Library zoo_saturn.mpmc_queue_1
From iris.base_logic Require Import
lib.ghost_map.
From zoo Require Import
prelude.
From zoo.common Require Import
countable.
From zoo.iris.bi Require Import
big_op.
From zoo.iris.base_logic Require Import
lib.mono_list
lib.auth_nat_max
lib.twins
lib.saved_pred.
From zoo.language Require Import
notations.
From zoo.diaframe Require Import
diaframe.
From zoo_std Require Import
option
xtchain
domain.
From zoo_saturn Require Export
base
mpmc_queue_1__code.
From zoo_saturn Require Import
mpmc_queue_1__types.
From zoo Require Import
options.
Implicit Types b : bool.
Implicit Types front node back new_back : location.
Implicit Types hist past nodes : list location.
Implicit Types v : val.
Implicit Types vs : list val.
Implicit Types waiter : gname.
Implicit Types waiters : gmap gname nat.
Class MpmcQueue1G Σ `{zoo_G : !ZooG Σ} :=
{ #[local] mpmc_queue_1_G_history_G :: MonoListG Σ location
; #[local] mpmc_queue_1_G_front_G :: AuthNatMaxG Σ
; #[local] mpmc_queue_1_G_model_G :: TwinsG Σ (leibnizO (list val))
; #[local] mpmc_queue_1_G_waiters_G :: ghost_mapG Σ gname nat
; #[local] mpmc_queue_1_G_saved_pred_G :: SavedPredG Σ bool
}.
Definition mpmc_queue_1_Σ :=
#[mono_list_Σ location
; auth_nat_max_Σ
; twins_Σ (leibnizO (list val))
; ghost_mapΣ gname nat
; saved_pred_Σ bool
].
#[global] Instance subG_mpmc_queue_1_Σ Σ `{zoo_G : !ZooG Σ} :
subG mpmc_queue_1_Σ Σ →
MpmcQueue1G Σ.
Module base.
Section mpmc_queue_1_G.
Context `{mpmc_queue_1_G : MpmcQueue1G Σ}.
Implicit Types t : location.
Record mpmc_queue_1_name :=
{ mpmc_queue_1_name_inv : namespace
; mpmc_queue_1_name_history : gname
; mpmc_queue_1_name_front : gname
; mpmc_queue_1_name_model : gname
; mpmc_queue_1_name_waiters : gname
}.
Implicit Type γ : mpmc_queue_1_name.
#[global] Instance mpmc_queue_1_name_eq_dec : EqDecision mpmc_queue_1_name :=
ltac:(solve_decision).
#[global] Instance mpmc_queue_1_name_countable :
Countable mpmc_queue_1_name.
#[local] Definition history_auth' γ_history hist :=
mono_list_auth γ_history (DfracOwn 1) hist.
#[local] Definition history_auth γ hist :=
history_auth' γ.(mpmc_queue_1_name_history) hist.
#[local] Definition history_at γ i node :=
mono_list_at γ.(mpmc_queue_1_name_history) i node.
#[local] Definition front_auth' γ_front i :=
auth_nat_max_auth γ_front (DfracOwn 1) i.
#[local] Definition front_auth γ i :=
front_auth' γ.(mpmc_queue_1_name_front) i.
#[local] Definition front_lb γ i :=
auth_nat_max_lb γ.(mpmc_queue_1_name_front) i.
#[local] Definition model₁' γ_model vs :=
twins_twin1 γ_model (DfracOwn 1) vs.
#[local] Definition model₁ γ vs :=
model₁' γ.(mpmc_queue_1_name_model) vs.
#[local] Definition model₂' γ_model vs :=
twins_twin2 γ_model vs.
#[local] Definition model₂ γ vs :=
model₂' γ.(mpmc_queue_1_name_model) vs.
#[local] Definition waiters_auth' γ_waiters waiters :=
ghost_map_auth γ_waiters 1 waiters.
#[local] Definition waiters_auth γ waiters :=
waiters_auth' γ.(mpmc_queue_1_name_waiters) waiters.
#[local] Definition waiters_at γ waiter i :=
ghost_map_elem γ.(mpmc_queue_1_name_waiters) waiter (DfracOwn 1) i.
#[local] Definition node_model γ node i b : iProp Σ :=
node ↦ₕ Header §Node 2 ∗
history_at γ i node ∗
if b then front_lb γ i else True%I.
#[local] Instance : CustomIpat "node_model" :=
" ( #H{}_header & #Hhistory_at_{} & {{front}#Hfront_lb_{};_} ) ".
#[local] Definition waiter_au γ (Ψ : bool → iProp Σ) : iProp Σ :=
AU <{
∃∃ vs,
model₁ γ vs
}> @ ⊤ ∖ ↑γ.(mpmc_queue_1_name_inv), ∅ <{
model₁ γ vs
, COMM
Ψ (bool_decide (vs = []))
}>.
#[local] Definition waiter_model γ past waiter i : iProp Σ :=
∃ Ψ,
saved_pred waiter Ψ ∗
if decide (i < length past) then
Ψ false
else
waiter_au γ Ψ.
