Library zoo_saturn.inf_ws_deque_1
Require Import zoo.prelude.
Require Import zoo.common.countable.
Require Import zoo.common.function.
Require Import zoo.common.relations.
Require Import zoo.iris.base_logic.lib.excl.
Require Import zoo.iris.base_logic.lib.twins.
Require Import zoo.iris.base_logic.lib.auth_twins.
Require Import zoo.iris.base_logic.lib.auth_nat_max.
Require Import zoo.iris.base_logic.lib.mono_list.
Require Import zoo.base.
Require Import zoo.program_logic.prophet_identifier.
Require Import zoo.program_logic.prophet_multi.
Require Import zoo_std.option.
Require Export zoo_saturn.inf_ws_deque_1__code.
Require Import zoo_saturn.inf_ws_deque_1__types.
Require Import zoo.options.
Implicit Type front back : nat.
Implicit Type id : prophet_id.
Implicit Type v : val.
Implicit Type vs ws hist lhist : list val.
Implicit Type priv : nat → val.
Implicit Type past prophs : list prophet_identifier.(prophet_typed۰type).
Implicit Type pasts prophss : nat → list prophet_identifier.(prophet_typed۰type).
Variant state :=
| Empty
| Nonempty
| Emptyish
| Superempty.
Implicit Type state : state.
#[local] Instance stateｰinhabited : Inhabited state :=
populate Empty.
Variant stability :=
| Stable
| Unstable.
Implicit Type stable : stability.
#[local] Instance stabilityｰinhabited : Inhabited stability :=
populate Stable.
Class InfWsDeque1G Σ `{zoo۰G : !ZooG Σ} :=
{ #[local] inf_ws_deque_1۰G۰inf_array۰G :: InfArrayG Σ
; #[local] inf_ws_deque_1۰G۰prophet۰G :: ProphetMultiG Σ prophet_identifier
; #[local] inf_ws_deque_1۰G۰model۰G :: AuthTwinsG Σ (leibnizO (list val)) suffix
; #[local] inf_ws_deque_1۰G۰owner۰G :: TwinsG Σ (leibnizO (stability × nat × (nat → val)))
; #[local] inf_ws_deque_1۰G۰front۰G :: AuthNatMaxG Σ
; #[local] inf_ws_deque_1۰G۰history۰G :: MonoListG Σ val
; #[local] inf_ws_deque_1۰G۰winner۰G :: TwinsG Σ (natO × ▶ ∙)
}.
Definition inf_ws_deque_1۰Σ :=
#[inf_array۰Σ
; prophet_multi۰Σ prophet_identifier
; auth_twins۰Σ (leibnizO (list val)) suffix
; twins۰Σ (leibnizO (stability × nat × (nat → val)))
; auth_nat_max۰Σ
; mono_list۰Σ val
; twins۰Σ (natO × ▶ ∙)
].
#[global] Instance subGｰinf_ws_deque_1۰Σ Σ `{zoo۰G : !ZooG Σ} :
subG inf_ws_deque_1۰Σ Σ →
InfWsDeque1G Σ .
Module base.
Section inf_ws_deque_1۰G.
Context `{inf_ws_deque_1۰G : InfWsDeque1G Σ}.
Implicit Type t : location.
Implicit Type P : iProp Σ.
Record inf_ws_deque_1۰name :=
{ inf_ws_deque_1۰name۰data : val
; inf_ws_deque_1۰name۰inv : namespace
; inf_ws_deque_1۰name۰prophet : prophet_id
; inf_ws_deque_1۰name۰prophet_name : prophet_multi۰name
; inf_ws_deque_1۰name۰model : auth_twins۰name
; inf_ws_deque_1۰name۰owner : gname
; inf_ws_deque_1۰name۰front : gname
; inf_ws_deque_1۰name۰history : gname
; inf_ws_deque_1۰name۰winner : gname
}.
Implicit Type γ : inf_ws_deque_1۰name.
#[global] Instance inf_ws_deque_1۰nameｰeq_dec : EqDecision inf_ws_deque_1۰name :=
ltac:(solve_decision).
#[global] Instance inf_ws_deque_1۰nameｰcountable :
Countable inf_ws_deque_1۰name.
#[local] Definition model₁' γ_model vs :=
auth_twins۰twin₁ _ γ_model vs.
#[local] Definition model₁ γ :=
model₁' γ.(inf_ws_deque_1۰name۰model).
#[local] Definition model₂' γ_model vs :=
auth_twins۰twin₂ _ γ_model vs.
#[local] Definition model₂ γ :=
model₂' γ.(inf_ws_deque_1۰name۰model).
#[local] Definition owner₁' γ_owner γ_model stable back priv ws : iProp Σ :=
twins۰twin₁ (twins۰G := inf_ws_deque_1۰G۰owner۰G) γ_owner (DfracOwn 1) (stable, back, priv) ∗
auth_twins۰auth _ γ_model ws.
#[local] Definition owner₁ γ :=
owner₁' γ.(inf_ws_deque_1۰name۰owner) γ.(inf_ws_deque_1۰name۰model).
#[local] Instance : CustomIpat "owner₁" :=
" ( Howner₁{_{}} & Hmodel_auth{_{}} ) ".
#[local] Definition owner₂' γ_owner stable back priv :=
twins۰twin₂ (twins۰G := inf_ws_deque_1۰G۰owner۰G) γ_owner (stable, back, priv).
#[local] Definition owner₂ γ :=
owner₂' γ.(inf_ws_deque_1۰name۰owner).
#[local] Definition front۰auth' γ_front :=
auth_nat_max۰auth γ_front (DfracOwn 1).
#[local] Definition front۰auth γ :=
front۰auth' γ.(inf_ws_deque_1۰name۰front).
#[local] Definition front۰lb γ :=
auth_nat_max۰lb γ.(inf_ws_deque_1۰name۰front).
#[local] Definition history۰auth' γ_history :=
mono_list۰auth γ_history (DfracOwn 1).
#[local] Definition history۰auth γ :=
history۰auth' γ.(inf_ws_deque_1۰name۰history).
#[local] Definition history۰at γ :=
mono_list۰at γ.(inf_ws_deque_1۰name۰history).
#[local] Definition winner۰pop' γ_winner front P : iProp Σ :=
twins۰twin₁ γ_winner (DfracOwn 1) (front, Next P).
#[local] Definition winner۰pop γ :=
winner۰pop' γ.(inf_ws_deque_1۰name۰winner).
#[local] Definition winner۰steal' γ_winner front P :=
twins۰twin₂ γ_winner (front, Next P).
#[local] Definition winner۰steal γ :=
winner۰steal' γ.(inf_ws_deque_1۰name۰winner).
