Library zoo_saturn.queue_spmc
Require Import iris.base_logic.lib.ghost_map.
Require Import zoo.prelude.
Require Import zoo.common.relations.
Require Import zoo.common.countable.
Require Import zoo.iris.bi.big_op.
Require Import zoo.iris.base_logic.lib.mono_list.
Require Import zoo.iris.base_logic.lib.auth_nat_max.
Require Import zoo.iris.base_logic.lib.auth_twins.
Require Import zoo.iris.base_logic.lib.saved_pred.
Require Import zoo.base.
Require Import zoo_std.option.
Require Import zoo_std.xtchain.
Require Export zoo_saturn.queue_spmc__code.
Require Import zoo_saturn.queue_spmc__types.
Require Import zoo.options.
Implicit Type b : bool.
Implicit Type front node back new_back : location.
Implicit Type hist past nodes : list location.
Implicit Type v : val.
Implicit Type vs ws : list val.
Implicit Type waiter : gname.
Implicit Type waiters : gmap gname nat.
Class QueueSpmcG Σ `{zoo۰G : !ZooG Σ} :=
{ #[local] queue_spmc۰G۰history۰G :: MonoListG Σ location
; #[local] queue_spmc۰G۰front۰G :: AuthNatMaxG Σ
; #[local] queue_spmc۰G۰model۰G :: AuthTwinsG Σ (leibnizO (list val)) suffix
; #[local] queue_spmc۰G۰waiters۰G :: ghost_mapG Σ gname nat
; #[local] queue_spmc۰G۰saved_pred۰G :: SavedPredG Σ bool
}.
Definition queue_spmc۰Σ :=
#[mono_list۰Σ location
; auth_nat_max۰Σ
; auth_twins۰Σ (leibnizO (list val)) suffix
; ghost_mapΣ gname nat
; saved_pred۰Σ bool
].
#[global] Instance subGｰqueue_spmc۰Σ Σ `{zoo۰G : !ZooG Σ} :
subG queue_spmc۰Σ Σ →
QueueSpmcG Σ.
Module base.
Section queue_spmc۰G.
Context `{queue_spmc۰G : QueueSpmcG Σ}.
Implicit Type t : location.
Record metadata :=
{ metadata۰inv : namespace
; metadata۰history : gname
; metadata۰front : gname
; metadata۰model : auth_twins۰name
; metadata۰waiters : gname
}.
Implicit Type γ : metadata.
#[global] Instance metadataｰeq_dec : EqDecision metadata :=
ltac:(solve_decision).
#[global] Instance metadataｰcountable :
Countable metadata.
#[local] Definition history۰auth' γ_history hist :=
mono_list۰auth γ_history (DfracOwn (1/2)) hist.
#[local] Definition history۰auth γ hist :=
history۰auth' γ.(metadata۰history) hist.
#[local] Definition history۰last' γ_history node : iProp Σ :=
∃ hist,
mono_list۰auth γ_history (DfracOwn (1/2)) hist ∗
⌜last hist = Some node⌝.
#[local] Instance : CustomIpat "history۰last" :=
" ( %hist{} & Hauth{_{}} & %Hlast ) ".
#[local] Definition history۰last γ :=
history۰last' γ.(metadata۰history).
#[local] Definition history۰at γ i node :=
mono_list۰at γ.(metadata۰history) i node.
#[local] Definition front۰auth' γ_front i :=
auth_nat_max۰auth γ_front (DfracOwn 1) i.
#[local] Definition front۰auth γ i :=
front۰auth' γ.(metadata۰front) i.
#[local] Definition front۰lb γ i :=
auth_nat_max۰lb γ.(metadata۰front) i.
#[local] Definition producer' γ_model ws :=
auth_twins۰auth _ γ_model ws.
#[local] Definition producer γ :=
producer' γ.(metadata۰model).
#[local] Definition model₁' γ_model vs :=
auth_twins۰twin₁ _ γ_model vs.
#[local] Definition model₁ γ :=
model₁' γ.(metadata۰model).
#[local] Definition model₂' γ_model vs :=
auth_twins۰twin₂ _ γ_model vs.
#[local] Definition model₂ γ :=
model₂' γ.(metadata۰model).
#[local] Definition waiters۰auth' γ_waiters waiters :=
ghost_map_auth γ_waiters 1 waiters.
#[local] Definition waiters۰auth γ waiters :=
waiters۰auth' γ.(metadata۰waiters) waiters.
#[local] Definition waiters۰at γ waiter i :=
ghost_map_elem γ.(metadata۰waiters) waiter (DfracOwn 1) i.
#[local] Definition node۰model γ node i b : iProp Σ :=
node ↦ₕ Header §Node 2 ∗
history۰at γ i node ∗
if b then front۰lb γ i else True%I.
#[local] Instance : CustomIpat "node۰model" :=
" ( #H{}_header & #Hhistory_at_{} & {{front}#Hfront_lb_{};_} ) ".
#[local] Definition waiter۰au γ (Ψ : bool → iProp Σ) : iProp Σ :=
AU <{
∃∃ vs,
model₁ γ vs
}> @ ⊤ ∖ ↑γ.(metadata۰inv), ∅ <{
model₁ γ vs
, COMM
Ψ (bool_decide (vs = []))
}>.
