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arc.v
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arc.v
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From iris.bi Require Import fractional.
From iris.proofmode Require Import tactics.
From iris.algebra Require Import excl csum frac auth agree.
From iris.base_logic.lib Require Import invariants.
From gpfsl.gps Require Import middleware protocols.
From gpfsl.logic Require Import view_invariants.
From lrust.lang Require Import notation new_delete arc_cmra.
From iris.prelude Require Import options.
(* Rust Arc has a maximum number of clones to prevent overflow. We currently
don't support this. *)
Definition strong_count : val :=
λ: ["l"], !ᵃᶜ"l".
Definition weak_count : val :=
λ: ["l"], !ᵃᶜ("l" +ₗ #1).
Definition clone_arc : val :=
rec: "clone" ["l"] :=
let: "strong" := !ʳˡˣ"l" in
(* fully relaxed CAS *)
if: CAS "l" "strong" ("strong" + #1) Relaxed Relaxed Relaxed then #☠ else "clone" ["l"].
Definition clone_weak : val :=
rec: "clone" ["l"] :=
let: "weak" := !ʳˡˣ("l" +ₗ #1) in
if: CAS ("l" +ₗ #1) "weak" ("weak" + #1) Relaxed Relaxed Relaxed then #☠ else "clone" ["l"].
(* // the value usize::MAX acts as a sentinel for temporarily "locking" the
// ability to upgrade weak pointers or downgrade strong ones; this is used
// to avoid races in `make_mut` and `get_mut`.
Her we use -1 for usize::MAX.
*)
Definition downgrade : val :=
rec: "downgrade" ["l"] :=
let: "weak" := !ʳˡˣ("l" +ₗ #1) in
if: "weak" = #(-1) then
(* -1 in weak indicates that somebody locked the counts; let's spin. *)
"downgrade" ["l"]
else
(*
// Unlike with Clone(), we need this to be an Acquire read to
// synchronize with the write coming from `is_unique`, so that the
// events prior to that write happen before this read.
*)
(* acquire CAS, i.e. relaxed fail read, acquire success read, relaxed write *)
if: CAS ("l" +ₗ #1) "weak" ("weak" + #1) Relaxed AcqRel Relaxed then #☠
else "downgrade" ["l"].
Definition upgrade : val :=
rec: "upgrade" ["l"] :=
let: "strong" := !ʳˡˣ"l" in
if: "strong" = #0 then #false
else
if: CAS "l" "strong" ("strong" + #1) Relaxed Relaxed Relaxed then #true
else "upgrade" ["l"].
Definition drop_weak : val :=
rec: "drop" ["l"] :=
(* This can ignore the lock because the lock is only held when there
are no other weak refs in existence, so this code will never be called
with the lock held. *)
let: "weak" := !ʳˡˣ("l" +ₗ #1) in
(* release CAS, i.e. relaxed fail read, relaxed success read, release write *)
if: CAS ("l" +ₗ #1) "weak" ("weak" - #1) Relaxed Relaxed AcqRel
then if: "weak" = #1 then FenceAcq ;; #true else #false
else "drop" ["l"].
Definition drop_arc : val :=
rec: "drop" ["l"] :=
let: "strong" := !ʳˡˣ"l" in
if: CAS "l" "strong" ("strong" - #1) Relaxed Relaxed AcqRel
then if: "strong" = #1 then FenceAcq ;; #true else #false
else "drop" ["l"].
Definition try_unwrap : val :=
λ: ["l"],
(* release CAS *)
if: CAS "l" #1 #0 Relaxed Relaxed AcqRel
then FenceAcq ;; #true
else #false.
(* A bug was discovered where the read of strong was relaxed.
It can be fixed by using an acquire read.
See https://github.com/rust-lang/rust/issues/51780 *)
Definition is_unique : val :=
λ: ["l"],
(* Try to acquire the lock for the last ref by CASing weak count from 1 to -1. *)
(* acquire CAS *)
if: CAS ("l" +ₗ #1) #1 #(-1) Relaxed AcqRel Relaxed then
let: "strong" := !ᵃᶜ"l" in
let: "unique" := "strong" = #1 in
"l" +ₗ #1 <-ʳᵉˡ #1;;
"unique"
else
#false.
(* to be used in make_mut *)
(*// Note that we hold both a strong reference and a weak reference.
// Thus, releasing our strong reference only will not, by itself, cause
// the memory to be deallocated.
//
// Use Acquire to ensure that we see any writes to `weak` that happen
// before release writes (i.e., decrements) to `strong`. Since we hold a
// weak count, there's no chance the ArcInner itself could be
// deallocated. *)
(* - Returns #2 to signal that there is another strong pointer,
and trigger a deep copy and make a new Arc pointing to the new copy.
- Returns #1 to signal that we have the last strong pointer and
the main content, but there are more than 1 weak pointers left.
So we will move the main content to a new Arc, and leave the job
of dropping to old Arc to its remaining weak pointers.
- Returns #0 to signal that we have collected both the last strong and weak pointer,
so it's time to clean up everything. We reset "strong" back to 1 to unlock.
The write "l" <- #1 seems to be good enough, but C/Rust
uses a release write, may be because the strict distinction between atomics
and non-atomics. (TODO: is a relaxed write enough?) *)
(* we use acquire read instead of relaxed read on weak: we do not know how to
prove or disprove the relaxed read yet *)
Definition try_unwrap_full : val :=
λ: ["l"],
(* acquire CAS *)
if: CAS "l" #1 #0 Relaxed AcqRel Relaxed then
if: !ᵃᶜ("l" +ₗ #1) = #1 then "l" <- #1;; #0
else #1
else #2.
Section ArcInv.
(* P1 is the thing that is shared by strong reference owners (in practice,
this is the lifetime token), and P2 is the thing that is owned by the
protocol when only weak references are left (in practice, P2 is the
ownership of the underlying memory incl. deallocation). *)
Context `{!noprolG Σ, !arcG Σ, !view_invG Σ}
(P1 : Qp → vProp Σ) (P2 : vProp Σ) (N : namespace).
Local Notation vProp := (vProp Σ).
Definition SeenActs γ l (Mt: gset time) : vProp :=
(∃ t v, GPS_Reader l t () v γ ∗ ⌜∀ t', t' ∈ Mt → t' ⊑ t⌝)%I.
Definition WkResources lw ls t v γi γw γs γsw q Ut : vProp :=
(* writer permission and remaining reader permission *)
(view_tok γi (q/2) ∗ GPS_SWSharedReader lw t () v q γw ∗
(* I have seen all upgrades by strong that I'm involved in --- γs *)
WkUps γsw q Ut ∗ SeenActs γs ls Ut)%I.
Definition weak_interp ls γi γs γsw : interpO Σ unitProtocol :=
(λ pb lw γw t _ v, ∃ st, ⌜v = #(Z_of_arcweak_st st)⌝ ∗
if pb
(* Full interp *)
then (* the weak state st, the move authority Mt, the upgrade authority Ut *)
WeakAuth γsw st ∗
match st return _ with
| (None, None) => (* nothing *) True
| (Some qs, None) => ⌜qs = 1%Qp⌝ ∗ (* some strongs, no weak *)
(* weak resources are returned *)
GPS_SWSharedWriter lw t () v γw ∗
(∃ Ut, WkResources lw ls t v γi γw γs γsw 1%Qp Ut)
| (None, Some (Cinl (q, n))) => (* no strong, n weaks *)
(* remaining weak resources *)
GPS_SWSharedWriter lw t () v γw ∗
(∃ q' Ut, ⌜(q + q')%Qp = 1%Qp⌝ ∗
WkResources lw ls t v γi γw γs γsw q' Ut) ∗
P2 ∗
(* strong resources *)
view_tok γi (1/2)%Qp ∗
(∃ ts, GPS_SWWriter ls ts () #0 γs) ∗
(∃ Mts, StrMoves γsw 1%Qp Mts) ∗
(∃ Dts, StrDowns γsw 1%Qp Dts)
| (Some qs, Some (Cinl (q, n))) => ⌜qs = 1%Qp⌝ ∗
(* some strongs, n weaks *)
GPS_SWSharedWriter lw t () v γw ∗
(∃ q' Ut, ⌜(q + q')%Qp = 1%Qp⌝ ∗
WkResources lw ls t v γi γw γs γsw q' Ut)
| (Some qs, Some (Cinr (_, agq))) => ⌜qs = 1%Qp⌝ ∗
(* some strong arcs, no weak arcs, locked,
agown γsw (awk_n ((Some q, ε))) is the payload *)
∃ q, ⌜agq ≡ to_agree q⌝ ∗ StrWkTok γsw q
| _ => False (* anything else is impossible *)
end
(* Read interp *)
else ⌜0 < Z_of_arcweak_st st⌝ ∗ (* CASable values > 0 *)
match st with
| (Some _, None) | (None, Some (Cinl (_, 1%positive))) =>
(* move from 1 *)
∃ t' : time, ⌜(t < t')%positive⌝ ∗ StrDownCert γsw t'
| _ => True
end
)%I.
