_decide; auto. rewrite mod_set_get_ne; auto.
apply neq_symm, close_mod_neq; lia.
Qed.
Lemma dqst_auth_update γdqst era C l t b :
t < b →
dqst_auth γdqst era C l t b ==∗
dqst_auth γdqst era C l (S t) b.
Proof with extended_auto.
desγ γdqst.
iIntros (Hlt) "Auth".
iDestruct "Auth" as (γtbeO) "(%HltO & [tbO tbeO] & eltO & museO)".
iDestruct "eltO" as (elts) "[Elts %Heltslen]".
iMod (mono_nat_own_update (top_bot_state (S t) b)
with "tbO") as "[tbO _]".
{ unfold top_bot_state. do 2 case_bool_decide... }
iMod (mono_nat_own_update (top_bot_state (S t) b)
with "tbeO") as "[tbeO _]".
{ unfold top_bot_state. do 2 case_bool_decide... }
destruct (decide (S t = b)).
{ iModIntro. fr. fr. do 2 case_bool_decide... }
destruct (mod_get_is_Some l (S t)) as [v Hv]...
iMod (mono_list_auth_own_update_app [v] with "Elts") as "[Elts _]".
iModIntro. fr. fr. do 2 case_bool_decide...
rewrite length_app lookup_app_r; simpl...
replace (S t - length elts) with 0; simpl... fr.
Qed.
Lemma dqst_auth_push γdqst era C l t b :
S b < t + length l →
dqst_auth γdqst era C l t b ==∗
dqst_auth γdqst era C l t (S b).
Proof with extended_auto.
desγ γdqst.
iIntros (Hlt) "Auth".
iDestruct "Auth" as (γtbeO) "(%HltO & [tbO tbeO] & eltO & museO)".
iDestruct "eltO" as (elts) "[Elts %Heltslen]".
iMod (mono_nat_own_update (top_bot_state t (S b))
with "tbO") as "[tbO _]".
{ unfold top_bot_state. do 2 case_bool_decide... }
iMod (mono_nat_own_update (top_bot_state t (S b))
with "tbeO") as "[tbeO _]".
{ unfold top_bot_state. do 2 case_bool_decide... }
case_bool_decide; last first.
{ iModIntro. fr. fr. }
destruct (mod_get_is_Some l t) as [v Hv]...
iMod (mono_list_auth_own_update_app [v] with "Elts") as "[Elts _]".
iModIntro. fr. fr. case_bool_decide...
rewrite length_app lookup_app_r; simpl...
replace (t - length elts) with 0; simpl... fr.
Qed.
Lemma dqst_auth_pop γdqst era C l t b :
t < b - 1 →
dqst_auth γdqst era C l t b ==∗
dqst_auth γdqst era C l t (b - 1).
Proof with extended_auto.
desγ γdqst.
iIntros (Hlt) "Auth".
iDestruct "Auth" as (γtbeO) "(%HltO & [tbO tbeO] & eltO & museO)".
iDestruct "eltO" as (elts) "[Elts %Heltslen]".
replace (top_bot_state t b) with (top_bot_state t (b-1)).
2: unfold top_bot_state; repeat case_bool_decide...
iModIntro. fr. fr. case_bool_decide...
Qed.
Lemma dqst_auth_archive ca' l' γdqst era (C : blk) l t b :
length l ≤ length l' →
circ_slice l t b = circ_slice l' t b →
(C +ₗ carr) ↦∗ l -∗
dqst_auth γdqst era C l t b ==∗
dqst_archived γdqst era C l t b ∗
dqst_auth γdqst (S era) ca' l' t b.
Proof with extended_auto.
desγ γdqst.
iIntros (Hlong Heqs) "Own Auth".
iDestruct "Auth" as (γtbeO) "(%HltO & [tbO tbeO] & eltO & museO)".
iDestruct "eltO" as (elts) "[Elts %Heltslen]".
iDestruct "museO" as (proom museum) "museO".
iDestruct "museO" as "([%Hproomlen %Hmuslen] & museO)".
iDestruct "museO" as "(Room & Museum & Archives)".