#[local] Definition waiter۰model γ past waiter i : iProp Σ :=
∃ Ψ,
saved_pred waiter Ψ ∗
if decide (i < length past) then
Ψ false
else
waiter۰au γ Ψ.
#[local] Definition inv۰inner t γ : iProp Σ :=
∃ hist past front nodes vs waiters,
⌜hist = past ++ front :: nodes⌝ ∗
t.[front] ↦ #front ∗
xtchain (Header §Node 2) (DfracOwn 1) hist §Null ∗
([∗ list] node; v ∈ nodes; vs, node.[data] ↦ v) ∗
history۰auth γ hist ∗
front۰auth γ (length past) ∗
model₂ γ vs ∗
waiters۰auth γ waiters ∗
([∗ map] waiter ↦ i ∈ waiters, waiter۰model γ past waiter i).
#[local] Instance : CustomIpat "inv۰inner" :=
" ( %hist{} & %past{} & %front{} & %nodes{} & %vs{} & %waiters{} & >%Hhist{} & >Ht_front & >Hhist & >Hnodes & >Hhistory_auth & >Hfront_auth & >Hmodel₂ & >Hwaiters_auth & Hwaiters ) ".
#[local] Definition inv' t γ :=
inv γ.(metadata۰inv) (inv۰inner t γ).
Definition queue_spmc۰inv t γ ι : iProp Σ :=
⌜ι = γ.(metadata۰inv)⌝ ∗
inv' t γ.
#[local] Instance : CustomIpat "inv" :=
" ( -> & #Hinv ) ".
Definition queue_spmc۰producer t γ ws : iProp Σ :=
∃ back,
t.[back] ↦ #back ∗
back ↦ₕ Header §Node 2 ∗
history۰last γ back ∗
producer γ ws.
#[local] Instance : CustomIpat "producer" :=
" ( %back{} & Ht_back{_{}} & #Hback{}_header & Hhistory_last{_{}} & Hproducer{_{}} ) ".
Definition queue_spmc۰model :=
model₁.
#[local] Instance : CustomIpat "model" :=
" Hmodel₁{_{}} ".
#[global] Instance queue_spmc۰modelｰtimeless γ vs :
Timeless (queue_spmc۰model γ vs).
#[global] Instance queue_spmc۰producerｰtimeless t γ ws :
Timeless (queue_spmc۰producer t γ ws).
#[global] Instance queue_spmc۰invｰpersistent t γ ι :
Persistent (queue_spmc۰inv t γ ι).
#[local] Lemma historyｰalloc front :
⊢ |==>
∃ γ_history,
history۰auth' γ_history [front] ∗
history۰last' γ_history front.
#[local] Lemma history۰atｰget {γ hist} i node :
hist !! i = Some node →
history۰auth γ hist ⊢
history۰at γ i node.
#[local] Lemma history۰atｰlookup γ hist i node :
history۰auth γ hist -∗
history۰at γ i node -∗
⌜hist !! i = Some node⌝.
#[local] Lemma historyｰauthｰlast γ hist node :
history۰auth γ hist -∗
history۰last γ node -∗
⌜last hist = Some node⌝.
#[local] Lemma historyｰupdate {γ hist node} node' :
history۰auth γ hist -∗
history۰last γ node ==∗
history۰auth γ (hist ++ [node']) ∗
history۰last γ node'.
Opaque history۰last'.
#[local] Lemma frontｰalloc :
⊢ |==>
∃ γ_front,
front۰auth' γ_front 0.
#[local] Lemma front۰lbｰget γ i :
front۰auth γ i ⊢
front۰lb γ i.
#[local] Lemma front۰lbｰvalid γ i1 i2 :
front۰auth γ i1 -∗
front۰lb γ i2 -∗
⌜i2 ≤ i1⌝.
#[local] Lemma frontｰupdate {γ i} i' :
i ≤ i' →
front۰auth γ i ⊢ |==>
front۰auth γ i'.
#[local] Lemma producerｰvalid γ ws vs :
producer γ ws -∗
model₁ γ vs -∗
⌜vs `suffix_of` ws⌝.
#[local] Lemma producerｰexclusive γ ws1 ws2 :
producer γ ws1 -∗
producer γ ws2 -∗
False.
#[local] Lemma modelｰproducerｰalloc :
⊢ |==>
∃ γ_model,
producer' γ_model [] ∗
model₁' γ_model [] ∗
model₂' γ_model [].
#[local] Lemma model₁ｰexclusive γ vs1 vs2 :
model₁ γ vs1 -∗
model₁ γ vs2 -∗
False.
#[local] Lemma modelｰagree γ vs1 vs2 :
model₁ γ vs1 -∗
model₂ γ vs2 -∗
⌜vs1 = vs2⌝.
#[local] Lemma modelｰpush {γ ws vs1 vs2} v :
producer γ ws -∗
model₁ γ vs1 -∗
model₂ γ vs2 ==∗
producer γ (vs1 ++ [v]) ∗
model₁ γ (vs1 ++ [v]) ∗
model₂ γ (vs1 ++ [v]).