#[local] Definition inv_inner t γ : iProp Σ :=
∃ hist past front nodes back vs waiters,
⌜hist = past ++ front :: nodes⌝ ∗
⌜back ∈ hist⌝ ∗
t.[front] ↦ #front ∗
t.[back] ↦ #back ∗
xtchain (Header §Node 2) (DfracOwn 1) hist §Null ∗
([∗ list] node; v ∈ nodes; vs, node.[data] ↦ v) ∗
history_auth γ hist ∗
front_auth γ (length past) ∗
model₂ γ vs ∗
waiters_auth γ waiters ∗
([∗ map] waiter ↦ i ∈ waiters, waiter_model γ past waiter i).
#[local] Instance : CustomIpat "inv_inner" :=
" ( %hist{} & %past{} & %front{} & %nodes{} & %back{} & %vs{} & %waiters{} & >%Hhist{} & >%Hback{} & >Ht_front & >Ht_back & >Hhist & >Hnodes & >Hhistory_auth & >Hfront_auth & >Hmodel₂ & >Hwaiters_auth & Hwaiters ) ".
#[local] Definition inv' t γ :=
inv γ.(mpmc_queue_1_name_inv) (inv_inner t γ).
Definition mpmc_queue_1_inv t γ ι : iProp Σ :=
⌜ι = γ.(mpmc_queue_1_name_inv)⌝ ∗
inv' t γ.
#[local] Instance : CustomIpat "inv" :=
" ( -> & #Hinv ) ".
Definition mpmc_queue_1_model :=
model₁.
#[local] Instance : CustomIpat "model" :=
" Hmodel₁{_{}} ".
#[global] Instance mpmc_queue_1_model_timeless γ vs :
Timeless (mpmc_queue_1_model γ vs).
#[global] Instance mpmc_queue_1_inv_persistent t γ ι :
Persistent (mpmc_queue_1_inv t γ ι).
#[local] Lemma history_alloc front :
⊢ |==>
∃ γ_history,
history_auth' γ_history [front].
#[local] Lemma history_at_get {γ hist} i node :
hist !! i = Some node →
history_auth γ hist ⊢
history_at γ i node.
#[local] Lemma history_at_lookup γ hist i node :
history_auth γ hist -∗
history_at γ i node -∗
⌜hist !! i = Some node⌝.
#[local] Lemma history_update {γ hist} node :
history_auth γ hist ⊢ |==>
history_auth γ (hist ++ [node]) ∗
history_at γ (length hist) node.
#[local] Lemma front_alloc :
⊢ |==>
∃ γ_front,
front_auth' γ_front 0.
#[local] Lemma front_lb_get γ i :
front_auth γ i ⊢
front_lb γ i.
#[local] Lemma front_lb_valid γ i1 i2 :
front_auth γ i1 -∗
front_lb γ i2 -∗
⌜i2 ≤ i1⌝.
#[local] Lemma front_update {γ i} i' :
i ≤ i' →
front_auth γ i ⊢ |==>
front_auth γ i'.
#[local] Lemma model_alloc :
⊢ |==>
∃ γ_model,
model₁' γ_model [] ∗
model₂' γ_model [].
#[local] Lemma model₁_exclusive γ vs1 vs2 :
model₁ γ vs1 -∗
model₁ γ vs2 -∗
False.
#[local] Lemma model_agree γ vs1 vs2 :
model₁ γ vs1 -∗
model₂ γ vs2 -∗
⌜vs1 = vs2⌝.
#[local] Lemma model_update {γ vs1 vs2} vs :
model₁ γ vs1 -∗
model₂ γ vs2 ==∗
model₁ γ vs ∗
model₂ γ vs.
#[local] Lemma waiters_alloc :
⊢ |==>
∃ γ_waiters,
waiters_auth' γ_waiters ∅.
#[local] Lemma waiters_insert {γ waiters} i Ψ :
waiters_auth γ waiters ⊢ |==>
∃ waiter,
waiters_auth γ (<[waiter := i]> waiters) ∗
saved_pred waiter Ψ ∗
waiters_at γ waiter i.
#[local] Lemma waiters_delete γ waiters waiter i :
waiters_auth γ waiters -∗
waiters_at γ waiter i ==∗
⌜waiters !! waiter = Some i⌝ ∗
waiters_auth γ (delete waiter waiters).
Lemma mpmc_queue_1_model_exclusive γ vs1 vs2 :
mpmc_queue_1_model γ vs1 -∗
mpmc_queue_1_model γ vs2 -∗
False.
Lemma mpmc_queue_1٠create𑁒spec ι :
{{{
True
}}}
mpmc_queue_1٠create ()
{{{
t γ
, RET #t;
meta_token t ⊤ ∗
mpmc_queue_1_inv t γ ι ∗
mpmc_queue_1_model γ []
}}}.
#[local] Lemma front𑁒spec_strong Ψ t γ :
{{{
inv' t γ ∗
if Ψ is Some Ψ then
waiter_au γ Ψ
else
True
}}}
(#t).{front}
{{{
front i
, RET #front;
node_model γ front i true ∗
if Ψ is Some Ψ then
∃ waiter,
saved_pred waiter Ψ ∗
waiters_at γ waiter i
else
True
}}}.