#[local] Definition winner γ : iProp Σ :=
∃ front P1 P2,
winner۰pop γ front P1 ∗
winner۰steal γ front P2.
#[local] Instance : CustomIpat "winner" :=
" ( %front_winner & %P_winner_1 & %P_winner_2 & Hwinner_pop{_{}} & Hwinner_steal{_{}} ) ".
#[local] Definition winner۰au γ front P : iProp Σ :=
AU <{
∃∃ vs,
model₁ γ vs
}> @ ⊤ ∖ ↑γ.(inf_ws_deque_1۰name۰inv), ∅ <{
∀∀ v vs',
⌜vs = v :: vs'⌝ ∗
model₁ γ vs' ∗
history۰at γ front v
, COMM
P
}>.
#[local] Definition winner۰pending₁ γ front P id : iProp Σ :=
winner۰steal γ front P ∗
identifier۰model id ∗
winner۰au γ front P.
#[local] Instance : CustomIpat "winner۰pending₁" :=
" ( Hwinner_steal{_{!}} & Hid{_{!}} & HP ) ".
#[local] Definition winner۰pending₂ γ front id : iProp Σ :=
∃ P,
winner۰pending₁ γ front P id.
#[local] Instance : CustomIpat "winner۰pending₂" :=
" ( %P{} & (:winner۰pending₁) ) ".
#[local] Definition winner۰linearized γ front P : iProp Σ :=
winner۰steal γ front P ∗
P.
#[local] Instance : CustomIpat "winner۰linearized" :=
" ( Hwinner_steal{_{!}} & HP ) ".
#[local] Definition inv۰state۰empty γ stable front back hist lhist : iProp Σ :=
⌜stable = Stable⌝ ∗
⌜front = back⌝ ∗
⌜lhist = hist⌝ ∗
⌜length hist = front⌝ ∗
winner γ.
#[local] Instance : CustomIpat "inv۰state۰empty" :=
" ( { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}-> & {>;}-> & {>;}%Hhist{} & Hwinner ) ".
#[local] Definition inv۰state۰nonempty γ stable front back hist lhist vs prophs : iProp Σ :=
⌜stable = Stable⌝ ∗
⌜front < back⌝ ∗
⌜lhist = hist ++ take 1 vs⌝ ∗
⌜length hist = front⌝ ∗
( winner γ
∨ match prophs with
| [] ⇒
False
| id :: _ ⇒
winner۰pending₂ γ front id
end
).
#[local] Instance : CustomIpat "inv۰state۰nonempty" :=
" ( { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}% & {>;}-> & {>;}%Hhist{} & Hwinner ) ".
#[local] Definition inv۰state۰nonempty۰steal γ state stable front back hist lhist vs prophs P : iProp Σ :=
⌜state = Nonempty⌝ ∗
⌜stable = Stable⌝ ∗
⌜front < back⌝ ∗
⌜lhist = hist ++ take 1 vs⌝ ∗
⌜length hist = front⌝ ∗
match prophs with
| [] ⇒
False
| id :: _ ⇒
winner۰pending₁ γ front P id
end.
#[local] Instance : CustomIpat "inv۰state۰nonempty۰steal" :=
" ( {>;}-> & { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}% & {>;}-> & {>;}%Hhist{} & (:winner۰pending₁) ) ".
#[local] Definition inv۰state۰emptyish γ stable front back hist lhist : iProp Σ :=
∃ P,
⌜stable = Unstable⌝ ∗
⌜front = back⌝ ∗
⌜lhist = hist⌝ ∗
⌜length hist = ˖front⌝ ∗
( winner۰pop γ front P
∨ winner۰linearized γ front P
).
#[local] Instance : CustomIpat "inv۰state۰emptyish" :=
" ( %P_ & { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}-> & {>;}-> & {>;}%Hhist{} & Hwinner ) ".
#[local] Definition inv۰state۰emptyish۰pop γ state stable front back hist lhist P : iProp Σ :=
⌜state = Emptyish⌝ ∗
⌜stable = Unstable⌝ ∗
⌜front = back⌝ ∗
⌜lhist = hist⌝ ∗
⌜length hist = ˖front⌝ ∗
winner۰pop γ front P.
#[local] Instance : CustomIpat "inv۰state۰emptyish۰pop" :=
" ( {>;}-> & { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}-> & {>;}-> & {>;}%Hhist{} & Hwinner_pop ) ".
#[local] Definition inv۰state۰emptyish۰steal γ state stable front back hist lhist P : iProp Σ :=
⌜state = Emptyish⌝ ∗
⌜stable = Unstable⌝ ∗
⌜front = back⌝ ∗
⌜lhist = hist⌝ ∗
⌜length hist = ˖front⌝ ∗
winner۰linearized γ front P.
#[local] Instance : CustomIpat "inv۰state۰emptyish۰steal" :=
" ( {>;}-> & { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}-> & {>;}-> & {>;}%Hhist{} & (:winner۰linearized) ) ".
#[local] Definition inv۰state۰superempty γ stable front back hist lhist : iProp Σ :=
⌜stable = Unstable⌝ ∗
⌜front = ˖back⌝ ∗
⌜lhist = hist⌝ ∗
⌜length hist = front⌝ ∗
winner γ.
#[local] Instance : CustomIpat "inv۰state۰superempty" :=
" ( { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}-> & {>;}-> & {>;}%Hhist{} & Hwinner ) ".
#[local] Definition inv۰state γ state stable front back hist lhist vs prophs : iProp Σ :=
match state with
| Empty ⇒
inv۰state۰empty γ stable front back hist lhist
| Nonempty ⇒
inv۰state۰nonempty γ stable front back hist lhist vs prophs
| Emptyish ⇒
inv۰state۰emptyish γ stable front back hist lhist
| Superempty ⇒
inv۰state۰superempty γ stable front back hist lhist
end.
#[local] Definition inv۰inner t γ : iProp Σ :=
∃ state stable front back hist lhist vs priv pasts prophss,
t.[front] ↦ #front ∗
t.[back] ↦ #back ∗
owner₂ γ stable back priv ∗
front۰auth γ front ∗
⌜0 < front⌝ ∗
model₂ γ vs ∗
⌜length vs = back - front⌝ ∗
inf_array۰model' γ.(inf_ws_deque_1۰name۰data) (hist ++ vs) priv ∗
history۰auth γ lhist ∗
prophet_multi۰model prophet_identifier γ.(inf_ws_deque_1۰name۰prophet) γ.(inf_ws_deque_1۰name۰prophet_name) pasts prophss ∗
⌜∀ i, front ≤ i → pasts i = []⌝ ∗
inv۰state γ state stable front back hist lhist vs (prophss front).