#[local] Lemma modelｰpop γ v vs1 vs2 :
model₁ γ (v :: vs1) -∗
model₂ γ vs2 ==∗
model₁ γ vs1 ∗
model₂ γ vs1.
#[local] Lemma waitersｰalloc :
⊢ |==>
∃ γ_waiters,
waiters۰auth' γ_waiters ∅.
#[local] Lemma waitersｰinsert {γ waiters} i Ψ :
waiters۰auth γ waiters ⊢ |==>
∃ waiter,
waiters۰auth γ (<[waiter := i]> waiters) ∗
saved_pred waiter Ψ ∗
waiters۰at γ waiter i.
#[local] Lemma waitersｰdelete γ waiters waiter i :
waiters۰auth γ waiters -∗
waiters۰at γ waiter i ==∗
⌜waiters !! waiter = Some i⌝ ∗
waiters۰auth γ (delete waiter waiters).
Lemma queue_spmc۰modelｰexclusive γ vs1 vs2 :
queue_spmc۰model γ vs1 -∗
queue_spmc۰model γ vs2 -∗
False.
Lemma queue_spmc۰producerｰvalid t γ vs ws :
queue_spmc۰producer t γ ws -∗
queue_spmc۰model γ vs -∗
⌜vs `suffix_of` ws⌝.
Lemma queue_spmc۰producerｰexclusive t γ ws1 ws2 :
queue_spmc۰producer t γ ws1 -∗
queue_spmc۰producer t γ ws2 -∗
False.
Lemma queue_spmc٠createｰspec ι :
{{{
True
}}}
queue_spmc٠create ()
{{{
t γ
, RET #t;
meta_token t ⊤ ∗
queue_spmc۰inv t γ ι ∗
queue_spmc۰model γ [] ∗
queue_spmc۰producer t γ []
}}}.
#[local] Lemma frontｰspecｰstrong Ψ t γ :
{{{
inv' t γ ∗
if Ψ is Some Ψ then
waiter۰au γ Ψ
else
True
}}}
(#t).{front}
{{{
front i
, RET #front;
node۰model γ front i true ∗
if Ψ is Some Ψ then
∃ waiter,
saved_pred waiter Ψ ∗
waiters۰at γ waiter i
else
True
}}}.
#[local] Lemma frontｰspec t γ :
{{{
inv' t γ
}}}
(#t).{front}
{{{
front i
, RET #front;
node۰model γ front i true
}}}.
Variant operation :=
| IsEmpty waiter (Ψ : bool → iProp Σ)
| Pop (Ψ : option val → iProp Σ)
| Other.
Implicit Type op : operation.
Variant operation' :=
| IsEmpty'
| Pop'
| Other'.
#[local] Instance operation'ｰeq_dec : EqDecision operation' :=
ltac:(solve_decision).
#[local] Coercion operation۰to_operation' op :=
match op with
| IsEmpty _ _ ⇒
IsEmpty'
| Pop _ ⇒
Pop'
| Other ⇒
Other'
end.
#[local] Definition pop۰au γ (Ψ : option val → iProp Σ) : iProp Σ :=
AU <{
∃∃ vs,
model₁ γ vs
}> @ ⊤ ∖ ↑γ.(metadata۰inv), ∅ <{
model₁ γ (tail vs)
, COMM
Ψ (head vs)
}>.
#[local] Lemma nextｰspecｰaux op t γ i node :
{{{
inv' t γ ∗
history۰at γ i node ∗
( if decide (op = Other' :> operation') then True else
front۰lb γ i
) ∗
match op with
| IsEmpty waiter Ψ ⇒
saved_pred waiter Ψ ∗
waiters۰at γ waiter i ∗
£ 1
| Pop Ψ ⇒
pop۰au γ Ψ
| Other ⇒
True
end
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝ ∗
match op with
| IsEmpty waiter Ψ ⇒
Ψ true
| Pop Ψ ⇒
Ψ None
| Other ⇒
True
end
∨ ∃ node',
⌜res = #node'⌝ ∗
node۰model γ node' ˖i false ∗
match op with
| IsEmpty waiter Ψ ⇒
Ψ false
| Pop Ψ ⇒
pop۰au γ Ψ
| Other ⇒
True
end
}}}.
#[local] Lemma nextｰspec t γ i node :
{{{
inv' t γ ∗
history۰at γ i node
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝
∨ ∃ node',
⌜res = #node'⌝ ∗
node۰model γ node' ˖i false
}}}.
#[local] Lemma nextｰspecｰis_empty {t γ i node} waiter Ψ :
{{{
inv' t γ ∗
history۰at γ i node ∗
front۰lb γ i ∗
saved_pred waiter Ψ ∗
waiters۰at γ waiter i ∗
£ 1
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝ ∗
Ψ true
∨ ∃ node',
⌜res = #node'⌝ ∗
node۰model γ node' ˖i false ∗
Ψ false
}}}.