Definition weak_noCAS γsw : interpCasO Σ unitProtocol :=
(λ _ _ t _ v, ⌜v = #(-1)⌝ ∗ ∃ t' : time, ⌜(t < t')%positive⌝ ∗ StrDownCert γsw t')%I.
Definition StrResources lw ls t v γi γs γw γsw q Mt Dt :=
(* writer permission and remaining reader permission *)
(view_tok γi (q/2) ∗ GPS_SWSharedReader ls t () v q γs ∗
(* I have seen all my moves --- γ *)
StrMoves γsw q Mt ∗ SeenActs γs ls Mt ∗
(* I have seen all downgrads by weak that I'm involved in --- γs *)
StrDowns γsw q Dt ∗ SeenActs γw lw Dt ∗
(* weak resources I need to keep to use weak *)
StrWkTok γsw q)%I.
Definition strong_interp lw γi γw γsw: interpO Σ unitProtocol :=
(λ pb ls γs t _ v, ∃ st, ⌜v = #(Z_of_arcstrong_st st)⌝ ∗
if pb
(* Full interp *)
then (* the strong state st *)
StrongAuth γsw st ∗
match st return _ with
| None => (* nothing *) True
| Some (q, n) => (* n strongs *)
GPS_SWSharedWriter ls t () v γs ∗ ∃ q', ⌜(q + q')%Qp = 1%Qp⌝ ∗
P1 q' ∗
(∃ Mt Dt, StrResources lw ls t v γi γs γw γsw q' Mt Dt)
end
(* Read interp *)
else match st with
| None => False
| Some (_, 1%positive) =>
(* move from 1 *)
∃ t' : time, ⌜(t < t')%positive⌝ ∗ (StrMoveCert γsw t' ∨ WkUpCert γsw t')
| _ => True
end
)%I.
Definition strong_noCAS : interpCasO Σ unitProtocol :=
(λ _ _ _ _ v, ⌜v = #0⌝)%I.
Definition strong_arc t q l γi γs γw γsw : vProp :=
(∃ v Mt Dt, StrongTok γsw q ∗ StrResources (l >> 1) l t v γi γs γw γsw q Mt Dt)%I.
Definition weak_arc t q l γi γs γw γsw : vProp :=
(∃ v Ut, WeakTok γsw q ∗ WkResources (l >> 1) l t v γi γw γs γsw q Ut)%I.
Definition fake_arc l γi γs γw γsw :=
(view_tok γi (1/2)%Qp ∗ (∃ ts, GPS_SWWriter l ts () #0 γs) ∗
(∃ Mts, StrMoves γsw 1%Qp Mts ∗ SeenActs γs l Mts) ∗
(∃ Dts, StrDowns γsw 1%Qp Dts ∗ SeenActs γw (l >> 1) Dts) ∗
StrWkTok γsw 1%Qp)%I.
Global Instance strong_arc_timeless t q l γi γs γw γsw : Timeless (strong_arc t q l γi γs γw γsw) := _.
Global Instance weak_arc_timeless t q l γi γs γw γsw : Timeless (weak_arc t q l γi γs γw γsw) := _.
Global Instance fake_arc_timeless l γi γs γw γsw : Timeless (fake_arc l γi γs γw γsw) := _.
Definition strong_arc_acc l t γi γs γw γsw P E : vProp :=
(□ (P ={E,↑histN}=∗ ∃ Vb q,
(⊒Vb → ▷ strong_arc t q l γi γs γw γsw) ∗
((⊒Vb → ▷ strong_arc t q l γi γs γw γsw) ={↑histN,E}=∗ P)))%I.
Definition weak_arc_acc l t γi γs γw γsw P E : vProp :=
(□ (P ={E,↑histN}=∗ ∃ Vb q,
(⊒Vb → ▷ weak_arc t q l γi γs γw γsw) ∗
((⊒Vb → ▷ weak_arc t q l γi γs γw γsw) ={↑histN,E}=∗ P)))%I.
Definition arc_inv γi γs γw γsw (l : loc) : vProp :=
(((∃ Mt, StrongMoveAuth γsw Mt) ∗
(∃ Dt, StrongDownAuth γsw Dt) ∗ (∃ Ut, WeakUpAuth γsw Ut)) ∗
GPS_INV (strong_interp (l >> 1) γi γw γsw) l strong_noCAS γs ∗
GPS_INV (weak_interp l γi γs γsw) (l >> 1) (weak_noCAS γsw) γw)%I.
Definition is_arc γi γs γw γsw l : vProp :=
view_inv γi N (arc_inv γi γs γw γsw l).
Global Instance is_arc_persistent γi γs γw γsw l : Persistent (is_arc γi γs γw γsw l) := _.
End ArcInv.
Global Typeclasses Opaque StrongTok WeakTok StrMoves StrDowns WkUps.
Section arc.
(* P1 is the thing that is shared by strong reference owners (in practice,
this is the lifetime token), and P2 is the thing that is owned by the
protocol when only weak references are left (in practice, P2 is the
ownership of the underlying memory incl. deallocation). *)
Context `{!noprolG Σ, !arcG Σ, !view_invG Σ}
(P1 : Qp → vProp Σ)
(P2 : vProp Σ) (N : namespace).
Local Notation vProp := (vProp Σ).
Local Notation HP1 := (∀ q1 q2, P1 (q1 + q2)%Qp -∗ P1 q1 ∗ <obj> P1 q2).
Local Notation HP2 := (∀ q1 q2, P1 q1 ∗ P1 q2 -∗ P1 (q1 + q2)%Qp).
Global Instance arc_inv_ne n :
Proper (pointwise_relation _ (dist n) ==> dist n ==>
eq ==> eq ==> eq ==> eq ==> eq ==> dist n) arc_inv.
Proof.
repeat intro. subst.
apply bi.sep_ne; [done|]. apply bi.sep_ne; apply GPS_INV_ne; [|done| |done].
- move => b ?????. apply bi.exist_ne => st.
apply bi.sep_ne; [done|]. destruct b; [|done]. apply bi.sep_ne; [done|].
destruct st as [[]|]; [|done]. solve_proper.
- move => b ?????. apply bi.exist_ne => st.
apply bi.sep_ne; [done|]. destruct b; [|done]. apply bi.sep_ne; [done|].
destruct st as [[|] [[[]| |]|]]; simpl; try done. solve_proper.
Qed.
Global Instance arc_inv_proper :
Proper (pointwise_relation _ (≡) ==> (≡) ==>
eq ==> eq ==> eq ==> eq ==> eq ==> (≡))
arc_inv.
Proof.
repeat intro. subst.
apply bi.sep_proper; [done|].
apply bi.sep_proper; apply GPS_INV_proper; auto;
move => b ?????; apply bi.exist_proper => ?; solve_proper.
Qed.
Global Instance is_arc_contractive n :
Proper (pointwise_relation _ (dist_later n) ==> dist_later n ==>
eq ==> eq ==> eq ==> eq ==> eq ==> eq ==> dist n)
is_arc.
Proof.
repeat intro. subst. constructor => ?.
rewrite /is_arc. apply view_inv_contractive.
dist_later_intro; by apply arc_inv_ne.
Qed.
Global Instance is_arc_proper :
Proper (pointwise_relation _ (≡) ==> (≡) ==> eq
==> eq ==> eq ==> eq ==> eq ==> eq ==> (≡)) is_arc.
Proof.
repeat intro. subst.
by apply view_inv_proper, arc_inv_proper.
Qed.