(* archive *)
iMod (array_persist with "Own") as "#PC".
iDestruct (mono_nat_lb_own_get with "tbO") as "#tb".
iMod (mono_nat_own_persist with "tbeO") as "#tbe".
iDestruct (mono_list_lb_own_get with "Elts") as "#eltslb".
iDestruct (mono_list_lb_own_get with "Room") as "#roomlb".
iMod (mono_list_auth_own_update_app [(l, t, b)] with "Museum") as "[Museum #muslb]".
iSplitR.
{ iModIntro. fr. fr. all: fr.
- case_bool_decide... iRight. destruct Heltslen...
- rewrite lookup_app_r... replace (era - length proom) with 0...
- rewrite lookup_app_r... replace (era - length museum) with 0...
}
(* new era *)
iMod (mono_nat_own_alloc (top_bot_state t b)) as (γtbe') "[tbeO _]".
iMod (mono_list_auth_own_update_app [(γtbe', ca', length l')]
with "Room") as "[Room #rlb]".
(* frame *)
iModIntro. fr. fr.
{ case_bool_decide... fr.
destruct Heltslen as [_ Hget].
apply (circ_slice_split_eq (S t)) in Heqs as [Heqs _]...
destruct (circ_slice_singleton l t) as [x [Hx Hsx]]...
destruct (circ_slice_singleton l' t) as [y [Hy Hsy]]...
rewrite Hsx Hsy in Heqs. injection Heqs as [= <-].
rewrite Hy -Hget Hx...
}
{ rewrite length_app -Hproomlen. simpl... }
{ rewrite length_app -Hmuslen. simpl... }
simpl. fr. all: fr.
- case_bool_decide... iRight. destruct Heltslen...
- rewrite lookup_app_l...
2: rewrite length_app; simpl...
rewrite lookup_app_r...
replace (length proom + 0 - length proom) with 0...
- rewrite lookup_app_r...
replace (length proom + 0 - length museum) with 0...
Qed.
End dqst.
Section proof.
Context `{!heapGS Σ, !cldequeG Σ} (dequeN : namespace).
Notation iProp := (iProp Σ).
Definition deque_inv (γq γera : gname) (γdqst : dqst_gnames) (q : blk) : iProp :=
∃ (era : nat) (C : blk) (l : list val) (t b : nat) (pop : bool),
⌜1 ≤ t ≤ b ∧ b < t + length l ∧ length l ≠ 0⌝ ∗
(C +ₗ csz) ↦□ #(length l) ∗
(* abstract *)
own γq (●E (circ_slice l t b)) ∗
ghost_var γera (1/2) (era, C, l, b, pop) ∗
dqst_auth γdqst era C l t b ∗
(* physical *)
(q +ₗ circle) ↦{#1/2} #C ∗
(C +ₗ carr) ↦∗{#1/2} l ∗
(q +ₗ qtop) ↦ #t ∗
(q +ₗ qbot) ↦{#1/2} #(if pop then b-1 else b).
Definition IsDeque (γ : gname) (q : blk) : iProp :=
∃ (γq γera : gname) (γdqst : dqst_gnames),
⌜γ = encode (γq, γera, γdqst)⌝ ∗
inv dequeN (deque_inv γq γera γdqst q).
Global Instance IsDeque_persistent γ q :
Persistent (IsDeque γ q) := _.
Definition Deque (γ : gname) (frag : list val) : iProp :=
∃ (γq γera : gname) (γdqst : dqst_gnames),
⌜γ = encode (γq, γera, γdqst)⌝ ∗
own γq (◯E frag).
Global Instance Deque_timeless γ frag :
Timeless (Deque γ frag) := _.
(* owner of the deque who can call push and pop *)
Definition OwnDeque (γ : gname) (q : blk) : iProp :=
∃ (γq γera : gname) (γdqst : dqst_gnames) (era : nat)
(l : list val) (C : blk) (b : nat),
⌜γ = encode (γq, γera, γdqst)⌝ ∗
(C +ₗ csz) ↦□ #(length l) ∗
ghost_var γera (1/2) (era, C, l, b, false) ∗
(q +ₗ circle) ↦{#1/2} #C ∗
(C +ₗ carr) ↦∗{#1/2} l ∗
(q +ₗ qbot) ↦{#1/2} #b.