#[local] Lemma nextｰspecｰpop {t γ i node} Ψ :
{{{
inv' t γ ∗
history۰at γ i node ∗
front۰lb γ i ∗
pop۰au γ Ψ
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝ ∗
Ψ None
∨ ∃ node',
⌜res = #node'⌝ ∗
node۰model γ node' ˖i false ∗
pop۰au γ Ψ
}}}.
Lemma queue_spmc٠is_emptyｰspec t γ ι :
<<<
queue_spmc۰inv t γ ι
| ∀∀ vs,
queue_spmc۰model γ vs
>>>
queue_spmc٠is_empty #t @ ↑ι
<<<
queue_spmc۰model γ vs
| RET #(bool_decide (vs = []%list));
True
>>>.
Lemma queue_spmc٠pushｰspec t γ ι ws v :
<<<
queue_spmc۰inv t γ ι ∗
queue_spmc۰producer t γ ws
| ∀∀ vs,
queue_spmc۰model γ vs
>>>
queue_spmc٠push #t v @ ↑ι
<<<
queue_spmc۰model γ (vs ++ [v])
| RET ();
queue_spmc۰producer t γ (vs ++ [v])
>>>.
#[local] Lemma queue_spmc٠popｰspecｰaux t γ :
<<<
inv' t γ
| ∀∀ vs,
model₁ γ vs
>>>
queue_spmc٠pop #t @ ↑γ.(metadata۰inv)
<<<
model₁ γ (tail vs)
| RET head vs;
True
>>>.
Lemma queue_spmc٠popｰspec t γ ι :
<<<
queue_spmc۰inv t γ ι
| ∀∀ vs,
queue_spmc۰model γ vs
>>>
queue_spmc٠pop #t @ ↑ι
<<<
queue_spmc۰model γ (tail vs)
| RET head vs;
True
>>>.
End queue_spmc۰G.
#[global] Opaque queue_spmc۰inv.
#[global] Opaque queue_spmc۰producer.
#[global] Opaque queue_spmc۰model.
End base.
Require zoo_saturn.queue_spmc__opaque.
Section queue_spmc۰G.
Context `{queue_spmc۰G : QueueSpmcG Σ}.
Implicit Type 𝑡 : location.
Implicit Type t : val.
Definition queue_spmc۰inv t ι : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
𝑡 ↪ γ ∗
base.queue_spmc۰inv 𝑡 γ ι.
#[local] Instance : CustomIpat "inv" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hinv{_{}} ) ".
Definition queue_spmc۰producer t ws : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
𝑡 ↪ γ ∗
base.queue_spmc۰producer 𝑡 γ ws.
#[local] Instance : CustomIpat "producer" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hproducer{_{}} ) ".
Definition queue_spmc۰model t vs : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
𝑡 ↪ γ ∗
base.queue_spmc۰model γ vs.
#[local] Instance : CustomIpat "model" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hmodel{_{}} ) ".
#[global] Instance queue_spmc۰modelｰtimeless t vs :
Timeless (queue_spmc۰model t vs).
#[global] Instance queue_spmc۰producerｰtimeless t ws :
Timeless (queue_spmc۰producer t ws).
#[global] Instance queue_spmc۰invｰpersistent t ι :
Persistent (queue_spmc۰inv t ι).
Lemma queue_spmc۰modelｰexclusive t vs1 vs2 :
queue_spmc۰model t vs1 -∗
queue_spmc۰model t vs2 -∗
False.
Lemma queue_spmc۰producerｰvalid t vs ws :
queue_spmc۰producer t ws -∗
queue_spmc۰model t vs -∗
⌜vs `suffix_of` ws⌝.
Lemma queue_spmc۰producerｰexclusive t ws1 ws2 :
queue_spmc۰producer t ws1 -∗
queue_spmc۰producer t ws2 -∗
False.
Lemma queue_spmc٠createｰspec ι :
{{{
True
}}}
queue_spmc٠create ()
{{{
t
, RET t;
queue_spmc۰inv t ι ∗
queue_spmc۰model t [] ∗
queue_spmc۰producer t []
}}}.
Lemma queue_spmc٠is_emptyｰspec t ι :
<<<
queue_spmc۰inv t ι
| ∀∀ vs,
queue_spmc۰model t vs
>>>
queue_spmc٠is_empty t @ ↑ι
<<<
queue_spmc۰model t vs
| RET #(bool_decide (vs = []%list));
True
>>>.
Lemma queue_spmc٠pushｰspec t ι ws v :
<<<
queue_spmc۰inv t ι ∗
queue_spmc۰producer t ws
| ∀∀ vs,
queue_spmc۰model t vs
>>>
queue_spmc٠push t v @ ↑ι
<<<
queue_spmc۰model t (vs ++ [v])
| RET ();
queue_spmc۰producer t (vs ++ [v])
>>>.
Lemma queue_spmc٠popｰspec t ι :
<<<
queue_spmc۰inv t ι
| ∀∀ vs,
queue_spmc۰model t vs
>>>
queue_spmc٠pop t @ ↑ι
<<<
queue_spmc۰model t (tail vs)
| RET head vs;
True
>>>.
End queue_spmc۰G.
#[global] Opaque queue_spmc۰inv.
#[global] Opaque queue_spmc۰producer.