#[local] Instance : CustomIpat "inv۰inner" :=
" ( %state{} & %stable{} & %front{} & %back{} & %hist{} & %lhist{} & %vs{} & %priv{} & %pasts{} & %prophss{} & >Ht_front & >Ht_back & >Howner₂ & >Hfront_auth & >%Hfront{} & >Hmodel₂ & >%Hvs{} & >Hdata_model & >Hhistory_auth & >Hprophet_model & >%Hpasts{} & Hstate ) ".
#[local] Definition inv' t γ : iProp Σ :=
t.[data] ↦□ γ.(inf_ws_deque_1۰name۰data) ∗
t.[proph] ↦□ #γ.(inf_ws_deque_1۰name۰prophet) ∗
inf_array۰inv γ.(inf_ws_deque_1۰name۰data) ∗
inv γ.(inf_ws_deque_1۰name۰inv) (inv۰inner t γ).
#[local] Instance : CustomIpat "inv'" :=
" ( #Ht_data & #Ht_proph & #Hdata_inv & #Hinv ) ".
Definition inf_ws_deque_1۰inv t γ ι : iProp Σ :=
⌜ι = γ.(inf_ws_deque_1۰name۰inv)⌝ ∗
inv' t γ.
#[local] Instance : CustomIpat "inv" :=
" ( -> & (:inv') ) ".
Definition inf_ws_deque_1۰model :=
model₁.
#[local] Instance : CustomIpat "model" :=
" Hmodel₁{_{}} ".
Definition inf_ws_deque_1۰owner γ ws : iProp Σ :=
∃ back priv,
owner₁ γ Stable back priv ws.
#[local] Instance : CustomIpat "owner" :=
" ( %back{} & %priv{} & Howner₁{_{}} ) ".
#[global] Instance inf_ws_deque_1۰modelｰtimeless γ vs :
Timeless (inf_ws_deque_1۰model γ vs).
#[global] Instance inf_ws_deque_1۰ownerｰtimeless γ ws :
Timeless (inf_ws_deque_1۰owner γ ws).
#[global] Instance inf_ws_deque_1۰invｰpersistent t γ ι :
Persistent (inf_ws_deque_1۰inv t γ ι).
#[local] Lemma modelｰownerｰalloc :
⊢ |==>
∃ γ_model γ_owner,
model₁' γ_model [] ∗
model₂' γ_model [] ∗
owner₁' γ_owner γ_model Stable 1 (λ _, ()%V) [] ∗
owner₂' γ_owner Stable 1 (λ _, ()%V).
#[local] Lemma model₁ｰvalid γ stable back priv ws vs :
owner₁ γ stable back priv ws -∗
model₁ γ vs -∗
⌜vs `suffix_of` ws⌝.
#[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 model۰owner₁ｰagree γ stable back priv ws vs1 vs2 :
owner₁ γ stable back priv ws -∗
model₁ γ vs1 -∗
model₂ γ vs2 -∗
⌜vs1 `suffix_of` ws⌝ ∗
⌜vs1 = vs2⌝.
#[local] Lemma modelｰempty {γ stable back priv ws vs1 vs2} :
owner₁ γ stable back priv ws -∗
model₁ γ vs1 -∗
model₂ γ vs2 ==∗
owner₁ γ stable back priv [] ∗
model₁ γ [] ∗
model₂ γ [].
#[local] Lemma modelｰpush {γ stable back priv ws vs1 vs2} v :
owner₁ γ stable back priv ws -∗
model₁ γ vs1 -∗
model₂ γ vs2 ==∗
owner₁ γ stable back priv (vs1 ++ [v]) ∗
model₁ γ (vs1 ++ [v]) ∗
model₂ γ (vs1 ++ [v]).
#[local] Lemma modelｰsteal γ vs1 vs2 :
model₁ γ vs1 -∗
model₂ γ vs2 ==∗
model₁ γ (tail vs1) ∗
model₂ γ (tail vs1).
#[local] Lemma modelｰpop γ stable back priv ws vs1 vs2 :
owner₁ γ stable back priv ws -∗
model₁ γ vs1 -∗
model₂ γ vs2 ==∗
owner₁ γ stable back priv (removelast vs1) ∗
model₁ γ (removelast vs1) ∗
model₂ γ (removelast vs1).
#[local] Lemma modelｰpop' γ stable back priv ws vs1 v vs2 :
owner₁ γ stable back priv ws -∗
model₁ γ (vs1 ++ [v]) -∗
model₂ γ vs2 ==∗
owner₁ γ stable back priv vs1 ∗
model₁ γ vs1 ∗
model₂ γ vs1.
#[local] Lemma owner₁ｰexclusive γ stable1 back1 priv1 ws1 stable2 back2 priv2 ws2 :
owner₁ γ stable1 back1 priv1 ws1 -∗
owner₁ γ stable2 back2 priv2 ws2 -∗
False.
#[local] Lemma ownerｰagree γ stable1 back1 priv1 ws stable2 back2 priv2 :
owner₁ γ stable1 back1 priv1 ws -∗
owner₂ γ stable2 back2 priv2 -∗
⌜stable1 = stable2⌝ ∗
⌜back1 = back2⌝ ∗
⌜priv1 = priv2⌝.
#[local] Lemma owner₁ｰupdate γ stable back priv ws vs :
owner₁ γ stable back priv ws -∗
model₁ γ vs -∗
model₂ γ vs ==∗
owner₁ γ stable back priv vs ∗
model₁ γ vs ∗
model₂ γ vs.
#[local] Lemma ownerｰupdate {γ stable1 back1 priv1 ws stable2 back2 priv2} stable back priv :
owner₁ γ stable1 back1 priv1 ws -∗
owner₂ γ stable2 back2 priv2 ==∗
owner₁ γ stable back priv ws ∗
owner₂ γ stable back priv.
#[local] Lemma frontｰalloc :
⊢ |==>
∃ γ_front,
front۰auth' γ_front 1.
#[local] Lemma front۰lbｰget γ front :
front۰auth γ front ⊢
front۰lb γ front.
#[local] Lemma front۰lbｰle {γ front} front' :
front' ≤ front →
front۰lb γ front ⊢
front۰lb γ front'.
#[local] Lemma front۰lbｰvalid γ front1 front2 :
front۰auth γ front1 -∗
front۰lb γ front2 -∗
⌜front2 ≤ front1⌝.
#[local] Lemma frontｰupdate γ front :
front۰auth γ front ⊢ |==>
front۰auth γ ˖front.
#[local] Lemma historyｰalloc :
⊢ |==>
∃ γ_hist,
history۰auth' γ_hist [()%V].