Local Instance strong_interp_read_persistent γi γw γsw l γs t s v :
Persistent (strong_interp P1 (l >> 1) γi γw γsw false l γs t s v).
Proof. apply bi.exist_persistent. intros [[? []]|]; simpl; by apply _. Qed.
Local Instance weak_interp_read_persistent i γs γsw l γw t s v :
Persistent (weak_interp P2 l i γs γsw false (l >> 1) γw t s v).
Proof.
intros. apply bi.exist_persistent.
intros [[?|] [[[?[]]|?|]|]]; simpl; apply _.
Qed.
Lemma SeenActs_join γ l Mt Mt' :
⊢ (SeenActs γ l Mt ∗ SeenActs γ l Mt') ∗-∗ SeenActs γ l (Mt ∪ Mt').
Proof.
iSplit.
- iIntros "[S1 S2]". iDestruct "S1" as (t1 v1) "[R1 MAX1]".
iDestruct "S2" as (t2 v2) "[R2 MAX2]".
iDestruct "MAX1" as %MAX1. iDestruct "MAX2" as %MAX2.
case (decide (t1 ≤ t2)%positive) => [Le|/Pos.lt_nle /Pos.lt_le_incl Le].
+ iExists t2, v2. iFrame "R2". iPureIntro.
move => ? /elem_of_union [/MAX1 -> //|/MAX2//].
+ iExists t1, v1. iFrame "R1". iPureIntro.
move => ? /elem_of_union [/MAX1//|/MAX2 ->//].
- iDestruct 1 as (t v) "[#R MAX]". iDestruct "MAX" as %MAX.
iSplitL""; iExists t, v; iFrame "R"; iPureIntro; move => ? In;
apply MAX, elem_of_union; [by left|by right].
Qed.
Lemma arc_inv_split Vb γi γs γw γsw l E:
(⊒Vb → ▷ arc_inv P1 P2 γi γs γw γsw l)
={E}=∗ ((∃ Mt, StrongMoveAuth γsw Mt) ∗
(∃ Dt, StrongDownAuth γsw Dt) ∗
(∃ Ut, WeakUpAuth γsw Ut)) ∗
(⊒Vb →
▷ GPS_INV (strong_interp P1 (l >> 1) γi γw γsw) l strong_noCAS γs) ∗
(⊒Vb →
▷ GPS_INV (weak_interp P2 l γi γs γsw) (l >> 1) (weak_noCAS γsw) γw).
Proof.
rewrite -!view_join_unfold !view_join_later.
iDestruct 1 as "((SMA & SDA & WUA) & $ & $)".
iDestruct "SMA" as (Mt) ">SMA". iDestruct "SDA" as (Dt) ">SDA".
iDestruct "WUA" as (Ut) ">WUA".
iIntros "!>".
iSplitL "SMA"; last iSplitL "SDA"; iExists _;
rewrite view_join_embed; iFrame.
Qed.
Lemma arc_inv_join Vb γi γs γw γsw l :
(∃ Mt, StrongMoveAuth γsw Mt) -∗ (∃ Dt, StrongDownAuth γsw Dt) -∗
(∃ Ut, WeakUpAuth γsw Ut) -∗
(⊒Vb →
▷ GPS_INV (strong_interp P1 (l >> 1) γi γw γsw) l strong_noCAS γs) -∗
(⊒Vb →
▷ GPS_INV (weak_interp P2 l γi γs γsw) (l >> 1) (weak_noCAS γsw) γw) -∗
(⊒Vb → ▷ arc_inv P1 P2 γi γs γw γsw l).
Proof.
iIntros "SMA SDA WUA IS IW #In". iFrame "SMA SDA WUA".
iDestruct ("IS" with "In") as "$". by iDestruct ("IW" with "In") as "$".
Qed.
Lemma strong_arc_acc_open l t γi γs γw γsw P E :
P -∗ strong_arc_acc l t γi γs γw γsw P E ={E,↑histN}=∗
∃ Vb q Mt Dt, (StrongTok γsw q ∗
(⊒Vb → view_tok γi (q / 2)) ∗
(∃ v, ⊒Vb → GPS_SWSharedReader l t ((): unitProtocol) v q γs) ∗
StrMoves γsw q Mt ∗ StrDowns γsw q Dt ∗ StrWkTok γsw q) ∗
(∀ Mt' Dt',
StrongTok γsw q -∗
(⊒Vb → view_tok γi (q / 2)) -∗
(∃ v, ⊒Vb → GPS_SWSharedReader l t ((): unitProtocol) v q γs) -∗
StrMoves γsw q (Mt ∪ Mt') -∗
(if decide (Mt' = ∅) then True else SeenActs γs l Mt') -∗
StrDowns γsw q (Dt ∪ Dt') -∗
(if decide (Dt' = ∅) then True else SeenActs γw (l >> 1) Dt') -∗
StrWkTok γsw q ={↑histN,E}=∗ P).
Proof.
iIntros "HP Hacc". iMod ("Hacc" with "HP") as (Vb q) "[Hown Hclose2]".
iExists Vb, q. iMod (monPred_in_later_timeless with "Hown") as "Hown".
rewrite monPred_in_exist'. iDestruct "Hown" as (v) "Hown".
rewrite monPred_in_exist'. iDestruct "Hown" as (Mt) "Hown".
rewrite monPred_in_exist'. iDestruct "Hown" as (Dt) "Hown".
rewrite 7!monPred_in_sep'.
iDestruct "Hown" as "(St & tok & SR & SM & SeenM & SD & SeenD & oW)".
iExists Mt, Dt. iFrame "tok". iSplitR "Hclose2 SeenM SeenD".
- iSplitL "St". { by rewrite monPred_in_embed' /StrongTok. }
iSplitL "SR". { by iExists _. } iSplitL "SM".
{ by rewrite monPred_in_embed' /StrMoves. } iSplitL "SD".
{ by rewrite monPred_in_embed' /StrDowns. }
by rewrite monPred_in_embed'.
- iIntros "!>" (Mt' Dt') "St tok SR SM SeenM' SD SeenD' oW".
iMod ("Hclose2" with "[-]") as "$"; [|done]. iIntros "#mb !>".
iSpecialize ("tok" with "mb").
iDestruct "SR" as (v') "SR". iSpecialize ("SR" with "mb").
iExists _,_,_. iFrame "St tok SR SM SD oW".
iSpecialize ("SeenM" with "mb"). iSpecialize ("SeenD" with "mb").
iSplitL "SeenM SeenM'"; (case decide => [->|?];
[by rewrite right_id_L| iApply SeenActs_join; by iFrame]).
Qed.
Lemma strong_arc_acc_open_no_change l t γi γs γw γsw P E :
P -∗ strong_arc_acc l t γi γs γw γsw P E ={E,↑histN}=∗
∃ Vb q Mt Dt, (StrongTok γsw q ∗
(⊒Vb → view_tok γi (q / 2)) ∗
(∃ v, ⊒Vb → GPS_SWSharedReader l t ((): unitProtocol) v q γs) ∗
StrMoves γsw q Mt ∗ StrDowns γsw q Dt ∗ StrWkTok γsw q) ∗
(StrongTok γsw q -∗
(⊒Vb → view_tok γi (q / 2)) -∗
(∃ v, ⊒Vb → GPS_SWSharedReader l t ((): unitProtocol) v q γs) -∗
StrMoves γsw q Mt -∗ StrDowns γsw q Dt -∗
StrWkTok γsw q ={↑histN,E}=∗ P).
Proof.
iIntros "HP Hacc". iMod (strong_arc_acc_open with "HP Hacc")
as (Vs q Mt Dt) "[(St & tok & SR & SM & SD & oW) Hclose2]".
iExists Vs, q, Mt, Dt. iFrame "St tok SR SM SD oW".
iIntros "!> St tok SR SM SD oW".
iApply ("Hclose2" $! ∅ ∅ with "St tok SR [SM] [//] [SD] [//] oW");
by rewrite right_id_L.
Qed.