#[global] Opaque queue_spmc۰model.
Require Import zoo.prelude.
Require Import zoo.common.relations.
Require Import zoo.common.countable.
Require Import zoo.iris.bi.big_op.
Require Import zoo.iris.base_logic.lib.mono_list.
Require Import zoo.iris.base_logic.lib.auth_nat_max.
Require Import zoo.iris.base_logic.lib.auth_twins.
Require Import zoo.iris.base_logic.lib.saved_pred.
Require Import zoo.base.
Require Import zoo_std.option.
Require Import zoo_std.xtchain.
Require Export zoo_saturn.queue_spmc__code.
Require Import zoo_saturn.queue_spmc__types.
Require Import zoo.options.
Implicit Type b : bool.
Implicit Type front node back new_back : location.
Implicit Type hist past nodes : list location.
Implicit Type v : val.
Implicit Type vs ws : list val.
Implicit Type waiter : gname.
Implicit Type waiters : gmap gname nat.
Class QueueSpmcG Σ `{zoo۰G : !ZooG Σ} :=
{ #[local] queue_spmc۰G۰history۰G :: MonoListG Σ location
; #[local] queue_spmc۰G۰front۰G :: AuthNatMaxG Σ
; #[local] queue_spmc۰G۰model۰G :: AuthTwinsG Σ (leibnizO (list val)) suffix
; #[local] queue_spmc۰G۰waiters۰G :: ghost_mapG Σ gname nat
; #[local] queue_spmc۰G۰saved_pred۰G :: SavedPredG Σ bool
}.
Definition queue_spmc۰Σ :=
#[mono_list۰Σ location
; auth_nat_max۰Σ
; auth_twins۰Σ (leibnizO (list val)) suffix
; ghost_mapΣ gname nat
; saved_pred۰Σ bool
].
#[global] Instance subGｰqueue_spmc۰Σ Σ `{zoo۰G : !ZooG Σ} :
subG queue_spmc۰Σ Σ →
QueueSpmcG Σ.
Module base.
Section queue_spmc۰G.
Context `{queue_spmc۰G : QueueSpmcG Σ}.
Implicit Type t : location.
Record metadata :=
{ metadata۰inv : namespace
; metadata۰history : gname
; metadata۰front : gname
; metadata۰model : auth_twins۰name
; metadata۰waiters : gname
}.
Implicit Type γ : metadata.
#[global] Instance metadataｰeq_dec : EqDecision metadata :=
ltac:(solve_decision).
#[global] Instance metadataｰcountable :
Countable metadata.
#[local] Definition history۰auth' γ_history hist :=
mono_list۰auth γ_history (DfracOwn (1/2)) hist.
#[local] Definition history۰auth γ hist :=
history۰auth' γ.(metadata۰history) hist.
#[local] Definition history۰last' γ_history node : iProp Σ :=
∃ hist,
mono_list۰auth γ_history (DfracOwn (1/2)) hist ∗
⌜last hist = Some node⌝.
#[local] Instance : CustomIpat "history۰last" :=
" ( %hist{} & Hauth{_{}} & %Hlast ) ".
#[local] Definition history۰last γ :=
history۰last' γ.(metadata۰history).
#[local] Definition history۰at γ i node :=
mono_list۰at γ.(metadata۰history) i node.
#[local] Definition front۰auth' γ_front i :=
auth_nat_max۰auth γ_front (DfracOwn 1) i.
#[local] Definition front۰auth γ i :=
front۰auth' γ.(metadata۰front) i.
#[local] Definition front۰lb γ i :=
auth_nat_max۰lb γ.(metadata۰front) i.
#[local] Definition producer' γ_model ws :=
auth_twins۰auth _ γ_model ws.
#[local] Definition producer γ :=
producer' γ.(metadata۰model).
#[local] Definition model₁' γ_model vs :=
auth_twins۰twin₁ _ γ_model vs.
#[local] Definition model₁ γ :=
model₁' γ.(metadata۰model).
#[local] Definition model₂' γ_model vs :=
auth_twins۰twin₂ _ γ_model vs.
#[local] Definition model₂ γ :=
model₂' γ.(metadata۰model).
#[local] Definition waiters۰auth' γ_waiters waiters :=
ghost_map_auth γ_waiters 1 waiters.
#[local] Definition waiters۰auth γ waiters :=
waiters۰auth' γ.(metadata۰waiters) waiters.
#[local] Definition waiters۰at γ waiter i :=
ghost_map_elem γ.(metadata۰waiters) waiter (DfracOwn 1) i.
#[local] Definition node۰model γ node i b : iProp Σ :=
node ↦ₕ Header §Node 2 ∗
history۰at γ i node ∗
if b then front۰lb γ i else True%I.
#[local] Instance : CustomIpat "node۰model" :=
" ( #H{}_header & #Hhistory_at_{} & {{front}#Hfront_lb_{};_} ) ".
#[local] Definition waiter۰au γ (Ψ : bool → iProp Σ) : iProp Σ :=
AU <{
∃∃ vs,
model₁ γ vs
}> @ ⊤ ∖ ↑γ.(metadata۰inv), ∅ <{
model₁ γ vs
, COMM
Ψ (bool_decide (vs = []))
}>.