Lemma own_ea_agree γ a b :
own γ (●E a) -∗ own γ (◯E b) -∗ ⌜a = b⌝.
Proof.
iIntros "● ◯".
by iCombine "● ◯" gives %?%excl_auth_agree_L.
Qed.
Lemma own_ea_update a' γ a b :
own γ (●E a) -∗ own γ (◯E b) ==∗ own γ (●E a') ∗ own γ (◯E a').
Proof.
iIntros "● ◯".
iMod (own_update_2 γ _ _ (●E a' ⋅ ◯E a') with "● ◯") as "[● ◯]".
- apply excl_auth_update.
- by iFrame.
Qed.
Lemma deque_new_spec :
deque_new_spec' dequeN deque_new Deque IsDeque OwnDeque.
Proof with extended_auto.
iIntros (n Hsz Φ) "!> _ HΦ". wp_lam.
(* allocate *)
wp_alloc q as "q↦" "†q". wp_pures.
do 3 rewrite array_cons. iDestruct "q↦" as "(C & T & B & _)".
wp_lam. wp_alloc C as "SzA" "†SzA"... wp_pures.
replace (Z.to_nat (1 + n)) with (1 + n)...
rewrite replicate_add array_cons. iDestruct "SzA" as "[Sz A]".
wp_store. wp_pures... wp_store. wp_pures. wp_store. wp_store.
iDestruct "C" as "[C1 C2]". iDestruct "B" as "[B1 B2]".
iDestruct "A" as "[A1 A2]".
(* make resources *)
iMod (pointsto_persist with "Sz") as "#Sz".
iMod (own_alloc (●E [] ⋅ ◯E [])) as (γq) "[γq● γq◯]". 1: apply excl_auth_valid.
iMod (ghost_var_alloc (0, C, (replicate n #0), 1, false)) as (γera) "[Era1 Era2]".
iMod (dqst_auth_alloc C (replicate n #0)) as (γdqst) "Auth"...
iMod (inv_alloc dequeN _ (deque_inv γq γera γdqst q)
with "[C1 T B1 A1 γq● Auth Era1]") as "Inv".
{ fr. fr. }
(* apply Φ *)
iApply "HΦ". iModIntro. iSplitL "Inv"; first fr.
iSplitL "γq◯"; fr. fr.
Qed.
Lemma circle_grow_rec_spec (arr arr' : loc)
(l l' : list val) (n m t b : nat) :
length l = n → length l' = m →
0 < n < m →
t ≤ b < t + n →
arr' ↦∗ l' -∗
<<{ ∀∀ (_ : ()), arr ↦∗ l }>>
circle_grow_rec #arr #n #arr' #m #t #b @ ⊤,∅,∅
<<{ ∃∃ (l2' : list val),
⌜length l2' = m⌝ ∗
⌜circ_slice l t b = circ_slice l2' t b⌝ ∗
⌜∀ i, b ≤ i < t + length l → mod_get l' i = mod_get l2' i⌝ ∗
arr ↦∗ l |
RET #(); arr' ↦∗ l2'
}>>.
Proof with extended_auto.
iIntros "%Hn %Hm %Hlen %Hlt A'" (Φ) "AU".
iRevert "A' AU". iRevert (b l' Hm Hlt).
iLöb as "IH". iIntros (b l') "%Hm %Hlt A' AU".
wp_lam. unfold circ_access. wp_pures.
case_bool_decide; last first; wp_pures.
{ (* end loop *)
iMod "AU" as (_) "[Cont [_ Commit]]".
iMod ("Commit" $! l' with "[Cont]") as "HΦ"; fr.
1: repeat rewrite circ_slice_nil...
iApply "HΦ"...
}
(* read b *)
destruct b as [|b]...
replace (Z.of_nat (S b) - 1)%Z with (Z.of_nat b)...
rewrite rem_mod_eq...
wp_bind (! _)%E.
iMod "AU" as (_) "[A [Abort _]]".
destruct (mod_get_is_Some l b) as [v Hv]...
wp_apply (wp_load_offset with "A") as "A". 1: rewrite -Hn...
iMod ("Abort" with "A") as "AU". 1: fr.
iModIntro. wp_pures.