Lemma strong_arc_acc_open_2 l t γi γs γw γsw P E :
P -∗ strong_arc_acc l t γi γs γw γsw P E ={E,↑histN}=∗
∃ Vb q Mt Dt, (StrongTok γsw q ∗
(⊒Vb → view_tok γi (q / 2)) ∗
(∃ v, ⊒Vb → GPS_SWSharedReader l t ((): unitProtocol) v q γs) ∗
StrMoves γsw q Mt ∗ StrDowns γsw q Dt ∗ StrWkTok γsw q) ∗
(∀ Mt' Dt',
StrongTok γsw q -∗
(⊒Vb → view_tok γi (q / 2)) -∗
(∃ v, GPS_SWSharedReader l t ((): unitProtocol) v q γs) -∗
StrMoves γsw q (Mt ∪ Mt') -∗
(if decide (Mt' = ∅) then True else SeenActs γs l Mt') -∗
StrDowns γsw q (Dt ∪ Dt') -∗
(if decide (Dt' = ∅) then True else SeenActs γw (l >> 1) Dt') -∗
StrWkTok γsw q ={↑histN,E}=∗ P).
Proof.
iIntros "HP Hacc". iMod (strong_arc_acc_open with "HP Hacc")
as (Vs q Mt Dt) "[(St & tok & SR & SM & SD & oW) Hclose2]".
iExists Vs, q, Mt, Dt. iFrame "St tok SR SM SD oW".
iIntros "!>" (Mt' Dt') "St tok SR SM SeenM SD SeenD oW".
iApply ("Hclose2" $! Mt' Dt' with "St tok [SR] SM SeenM SD SeenD oW").
iDestruct "SR" as (?) "SR". iExists _. by iIntros "_".
Qed.
Lemma strong_arc_acc_open_no_change_2 l t γi γs γw γsw P E :
P -∗ strong_arc_acc l t γi γs γw γsw P E ={E,↑histN}=∗
∃ Vb q Mt Dt, (StrongTok γsw q ∗
(⊒Vb → view_tok γi (q / 2)) ∗
(∃ v, ⊒Vb → GPS_SWSharedReader l t ((): unitProtocol) v q γs) ∗
StrMoves γsw q Mt ∗ StrDowns γsw q Dt ∗ StrWkTok γsw q) ∗
(StrongTok γsw q -∗
(⊒Vb → view_tok γi (q / 2)) -∗
(∃ v, GPS_SWSharedReader l t ((): unitProtocol) v q γs) -∗
StrMoves γsw q Mt -∗ StrDowns γsw q Dt -∗
StrWkTok γsw q ={↑histN,E}=∗ P).
Proof.
iIntros "HP Hacc". iMod (strong_arc_acc_open_no_change with "HP Hacc")
as (Vs q Mt Dt) "[(St & tok & SR & SM & SD & oW) Hclose2]".
iExists Vs, q, Mt, Dt. iFrame "St tok SR SM SD oW".
iIntros "!> St tok SR SM SD oW". iApply ("Hclose2" with "St tok [SR] SM SD oW").
iDestruct "SR" as (?) "SR". iExists _. by iIntros "_".
Qed.
Lemma weak_arc_acc_open l t γi γs γw γsw P E :
P -∗ weak_arc_acc l t γi γs γw γsw P E ={E,↑histN}=∗
∃ Vb q Ut, (WeakTok γsw q ∗
(⊒Vb → view_tok γi (q / 2)) ∗
(∃ v, ⊒Vb →
GPS_SWSharedReader (l >> 1) t ((): unitProtocol) v q γw) ∗
WkUps γsw q Ut) ∗
(∀ Ut', WeakTok γsw q -∗
(⊒Vb → view_tok γi (q / 2)) -∗
(∃ v, ⊒Vb →
GPS_SWSharedReader (l >> 1) t ((): unitProtocol) v q γw) -∗
WkUps γsw q (Ut ∪ Ut') -∗
(if decide (Ut' = ∅) then True else SeenActs γs l Ut') ={↑histN,E}=∗ P).
Proof.
iIntros "HP Hacc". iMod ("Hacc" with "HP") as (Vb q) "[Hown Hclose2]".
iExists Vb, q. iMod (monPred_in_later_timeless with "Hown") as "Hown".
rewrite monPred_in_exist'. iDestruct "Hown" as (v) "Hown".
rewrite monPred_in_exist'. iDestruct "Hown" as (Ut) "Hown". iExists Ut.
rewrite 4!monPred_in_sep'. iDestruct "Hown" as "(Wt & $ & WR & WU & SeenU)".
iSplitR "Hclose2 SeenU".
- iSplitL "Wt". { by rewrite monPred_in_embed' /WeakTok. }
iSplitL "WR". { by iExists _. } by rewrite monPred_in_embed' /WkUps.
- iIntros "!>" (Ut') "Wt tok WR WU SeenU'".
iMod ("Hclose2" with "[-]") as "$"; [|done]. iIntros "#mb !>".
iDestruct "WR" as (v') "WR". iSpecialize ("WR" with "mb").
iSpecialize ("tok" with "mb").
iExists _,_. iFrame "Wt tok WR WU".
iSpecialize ("SeenU" with "mb"). case decide => [->|?];
[by rewrite right_id_L| iApply SeenActs_join; by iFrame].
Qed.
Lemma weak_arc_acc_open_no_change l t γi γs γw γsw P E :
P -∗ weak_arc_acc l t γi γs γw γsw P E ={E,↑histN}=∗
∃ Vb q Ut, (WeakTok γsw q ∗
(⊒Vb → view_tok γi (q / 2)) ∗
(∃ v, ⊒Vb →
GPS_SWSharedReader (l >> 1) t ((): unitProtocol) v q γw) ∗
WkUps γsw q Ut) ∗
(WeakTok γsw q -∗
(⊒Vb → view_tok γi (q / 2)) -∗
(∃ v, ⊒Vb →
GPS_SWSharedReader (l >> 1) t ((): unitProtocol) v q γw) -∗
WkUps γsw q Ut ={↑histN,E}=∗ P).
Proof.
iIntros "HP Hacc". iMod (weak_arc_acc_open with "HP Hacc")
as (Vs q Ut) "[(Wt & tok & WR & WU) Hclose2]".
iExists Vs, q, Ut. iFrame "Wt tok WR WU".
iIntros "!> Wt tok WR WU".
iApply ("Hclose2" $! ∅ with "Wt tok WR [WU] [//]"); by rewrite right_id_L.
Qed.
Lemma weak_arc_acc_open_2 l t γi γs γw γsw P E :
P -∗ weak_arc_acc l t γi γs γw γsw P E ={E,↑histN}=∗
∃ Vb q Ut, (WeakTok γsw q ∗
(⊒Vb → view_tok γi (q / 2)) ∗
(∃ v, ⊒Vb →
GPS_SWSharedReader (l >> 1) t ((): unitProtocol) v q γw) ∗
WkUps γsw q Ut) ∗
(∀ Ut', WeakTok γsw q -∗
(⊒Vb → view_tok γi (q / 2)) -∗
(∃ v, GPS_SWSharedReader (l >> 1) t ((): unitProtocol) v q γw) -∗
WkUps γsw q (Ut ∪ Ut') -∗
(if decide (Ut' = ∅) then True else SeenActs γs l Ut') ={↑histN,E}=∗ P).
Proof.
iIntros "HP Hacc". iMod (weak_arc_acc_open with "HP Hacc")
as (Vs q Ut) "[(Wt & tok & WR & WU) Hclose2]".
iExists Vs, q, Ut. iFrame "Wt tok WR WU".
iIntros "!>" (Ut') "Wt tok WR WU SeenU".
iApply ("Hclose2" $! Ut' with "Wt tok [WR] WU SeenU").
iDestruct "WR" as (?) "WR". iExists _. by iIntros "_".
Qed.
Lemma weak_arc_acc_open_no_change_2 l t γi γs γw γsw P E :
P -∗ weak_arc_acc l t γi γs γw γsw P E ={E,↑histN}=∗
∃ Vb q Ut, (WeakTok γsw q ∗
(⊒Vb → view_tok γi (q / 2)) ∗
(∃ v, ⊒Vb →
GPS_SWSharedReader (l >> 1) t ((): unitProtocol) v q γw) ∗
WkUps γsw q Ut) ∗
(WeakTok γsw q -∗
(⊒Vb → view_tok γi (q / 2)) -∗
(∃ v, GPS_SWSharedReader (l >> 1) t ((): unitProtocol) v q γw) -∗
WkUps γsw q Ut ={↑histN,E}=∗ P).