#[local] Definition waiter۰model γ past waiter i : iProp Σ :=
∃ Ψ,
saved_pred waiter Ψ ∗
if decide (i < length past) then
Ψ false
else
waiter۰au γ Ψ.
#[local] Definition inv۰inner t γ : iProp Σ :=
∃ hist past front nodes vs waiters,
⌜hist = past ++ front :: nodes⌝ ∗
t.[front] ↦ #front ∗
xtchain (Header §Node 2) (DfracOwn 1) hist §Null ∗
([∗ list] node; v ∈ nodes; vs, node.[data] ↦ v) ∗
history۰auth γ hist ∗
front۰auth γ (length past) ∗
model₂ γ vs ∗
waiters۰auth γ waiters ∗
([∗ map] waiter ↦ i ∈ waiters, waiter۰model γ past waiter i).
#[local] Instance : CustomIpat "inv۰inner" :=
" ( %hist{} & %past{} & %front{} & %nodes{} & %vs{} & %waiters{} & >%Hhist{} & >Ht_front & >Hhist & >Hnodes & >Hhistory_auth & >Hfront_auth & >Hmodel₂ & >Hwaiters_auth & Hwaiters ) ".
#[local] Definition inv' t γ :=
inv γ.(metadata۰inv) (inv۰inner t γ).
Definition queue_spmc۰inv t γ ι : iProp Σ :=
⌜ι = γ.(metadata۰inv)⌝ ∗
inv' t γ.
#[local] Instance : CustomIpat "inv" :=
" ( -> & #Hinv ) ".
Definition queue_spmc۰producer t γ ws : iProp Σ :=
∃ back,
t.[back] ↦ #back ∗
back ↦ₕ Header §Node 2 ∗
history۰last γ back ∗
producer γ ws.
#[local] Instance : CustomIpat "producer" :=
" ( %back{} & Ht_back{_{}} & #Hback{}_header & Hhistory_last{_{}} & Hproducer{_{}} ) ".
Definition queue_spmc۰model :=
model₁.
#[local] Instance : CustomIpat "model" :=
" Hmodel₁{_{}} ".
#[global] Instance queue_spmc۰modelｰtimeless γ vs :
Timeless (queue_spmc۰model γ vs).
#[global] Instance queue_spmc۰producerｰtimeless t γ ws :
Timeless (queue_spmc۰producer t γ ws).
#[global] Instance queue_spmc۰invｰpersistent t γ ι :
Persistent (queue_spmc۰inv t γ ι).
#[local] Lemma historyｰalloc front :
⊢ |==>
∃ γ_history,
history۰auth' γ_history [front] ∗
history۰last' γ_history front.
#[local] Lemma history۰atｰget {γ hist} i node :
hist !! i = Some node →
history۰auth γ hist ⊢
history۰at γ i node.
#[local] Lemma history۰atｰlookup γ hist i node :
history۰auth γ hist -∗
history۰at γ i node -∗
⌜hist !! i = Some node⌝.
#[local] Lemma historyｰauthｰlast γ hist node :
history۰auth γ hist -∗
history۰last γ node -∗
⌜last hist = Some node⌝.
#[local] Lemma historyｰupdate {γ hist node} node' :
history۰auth γ hist -∗
history۰last γ node ==∗
history۰auth γ (hist ++ [node']) ∗
history۰last γ node'.
Opaque history۰last'.
#[local] Lemma frontｰalloc :
⊢ |==>
∃ γ_front,
front۰auth' γ_front 0.
#[local] Lemma front۰lbｰget γ i :
front۰auth γ i ⊢
front۰lb γ i.
#[local] Lemma front۰lbｰvalid γ i1 i2 :
front۰auth γ i1 -∗
front۰lb γ i2 -∗
⌜i2 ≤ i1⌝.
#[local] Lemma frontｰupdate {γ i} i' :
i ≤ i' →
front۰auth γ i ⊢ |==>
front۰auth γ i'.
#[local] Lemma producerｰvalid γ ws vs :
producer γ ws -∗
model₁ γ vs -∗
⌜vs `suffix_of` ws⌝.
#[local] Lemma producerｰexclusive γ ws1 ws2 :
producer γ ws1 -∗
producer γ ws2 -∗
False.
#[local] Lemma modelｰproducerｰalloc :
⊢ |==>
∃ γ_model,
producer' γ_model [] ∗
model₁' γ_model [] ∗
model₂' γ_model [].
#[local] Lemma model₁ｰexclusive γ vs1 vs2 :
model₁ γ vs1 -∗
model₁ γ vs2 -∗
False.
#[local] Lemma modelｰagree γ vs1 vs2 :
model₁ γ vs1 -∗
model₂ γ vs2 -∗
⌜vs1 = vs2⌝.
#[local] Lemma modelｰpush {γ ws vs1 vs2} v :
producer γ ws -∗
model₁ γ vs1 -∗
model₂ γ vs2 ==∗
producer γ (vs1 ++ [v]) ∗
model₁ γ (vs1 ++ [v]) ∗
model₂ γ (vs1 ++ [v]).