#[local] Lemma history۰atｰget {γ hist v} i :
i = length hist →
history۰auth γ (hist ++ [v]) ⊢
history۰at γ i v.
#[local] Lemma history۰atｰlookup γ hist i v :
history۰auth γ hist -∗
history۰at γ i v -∗
⌜hist !! i = Some v⌝.
#[local] Lemma history۰atｰagree γ i v1 v2 :
history۰at γ i v1 -∗
history۰at γ i v2 -∗
⌜v1 = v2⌝.
#[local] Lemma historyｰupdate {γ hist} i v :
i = length hist →
history۰auth γ hist ⊢ |==>
history۰auth γ (hist ++ [v]) ∗
history۰at γ i v.
#[local] Lemma winnerｰalloc :
⊢ |==>
∃ γ_winner,
winner۰pop' γ_winner 1 True ∗
winner۰steal' γ_winner 1 True.
#[local] Lemma winner۰popｰexclusive γ front1 P1 front2 P2 :
winner۰pop γ front1 P1 -∗
winner۰pop γ front2 P2 -∗
False.
#[local] Lemma winner۰popｰexclusive' γ front P :
winner۰pop γ front P -∗
winner γ -∗
False.
#[local] Lemma winner۰stealｰexclusive γ front1 P1 front2 P2 :
winner۰steal γ front1 P1 -∗
winner۰steal γ front2 P2 -∗
False.
#[local] Lemma winner۰stealｰexclusive' γ front P :
winner۰steal γ front P -∗
winner γ -∗
False.
#[local] Lemma winnerｰagree γ front1 P1 front2 P2 :
winner۰pop γ front1 P1 -∗
winner۰steal γ front2 P2 -∗
⌜front1 = front2⌝ ∗
▷ (P1 ≡ P2).
#[local] Lemma winnerｰupdate {γ front1 P1 front2 P2} front P :
winner۰pop γ front1 P1 -∗
winner۰steal γ front2 P2 ==∗
winner۰pop γ front P ∗
winner۰steal γ front P.
Opaque owner₁'.
Lemma inf_ws_deque_1۰modelｰexclusive γ vs1 vs2 :
inf_ws_deque_1۰model γ vs1 -∗
inf_ws_deque_1۰model γ vs2 -∗
False.
Lemma inf_ws_deque_1۰ownerｰexclusive γ ws1 ws2 :
inf_ws_deque_1۰owner γ ws1 -∗
inf_ws_deque_1۰owner γ ws2 -∗
False.
Lemma inf_ws_deque_1۰ownerｰmodel γ ws vs :
inf_ws_deque_1۰owner γ ws -∗
inf_ws_deque_1۰model γ vs -∗
⌜vs `suffix_of` ws⌝.
#[local] Lemma inv۰stateｰStable γ state front back hist lhist vs prophs :
length vs = back - front →
inv۰state γ state Stable front back hist lhist vs prophs ⊢
⌜state = Empty ∨ state = Nonempty⌝ ∗
⌜front ≤ back⌝ ∗
⌜length (hist ++ vs) = back⌝.
#[local] Lemma inv۰stateｰUnstable γ state front back hist lhist vs prophs :
inv۰state γ state Unstable front back hist lhist vs prophs ⊢
⌜state = Emptyish ∨ state = Superempty⌝ ∗
⌜front = back ∨ front = ˖back⌝.
#[local] Lemma inv۰stateｰNonempty γ state stable front back hist lhist vs prophs :
front < back →
inv۰state γ state stable front back hist lhist vs prophs ⊢
⌜state = Nonempty⌝.
#[local] Lemma inv۰stateｰSuperempty γ state front back hist lhist vs prophs :
back < front →
inv۰state γ state Unstable front back hist lhist vs prophs -∗
⌜state = Superempty⌝.
#[local] Lemma inv۰stateｰwinner۰pop γ state stable front1 back hist lhist vs prophs front2 P :
inv۰state γ state stable front1 back hist lhist vs prophs -∗
winner۰pop γ front2 P -∗
∃ P_,
⌜front1 = front2⌝ ∗
▷ (P ≡ P_) ∗
( inv۰state۰nonempty۰steal γ state stable front2 back hist lhist vs prophs P_
∨ inv۰state۰emptyish۰steal γ state stable front2 back hist lhist P_
) ∗
winner۰pop γ front2 P.
#[local] Lemma inv۰stateｰwinner۰steal γ state stable front1 back hist lhist vs prophs front2 P :
inv۰state γ state stable front1 back hist lhist vs prophs -∗
winner۰steal γ front2 P -∗
∃ P_,
⌜front1 = front2⌝ ∗
▷ (P_ ≡ P) ∗
inv۰state۰emptyish۰pop γ state stable front2 back hist lhist P_ ∗
winner۰steal γ front2 P.
Lemma inf_ws_deque_1٠createｰspec ι :
{{{
True
}}}
inf_ws_deque_1٠create ()
{{{
t γ
, RET #t;
meta_token t ⊤ ∗
inf_ws_deque_1۰inv t γ ι ∗
inf_ws_deque_1۰model γ [] ∗
inf_ws_deque_1۰owner γ []
}}}.
#[local] Lemma frontｰspec t γ :
{{{
inv' t γ
}}}
(#t).{front}
{{{
front
, RET #front;
front۰lb γ front
}}}.
#[local] Lemma frontｰspecｰownerｰStable t γ back priv ws :
{{{
inv' t γ ∗
owner₁ γ Stable back priv ws
}}}
(#t).{front}
{{{
front
, RET #front;
owner₁ γ Stable back priv ws ∗
front۰lb γ front ∗
⌜front ≤ back⌝
}}}.
#[local] Lemma frontｰspecｰownerｰUnstable t γ back priv ws :
{{{
inv' t γ ∗
owner₁ γ Unstable back priv ws
}}}
(#t).{front}
{{{
front
, RET #front;
owner₁ γ Unstable back priv ws ∗
front۰lb γ front ∗
⌜front = back ∨ front = ˖back⌝
}}}.
#[local] Lemma frontｰspecｰSuperempty t γ back priv ws front :
back < front →
{{{
inv' t γ ∗
owner₁ γ Unstable back priv ws ∗
front۰lb γ front
}}}
(#t).{front}
{{{
RET #front;
owner₁ γ Unstable back priv ws
}}}.
#[local] Lemma frontｰspecｰwinner۰steal t γ front P :
{{{
inv' t γ ∗
winner۰steal γ front P
}}}
(#t).{front}
{{{
RET #front;
winner۰steal γ front P
}}}.