#[local] Lemma front𑁒spec t γ :
{{{
inv' t γ
}}}
(#t).{front}
{{{
front i
, RET #front;
node_model γ front i true
}}}.
#[local] Lemma back𑁒spec t γ :
{{{
inv' t γ
}}}
(#t).{back}
{{{
back i
, RET #back;
node_model γ back i false
}}}.
Variant operation :=
| IsEmpty waiter (Ψ : bool → iProp Σ)
| Pop (Ψ : option val → iProp Σ)
| Other.
Implicit Types op : operation.
Variant operation' :=
| IsEmpty'
| Pop'
| Other'.
#[local] Instance operation'_eq_dec : EqDecision operation' :=
ltac:(solve_decision).
#[local] Coercion operation_to_operation' op :=
match op with
| IsEmpty _ _ ⇒
IsEmpty'
| Pop _ ⇒
Pop'
| Other ⇒
Other'
end.
#[local] Definition pop_au γ (Ψ : option val → iProp Σ) : iProp Σ :=
AU <{
∃∃ vs,
model₁ γ vs
}> @ ⊤ ∖ ↑γ.(mpmc_queue_1_name_inv), ∅ <{
model₁ γ (tail vs)
, COMM
Ψ (head vs)
}>.
#[local] Lemma next𑁒spec_aux op t γ i node :
{{{
inv' t γ ∗
history_at γ i node ∗
( if decide (op = Other' :> operation') then True else
front_lb γ i
) ∗
match op with
| IsEmpty waiter Ψ ⇒
saved_pred waiter Ψ ∗
waiters_at γ waiter i ∗
£ 1
| Pop Ψ ⇒
pop_au 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'⌝ ∗
node_model γ node' (S i) false ∗
match op with
| IsEmpty waiter Ψ ⇒
Ψ false
| Pop Ψ ⇒
pop_au γ Ψ
| Other ⇒
True
end
}}}.
#[local] Lemma next𑁒spec {t γ i} node :
{{{
inv' t γ ∗
history_at γ i node
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝
∨ ∃ node',
⌜res = #node'⌝ ∗
node_model γ node' (S i) false
}}}.
#[local] Lemma next𑁒spec_is_empty {t γ i node} waiter Ψ :
{{{
inv' t γ ∗
history_at γ i node ∗
front_lb γ i ∗
saved_pred waiter Ψ ∗
waiters_at γ waiter i ∗
£ 1
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝ ∗
Ψ true
∨ ∃ node',
⌜res = #node'⌝ ∗
node_model γ node' (S i) false ∗
Ψ false
}}}.
#[local] Lemma next𑁒spec_pop {t γ i node} Ψ :
{{{
inv' t γ ∗
history_at γ i node ∗
front_lb γ i ∗
pop_au γ Ψ
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝ ∗
Ψ None
∨ ∃ node',
⌜res = #node'⌝ ∗
node_model γ node' (S i) false ∗
pop_au γ Ψ
}}}.
Lemma mpmc_queue_1٠is_empty𑁒spec t γ ι :
<<<
mpmc_queue_1_inv t γ ι
| ∀∀ vs,
mpmc_queue_1_model γ vs
>>>
mpmc_queue_1٠is_empty #t @ ↑ι
<<<
mpmc_queue_1_model γ vs
| RET #(bool_decide (vs = []%list));
£ 1
>>>.
Lemma mpmc_queue_1٠is_empty𑁒spec' t γ ι :
{{{
mpmc_queue_1_inv t γ ι
}}}
mpmc_queue_1٠is_empty #t
{{{
b
, RET #b;
True
}}}.
#[local] Lemma mpmc_queue_1٠push₀𑁒spec t γ i node new_back v :
<<<
inv' t γ ∗
node_model γ node i false ∗
new_back ↦ₕ Header §Node 2 ∗
new_back.[next] ↦ §Null ∗
new_back.[data] ↦ v
| ∀∀ vs,
mpmc_queue_1_model γ vs
>>>
mpmc_queue_1٠push₀ #node #new_back @ ↑γ.(mpmc_queue_1_name_inv)
<<<
mpmc_queue_1_model γ (vs ++ [v])
| RET ();
∃ j,
history_at γ j new_back
>>>.
#[local] Lemma mpmc_queue_1٠fix_back𑁒spec t γ i back j new_back :
{{{
inv' t γ ∗
history_at γ i back ∗
node_model γ new_back j false
}}}
mpmc_queue_1٠fix_back #t #back #new_back
{{{
RET ();
True
}}}.
Lemma mpmc_queue_1٠push𑁒spec t γ ι v :
<<<
mpmc_queue_1_inv t γ ι
| ∀∀ vs,
mpmc_queue_1_model γ vs
>>>
mpmc_queue_1٠push #t v @ ↑ι
<<<
mpmc_queue_1_model γ (vs ++ [v])
| RET ();
£ 1
>>>.