#[local] Lemma modelｰpop γ v vs1 vs2 :
model₁ γ (v :: vs1) -∗
model₂ γ vs2 ==∗
model₁ γ vs1 ∗
model₂ γ vs1.
#[local] Lemma waitersｰalloc :
⊢ |==>
∃ γ_waiters,
waiters۰auth' γ_waiters ∅.
#[local] Lemma waitersｰinsert {γ waiters} i Ψ :
waiters۰auth γ waiters ⊢ |==>
∃ waiter,
waiters۰auth γ (<[waiter := i]> waiters) ∗
saved_pred waiter Ψ ∗
waiters۰at γ waiter i.
#[local] Lemma waitersｰdelete γ waiters waiter i :
waiters۰auth γ waiters -∗
waiters۰at γ waiter i ==∗
⌜waiters !! waiter = Some i⌝ ∗
waiters۰auth γ (delete waiter waiters).
Lemma queue_spmc۰modelｰexclusive γ vs1 vs2 :
queue_spmc۰model γ vs1 -∗
queue_spmc۰model γ vs2 -∗
False.
Lemma queue_spmc۰producerｰvalid t γ vs ws :
queue_spmc۰producer t γ ws -∗
queue_spmc۰model γ vs -∗
⌜vs `suffix_of` ws⌝.
Lemma queue_spmc۰producerｰexclusive t γ ws1 ws2 :
queue_spmc۰producer t γ ws1 -∗
queue_spmc۰producer t γ ws2 -∗
False.
Lemma queue_spmc٠createｰspec ι :
{{{
True
}}}
queue_spmc٠create ()
{{{
t γ
, RET #t;
meta_token t ⊤ ∗
queue_spmc۰inv t γ ι ∗
queue_spmc۰model γ [] ∗
queue_spmc۰producer t γ []
}}}.
#[local] Lemma frontｰspecｰstrong Ψ t γ :
{{{
inv' t γ ∗
if Ψ is Some Ψ then
waiter۰au γ Ψ
else
True
}}}
(#t).{front}
{{{
front i
, RET #front;
node۰model γ front i true ∗
if Ψ is Some Ψ then
∃ waiter,
saved_pred waiter Ψ ∗
waiters۰at γ waiter i
else
True
}}}.
#[local] Lemma frontｰspec t γ :
{{{
inv' t γ
}}}
(#t).{front}
{{{
front i
, RET #front;
node۰model γ front i true
}}}.
Variant operation :=
| IsEmpty waiter (Ψ : bool → iProp Σ)
| Pop (Ψ : option val → iProp Σ)
| Other.
Implicit Type op : operation.
Variant operation' :=
| IsEmpty'
| Pop'
| Other'.
#[local] Instance operation'ｰeq_dec : EqDecision operation' :=
ltac:(solve_decision).
#[local] Coercion operation۰to_operation' op :=
match op with
| IsEmpty _ _ ⇒
IsEmpty'
| Pop _ ⇒
Pop'
| Other ⇒
Other'
end.
#[local] Definition pop۰au γ (Ψ : option val → iProp Σ) : iProp Σ :=
AU <{
∃∃ vs,
model₁ γ vs
}> @ ⊤ ∖ ↑γ.(metadata۰inv), ∅ <{
model₁ γ (tail vs)
, COMM
Ψ (head vs)
}>.
#[local] Lemma nextｰspecｰaux op t γ i node :
{{{
inv' t γ ∗
history۰at γ i node ∗
( if decide (op = Other' :> operation') then True else
front۰lb γ i
) ∗
match op with
| IsEmpty waiter Ψ ⇒
saved_pred waiter Ψ ∗
waiters۰at γ waiter i ∗
£ 1
| Pop Ψ ⇒
pop۰au γ Ψ
| Other ⇒
True
end
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝ ∗
match op with
| IsEmpty waiter Ψ ⇒
Ψ true
| Pop Ψ ⇒
Ψ None
| Other ⇒
True
end
∨ ∃ node',
⌜res = #node'⌝ ∗
node۰model γ node' ˖i false ∗
match op with
| IsEmpty waiter Ψ ⇒
Ψ false
| Pop Ψ ⇒
pop۰au γ Ψ
| Other ⇒
True
end
}}}.
#[local] Lemma nextｰspec t γ i node :
{{{
inv' t γ ∗
history۰at γ i node
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝
∨ ∃ node',
⌜res = #node'⌝ ∗
node۰model γ node' ˖i false
}}}.
#[local] Lemma nextｰspecｰis_empty {t γ i node} waiter Ψ :
{{{
inv' t γ ∗
history۰at γ i node ∗
front۰lb γ i ∗
saved_pred waiter Ψ ∗
waiters۰at γ waiter i ∗
£ 1
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝ ∗
Ψ true
∨ ∃ node',
⌜res = #node'⌝ ∗
node۰model γ node' ˖i false ∗
Ψ false
}}}.