#[local] Lemma backｰspec t γ stable back priv ws :
{{{
inv' t γ ∗
owner₁ γ stable back priv ws
}}}
(#t).{back}
{{{
RET #back;
owner₁ γ stable back priv ws
}}}.
#[local] Lemma set_backｰspecｰSuperempty t γ back priv ws front (back' : Z) :
back < front →
back' = ˖back →
{{{
inv' t γ ∗
owner₁ γ Unstable back priv ws ∗
front۰lb γ front
}}}
#t <-{back} #back'
{{{
RET ();
owner₁ γ Stable ˖back priv ws
}}}.
#[local] Lemma inf_array٠getｰspecｰhistory t γ (i : nat) (i_ : Z) v :
i_ = i →
{{{
inv' t γ ∗
history۰at γ i v
}}}
inf_array٠get γ.(inf_ws_deque_1۰name۰data) #i_
{{{
RET v;
True
}}}.
#[local] Lemma inf_array٠getｰspecｰowner t γ back (back_ : Z) priv ws v :
back_ = back →
priv 0 = v →
{{{
inv' t γ ∗
owner₁ γ Stable back priv ws
}}}
inf_array٠get γ.(inf_ws_deque_1۰name۰data) #back_
{{{
RET v;
owner₁ γ Stable back priv ws
}}}.
#[local] Lemma inf_array٠setｰspecｰowner t γ back priv ws v :
{{{
inv' t γ ∗
owner₁ γ Stable back priv ws
}}}
inf_array٠set γ.(inf_ws_deque_1۰name۰data) #back v
{{{
RET ();
owner₁ γ Stable back (<[0 := v]> priv) ws
}}}.
#[local] Lemma resolveｰspecｰloser₁ t γ front1 front2 id :
front1 < front2 →
{{{
inv' t γ ∗
front۰lb γ front2
}}}
Resolve (CAS (#t).[front]%V #front1 #(front1 + 1)) #γ.(inf_ws_deque_1۰name۰prophet) (#front1, #id)%V
{{{
RET false;
True
}}}.
#[local] Lemma resolveｰspecｰloser₂ t γ front id prophs0 :
head prophs0 ≠ Some id →
{{{
inv' t γ ∗
front۰lb γ front ∗
prophet_multi۰full prophet_identifier γ.(inf_ws_deque_1۰name۰prophet_name) front prophs0
}}}
Resolve (CAS (#t).[front]%V #front #(front + 1)) #γ.(inf_ws_deque_1۰name۰prophet) (#front, #id)%V
{{{
RET false;
front۰lb γ ˖front
}}}.
#[local] Lemma resolveｰspecｰwinnerｰpop t γ front P id :
{{{
inv' t γ ∗
winner۰pop γ front P
}}}
Resolve (CAS (#t).[front]%V #front #(front + 1)) #γ.(inf_ws_deque_1۰name۰prophet) (#front, #id)%V
{{{
RET true;
▷ P
}}}.
#[local] Lemma resolveｰspecｰwinnerｰsteal t γ front P id :
{{{
inv' t γ ∗
winner۰steal γ front P
}}}
Resolve (CAS (#t).[front]%V #front #(front + 1)) #γ.(inf_ws_deque_1۰name۰prophet) (#front, #id)%V
{{{
RET true;
front۰lb γ ˖front
}}}.
#[local] Lemma resolveｰspecｰEmpty t γ back priv ws id :
{{{
inv' t γ ∗
owner₁ γ Stable back priv ws ∗
front۰lb γ back
}}}
Resolve (CAS (#t).[front]%V #back #(back + 1)) #γ.(inf_ws_deque_1۰name۰prophet) (#back, #id)%V
{{{
RET true;
owner₁ γ Unstable back (priv ∘ S) ws ∗
front۰lb γ ˖back ∗
history۰at γ back (priv 0)
}}}.
Lemma inf_ws_deque_1٠sizeｰspec t γ ι ws :
<<<
inf_ws_deque_1۰inv t γ ι ∗
inf_ws_deque_1۰owner γ ws
| ∀∀ vs,
inf_ws_deque_1۰model γ vs
>>>
inf_ws_deque_1٠size #t @ ↑ι
<<<
⌜vs `suffix_of` ws⌝ ∗
inf_ws_deque_1۰model γ vs
| RET #(length vs);
inf_ws_deque_1۰owner γ vs
>>>.
Lemma inf_ws_deque_1٠is_emptyｰspec t γ ι ws :
<<<
inf_ws_deque_1۰inv t γ ι ∗
inf_ws_deque_1۰owner γ ws
| ∀∀ vs,
inf_ws_deque_1۰model γ vs
>>>
inf_ws_deque_1٠is_empty #t @ ↑ι
<<<
⌜vs `suffix_of` ws⌝ ∗
inf_ws_deque_1۰model γ vs
| RET #(bool_decide (vs = []%list));
inf_ws_deque_1۰owner γ vs
>>>.
Lemma inf_ws_deque_1٠pushｰspec t γ ι ws v :
<<<
inf_ws_deque_1۰inv t γ ι ∗
inf_ws_deque_1۰owner γ ws
| ∀∀ vs,
inf_ws_deque_1۰model γ vs
>>>
inf_ws_deque_1٠push #t v @ ↑ι
<<<
⌜vs `suffix_of` ws⌝ ∗
inf_ws_deque_1۰model γ (vs ++ [v])
| RET ();
inf_ws_deque_1۰owner γ (vs ++ [v])
>>>.
Lemma inf_ws_deque_1٠stealｰspec t γ ι :
<<<
inf_ws_deque_1۰inv t γ ι
| ∀∀ vs,
inf_ws_deque_1۰model γ vs
>>>
inf_ws_deque_1٠steal #t @ ↑ι
<<<
inf_ws_deque_1۰model γ (tail vs)
| RET head vs;
True
>>>.
Variant pop۰state :=
| PopNonempty v
| PopEmptyishWinner v
| PopEmptyishLoser
| PopSuperempty.
#[local] Lemma inf_ws_deque_1٠pop₁ｰspec {t γ} (state : pop۰state) stable back (back_ : Z) priv ws id :
back_ = back →
{{{
inv' t γ ∗
owner₁ γ stable back priv ws ∗
match state with
| PopNonempty v ⇒
⌜stable = Stable⌝ ∗
⌜priv 0 = v⌝
| PopEmptyishWinner v ⇒
⌜stable = Unstable⌝ ∗
history۰at γ back v ∗
winner۰steal γ back inhabitant
| PopEmptyishLoser ⇒
∃ id_winner prophs,
⌜stable = Unstable⌝ ∗
prophet_multi۰full prophet_identifier γ.(inf_ws_deque_1۰name۰prophet_name) back (id_winner :: prophs) ∗
⌜head (id_winner :: prophs) ≠ Some id⌝
| PopSuperempty ⇒
∃ front,
⌜stable = Unstable⌝ ∗
front۰lb γ front ∗
⌜front = ˖back⌝
end
}}}
inf_ws_deque_1٠pop₁ #t #id #back_
{{{
o back priv
, RET o;
owner₁ γ Stable back priv ws ∗
match state with
| PopNonempty v ⇒
⌜o = Some v⌝
| PopEmptyishWinner v ⇒
⌜o = Some v⌝
| PopEmptyishLoser ⇒
⌜o = None⌝
| PopSuperempty ⇒
⌜o = None⌝
end
}}}.