Proof.
iIntros "HP Hacc". iMod (weak_arc_acc_open_no_change with "HP Hacc")
as (Vs q Ut) "[(Wt & tok & WR & WU) Hclose2]".
iExists Vs, q, Ut. iFrame "Wt tok WR WU".
iIntros "!> Wt tok WR WU". iApply ("Hclose2" with "Wt tok [WR] WU").
iDestruct "WR" as (?) "WR". iExists _. by iIntros "_".
Qed.
(* MAIN SPECS START HERE *)
Lemma create_strong {HPn: HP1} E l :
l ↦ #1 -∗ (l >> 1) ↦ #1 -∗ P1 1%Qp ={E}=∗
∃ γi γs γw γsw t q, is_arc P1 P2 N γi γs γw γsw l ∗ P1 q ∗
strong_arc t q l γi γs γw γsw .
Proof.
iIntros "Hl1 Hl2 HP1".
iDestruct (HPn (1/2)%Qp (1/2)%Qp with "[HP1]") as "[HP1 HP1']";
[by rewrite Qp.div_2|rewrite monPred_objectively_elim].
set stS : strong_stUR := Some ((1/2)%Qp, xH).
set stW : weak_stUR := (Some 1%Qp, None).
iAssert (|==> ∃ γ, StrongAuth γ stS ∗ StrongTok γ (1/2)%Qp ∗
WeakAuth γ stW ∗
StrongMoveAuth γ ∅ ∗ StrMoves γ 1%Qp ∅ ∗
StrongDownAuth γ ∅ ∗ StrDowns γ 1%Qp ∅ ∗
WeakUpAuth γ ∅ ∗ WkUps γ 1%Qp ∅ ∗
⎡ own γ ((ε, ◯ ((Some 1%Qp, ε)), ε) : arcUR) ⎤)%I as ">Own".
{ do 9 setoid_rewrite <- embed_sep. do 9 setoid_rewrite <- own_op.
iMod (own_alloc (A:=arcUR)) as (γ) "Own"; last by iExists γ.
split; split; simpl.
- rewrite !right_id. by apply auth_both_valid_discrete.
- rewrite !left_id. by apply auth_both_valid_discrete.
- split; split; simpl; rewrite !left_id !right_id.
+ by apply auth_both_valid_discrete.
+ by apply auth_auth_valid.
+ by apply auth_both_valid_discrete.
+ by apply auth_auth_valid.
- split; simpl; rewrite !left_id !right_id.
+ by apply auth_both_valid_discrete.
+ by apply auth_auth_valid. }
iDestruct "Own" as (γsw)
"(SA & St & WA & SMA & [SM1 SM2] & SDA & [SD1 SD2] & WUA & WU & [S1 S2])".
iMod (view_inv_alloc_frac N _ (1/2 + 1/2/2) (1/2/2)) as (γi) "[[tok2 tok1] VI]";
first by rewrite -Qp.add_assoc 2!Qp.div_2.
set Qs : time → time → gname → gname → vProp :=
(λ ts tw γs γw,
GPS_SWSharedReader l ts (() : unitProtocol) #1 (1/2)%Qp γs ∗
GPS_Reader l ts (() : unitProtocol) #1 γs ∗
GPS_Reader (l >> 1) tw (() : unitProtocol) #1 γw)%I.
iMod (GPS_SWRaw_Init_vs_strong_2 _ _
(λ γw, strong_interp P1 (l >> 1) γi γw γsw)
(λ γs, weak_interp P2 l γi γs γsw) true _ _ () () Qs
with "Hl1 Hl2 [-HP1' S2 SM2 SD2 SMA SDA WUA St VI]")
as (γs γw ts tw) "INV".
{ iIntros (ts tw γs γw) "Ws Ww". simpl.
iDestruct (GPS_SWWriter_shared with "Ws") as "[Ws [Rs1 Rs2]]".
iDestruct (GPS_SWWriter_shared with "Ww") as "[Ww [Rw1 Rw2]]".
iDestruct (GPS_SWSharedReader_Reader with "Rs1") as "#R1".
iDestruct (GPS_SWSharedReader_Reader with "Rw2") as "#R2".
iModIntro. iSplitL "SA SM1 SD1 Ws Rs1 HP1 S1 tok1"; last iSplitR "Rs2".
- iNext. iExists stS. iSplitL ""; [done|]. iFrame "SA Ws".
iExists (1/2)%Qp. rewrite Qp.div_2. iSplitL ""; [done|]. iFrame "HP1".
iExists ∅,∅. iFrame "tok1 Rs1 SM1 SD1 S1".
iSplitL ""; iExists _, _; [by iFrame "R1"|by iFrame "R2"].
- iNext. iExists stW. iSplitL ""; [done|]. iFrame "WA". iSplitL ""; [done|].
iFrame "Ww". iExists ∅. iFrame "tok2 Rw2 WU Rw1". iExists _,_. by iFrame "R1".
- by iFrame "Rs2 R1 R2". }
iExists γi, γs, γw, γsw, _, (1/2)%Qp. iFrame "HP1'".
iDestruct ("INV" $! strong_noCAS (weak_noCAS γsw)) as "(Invs & Invw & Rs & #R1 & #R2)".
iMod ("VI" $! (arc_inv P1 P2 γi γs γw γsw l) with "[SMA SDA WUA $Invs $Invw]")
as "[$ tok]".
{ iNext. iSplitL "SMA"; [by iExists _|]. iSplitL "SDA"; by iExists _. }
iModIntro. iExists _,∅,∅. iFrame "St tok Rs SM2 SD2 S2".
iSplitL""; iExists _,_; [by iFrame "R1"|by iFrame "R2"].
Qed.
Lemma create_weak E l :
l ↦ #0 -∗ (l >> 1) ↦ #1 -∗ P2 ={E}=∗ ∃ γi γs γw γsw t q,
is_arc P1 P2 N γi γs γw γsw l ∗ weak_arc t q l γi γs γw γsw.
Proof.
iIntros "Hl1 Hl2 HP2".
set stS : strong_stUR := None.
set stW : weak_stUR := (None, Some (Cinl ((1/2)%Qp, xH))).
iAssert (|==> ∃ γ, StrongAuth γ stS ∗ WeakAuth γ stW ∗ WeakTok γ (1/2)%Qp ∗
StrongMoveAuth γ ∅ ∗ StrMoves γ 1%Qp ∅ ∗
StrongDownAuth γ ∅ ∗ StrDowns γ 1%Qp ∅ ∗
WeakUpAuth γ ∅ ∗ WkUps γ 1%Qp ∅)%I as ">Own".
{ do 8 setoid_rewrite <- embed_sep. do 8 setoid_rewrite <- own_op.
iMod (own_alloc (A:=arcUR)) as (γ) "?"; last by iExists γ.
split; split; simpl.
- rewrite !right_id. by apply auth_auth_valid.
- rewrite !left_id !right_id. by apply auth_both_valid_discrete.
- split; split; simpl; rewrite !left_id !right_id.
+ by apply auth_both_valid_discrete.
+ by apply auth_auth_valid.
+ by apply auth_both_valid_discrete.
+ by apply auth_auth_valid.
- split; simpl; rewrite !left_id ?right_id.
+ by apply auth_both_valid_discrete.
+ by apply auth_auth_valid. }
iDestruct "Own" as (γsw) "(SA & WA & Wt & SMA & SM & SDA & SD & WUA & [WU1 WU2])".
iMod (view_inv_alloc_frac N _ (1/2 + 1/2/2) (1/2/2)) as (γi) "[[tok2 tok1] VI]";
first by rewrite -Qp.add_assoc 2!Qp.div_2.
set Qs : time → time → gname → gname → vProp :=
(λ ts tw γs γw,
GPS_SWSharedReader (l >> 1) tw (() : unitProtocol) #1 (1 / 2)%Qp γw ∗
GPS_Reader l ts (() : unitProtocol) #0 γs ∗
GPS_Reader (l >> 1) tw (() : unitProtocol) #1 γw)%I.
iMod (GPS_SWRaw_Init_vs_strong_2 _ _
(λ γw, strong_interp P1 (l >> 1) γi γw γsw)
(λ γs, weak_interp P2 l γi γs γsw) true _ _ () () Qs
with "Hl1 Hl2 [- SMA SDA WUA WU2 Wt VI]") as (γs γw ts tw) "INV".