#[local] Lemma nextｰspecｰpop {t γ i node} Ψ :
{{{
inv' t γ ∗
history۰at γ i node ∗
front۰lb γ i ∗
pop۰au γ Ψ
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝ ∗
Ψ None
∨ ∃ node',
⌜res = #node'⌝ ∗
node۰model γ node' ˖i false ∗
pop۰au γ Ψ
}}}.
Lemma queue_spmc٠is_emptyｰspec t γ ι :
<<<
queue_spmc۰inv t γ ι
| ∀∀ vs,
queue_spmc۰model γ vs
>>>
queue_spmc٠is_empty #t @ ↑ι
<<<
queue_spmc۰model γ vs
| RET #(bool_decide (vs = []%list));
True
>>>.
Lemma queue_spmc٠pushｰspec t γ ι ws v :
<<<
queue_spmc۰inv t γ ι ∗
queue_spmc۰producer t γ ws
| ∀∀ vs,
queue_spmc۰model γ vs
>>>
queue_spmc٠push #t v @ ↑ι
<<<
queue_spmc۰model γ (vs ++ [v])
| RET ();
queue_spmc۰producer t γ (vs ++ [v])
>>>.
#[local] Lemma queue_spmc٠popｰspecｰaux t γ :
<<<
inv' t γ
| ∀∀ vs,
model₁ γ vs
>>>
queue_spmc٠pop #t @ ↑γ.(metadata۰inv)
<<<
model₁ γ (tail vs)
| RET head vs;
True
>>>.
Lemma queue_spmc٠popｰspec t γ ι :
<<<
queue_spmc۰inv t γ ι
| ∀∀ vs,
queue_spmc۰model γ vs
>>>
queue_spmc٠pop #t @ ↑ι
<<<
queue_spmc۰model γ (tail vs)
| RET head vs;
True
>>>.
End queue_spmc۰G.
#[global] Opaque queue_spmc۰inv.
#[global] Opaque queue_spmc۰producer.
#[global] Opaque queue_spmc۰model.
End base.
Require zoo_saturn.queue_spmc__opaque.
Section queue_spmc۰G.
Context `{queue_spmc۰G : QueueSpmcG Σ}.
Implicit Type 𝑡 : location.
Implicit Type t : val.
Definition queue_spmc۰inv t ι : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
𝑡 ↪ γ ∗
base.queue_spmc۰inv 𝑡 γ ι.
#[local] Instance : CustomIpat "inv" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hinv{_{}} ) ".
Definition queue_spmc۰producer t ws : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
𝑡 ↪ γ ∗
base.queue_spmc۰producer 𝑡 γ ws.
#[local] Instance : CustomIpat "producer" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hproducer{_{}} ) ".
Definition queue_spmc۰model t vs : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
𝑡 ↪ γ ∗
base.queue_spmc۰model γ vs.
#[local] Instance : CustomIpat "model" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hmodel{_{}} ) ".
#[global] Instance queue_spmc۰modelｰtimeless t vs :
Timeless (queue_spmc۰model t vs).
#[global] Instance queue_spmc۰producerｰtimeless t ws :
Timeless (queue_spmc۰producer t ws).
#[global] Instance queue_spmc۰invｰpersistent t ι :
Persistent (queue_spmc۰inv t ι).
Lemma queue_spmc۰modelｰexclusive t vs1 vs2 :
queue_spmc۰model t vs1 -∗
queue_spmc۰model t vs2 -∗
False.
Lemma queue_spmc۰producerｰvalid t vs ws :
queue_spmc۰producer t ws -∗
queue_spmc۰model t vs -∗
⌜vs `suffix_of` ws⌝.
Lemma queue_spmc۰producerｰexclusive t ws1 ws2 :
queue_spmc۰producer t ws1 -∗
queue_spmc۰producer t ws2 -∗
False.
Lemma queue_spmc٠createｰspec ι :
{{{
True
}}}
queue_spmc٠create ()
{{{
t
, RET t;
queue_spmc۰inv t ι ∗
queue_spmc۰model t [] ∗
queue_spmc۰producer t []
}}}.
Lemma queue_spmc٠is_emptyｰspec t ι :
<<<
queue_spmc۰inv t ι
| ∀∀ vs,
queue_spmc۰model t vs
>>>
queue_spmc٠is_empty t @ ↑ι
<<<
queue_spmc۰model t vs
| RET #(bool_decide (vs = []%list));
True
>>>.
Lemma queue_spmc٠pushｰspec t ι ws v :
<<<
queue_spmc۰inv t ι ∗
queue_spmc۰producer t ws
| ∀∀ vs,
queue_spmc۰model t vs
>>>
queue_spmc٠push t v @ ↑ι
<<<
queue_spmc۰model t (vs ++ [v])
| RET ();
queue_spmc۰producer t (vs ++ [v])
>>>.
Lemma queue_spmc٠popｰspec t ι :
<<<
queue_spmc۰inv t ι
| ∀∀ vs,
queue_spmc۰model t vs
>>>
queue_spmc٠pop t @ ↑ι
<<<
queue_spmc۰model t (tail vs)
| RET head vs;
True
>>>.
End queue_spmc۰G.
#[global] Opaque queue_spmc۰inv.
#[global] Opaque queue_spmc۰producer.
#[global] Opaque queue_spmc۰model.