Lemma inf_ws_deque_1٠popｰspec t γ ι ws :
<<<
inf_ws_deque_1۰inv t γ ι ∗
inf_ws_deque_1۰owner γ ws
| ∀∀ vs,
inf_ws_deque_1۰model γ vs
>>>
inf_ws_deque_1٠pop #t @ ↑ι
<<<
∃∃ o ws',
⌜vs `suffix_of` ws⌝ ∗
match o with
| None ⇒
⌜vs = []⌝ ∗
⌜ws' = []⌝ ∗
inf_ws_deque_1۰model γ []
| Some v ⇒
∃ vs',
⌜vs = vs' ++ [v]⌝ ∗
⌜ws' = vs'⌝ ∗
inf_ws_deque_1۰model γ vs'
end
| RET o;
inf_ws_deque_1۰owner γ ws'
>>>.
End inf_ws_deque_1۰G.
#[global] Opaque inf_ws_deque_1۰inv.
#[global] Opaque inf_ws_deque_1۰model.
#[global] Opaque inf_ws_deque_1۰owner.
End base.
Require zoo_saturn.inf_ws_deque_1__opaque.
Section inf_ws_deque_1۰G.
Context `{inf_ws_deque_1۰G : InfWsDeque1G Σ}.
Implicit Type 𝑡 : location.
Implicit Type t : val.
Definition inf_ws_deque_1۰inv t ι : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
𝑡 ↪ γ ∗
base.inf_ws_deque_1۰inv 𝑡 γ ι.
#[local] Instance : CustomIpat "inv" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hinv{_{}} ) ".
Definition inf_ws_deque_1۰model t vs : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
𝑡 ↪ γ ∗
base.inf_ws_deque_1۰model γ vs.
#[local] Instance : CustomIpat "model" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hmodel{_{}} ) ".
Definition inf_ws_deque_1۰owner t ws : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
𝑡 ↪ γ ∗
base.inf_ws_deque_1۰owner γ ws.
#[local] Instance : CustomIpat "owner" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Howner{_{}} ) ".
#[global] Instance inf_ws_deque_1۰modelｰtimeless γ vs :
Timeless (inf_ws_deque_1۰model γ vs).
#[global] Instance inf_ws_deque_1۰ownerｰtimeless γ ws :
Timeless (inf_ws_deque_1۰owner γ ws).
#[global] Instance inf_ws_deque_1۰invｰpersistent t ι :
Persistent (inf_ws_deque_1۰inv t ι).
Lemma inf_ws_deque_1۰modelｰexclusive t vs1 vs2 :
inf_ws_deque_1۰model t vs1 -∗
inf_ws_deque_1۰model t vs2 -∗
False.
Lemma inf_ws_deque_1۰ownerｰexclusive t ws1 ws2 :
inf_ws_deque_1۰owner t ws1 -∗
inf_ws_deque_1۰owner t ws2 -∗
False.
Lemma inf_ws_deque_1ｰownerｰmodel γ ws vs :
inf_ws_deque_1۰owner γ ws -∗
inf_ws_deque_1۰model γ vs -∗
⌜vs `suffix_of` ws⌝.
Lemma inf_ws_deque_1٠createｰspec ι :
{{{
True
}}}
inf_ws_deque_1٠create ()
{{{
t
, RET t;
inf_ws_deque_1۰inv t ι ∗
inf_ws_deque_1۰model t [] ∗
inf_ws_deque_1۰owner t []
}}}.
Lemma inf_ws_deque_1٠sizeｰspec t ι ws :
<<<
inf_ws_deque_1۰inv t ι ∗
inf_ws_deque_1۰owner t ws
| ∀∀ vs,
inf_ws_deque_1۰model t vs
>>>
inf_ws_deque_1٠size t @ ↑ι
<<<
⌜vs `suffix_of` ws⌝ ∗
inf_ws_deque_1۰model t vs
| RET #(length vs);
inf_ws_deque_1۰owner t vs
>>>.
Lemma inf_ws_deque_1٠is_emptyｰspec t ι ws :
<<<
inf_ws_deque_1۰inv t ι ∗
inf_ws_deque_1۰owner t ws
| ∀∀ vs,
inf_ws_deque_1۰model t vs
>>>
inf_ws_deque_1٠is_empty t @ ↑ι
<<<
⌜vs `suffix_of` ws⌝ ∗
inf_ws_deque_1۰model t vs
| RET #(bool_decide (vs = []%list));
inf_ws_deque_1۰owner t vs
>>>.
Lemma inf_ws_deque_1٠pushｰspec t ι ws v :
<<<
inf_ws_deque_1۰inv t ι ∗
inf_ws_deque_1۰owner t ws
| ∀∀ vs,
inf_ws_deque_1۰model t vs
>>>
inf_ws_deque_1٠push t v @ ↑ι
<<<
⌜vs `suffix_of` ws⌝ ∗
inf_ws_deque_1۰model t (vs ++ [v])
| RET ();
inf_ws_deque_1۰owner t (vs ++ [v])
>>>.
Lemma inf_ws_deque_1٠stealｰspec t ι :
<<<
inf_ws_deque_1۰inv t ι
| ∀∀ vs,
inf_ws_deque_1۰model t vs
>>>
inf_ws_deque_1٠steal t @ ↑ι
<<<
inf_ws_deque_1۰model t (tail vs)
| RET head vs;
True
>>>.
Lemma inf_ws_deque_1٠popｰspec t ι ws :
<<<
inf_ws_deque_1۰inv t ι ∗
inf_ws_deque_1۰owner t ws
| ∀∀ vs,
inf_ws_deque_1۰model t vs
>>>
inf_ws_deque_1٠pop t @ ↑ι
<<<
∃∃ o ws',
⌜vs `suffix_of` ws⌝ ∗
match o with
| None ⇒
⌜vs = []⌝ ∗
⌜ws' = []⌝ ∗
inf_ws_deque_1۰model t []
| Some v ⇒
∃ vs',
⌜vs = vs' ++ [v]⌝ ∗
⌜ws' = vs'⌝ ∗
inf_ws_deque_1۰model t vs'
end
| RET o;
inf_ws_deque_1۰owner t ws'
>>>.