(* write b *)
wp_bind (_ <- _)%E.
rewrite rem_mod_eq...
destruct (mod_get_is_Some l' b) as [v' Hv']...
iApply (wp_store_offset with "A'"). 1: rewrite -Hm...
iIntros "!> A'". wp_pures.
(* recurse *)
iApply ("IH" $! b (<[b `mod` m:=v]> l') with "[] [] [A']")...
1: rewrite length_insert...
iAuIntro.
rewrite /atomic_acc /=.
iMod "AU" as (_) "[Cont AC]".
iModIntro. iExists (). iFrame "Cont".
iSplit.
{ iIntros "Cont".
iDestruct "AC" as "[Abort _]".
iMod ("Abort" with "Cont") as "AU". fr. }
iIntros (l2') "(%Hm' & %Heqs & %Hlast & A)".
iDestruct "AC" as "[_ Commit]".
iMod ("Commit" $! l2' with "[A]") as "HΦ".
{ iFrame. iPureIntro. repeat split...
- rewrite (circ_slice_shrink_right _ _ _ v)...
2: replace (S b - 1) with b...
rewrite (circ_slice_shrink_right _ _ (S b) v)...
all: replace (S b - 1) with b...
+ rewrite Heqs...
+ rewrite -Hlast... unfold mod_get.
rewrite length_insert Hm list_lookup_insert...
rewrite Hm. apply Nat.mod_upper_bound...
- intros i Hi. rewrite -Hlast... unfold mod_get.
rewrite length_insert Hm list_lookup_insert_ne...
apply close_mod_neq...
}
iIntros "!> A'". iApply "HΦ"...
Qed.
Lemma circle_grow_spec (C : blk) (l : list val) (t b : nat) :
0 < length l →
t ≤ b < t + length l →
(C +ₗ csz) ↦□ #(length l) -∗
<<{ ∀∀ (_ : ()), (C +ₗ carr) ↦∗ l }>>
circle_grow #C #t #b @ ⊤,∅,∅
<<{ ∃∃ (C' : blk) (l' : list val),
⌜length l < length l'⌝ ∗
⌜circ_slice l t b = circ_slice l' t b⌝ ∗
(C +ₗ carr) ↦∗ l |
RET #C';
(C' +ₗ csz) ↦ #(length l') ∗ (C' +ₗ carr) ↦∗ l' }>>.
Proof with extended_auto.
iIntros "%Hlen %Hlt Sz" (Φ) "AU".
wp_lam. wp_pures. wp_load. wp_pures.
wp_lam. wp_alloc SzA' as "SzA'" "†SzA'"... wp_pures.
replace (Z.to_nat (1 + 2 * length l)) with (1 + 2 * length l)...
rewrite replicate_add array_cons. iDestruct "SzA'" as "[Sz' A']".
wp_store. wp_pures.
replace (2 * Z.of_nat (length l))%Z with (Z.of_nat (2 * length l))...
(* make l' *)
awp_apply (circle_grow_rec_spec with "[A']")... unfold atomic_acc.
iMod "AU" as (_) "[A AC]".
iModIntro. iExists (). iFrame "A". iSplit.
{ iIntros "A". iDestruct "AC" as "[Abort _]". fr.
iApply ("Abort" with "A").
}
iIntros (l2') "(%Hlen' & %Heqs & %Hrest & A & A2)".
iCombine "A A2" as "A".
iDestruct "AC" as "[_ Commit]".
iMod ("Commit" $! SzA' l2' with "[A]") as "HΦ"; fr.
iIntros "!> A'".
wp_pures. iModIntro. iApply "HΦ". fr. rewrite Hlen'...
Qed.
Lemma deque_push_spec :
deque_push_spec' dequeN deque_push Deque IsDeque OwnDeque.
Proof with extended_auto.
iIntros (γ q v).
iIntros "#Is Own" (Φ) "AU".
iDestruct "Own" as (γq γera γdqst) "Own".
iDestruct "Own" as (era l C b) "Own".
iDestruct "Own" as "(%Enc & #Sz & eraOwn & COwn & AOwn & bOwn)".
replace (C +ₗ 1) with (C +ₗ Z.of_nat 1)...
remember (C +ₗ Z.of_nat 1) as Carr.
iDestruct "Is" as (γq' γera' γdqst') "Inv".
iDestruct "Inv" as "[%Enc' Inv]".
encode_agree Enc.
wp_lam. unfold circ_access. wp_pures.
wp_load. wp_pures.