{ iIntros (ts tw γs γw) "Ws Ww". simpl.
iDestruct (GPS_SWWriter_shared with "Ww") as "[Ww [Rw1 Rw2]]".
iDestruct (GPS_SWWriter_Reader with "Ws") as "#R1".
iDestruct (GPS_SWSharedReader_Reader with "Rw2") as "#R2".
iModIntro. iSplitL "SA"; last iSplitR "Rw2".
- iNext. iExists stS. iSplitL ""; [done|]. by iFrame "SA".
- iNext. iExists stW. iSplitL ""; [done|].
iFrame "WA HP2 Ww". iSplitL "tok1 Rw1 WU1".
+ iExists (1/2)%Qp,∅. rewrite Qp.div_2. iSplitL ""; [done|].
iFrame "tok1 Rw1 WU1". iExists _, _; by iFrame "R1".
+ iFrame "tok2". iSplitL "Ws"; [by iExists _|]. iSplitL "SM"; iExists _; by iFrame.
- by iFrame "Rw2 R1 R2". }
iExists γi, γs, γw, γsw, _, (1/2)%Qp.
iDestruct ("INV" $! strong_noCAS (weak_noCAS γsw)) as "(Invs & Invw & Rw & #R1 & #R2)".
iMod ("VI" $! (arc_inv P1 P2 γi γs γw γsw l) with "[SMA SDA WUA $Invs $Invw]")
as "[$ tok]".
{ iNext. iSplitL "SMA"; [by iExists _|]. iSplitL "SDA"; by iExists _. }
iModIntro. iExists _,∅. iFrame "Wt tok Rw WU2". iExists _,_; by iFrame "R1".
Qed.
Lemma strong_count_spec γi γs γw γsw l t v (P : vProp) tid :
is_arc P1 P2 N γi γs γw γsw l -∗ strong_arc_acc l t γi γs γw γsw P (⊤ ∖ ↑N) -∗
{{{ GPS_Reader l t () v γs ∗ P }}}
strong_count [ #l] @ tid; ⊤
{{{ (c : Z), RET #c; P ∗ ⌜0 < c⌝ }}}.
Proof.
iIntros "#INV #Hacc !# * [#RR HP] HΦ". iLöb as "IH". wp_rec.
iMod (view_inv_acc_base_1 γi N with "INV") as "VI"; [done|].
iMod (strong_arc_acc_open_no_change_2 with "HP Hacc")
as (Vs q Mt Dt) "[(St & tok & SR & SM & SD & oW) Hclose2]".
iSpecialize ("VI" $! Vs with "tok").
iMod (fupd_mask_mono with "VI") as "[tok VI]"; [set_solver|].
iDestruct "VI" as (Vb) "[VI Hclose1]".
iMod (arc_inv_split with "VI") as "((SMA & MDU) & IS & IW)".
iDestruct "SMA" as (Mt') "SMA".
iDestruct (StrongMoveAuth_agree with "[$SMA $SM]") as %?. subst Mt'.
iDestruct "SR" as (v0) "SR".
iDestruct (GPS_SWSharedReader_Reader_join_subjectively _ _ _ _ _ _ q
with "RR [SR]") as "[SR1 SR2]".
{ rewrite monPred_subjectively_unfold.
iDestruct (monPred_in_view_elim with "SR") as (V') "SR". by iExists V'. }
set R : time → unitProtocol → val → vProp :=
(λ t' _ v', (∃ z : Z, ⌜v' = #z ∧ 0 < z⌝) ∗
StrongTok γsw q ∗ StrongMoveAuth γsw Mt )%I.
iApply (GPS_SWRaw_SharedReader_Read (strong_interp P1 (l >> 1) γi γw γsw) _
strong_noCAS R with "[$SR1 $IS St SMA]"); [done|by right|..].
{ iIntros "!>" (t' [] v' [_ Ext']). rewrite /StrongTok. iIntros "!>". iSplit.
- iIntros "#SI !>". iFrame "SI SMA". rewrite /StrongTok. iFrame "St".
iDestruct "SI" as (st) "[% Eq]". subst v'. iExists (Z_of_arcstrong_st st).
destruct st as [[]|]; simpl; [|done]. iPureIntro. split; [done|split].
- iDestruct 1 as (st) "(Eq1 & SA & SI)". iDestruct "Eq1" as %?. subst v'.
destruct st as [[q' n]|]; simpl.
+ iModIntro. iFrame "SMA". rewrite /StrongTok. iFrame "St".
iSplitR""; [iExists (Some (q',n))|iExists (Z.pos n)].
* iSplitL""; [done|]. by iFrame.
* iPureIntro. split; [done|split].
+ iDestruct (StrongTok_Auth_agree with "[$SA St]") as %(?&?&EQ&?); [|done].
rewrite /StrongTok. iFrame "St". }
iIntros "!>" (t' [] v'). iDestruct 1 as "(_ & RR' & IS & Rs & St & SMA)".
iDestruct "Rs" as (z) "[% %]". subst v'.
iDestruct (GPS_SWSharedReaders_join with "SR2 RR'") as "SR". rewrite Qp.div_2.
iSpecialize ("Hclose1" with "tok [SMA MDU IW IS]").
{ iDestruct "MDU" as "[SDA WUA]".
iApply (arc_inv_join with "[SMA] SDA WUA IS IW"). by iExists _. }
iMod (fupd_mask_mono with "Hclose1") as "[tok Hclose1]"; [set_solver|].
iMod ("Hclose2" with "St tok [SR] SM SD oW") as "HP"; [by iExists _|].
iModIntro. iMod "Hclose1". iModIntro. iApply ("HΦ"). by iFrame "HP".
Qed.
(* FIXME : in the case the weak count is locked, we can possibly
return -1. This problem already exists in Rust. *)
Lemma weak_count_spec γi γs γw γsw l t v ts (P : vProp) tid :
is_arc P1 P2 N γi γs γw γsw l -∗
strong_arc_acc l ts γi γs γw γsw P (⊤ ∖ ↑N) -∗
GPS_Reader (l >> 1) t () v γw -∗
{{{ P }}}
weak_count [ #l] @ tid; ⊤
{{{ (c : Z), RET #c; P ∗ ⌜-1 ≤ c⌝ }}}.
Proof.
iIntros "#INV #Hacc #RR !# * HP HΦ". iLöb as "IH". wp_rec. wp_op.
iMod (view_inv_acc_base_1 γi N with "INV") as "VI"; [done|].
iMod (strong_arc_acc_open_no_change with "HP Hacc")
as (Vs q Mt Dt) "[(St & tok & SR & SM & SD & oW) Hclose2]".
iSpecialize ("VI" $! Vs with "tok").
iMod (fupd_mask_mono with "VI") as "[tok VI]"; [set_solver|].
iDestruct "VI" as (Vb) "[VI Hclose1]".
iMod (arc_inv_split with "VI") as "(MA & IS & IW)".
set R : time → unitProtocol → val → vProp :=
(λ _ _ v', ∃ z : Z, ⌜v' = #z ∧ -1 ≤ z⌝)%I.
iApply (GPS_SWRaw_Read (weak_interp P2 l γi γs γsw) _
(weak_noCAS γsw) R with "[$RR $IW]"); [done|by right|..].
{ iIntros "!>" (t' [] v' [_ Ext']) "!>". iSplit; last iSplit.
- iIntros "#WI !>". iFrame "WI". iDestruct "WI" as (st) "[Eq1 Eq2]".
iDestruct "Eq1" as %?. subst v'. iExists (Z_of_arcweak_st st).
iPureIntro. split; [done|]. by apply weak_stUR_value.
- iDestruct 1 as (st) "[Eq1 WI]". iDestruct "Eq1" as %?. subst v'.
iModIntro. iSplitR "". { iExists _. by iFrame "WI". }
iExists _. iPureIntro. split; [done|]. by apply weak_stUR_value.
- iDestruct 1 as "[% $]". subst v'. iIntros "!> !%". split; [done|].
by eexists. }
iIntros "!>" (t' [] v'). iDestruct 1 as "(_ & _ & IW & Rs)".
iDestruct "Rs" as (z) "Eq". iDestruct "Eq" as %[? Le]. subst v'.
iSpecialize ("Hclose1" with "tok [MA IW IS]").