End inf_ws_deque_1۰G.
#[global] Opaque inf_ws_deque_1۰inv.
#[global] Opaque inf_ws_deque_1۰model.
#[global] Opaque inf_ws_deque_1۰owner.
Require Import zoo.common.countable.
Require Import zoo.common.function.
Require Import zoo.common.relations.
Require Import zoo.iris.base_logic.lib.excl.
Require Import zoo.iris.base_logic.lib.twins.
Require Import zoo.iris.base_logic.lib.auth_twins.
Require Import zoo.iris.base_logic.lib.auth_nat_max.
Require Import zoo.iris.base_logic.lib.mono_list.
Require Import zoo.base.
Require Import zoo.program_logic.prophet_identifier.
Require Import zoo.program_logic.prophet_multi.
Require Import zoo_std.option.
Require Export zoo_saturn.inf_ws_deque_1__code.
Require Import zoo_saturn.inf_ws_deque_1__types.
Require Import zoo.options.
Implicit Type front back : nat.
Implicit Type id : prophet_id.
Implicit Type v : val.
Implicit Type vs ws hist lhist : list val.
Implicit Type priv : nat → val.
Implicit Type past prophs : list prophet_identifier.(prophet_typed۰type).
Implicit Type pasts prophss : nat → list prophet_identifier.(prophet_typed۰type).
Variant state :=
| Empty
| Nonempty
| Emptyish
| Superempty.
Implicit Type state : state.
#[local] Instance stateｰinhabited : Inhabited state :=
populate Empty.
Variant stability :=
| Stable
| Unstable.
Implicit Type stable : stability.
#[local] Instance stabilityｰinhabited : Inhabited stability :=
populate Stable.
Class InfWsDeque1G Σ `{zoo۰G : !ZooG Σ} :=
{ #[local] inf_ws_deque_1۰G۰inf_array۰G :: InfArrayG Σ
; #[local] inf_ws_deque_1۰G۰prophet۰G :: ProphetMultiG Σ prophet_identifier
; #[local] inf_ws_deque_1۰G۰model۰G :: AuthTwinsG Σ (leibnizO (list val)) suffix
; #[local] inf_ws_deque_1۰G۰owner۰G :: TwinsG Σ (leibnizO (stability × nat × (nat → val)))
; #[local] inf_ws_deque_1۰G۰front۰G :: AuthNatMaxG Σ
; #[local] inf_ws_deque_1۰G۰history۰G :: MonoListG Σ val
; #[local] inf_ws_deque_1۰G۰winner۰G :: TwinsG Σ (natO × ▶ ∙)
}.
Definition inf_ws_deque_1۰Σ :=
#[inf_array۰Σ
; prophet_multi۰Σ prophet_identifier
; auth_twins۰Σ (leibnizO (list val)) suffix
; twins۰Σ (leibnizO (stability × nat × (nat → val)))
; auth_nat_max۰Σ
; mono_list۰Σ val
; twins۰Σ (natO × ▶ ∙)
].
#[global] Instance subGｰinf_ws_deque_1۰Σ Σ `{zoo۰G : !ZooG Σ} :
subG inf_ws_deque_1۰Σ Σ →
InfWsDeque1G Σ .
Module base.
Section inf_ws_deque_1۰G.
Context `{inf_ws_deque_1۰G : InfWsDeque1G Σ}.
Implicit Type t : location.
Implicit Type P : iProp Σ.
Record inf_ws_deque_1۰name :=
{ inf_ws_deque_1۰name۰data : val
; inf_ws_deque_1۰name۰inv : namespace
; inf_ws_deque_1۰name۰prophet : prophet_id
; inf_ws_deque_1۰name۰prophet_name : prophet_multi۰name
; inf_ws_deque_1۰name۰model : auth_twins۰name
; inf_ws_deque_1۰name۰owner : gname
; inf_ws_deque_1۰name۰front : gname
; inf_ws_deque_1۰name۰history : gname
; inf_ws_deque_1۰name۰winner : gname
}.
Implicit Type γ : inf_ws_deque_1۰name.
#[global] Instance inf_ws_deque_1۰nameｰeq_dec : EqDecision inf_ws_deque_1۰name :=
ltac:(solve_decision).
#[global] Instance inf_ws_deque_1۰nameｰcountable :
Countable inf_ws_deque_1۰name.
#[local] Definition model₁' γ_model vs :=
auth_twins۰twin₁ _ γ_model vs.
#[local] Definition model₁ γ :=
model₁' γ.(inf_ws_deque_1۰name۰model).
#[local] Definition model₂' γ_model vs :=
auth_twins۰twin₂ _ γ_model vs.
#[local] Definition model₂ γ :=
model₂' γ.(inf_ws_deque_1۰name۰model).
#[local] Definition owner₁' γ_owner γ_model stable back priv ws : iProp Σ :=
twins۰twin₁ (twins۰G := inf_ws_deque_1۰G۰owner۰G) γ_owner (DfracOwn 1) (stable, back, priv) ∗
auth_twins۰auth _ γ_model ws.
#[local] Definition owner₁ γ :=
owner₁' γ.(inf_ws_deque_1۰name۰owner) γ.(inf_ws_deque_1۰name۰model).
#[local] Instance : CustomIpat "owner₁" :=
" ( Howner₁{_{}} & Hmodel_auth{_{}} ) ".
#[local] Definition owner₂' γ_owner stable back priv :=
twins۰twin₂ (twins۰G := inf_ws_deque_1۰G۰owner۰G) γ_owner (stable, back, priv).
#[local] Definition owner₂ γ :=
owner₂' γ.(inf_ws_deque_1۰name۰owner).
#[local] Definition front۰auth' γ_front :=
auth_nat_max۰auth γ_front (DfracOwn 1).
#[local] Definition front۰auth γ :=
front۰auth' γ.(inf_ws_deque_1۰name۰front).
#[local] Definition front۰lb γ :=
auth_nat_max۰lb γ.(inf_ws_deque_1۰name۰front).
#[local] Definition history۰auth' γ_history :=
mono_list۰auth γ_history (DfracOwn 1).
#[local] Definition history۰auth γ :=
history۰auth' γ.(inf_ws_deque_1۰name۰history).
#[local] Definition history۰at γ :=
mono_list۰at γ.(inf_ws_deque_1۰name۰history).
#[local] Definition winner۰pop' γ_winner front P : iProp Σ :=
twins۰twin₁ γ_winner (DfracOwn 1) (front, Next P).