(* 1. load top *)
wp_bind (! _)%E.
iInv "Inv" as (era1 C1 l1 t1 b1 pop1) ">Invs".
iDestruct "Invs" as "(%Htb1 & #Sz1 & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
iDestruct (dqst_get_frag with "Dqst") as "#F1".
wp_load.
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists _,_,l. fr. }
wp_pures... wp_load. wp_pures. wp_load. wp_pures.
case_bool_decide; last first; wp_pures...
- (* 2. write to circle *)
wp_load. wp_pures... wp_load. wp_pures.
rewrite -HeqCarr rem_mod_eq...
wp_bind (_ <- _)%E.
iInv "Inv" as (era2 C2 l2 t2 b2 pop2) ">Invs".
iDestruct "Invs" as "(%Htb2 & #Sz2 & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
rewrite -HeqCarr.
iMod (ghost_var_update_halves (era, C, (mod_set l b v), b, false)
with "eraOwn Era") as "[eraOwn Era]".
iDestruct (dqst_auth_write_bot v with "Dqst") as "Dqst".
iCombine "AOwn A" as "A".
iApply (wp_store_offset with "A"). 1: apply mod_get_is_Some...
iIntros "!> [AOwn A]".
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists _,_,(mod_set l b v),t2,b.
rewrite mod_set_length circ_slice_update_right... fr.
rewrite -HeqCarr. fr. }
wp_pures.
(* 3. increment bot *)
iInv "Inv" as (era3 C3 l3 t3 b3 pop3) ">Invs".
iDestruct "Invs" as "(%Htb3 & #Sz3 & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
iMod (ghost_var_update_halves (era, C, (mod_set l b v), S b, false)
with "eraOwn Era") as "[eraOwn Era]".
iDestruct (dqst_get_lb with "Dqst F1") as "%Ht13".
iMod (dqst_auth_push with "Dqst") as "Dqst". 1: rewrite mod_set_length...
iCombine "bOwn B" as "B". wp_store.
replace (Z.of_nat b + 1)%Z with (Z.of_nat (S b))...
iDestruct "B" as "[bOwn B]".
iMod "AU" as (l') "[Cont [_ Commit]]".
iDestruct "Cont" as (γq' γera' γbglob') "[%Enc' ◯]". encode_agree Enc.
iDestruct (own_ea_agree with "● ◯") as "%Hl'".
iMod (own_ea_update (l' ++ [v]) with "● ◯") as "[● ◯]".
iMod ("Commit" with "[◯]") as "HΦ". 1: fr.
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists era, C, (mod_set l b v), t3, (S b). fr.
rewrite (circ_slice_extend_right _ _ _ v)... 2: rewrite mod_set_get...
subst l'. fr. rewrite mod_set_length... }
iApply "HΦ".
iExists γq, γera, γdqst.
iExists era, (mod_set l b v), C, (S b).
rewrite -HeqCarr length_insert. fr.
- (* X. grow *)
wp_bind (circle_grow _ _ _)%E.
awp_apply (circle_grow_spec with "[Sz]")... all: try lia.
iInv "Inv" as (eraX CX lX tX bX popX) ">Invs".
iDestruct "Invs" as "(%HtbX & _ & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
rewrite -HeqCarr.
iCombine "AOwn A" as "A".
iAaccIntro with "A".
{ iIntros "[AOwn A]". iSplitL "● Era Dqst C A T B". 2: fr.
iExists _,_,l. fr. rewrite -HeqCarr. fr. }
iIntros (CX lX) "(%HlenX & %Heqs & [AOwn A])".
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists _,_,l. fr. rewrite -HeqCarr. fr. }
iIntros "(SzOwn & AX)". wp_pures...
iMod (pointsto_persist with "SzOwn") as "#SzX".
replace (CX +ₗ 1) with (CX +ₗ Z.of_nat 1)...
remember (CX +ₗ Z.of_nat 1) as CXarr.