Lemma mpmc_queue_1٠pop𑁒spec t γ ι :
<<<
mpmc_queue_1_inv t γ ι
| ∀∀ vs,
mpmc_queue_1_model γ vs
>>>
mpmc_queue_1٠pop #t @ ↑ι
<<<
mpmc_queue_1_model γ (tail vs)
| RET head vs;
£ 1
>>>.
End mpmc_queue_1_G.
#[global] Opaque mpmc_queue_1_inv.
#[global] Opaque mpmc_queue_1_model.
End base.
From zoo_saturn Require
mpmc_queue_1__opaque.
Section mpmc_queue_1_G.
Context `{mpmc_queue_1_G : MpmcQueue1G Σ}.
Implicit Types 𝑡 : location.
Implicit Types t : val.
Definition mpmc_queue_1_inv t ι : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
meta 𝑡 nroot γ ∗
base.mpmc_queue_1_inv 𝑡 γ ι.
#[local] Instance : CustomIpat "inv" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hinv{_{}} ) ".
Definition mpmc_queue_1_model t vs : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
meta 𝑡 nroot γ ∗
base.mpmc_queue_1_model γ vs.
#[local] Instance : CustomIpat "model" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hmodel{_{}} ) ".
#[global] Instance mpmc_queue_1_model_timeless t vs :
Timeless (mpmc_queue_1_model t vs).
#[global] Instance mpmc_queue_1_inv_persistent t ι :
Persistent (mpmc_queue_1_inv t ι).
Lemma mpmc_queue_1_model_exclusive t vs1 vs2 :
mpmc_queue_1_model t vs1 -∗
mpmc_queue_1_model t vs2 -∗
False.
Lemma mpmc_queue_1٠create𑁒spec ι :
{{{
True
}}}
mpmc_queue_1٠create ()
{{{
t
, RET t;
mpmc_queue_1_inv t ι ∗
mpmc_queue_1_model t []
}}}.
Lemma mpmc_queue_1٠is_empty𑁒spec t ι :
<<<
mpmc_queue_1_inv t ι
| ∀∀ vs,
mpmc_queue_1_model t vs
>>>
mpmc_queue_1٠is_empty t @ ↑ι
<<<
mpmc_queue_1_model t vs
| RET #(bool_decide (vs = []%list));
£ 1
>>>.
Lemma mpmc_queue_1٠is_empty𑁒spec' t ι :
{{{
mpmc_queue_1_inv t ι
}}}
mpmc_queue_1٠is_empty t
{{{
b
, RET #b;
True
}}}.
Lemma mpmc_queue_1٠push𑁒spec t ι v :
<<<
mpmc_queue_1_inv t ι
| ∀∀ vs,
mpmc_queue_1_model t vs
>>>
mpmc_queue_1٠push t v @ ↑ι
<<<
mpmc_queue_1_model t (vs ++ [v])
| RET ();
£ 1
>>>.
Lemma mpmc_queue_1٠pop𑁒spec t ι :
<<<
mpmc_queue_1_inv t ι
| ∀∀ vs,
mpmc_queue_1_model t vs
>>>
mpmc_queue_1٠pop t @ ↑ι
<<<
mpmc_queue_1_model t (tail vs)
| RET head vs;
£ 1
>>>.
End mpmc_queue_1_G.
#[global] Opaque mpmc_queue_1_inv.
#[global] Opaque mpmc_queue_1_model.
Section mpmc_queue_1_G.
Context `{mpmc_queue_1_G : MpmcQueue1G Σ}.
Context τ `{!iType (iProp Σ) τ}.
#[local] Definition itype_inner t : iProp Σ :=
∃ vs,
mpmc_queue_1_model t vs ∗
[∗ list] v ∈ vs, τ v.
#[local] Instance : CustomIpat "itype_inner" :=
" ( %vs & >Hmodel & #Hvs ) ".
Definition itype_mpmc_queue_1 t : iProp Σ :=
mpmc_queue_1_inv t (nroot.@"1") ∗
inv (nroot.@"2") (itype_inner t).
#[local] Instance : CustomIpat "itype" :=
" ( #Hinv1 & #Hinv2 ) ".
#[global] Instance itype_mpmc_queue_1_itype :
iType _ itype_mpmc_queue_1.
Lemma mpmc_queue_1٠create𑁒type :
{{{
True
}}}
mpmc_queue_1٠create ()
{{{
t
, RET t;
itype_mpmc_queue_1 t
}}}.
Lemma mpmc_queue_1٠is_empty𑁒type t :
{{{
itype_mpmc_queue_1 t
}}}
mpmc_queue_1٠is_empty t
{{{
b
, RET #b;
True
}}}.
Lemma mpmc_queue_1٠push𑁒type t v :
{{{
itype_mpmc_queue_1 t ∗
τ v
}}}
mpmc_queue_1٠push t v
{{{
RET ();
True
}}}.
Lemma mpmc_queue_1٠pop𑁒type t :
{{{
itype_mpmc_queue_1 t
}}}
mpmc_queue_1٠pop t
{{{
o
, RET o;
itype_option τ o
}}}.
End mpmc_queue_1_G.
#[global] Opaque itype_mpmc_queue_1.
lib.ghost_map.