{ iDestruct "MA" as "[SMA [SDA WUA]]".
by iApply (arc_inv_join with "SMA SDA WUA IS IW"). }
iMod (fupd_mask_mono with "Hclose1") as "[tok Hclose1]"; [set_solver|].
iMod ("Hclose2" with "St tok SR SM SD oW") as "HP".
iModIntro. iMod "Hclose1". iModIntro. iApply ("HΦ"). by iFrame "HP".
Qed.
Lemma arc_strong_interp_comparable l γi γs γw γsw t s v (n: Z):
strong_interp P1 (l >> 1) γi γw γsw true l γs t s v
∨ strong_interp P1 (l >> 1) γi γw γsw false l γs t s v
-∗ ⌜∃ vl : lit, v = #vl ∧ lit_comparable n vl⌝.
Proof.
iIntros "[F|F]"; iDestruct "F" as (st ?) "_"; subst v;
iExists _; iPureIntro; (split; [done|]); by constructor.
Qed.
Lemma arc_strong_interp_comparable_2 l γi γs γw γsw t s v (n: Z):
strong_interp P1 (l >> 1) γi γw γsw true l γs t s v
∨ strong_interp P1 (l >> 1) γi γw γsw false l γs t s v
∨ strong_noCAS l γs t s v
-∗ ⌜∃ vl : lit, v = #vl ∧ lit_comparable n vl⌝.
Proof.
iIntros "[F|[F|%]]";
last (subst; iPureIntro; eexists; split; [done|by constructor]);
iDestruct "F" as (st ?) "_"; subst;
iExists _; iPureIntro; (split; [done|]); by constructor.
Qed.
Lemma clone_arc_spec {HPn: HP1} γi γs γw γsw l t v (P : vProp) tid :
is_arc P1 P2 N γi γs γw γsw l -∗ strong_arc_acc l t γi γs γw γsw P (⊤ ∖ ↑N) -∗
(∃ t' v', GPS_Reader (l >> 1) t' () v' γw) -∗
GPS_Reader l t () v γs -∗
{{{ P }}} clone_arc [ #l] @ tid; ⊤
{{{ t' q', RET #☠; P ∗ strong_arc t' q' l γi γs γw γsw ∗ P1 q' }}}.
Proof.
iIntros "#INV #Hacc #WR #RR !# * HP HΦ". iLöb as "IH". wp_rec.
(* read the value *)
wp_bind (!ʳˡˣ_)%E.
(* opening invariant *)
iMod (view_inv_acc_base_1 γi N with "INV") as "VI"; [done|].
iMod (strong_arc_acc_open_no_change_2 with "HP Hacc")
as (Vs q Mt Dt) "[(St & tok & SR & SM & SD & oW) Hclose2]".
iSpecialize ("VI" $! Vs with "tok").
iMod (fupd_mask_mono with "VI") as "[tok VI]"; [set_solver|].
iDestruct "VI" as (Vb) "[VI Hclose1]".
iMod (arc_inv_split with "VI") as "(MA & IS & IW)".
iDestruct "SR" as (v0) "SR".
iDestruct (GPS_SWSharedReader_Reader_join_subjectively _ _ _ _ _ _ q
with "RR [SR]") as "[SR1 SR2]".
{ rewrite monPred_subjectively_unfold.
iDestruct (monPred_in_view_elim with "SR") as (V') "SR". by iExists V'. }
set R : time → unitProtocol → val → vProp := (λ t' _ v', ∃ z : Z, ⌜v' = #z⌝)%I.
(* actual read *)
iApply (GPS_SWRaw_SharedReader_Read (strong_interp P1 (l >> 1) γi γw γsw) _
strong_noCAS R with "[$SR1 $IS]"); [done|by left|..].
{ iIntros "!>" (t' [] v' [_ Ext']) "!>". iSplit.
- iIntros "#SI !>". iFrame "SI". iDestruct "SI" as (st) "[Eq1 _]".
iDestruct "Eq1" as %?. subst v'. by iExists _.
- iDestruct 1 as (st) "[Eq1 SI]". iDestruct "Eq1" as %?. subst v'.
iModIntro. iSplitR ""; [iExists _; by iFrame "SI"|by iExists _]. }
iIntros "!>" (t' [] v'). iDestruct 1 as "(_ & SR1 & IS & Rs)".
iDestruct "Rs" as (z) "%". subst v'.
iDestruct (GPS_SWSharedReaders_join with "SR2 SR1") as "SR". rewrite Qp.div_2.
(* closing invariant *)
iSpecialize ("Hclose1" with "tok [MA IW IS]").
{ iDestruct "MA" as "[SMA [SDA WUA]]".
by iApply (arc_inv_join with "SMA SDA WUA IS IW"). }
iMod (fupd_mask_mono with "Hclose1") as "[tok Hclose1]"; [set_solver|].
iMod ("Hclose2" with "St tok [SR] SM SD oW") as "HP"; [by iExists _|].
iModIntro. iMod "Hclose1" as "_".
iModIntro. wp_let. wp_op. clear Vs R v0 Mt Dt t' Vb q.
(* now we can CAS *)
wp_bind (CAS _ _ _ _ _ _).
(* not really, just preparing *)
(* opening invariant *)
iMod (view_inv_acc_base_1 γi N with "INV") as "VI"; [done|].
iMod (strong_arc_acc_open_2 with "HP Hacc")
as (Vs q Mt Dt) "[(St & tok & SR & SM & SD & oW) Hclose2]".
iSpecialize ("VI" $! Vs with "tok").
iMod (fupd_mask_mono with "VI") as "[tok VI]"; [set_solver|].
iDestruct "VI" as (Vb) "[VI Hclose1]".
iMod (arc_inv_split with "VI") as "((SMA & MDU) & IS & IW)".
iDestruct "SR" as (v0) "SR".
iDestruct (GPS_SWSharedReader_Reader_join_subjectively _ _ _ _ _ _ q
with "RR [SR]") as "[SR1 SR2]".
{ rewrite monPred_subjectively_unfold.
iDestruct (monPred_in_view_elim with "SR") as (V') "SR". by iExists V'. }
iDestruct "SMA" as (Mt') "SMA".
iDestruct (StrongMoveAuth_agree with "[$SMA $SM]") as %?. subst Mt'.
set Q: time → unitProtocol → vProp :=
(λ t'' s'',
let Mt' := if decide (Z.to_pos z = 1%positive) then (Mt ∪ {[t'']}) else Mt in
StrongTok γsw q ∗ StrongMoveAuth γsw Mt' ∗ StrMoves γsw q Mt' ∗
∃ q'', StrongTok γsw q'' ∗ (<obj> P1 q'') ∗
(<obj> view_tok γi (q''/2)) ∗
GPS_SWSharedReader l t'' ((): unitProtocol) #(z+1) q'' γs ∗
StrMoves γsw q'' Mt' ∗ ⌜∀ t' : time, t' ∈ Mt' → t' ⊑ t''⌝ ∗
StrDowns γsw q'' ∅ ∗ agown γsw (awk_n ((Some q'', ε))))%I.
set Q1: time → unitProtocol → vProp := (λ _ _, True)%I.
set Q2: time → unitProtocol → vProp :=
(λ tr sr, strong_interp P1 (l >> 1) γi γw γsw true l γs tr sr #z).
set R: time → unitProtocol → lit → vProp :=
(λ _ _ _, StrongTok γsw q ∗ StrongMoveAuth γsw Mt ∗ StrMoves γsw q Mt)%I.
iMod (rel_True_intro tid) as "#rTrue".
(* here comes the CAS *)
iApply (GPS_SWRaw_SharedWriter_CAS_int_strong (strong_interp P1 (l >> 1) γi γw γsw)
_ strong_noCAS false Relaxed Relaxed Relaxed z #(z+1) _ _ _ (q/2) _
True%I Q Q1 Q2 R _ Vb _ (↑histN) with "[$SR1 $IS St SMA SM $rTrue]");
[done|done|by left|by left|by left|..].