(* Y. replace array *)
wp_bind (_ <- _)%E.
iInv "Inv" as (eraY caY lY tY bY popY) ">Invs".
iDestruct "Invs" as "(%HtbY & #SzY & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
rewrite -HeqCarr.
iMod (ghost_var_update_halves (S era, CX, lX, b, false)
with "eraOwn Era") as "[eraOwn Era]".
iDestruct (dqst_get_lb with "Dqst F1") as "%Ht1Y".
apply (circ_slice_split_eq tY) in Heqs as Heqsd... destruct Heqsd as [_ HeqsR]...
iCombine "AOwn A" as "A".
iMod (dqst_auth_archive CX lX with "[A] [Dqst]") as "[#Arch Dqst]".
2: apply HeqsR. 3: fr. 2: rewrite HeqCarr... 1: lia.
iDestruct "AX" as "[AOwn A]".
iCombine "COwn C" as "C". wp_store.
iDestruct "C" as "[COwn C]".
iSplitL "● Era Dqst C A T B".
{ iModIntro; iNext. iExists _,_,lX.
fr. fr. rewrite -HeqsR -HeqCXarr. fr. }
iModIntro. wp_pures... wp_load. wp_pures... wp_load. wp_pures.
rewrite -HeqCXarr.
iRename "Sz" into "SzOld".
(* 2. write to circle *)
rewrite rem_mod_eq...
wp_bind (_ <- _)%E.
iInv "Inv" as (era2 C2 l2 t2 b2 pop2) ">Invs".
iDestruct "Invs" as "(%Htb2 & #Sz & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
rewrite -HeqCXarr.
iMod (ghost_var_update_halves (S era, CX, (mod_set lX b v), b, false)
with "eraOwn Era") as "[eraOwn Era]".
iDestruct (dqst_auth_write_bot v with "Dqst") as "Dqst".
iCombine "AOwn A" as "A".
iApply (wp_store_offset with "A"). 1: apply mod_get_is_Some...
iIntros "!> [AOwn A]".
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists _,_,(mod_set lX b v),t2,b.
rewrite mod_set_length circ_slice_update_right... fr.
fr. rewrite -HeqCXarr... }
wp_pures.
(* 3. increment bot *)
iInv "Inv" as (era3 C3 l3 t3 b3 pop3) ">Invs".
iDestruct "Invs" as "(%Htb3 & #Sz3 & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
rewrite -HeqCXarr.
iMod (ghost_var_update_halves (S era, CX, (mod_set lX b v), S b, false)
with "eraOwn Era") as "[eraOwn Era]".
iDestruct (dqst_get_lb with "Dqst F1") as "%Ht13".
iMod (dqst_auth_push with "Dqst") as "Dqst". 1: rewrite mod_set_length...
iCombine "bOwn B" as "B". wp_store.
replace (Z.of_nat b + 1)%Z with (Z.of_nat (S b))...
iDestruct "B" as "[bOwn B]".
iMod "AU" as (l') "[Cont [_ Commit]]".
iDestruct "Cont" as (γq' γera' γbglob') "[%Enc' ◯]". encode_agree Enc.
iDestruct (own_ea_agree with "● ◯") as "%Hl'".
iMod (own_ea_update (l' ++ [v]) with "● ◯") as "[● ◯]".
iMod ("Commit" with "[◯]") as "HΦ". 1: fr.
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists (S era), CX, (mod_set lX b v), t3, (S b), false. fr.
1: rewrite mod_set_length... subst l'.
rewrite (circ_slice_extend_right _ _ _ v)... 2: rewrite mod_set_get...
rewrite -HeqCXarr. fr. }
iApply "HΦ".
iExists γq, γera, γdqst.
iExists (S era), (mod_set lX b v). fr. fr. rewrite length_insert...
rewrite -HeqCXarr. fr.
Unshelve. done.
Qed.
Lemma deque_pop_spec :
deque_pop_spec' dequeN deque_pop Deque IsDeque OwnDeque.