From zoo Require Import
prelude.
From zoo.common Require Import
countable.
From zoo.iris.bi Require Import
big_op.
From zoo.iris.base_logic Require Import
lib.mono_list
lib.auth_nat_max
lib.twins
lib.saved_pred.
From zoo.language Require Import
notations.
From zoo.diaframe Require Import
diaframe.
From zoo_std Require Import
option
xtchain
domain.
From zoo_saturn Require Export
base
mpmc_queue_1__code.
From zoo_saturn Require Import
mpmc_queue_1__types.
From zoo Require Import
options.
Implicit Types b : bool.
Implicit Types front node back new_back : location.
Implicit Types hist past nodes : list location.
Implicit Types v : val.
Implicit Types vs : list val.
Implicit Types waiter : gname.
Implicit Types waiters : gmap gname nat.
Class MpmcQueue1G Σ `{zoo_G : !ZooG Σ} :=
{ #[local] mpmc_queue_1_G_history_G :: MonoListG Σ location
; #[local] mpmc_queue_1_G_front_G :: AuthNatMaxG Σ
; #[local] mpmc_queue_1_G_model_G :: TwinsG Σ (leibnizO (list val))
; #[local] mpmc_queue_1_G_waiters_G :: ghost_mapG Σ gname nat
; #[local] mpmc_queue_1_G_saved_pred_G :: SavedPredG Σ bool
}.
Definition mpmc_queue_1_Σ :=
#[mono_list_Σ location
; auth_nat_max_Σ
; twins_Σ (leibnizO (list val))
; ghost_mapΣ gname nat
; saved_pred_Σ bool
].
#[global] Instance subG_mpmc_queue_1_Σ Σ `{zoo_G : !ZooG Σ} :
subG mpmc_queue_1_Σ Σ →
MpmcQueue1G Σ.
Module base.
Section mpmc_queue_1_G.
Context `{mpmc_queue_1_G : MpmcQueue1G Σ}.
Implicit Types t : location.
Record mpmc_queue_1_name :=
{ mpmc_queue_1_name_inv : namespace
; mpmc_queue_1_name_history : gname
; mpmc_queue_1_name_front : gname
; mpmc_queue_1_name_model : gname
; mpmc_queue_1_name_waiters : gname
}.
Implicit Type γ : mpmc_queue_1_name.
#[global] Instance mpmc_queue_1_name_eq_dec : EqDecision mpmc_queue_1_name :=
ltac:(solve_decision).
#[global] Instance mpmc_queue_1_name_countable :
Countable mpmc_queue_1_name.
#[local] Definition history_auth' γ_history hist :=
mono_list_auth γ_history (DfracOwn 1) hist.
#[local] Definition history_auth γ hist :=
history_auth' γ.(mpmc_queue_1_name_history) hist.
#[local] Definition history_at γ i node :=
mono_list_at γ.(mpmc_queue_1_name_history) i node.
#[local] Definition front_auth' γ_front i :=
auth_nat_max_auth γ_front (DfracOwn 1) i.
#[local] Definition front_auth γ i :=
front_auth' γ.(mpmc_queue_1_name_front) i.
#[local] Definition front_lb γ i :=
auth_nat_max_lb γ.(mpmc_queue_1_name_front) i.
#[local] Definition model₁' γ_model vs :=
twins_twin1 γ_model (DfracOwn 1) vs.
#[local] Definition model₁ γ vs :=
model₁' γ.(mpmc_queue_1_name_model) vs.
#[local] Definition model₂' γ_model vs :=
twins_twin2 γ_model vs.
#[local] Definition model₂ γ vs :=
model₂' γ.(mpmc_queue_1_name_model) vs.
#[local] Definition waiters_auth' γ_waiters waiters :=
ghost_map_auth γ_waiters 1 waiters.
#[local] Definition waiters_auth γ waiters :=
waiters_auth' γ.(mpmc_queue_1_name_waiters) waiters.
#[local] Definition waiters_at γ waiter i :=
ghost_map_elem γ.(mpmc_queue_1_name_waiters) waiter (DfracOwn 1) i.
#[local] Definition node_model γ node i b : iProp Σ :=
node ↦ₕ Header §Node 2 ∗
history_at γ i node ∗
if b then front_lb γ i else True%I.
#[local] Instance : CustomIpat "node_model" :=
" ( #H{}_header & #Hhistory_at_{} & {{front}#Hfront_lb_{};_} ) ".
#[local] Definition waiter_au γ (Ψ : bool → iProp Σ) : iProp Σ :=
AU <{
∃∃ vs,
model₁ γ vs
}> @ ⊤ ∖ ↑γ.(mpmc_queue_1_name_inv), ∅ <{
model₁ γ vs
, COMM
Ψ (bool_decide (vs = []))
}>.
#[local] Definition waiter_model γ past waiter i : iProp Σ :=
∃ Ψ,
saved_pred waiter Ψ ∗
if decide (i < length past) then
Ψ false
else
waiter_au γ Ψ.