{ iSplitL "". { iIntros "!>!>" (??? _). by iApply arc_strong_interp_comparable. }
rewrite /= -(bi.True_sep' (∀ _ (_ : unitProtocol), _)%I).
iApply (rel_sep_objectively with "[$rTrue St SMA SM]").
rewrite /StrongTok /StrMoves. iIntros "!>" (t' [] [_ Ext']). iSplit; last first.
{ iIntros (??) "!> !>". iSplit; iIntros "$ !>"; iFrame "SMA";
rewrite /StrongTok /StrMoves; by iFrame. }
iSplitL "". { by iIntros "!> $". } iIntros "_".
iDestruct 1 as (st) "[>Eq [>SA SI]]".
iDestruct (StrongTok_Auth_agree with "[$SA St]") as %(q'&n&EQ&Eq');
[by rewrite /StrongTok|]. subst st. iDestruct "Eq" as %Eq.
inversion Eq. subst z. clear Eq. iDestruct "SI" as "[$ SI]".
iDestruct "SI" as (q'') "(>Eq'' & HP & SI)". iDestruct "Eq''" as %Eq''.
iAssert (▷ (P1 (q''/2) ∗ <obj> P1 (q''/2)))%I with "[HP]" as "[HP1 HP2]".
{ iNext. iApply HPn. by rewrite Qp.div_2. }
iDestruct "SI" as (Mt' Dt') "(>tok & R & SM' & RestS)". rewrite /StrMoves.
iDestruct "SM'" as ">SM'".
iDestruct (StrongMoveAuth_agree with "[$SMA SM']") as %?;
[by rewrite /StrMoves|]. subst Mt'.
iDestruct (view_tok_split_unit γi (q''/2/2) (q''/2/2) with "[tok]")
as "[tok1 tok2]"; first by rewrite Qp.div_2.
iModIntro. iExists (). iSplitL""; [done|]. iIntros (t'' Lt'') "W' _".
set Mt' : gset time := if decide (n = 1%positive) then (Mt ∪ {[t'']}) else Mt.
iAssert (|==> StrongMoveAuth γsw Mt' ∗ StrMoves γsw q Mt' ∗
StrMoves γsw q'' Mt' ∗
<obj> if decide (n = 1%positive)
then StrMoveCert γsw t'' else True)%I
with "[SMA SM SM']" as ">(SMA & SM & [SM1 SM2] & Cert)".
{ case decide => Eq; last first.
{ rewrite /Mt' decide_False; [|done]. rewrite /StrMoves.
iFrame; by iIntros "!> !>". }
rewrite /Mt' (decide_True _ _ Eq). rewrite (decide_True _ _ Eq) in Eq'.
iDestruct (StrMoves_fractional _ _ q with "[SM SM']") as "SM";
[rewrite /StrMoves; by iFrame|]. subst q'. rewrite Eq''.
iMod (StrMoves_update _ _ {[t'']} with "[$SMA $SM]") as "($ & SM & Cert)".
iIntros. rewrite bi.sep_assoc -StrMoves_fractional Eq''. iFrame "SM".
iIntros "!> !>". by iFrame "Cert". }
iModIntro. iSplitL "Cert".
{ iIntros "!> _ !> !>". iExists (Some (q', n)). iSplit; [done|].
destruct n; simpl; [done..|]. iExists t''. iFrame (Lt'').
iDestruct (monPred_objectively_elim with "Cert") as "$". }
iIntros "!> !>".
iDestruct (GPS_SWSharedWriter_Reader_update with "W' R") as "[W' [R1 R2]]".
iDestruct "RestS" as "(#SeenM & SD & #SeenD & [OS1 OS2])".
iAssert (⌜∀ t' : time, t' ∈ Mt' → t' ⊑ t''⌝)%I with "[W']" as "#MAX".
{ iDestruct "SeenM" as (t4 v4) "[R4 MAX]". iDestruct "MAX" as %MAX'.
iDestruct (GPS_SWSharedWriter_latest_time_1 with "W' R4") as %Le4.
iPureIntro. have MAX2: ∀ t2 : time, t2 ∈ Mt → t2 ⊑ t'' by move =>? /MAX'->.
rewrite /Mt'. case decide => ?; [|done].
move => t5 /elem_of_union [/MAX2//|/elem_of_singleton -> //]. }
iDestruct (GPS_SWSharedReader_Reader with "R1") as "#R''".
iAssert (SeenActs γs l Mt')%I as "SeenM'". { iExists _,_. iFrame "R'' MAX". }
iMod (StrongAuth_new_tok _ q' q'' _ Eq'' with "SA") as "[SA St']".
iAssert (StrDowns γsw (q'' / 2) Dt' ∗ StrDowns γsw (q'' / 2) ∅)%I
with "[SD]" as "[SD1 SD2]".
{ rewrite -(right_id_L ∅ union Dt') -{1}(Qp.div_2 q'').
by iApply (StrDowns_forget with "SD"). }
iModIntro. iSplitR "SA W' tok1 R1 SM1 OS1 SD1 HP1 SeenM'".
- rewrite /Q Pos2Z.id. iFrame "SMA SM". iSplitL "St"; [by rewrite /StrongTok|].
iExists _. by iFrame "St' tok2 R2 SM2 HP2 MAX SD2 OS2".
- iExists (Some ((q' + q'' / 2)%Qp, (n + 1)%positive)). iSplitL ""; [done|].
iFrame "SA W'". iExists (q''/2)%Qp.
iSplitL "". { iPureIntro. by rewrite -Qp.add_assoc Qp.div_2. }
iFrame "HP1". iExists Mt', Dt'. iFrame "tok1 R1 SM1 SeenM' SD1 SeenD OS1". }
(* finally done with the CAS *)
iIntros "!>" (b t' [] v') "(IS &%& [([%[%%]]&R'&Q) | ([%[_ %]] &R'&Rs&_)])".
- subst b v'.
iDestruct "Q" as "(St & SMA & SM & Q)". iDestruct "Q" as (q'') "Q".
iDestruct "Q" as "(St'' & HP1 & tok'' & R'' & SM'' & %MAX'' & oW'' & SD'')".
iDestruct (GPS_SWSharedReaders_join_subjectively _ _ _ _ _ _ _ _ q''
with "R' [R'']") as "[R' R'']". { by iApply acq_subjective. }
iDestruct (GPS_SWSharedReader_Reader with "R'") as "#RR'".
set Mt' := if decide (Z.to_pos z = 1%positive) then Mt ∪ {[t']} else Mt.
iAssert (SeenActs γs l Mt')%I with "[]" as "#SeenM'".
{ iExists _, _. iFrame (MAX'') "RR'". }
iDestruct (GPS_SWSharedReaders_join with "SR2 R'") as "SR".
(* tedious removal of acq_mod *)
rewrite Qp.div_2 2!acq_objectively_elim 2!monPred_objectively_elim.
rewrite 7!acq_embed_elim.
(* closing *)
iSpecialize ("Hclose1" with "tok [SMA MDU IW IS]").
{ iClear "#". iDestruct "MDU" as "[SDA WUA]".
iApply (arc_inv_join with "[SMA] SDA WUA IS IW"). by iExists _. }
iMod (fupd_mask_mono with "Hclose1") as "[tok Hclose1]"; [set_solver|].
iMod ("Hclose2" $! (if decide (Z.to_pos z = 1%positive) then {[t']} else ∅) ∅
with "[St] tok [SR] [SM] [] [SD] [] oW") as "HP";
[by iExact "St"|by iExists _|..|by rewrite right_id_L|done|].
{ rewrite /Mt'. case decide => _; [done|by rewrite right_id_L]. }
{ destruct (decide (Z.to_pos _ = _)); [|done].
iExists _,_. iFrame "RR'". iPureIntro. move => ? /elem_of_singleton -> //. }
iModIntro. iMod "Hclose1" as "_". iModIntro.
wp_case. iApply ("HΦ" with "[- $HP]"). iFrame "HP1".
iExists _,_,∅. rewrite /StrongTok. iFrame "St'' tok'' R''".
rewrite /StrMoves /StrDowns. iFrame "SM'' SeenM' oW'' SD''".
iDestruct "WR" as (??) "WR". iExists _, _. by iFrame "WR".
- subst b.
iDestruct "Rs" as "(St & SMA & SM)". rewrite 3!acq_embed_elim.
iDestruct (GPS_SWSharedReaders_join with "SR2 R'") as "SR". rewrite Qp.div_2.