Proof with extended_auto.
iIntros (γ q).
iIntros "#Is Own" (Φ) "AU".
iDestruct "Own" as (γq γera γdqst) "Own".
iDestruct "Own" as (era l C b) "Own".
iDestruct "Own" as "(%Enc & #Sz & eraOwn & COwn & AOwn & bOwn)".
replace (C +ₗ 1) with (C +ₗ Z.of_nat 1)...
remember (C +ₗ Z.of_nat 1) as Carr.
iDestruct "Is" as (γq' γera' γdqst') "Inv".
iDestruct "Inv" as "[%Enc' Inv]".
encode_agree Enc.
wp_lam. unfold circ_access. wp_pures.
wp_load. wp_pures... wp_load. wp_pures... wp_load. wp_pures.
(* 1. decrement bot *)
wp_bind (_ <- _)%E.
iInv "Inv" as (era1 C1 l1 t1 b1 pop1) ">Invs".
iDestruct "Invs" as "(%Htb1 & #Sz1 & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
rewrite -HeqCarr.
iMod (ghost_var_update_halves (era, C, l, b, true)
with "eraOwn Era") as "[eraOwn Era]".
iCombine "bOwn B" as "B". wp_store.
iDestruct "B" as "[bOwn B]".
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists _,_,l. fr. rewrite -HeqCarr. fr. }
wp_pures.
(* 2. load top *)
wp_bind (! _)%E.
iInv "Inv" as (era2 C2 l2 t2 b2 pop2) ">Invs".
iDestruct "Invs" as "(%Htb2 & #Sz2 & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
rewrite -HeqCarr.
wp_load.
destruct (decide (t2 < b-1)).
{ (* normal case, this point is the commit point *)
iMod "AU" as (l') "[Cont [_ Commit]]".
iDestruct "Cont" as (γq' γera' γdqst') "[%Enc' ◯]". encode_agree Enc.
destruct (mod_get_is_Some l (b-1)) as [x Hx]...
iDestruct (own_ea_agree with "● ◯") as "%Eq". subst l'.
iMod (own_ea_update (circ_slice l t2 (b-1)) with "● ◯") as "[● ◯]".
iMod (ghost_var_update_halves (era, C, l, b-1, false)
with "eraOwn Era") as "[eraOwn Era]".
iMod (dqst_auth_pop with "Dqst") as "Dqst"...
iMod ("Commit" $! (circ_slice l t2 (b-1)) (SOMEV x) with "[◯]") as "HΦ".
{ fr. rewrite -circ_slice_extend_right... replace (S (b-1)) with b... }
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists _,_,l,t2. fr. fr. replace (Z.of_nat b - 1)%Z with (Z.of_nat (b - 1))%Z...
rewrite -HeqCarr. fr. }
wp_pures.
case_bool_decide... wp_pures. rewrite -HeqCarr.
replace (Z.of_nat b - 1)%Z with (Z.of_nat (b - 1))%Z... rewrite rem_mod_eq...
wp_bind (! _)%E. iApply (wp_load_offset with "AOwn")...
iIntros "!> AOwn". wp_pures.
case_bool_decide... wp_pures.
iApply "HΦ". iExists _,_,_, _,l. fr. rewrite -HeqCarr. fr.
}
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists _,_,l. fr. rewrite -HeqCarr. fr. }
wp_pures.
case_bool_decide; wp_pures.
{ (* empty *)
assert (t2 = b)... subst t2.
(* 3. roll back bot *)
wp_bind (_ <- _)%E.
iInv "Inv" as (era3 C3 l3 t3 b3 pop3) ">Invs".
iDestruct "Invs" as "(%Htb3 & #Sz3 & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
iMod (ghost_var_update_halves (era, C, l, b, false)
with "eraOwn Era") as "[eraOwn Era]".
iCombine "bOwn B" as "B". wp_store.
iDestruct "B" as "[bOwn B]".
iMod "AU" as (l') "[Cont [_ Commit]]".
iMod ("Commit" $! l' NONEV with "[Cont]") as "HΦ"...
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists _,_,l. fr. }
wp_pures. iApply "HΦ". iExists _,_,_, _,l. fr.
rewrite -HeqCarr. fr.
}
(* non-empty *)
rewrite -HeqCarr.
replace (Z.of_nat b - 1)%Z with (Z.of_nat (b - 1))%Z... rewrite rem_mod_eq...
wp_bind (! _)%E.
destruct (mod_get_is_Some l (b-1)) as [v Hv]...
iApply (wp_load_offset with "AOwn")...
iIntros "!> AOwn". wp_pures.