#[local] Definition inv_inner t γ : iProp Σ :=
∃ hist past front nodes back vs waiters,
⌜hist = past ++ front :: nodes⌝ ∗
⌜back ∈ hist⌝ ∗
t.[front] ↦ #front ∗
t.[back] ↦ #back ∗
xtchain (Header §Node 2) (DfracOwn 1) hist §Null ∗
([∗ list] node; v ∈ nodes; vs, node.[data] ↦ v) ∗
history_auth γ hist ∗
front_auth γ (length past) ∗
model₂ γ vs ∗
waiters_auth γ waiters ∗
([∗ map] waiter ↦ i ∈ waiters, waiter_model γ past waiter i).
#[local] Instance : CustomIpat "inv_inner" :=
" ( %hist{} & %past{} & %front{} & %nodes{} & %back{} & %vs{} & %waiters{} & >%Hhist{} & >%Hback{} & >Ht_front & >Ht_back & >Hhist & >Hnodes & >Hhistory_auth & >Hfront_auth & >Hmodel₂ & >Hwaiters_auth & Hwaiters ) ".
#[local] Definition inv' t γ :=
inv γ.(mpmc_queue_1_name_inv) (inv_inner t γ).
Definition mpmc_queue_1_inv t γ ι : iProp Σ :=
⌜ι = γ.(mpmc_queue_1_name_inv)⌝ ∗
inv' t γ.
#[local] Instance : CustomIpat "inv" :=
" ( -> & #Hinv ) ".
Definition mpmc_queue_1_model :=
model₁.
#[local] Instance : CustomIpat "model" :=
" Hmodel₁{_{}} ".
#[global] Instance mpmc_queue_1_model_timeless γ vs :
Timeless (mpmc_queue_1_model γ vs).
#[global] Instance mpmc_queue_1_inv_persistent t γ ι :
Persistent (mpmc_queue_1_inv t γ ι).
#[local] Lemma history_alloc front :
⊢ |==>
∃ γ_history,
history_auth' γ_history [front].
#[local] Lemma history_at_get {γ hist} i node :
hist !! i = Some node →
history_auth γ hist ⊢
history_at γ i node.
#[local] Lemma history_at_lookup γ hist i node :
history_auth γ hist -∗
history_at γ i node -∗
⌜hist !! i = Some node⌝.
#[local] Lemma history_update {γ hist} node :
history_auth γ hist ⊢ |==>
history_auth γ (hist ++ [node]) ∗
history_at γ (length hist) node.
#[local] Lemma front_alloc :
⊢ |==>
∃ γ_front,
front_auth' γ_front 0.
#[local] Lemma front_lb_get γ i :
front_auth γ i ⊢
front_lb γ i.
#[local] Lemma front_lb_valid γ i1 i2 :
front_auth γ i1 -∗
front_lb γ i2 -∗
⌜i2 ≤ i1⌝.
#[local] Lemma front_update {γ i} i' :
i ≤ i' →
front_auth γ i ⊢ |==>
front_auth γ i'.
#[local] Lemma model_alloc :
⊢ |==>
∃ γ_model,
model₁' γ_model [] ∗
model₂' γ_model [].
#[local] Lemma model₁_exclusive γ vs1 vs2 :
model₁ γ vs1 -∗
model₁ γ vs2 -∗
False.
#[local] Lemma model_agree γ vs1 vs2 :
model₁ γ vs1 -∗
model₂ γ vs2 -∗
⌜vs1 = vs2⌝.
#[local] Lemma model_update {γ vs1 vs2} vs :
model₁ γ vs1 -∗
model₂ γ vs2 ==∗
model₁ γ vs ∗
model₂ γ vs.
#[local] Lemma waiters_alloc :
⊢ |==>
∃ γ_waiters,
waiters_auth' γ_waiters ∅.
#[local] Lemma waiters_insert {γ waiters} i Ψ :
waiters_auth γ waiters ⊢ |==>
∃ waiter,
waiters_auth γ (<[waiter := i]> waiters) ∗
saved_pred waiter Ψ ∗
waiters_at γ waiter i.
#[local] Lemma waiters_delete γ waiters waiter i :
waiters_auth γ waiters -∗
waiters_at γ waiter i ==∗
⌜waiters !! waiter = Some i⌝ ∗
waiters_auth γ (delete waiter waiters).
Lemma mpmc_queue_1_model_exclusive γ vs1 vs2 :
mpmc_queue_1_model γ vs1 -∗
mpmc_queue_1_model γ vs2 -∗
False.
Lemma mpmc_queue_1٠create𑁒spec ι :
{{{
True
}}}
mpmc_queue_1٠create ()
{{{
t γ
, RET #t;
meta_token t ⊤ ∗
mpmc_queue_1_inv t γ ι ∗
mpmc_queue_1_model γ []
}}}.
#[local] Lemma front𑁒spec_strong Ψ t γ :
{{{
inv' t γ ∗
if Ψ is Some Ψ then
waiter_au γ Ψ
else
True
}}}
(#t).{front}
{{{
front i
, RET #front;
node_model γ front i true ∗
if Ψ is Some Ψ then
∃ waiter,
saved_pred waiter Ψ ∗
waiters_at γ waiter i
else
True
}}}.
#[local] Lemma front𑁒spec t γ :
{{{
inv' t γ
}}}
(#t).{front}
{{{
front i
, RET #front;
node_model γ front i true
}}}.
#[local] Lemma back𑁒spec t γ :
{{{
inv' t γ
}}}
(#t).{back}
{{{
back i
, RET #back;
node_model γ back i false
}}}.
Variant operation :=
| IsEmpty waiter (Ψ : bool → iProp Σ)
| Pop (Ψ : option val → iProp Σ)
| Other.
Implicit Types op : operation.
Variant operation' :=
| IsEmpty'
| Pop'
| Other'.
#[local] Instance operation'_eq_dec : EqDecision operation' :=
ltac:(solve_decision).
#[local] Coercion operation_to_operation' op :=
match op with
| IsEmpty _ _ ⇒
IsEmpty'
| Pop _ ⇒
Pop'
| Other ⇒
Other'
end.
#[local] Definition pop_au γ (Ψ : option val → iProp Σ) : iProp Σ :=
AU <{
∃∃ vs,
model₁ γ vs
}> @ ⊤ ∖ ↑γ.(mpmc_queue_1_name_inv), ∅ <{
model₁ γ (tail vs)
, COMM
Ψ (head vs)
}>.
#[local] Lemma next𑁒spec_aux op t γ i node :
{{{
inv' t γ ∗
history_at γ i node ∗
( if decide (op = Other' :> operation') then True else
front_lb γ i
) ∗
match op with
| IsEmpty waiter Ψ ⇒
saved_pred waiter Ψ ∗
waiters_at γ waiter i ∗
£ 1
| Pop Ψ ⇒
pop_au 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'⌝ ∗
node_model γ node' (S i) false ∗
match op with
| IsEmpty waiter Ψ ⇒
Ψ false
| Pop Ψ ⇒
pop_au γ Ψ
| Other ⇒
True
end
}}}.
#[local] Lemma next𑁒spec {t γ i} node :
{{{
inv' t γ ∗
history_at γ i node
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝
∨ ∃ node',
⌜res = #node'⌝ ∗
node_model γ node' (S i) false
}}}.
#[local] Lemma next𑁒spec_is_empty {t γ i node} waiter Ψ :
{{{
inv' t γ ∗
history_at γ i node ∗
front_lb γ i ∗
saved_pred waiter Ψ ∗
waiters_at γ waiter i ∗
£ 1
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝ ∗
Ψ true
∨ ∃ node',
⌜res = #node'⌝ ∗
node_model γ node' (S i) false ∗
Ψ false
}}}.
#[local] Lemma next𑁒spec_pop {t γ i node} Ψ :
{{{
inv' t γ ∗
history_at γ i node ∗
front_lb γ i ∗
pop_au γ Ψ
}}}
(#node).{next}
{{{
res
, RET res;
⌜res = §Null%V⌝ ∗
Ψ None
∨ ∃ node',
⌜res = #node'⌝ ∗
node_model γ node' (S i) false ∗
pop_au γ Ψ
}}}.
Lemma mpmc_queue_1٠is_empty𑁒spec t γ ι :
<<<
mpmc_queue_1_inv t γ ι
| ∀∀ vs,
mpmc_queue_1_model γ vs
>>>
mpmc_queue_1٠is_empty #t @ ↑ι
<<<
mpmc_queue_1_model γ vs
| RET #(bool_decide (vs = []%list));
£ 1
>>>.
Lemma mpmc_queue_1٠is_empty𑁒spec' t γ ι :
{{{
mpmc_queue_1_inv t γ ι
}}}
mpmc_queue_1٠is_empty #t
{{{
b
, RET #b;
True
}}}.
#[local] Lemma mpmc_queue_1٠push₀𑁒spec t γ i node new_back v :
<<<
inv' t γ ∗
node_model γ node i false ∗
new_back ↦ₕ Header §Node 2 ∗
new_back.[next] ↦ §Null ∗
new_back.[data] ↦ v
| ∀∀ vs,
mpmc_queue_1_model γ vs
>>>
mpmc_queue_1٠push₀ #node #new_back @ ↑γ.(mpmc_queue_1_name_inv)
<<<
mpmc_queue_1_model γ (vs ++ [v])
| RET ();
∃ j,
history_at γ j new_back
>>>.
#[local] Lemma mpmc_queue_1٠fix_back𑁒spec t γ i back j new_back :
{{{
inv' t γ ∗
history_at γ i back ∗
node_model γ new_back j false
}}}
mpmc_queue_1٠fix_back #t #back #new_back
{{{
RET ();
True
}}}.
Lemma mpmc_queue_1٠push𑁒spec t γ ι v :
<<<
mpmc_queue_1_inv t γ ι
| ∀∀ vs,
mpmc_queue_1_model γ vs
>>>
mpmc_queue_1٠push #t v @ ↑ι
<<<
mpmc_queue_1_model γ (vs ++ [v])
| RET ();
£ 1
>>>.