(* we already did normal case *)
case_bool_decide... wp_pures.
(* 3. CAS for one-element case *)
assert (b = S t2)... subst b.
wp_bind (CmpXchg _ _ _)%E.
iInv "Inv" as (era3 C3 l3 t3 b3 pop3) ">Invs".
iDestruct "Invs" as "(%Htb3 & #Sz3 & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
rewrite -HeqCarr.
destruct (decide (t2 = t3)).
+ (* success *)
subst t3. wp_cmpxchg_suc.
replace (Z.of_nat t2 + 1)%Z with (Z.of_nat (S t2))...
iMod "AU" as (l') "[Cont [_ Commit]]".
iDestruct "Cont" as (γq' γera' γdqst') "[%Enc' ◯]". encode_agree Enc.
destruct (mod_get_is_Some l (S t2 - 1)) as [x Hx]...
rewrite Hv in Hx. injection Hx as [= <-].
iMod (dqst_auth_update with "Dqst") as "Dqst"...
iDestruct (own_ea_agree with "● ◯") as "%Eq". subst l'.
iMod (own_ea_update [] with "● ◯") as "[● ◯]".
iMod ("Commit" $! [] (SOMEV v) with "[◯]") as "HΦ".
{ fr. rewrite (circ_slice_extend_right _ _ _ v)...
1: rewrite circ_slice_nil... replace t2 with (S t2 - 1)... }
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists era, C, l, (S t2), (S t2).
rewrite circ_slice_nil... fr. fr. rewrite -HeqCarr... }
wp_pures.
(* 4. roll back bot *)
wp_bind (_ <- _)%E.
iInv "Inv" as (era4 C4 l4 t4 b4 pop4) ">Invs".
iDestruct "Invs" as "(%Htb4 & #Sz4 & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
replace (Z.of_nat (S t2 - 1))%Z with (Z.of_nat t2)%Z...
replace (Z.of_nat (S t2) - 1)%Z with (Z.of_nat t2)%Z...
iMod (ghost_var_update_halves (era, C, l, S t2, false)
with "eraOwn Era") as "[eraOwn Era]".
iCombine "bOwn B" as "B". wp_store.
replace (Z.of_nat t2 + 1)%Z with (Z.of_nat (S t2))%Z...
iDestruct "B" as "[bOwn B]".
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists _,_,l. fr. }
wp_pures. iApply "HΦ". iExists _,_,_, _,l. fr. rewrite -HeqCarr...
+ (* fail *)
wp_cmpxchg_fail.
iMod "AU" as (l') "[Cont [_ Commit]]".
iMod ("Commit" $! l' NONEV with "[Cont]") as "HΦ"...
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists _,_,l. fr. rewrite -HeqCarr. fr. }
wp_pures.
(* 4. roll back bot *)
wp_bind (_ <- _)%E.
iInv "Inv" as (era4 C4 l4 t4 b4 pop4) ">Invs".
iDestruct "Invs" as "(%Htb4 & #Sz4 & ● & Era & Dqst & C & A & T & B)".
iDestruct (ghost_var_agree with "eraOwn Era") as "%Eq"; injection Eq as [= <- <- <- <- <-].
replace (Z.of_nat (S t2 - 1))%Z with (Z.of_nat t2)%Z...
replace (Z.of_nat (S t2) - 1)%Z with (Z.of_nat t2)%Z...
iMod (ghost_var_update_halves (era, C, l, S t2, false)
with "eraOwn Era") as "[eraOwn Era]".
iCombine "bOwn B" as "B". wp_store.
replace (Z.of_nat t2 + 1)%Z with (Z.of_nat (S t2))%Z...
iDestruct "B" as "[bOwn B]".
iModIntro. iSplitL "● Era Dqst C A T B".
{ iExists _,_,l. fr. }
wp_pures. iApply "HΦ". iExists _,_,_, _,l. fr. rewrite -HeqCarr...
Qed.
Lemma deque_steal_spec :
deque_steal_spec' dequeN deque_steal Deque IsDeque.
Proof with extended_auto.
iIntros (γ q).
iIntros "#Is" (Φ) "AU".