Lemma mpmc_queue_1٠pop𑁒spec t γ ι :
<<<
mpmc_queue_1_inv t γ ι
| ∀∀ vs,
mpmc_queue_1_model γ vs
>>>
mpmc_queue_1٠pop #t @ ↑ι
<<<
mpmc_queue_1_model γ (tail vs)
| RET head vs;
£ 1
>>>.
End mpmc_queue_1_G.
#[global] Opaque mpmc_queue_1_inv.
#[global] Opaque mpmc_queue_1_model.
End base.
From zoo_saturn Require
mpmc_queue_1__opaque.
Section mpmc_queue_1_G.
Context `{mpmc_queue_1_G : MpmcQueue1G Σ}.
Implicit Types 𝑡 : location.
Implicit Types t : val.
Definition mpmc_queue_1_inv t ι : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
meta 𝑡 nroot γ ∗
base.mpmc_queue_1_inv 𝑡 γ ι.
#[local] Instance : CustomIpat "inv" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hinv{_{}} ) ".
Definition mpmc_queue_1_model t vs : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
meta 𝑡 nroot γ ∗
base.mpmc_queue_1_model γ vs.
#[local] Instance : CustomIpat "model" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hmodel{_{}} ) ".
#[global] Instance mpmc_queue_1_model_timeless t vs :
Timeless (mpmc_queue_1_model t vs).
#[global] Instance mpmc_queue_1_inv_persistent t ι :
Persistent (mpmc_queue_1_inv t ι).
Lemma mpmc_queue_1_model_exclusive t vs1 vs2 :
mpmc_queue_1_model t vs1 -∗
mpmc_queue_1_model t vs2 -∗
False.
Lemma mpmc_queue_1٠create𑁒spec ι :
{{{
True
}}}
mpmc_queue_1٠create ()
{{{
t
, RET t;
mpmc_queue_1_inv t ι ∗
mpmc_queue_1_model t []
}}}.
Lemma mpmc_queue_1٠is_empty𑁒spec t ι :
<<<
mpmc_queue_1_inv t ι
| ∀∀ vs,
mpmc_queue_1_model t vs
>>>
mpmc_queue_1٠is_empty t @ ↑ι
<<<
mpmc_queue_1_model t vs
| RET #(bool_decide (vs = []%list));
£ 1
>>>.
Lemma mpmc_queue_1٠is_empty𑁒spec' t ι :
{{{
mpmc_queue_1_inv t ι
}}}
mpmc_queue_1٠is_empty t
{{{
b
, RET #b;
True
}}}.
Lemma mpmc_queue_1٠push𑁒spec t ι v :
<<<
mpmc_queue_1_inv t ι
| ∀∀ vs,
mpmc_queue_1_model t vs
>>>
mpmc_queue_1٠push t v @ ↑ι
<<<
mpmc_queue_1_model t (vs ++ [v])
| RET ();
£ 1
>>>.
Lemma mpmc_queue_1٠pop𑁒spec t ι :
<<<
mpmc_queue_1_inv t ι
| ∀∀ vs,
mpmc_queue_1_model t vs
>>>
mpmc_queue_1٠pop t @ ↑ι
<<<
mpmc_queue_1_model t (tail vs)
| RET head vs;
£ 1
>>>.
End mpmc_queue_1_G.
#[global] Opaque mpmc_queue_1_inv.
#[global] Opaque mpmc_queue_1_model.
Section mpmc_queue_1_G.
Context `{mpmc_queue_1_G : MpmcQueue1G Σ}.
Context τ `{!iType (iProp Σ) τ}.
#[local] Definition itype_inner t : iProp Σ :=
∃ vs,
mpmc_queue_1_model t vs ∗
[∗ list] v ∈ vs, τ v.
#[local] Instance : CustomIpat "itype_inner" :=
" ( %vs & >Hmodel & #Hvs ) ".
Definition itype_mpmc_queue_1 t : iProp Σ :=
mpmc_queue_1_inv t (nroot.@"1") ∗
inv (nroot.@"2") (itype_inner t).
#[local] Instance : CustomIpat "itype" :=
" ( #Hinv1 & #Hinv2 ) ".
#[global] Instance itype_mpmc_queue_1_itype :
iType _ itype_mpmc_queue_1.
Lemma mpmc_queue_1٠create𑁒type :
{{{
True
}}}
mpmc_queue_1٠create ()
{{{
t
, RET t;
itype_mpmc_queue_1 t
}}}.
Lemma mpmc_queue_1٠is_empty𑁒type t :
{{{
itype_mpmc_queue_1 t
}}}
mpmc_queue_1٠is_empty t
{{{
b
, RET #b;
True
}}}.
Lemma mpmc_queue_1٠push𑁒type t v :
{{{
itype_mpmc_queue_1 t ∗
τ v
}}}
mpmc_queue_1٠push t v
{{{
RET ();
True
}}}.
Lemma mpmc_queue_1٠pop𑁒type t :
{{{
itype_mpmc_queue_1 t
}}}
mpmc_queue_1٠pop t
{{{
o
, RET o;
itype_option τ o
}}}.
End mpmc_queue_1_G.
#[global] Opaque itype_mpmc_queue_1.