#[local] Definition winner۰pop γ :=
winner۰pop' γ.(inf_ws_deque_1۰name۰winner).
#[local] Definition winner۰steal' γ_winner front P :=
twins۰twin₂ γ_winner (front, Next P).
#[local] Definition winner۰steal γ :=
winner۰steal' γ.(inf_ws_deque_1۰name۰winner).
#[local] Definition winner γ : iProp Σ :=
∃ front P1 P2,
winner۰pop γ front P1 ∗
winner۰steal γ front P2.
#[local] Instance : CustomIpat "winner" :=
" ( %front_winner & %P_winner_1 & %P_winner_2 & Hwinner_pop{_{}} & Hwinner_steal{_{}} ) ".
#[local] Definition winner۰au γ front P : iProp Σ :=
AU <{
∃∃ vs,
model₁ γ vs
}> @ ⊤ ∖ ↑γ.(inf_ws_deque_1۰name۰inv), ∅ <{
∀∀ v vs',
⌜vs = v :: vs'⌝ ∗
model₁ γ vs' ∗
history۰at γ front v
, COMM
P
}>.
#[local] Definition winner۰pending₁ γ front P id : iProp Σ :=
winner۰steal γ front P ∗
identifier۰model id ∗
winner۰au γ front P.
#[local] Instance : CustomIpat "winner۰pending₁" :=
" ( Hwinner_steal{_{!}} & Hid{_{!}} & HP ) ".
#[local] Definition winner۰pending₂ γ front id : iProp Σ :=
∃ P,
winner۰pending₁ γ front P id.
#[local] Instance : CustomIpat "winner۰pending₂" :=
" ( %P{} & (:winner۰pending₁) ) ".
#[local] Definition winner۰linearized γ front P : iProp Σ :=
winner۰steal γ front P ∗
P.
#[local] Instance : CustomIpat "winner۰linearized" :=
" ( Hwinner_steal{_{!}} & HP ) ".
#[local] Definition inv۰state۰empty γ stable front back hist lhist : iProp Σ :=
⌜stable = Stable⌝ ∗
⌜front = back⌝ ∗
⌜lhist = hist⌝ ∗
⌜length hist = front⌝ ∗
winner γ.
#[local] Instance : CustomIpat "inv۰state۰empty" :=
" ( { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}-> & {>;}-> & {>;}%Hhist{} & Hwinner ) ".
#[local] Definition inv۰state۰nonempty γ stable front back hist lhist vs prophs : iProp Σ :=
⌜stable = Stable⌝ ∗
⌜front < back⌝ ∗
⌜lhist = hist ++ take 1 vs⌝ ∗
⌜length hist = front⌝ ∗
( winner γ
∨ match prophs with
| [] ⇒
False
| id :: _ ⇒
winner۰pending₂ γ front id
end
).
#[local] Instance : CustomIpat "inv۰state۰nonempty" :=
" ( { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}% & {>;}-> & {>;}%Hhist{} & Hwinner ) ".
#[local] Definition inv۰state۰nonempty۰steal γ state stable front back hist lhist vs prophs P : iProp Σ :=
⌜state = Nonempty⌝ ∗
⌜stable = Stable⌝ ∗
⌜front < back⌝ ∗
⌜lhist = hist ++ take 1 vs⌝ ∗
⌜length hist = front⌝ ∗
match prophs with
| [] ⇒
False
| id :: _ ⇒
winner۰pending₁ γ front P id
end.
#[local] Instance : CustomIpat "inv۰state۰nonempty۰steal" :=
" ( {>;}-> & { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}% & {>;}-> & {>;}%Hhist{} & (:winner۰pending₁) ) ".
#[local] Definition inv۰state۰emptyish γ stable front back hist lhist : iProp Σ :=
∃ P,
⌜stable = Unstable⌝ ∗
⌜front = back⌝ ∗
⌜lhist = hist⌝ ∗
⌜length hist = ˖front⌝ ∗
( winner۰pop γ front P
∨ winner۰linearized γ front P
).
#[local] Instance : CustomIpat "inv۰state۰emptyish" :=
" ( %P_ & { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}-> & {>;}-> & {>;}%Hhist{} & Hwinner ) ".
#[local] Definition inv۰state۰emptyish۰pop γ state stable front back hist lhist P : iProp Σ :=
⌜state = Emptyish⌝ ∗
⌜stable = Unstable⌝ ∗
⌜front = back⌝ ∗
⌜lhist = hist⌝ ∗
⌜length hist = ˖front⌝ ∗
winner۰pop γ front P.
#[local] Instance : CustomIpat "inv۰state۰emptyish۰pop" :=
" ( {>;}-> & { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}-> & {>;}-> & {>;}%Hhist{} & Hwinner_pop ) ".
#[local] Definition inv۰state۰emptyish۰steal γ state stable front back hist lhist P : iProp Σ :=
⌜state = Emptyish⌝ ∗
⌜stable = Unstable⌝ ∗
⌜front = back⌝ ∗
⌜lhist = hist⌝ ∗
⌜length hist = ˖front⌝ ∗
winner۰linearized γ front P.
#[local] Instance : CustomIpat "inv۰state۰emptyish۰steal" :=
" ( {>;}-> & { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}-> & {>;}-> & {>;}%Hhist{} & (:winner۰linearized) ) ".
#[local] Definition inv۰state۰superempty γ stable front back hist lhist : iProp Σ :=
⌜stable = Unstable⌝ ∗
⌜front = ˖back⌝ ∗
⌜lhist = hist⌝ ∗
⌜length hist = front⌝ ∗
winner γ.
#[local] Instance : CustomIpat "inv۰state۰superempty" :=
" ( { {lazy}{>}% ; {lazy}% ; {>}-> ; -> } & {>;}-> & {>;}-> & {>;}%Hhist{} & Hwinner ) ".
#[local] Definition inv۰state γ state stable front back hist lhist vs prophs : iProp Σ :=
match state with
| Empty ⇒
inv۰state۰empty γ stable front back hist lhist
| Nonempty ⇒
inv۰state۰nonempty γ stable front back hist lhist vs prophs
| Emptyish ⇒
inv۰state۰emptyish γ stable front back hist lhist
| Superempty ⇒
inv۰state۰superempty γ stable front back hist lhist
end.
#[local] Definition inv۰inner t γ : iProp Σ :=
∃ state stable front back hist lhist vs priv pasts prophss,
t.[front] ↦ #front ∗
t.[back] ↦ #back ∗
owner₂ γ stable back priv ∗
front۰auth γ front ∗
⌜0 < front⌝ ∗
model₂ γ vs ∗
⌜length vs = back - front⌝ ∗
inf_array۰model' γ.(inf_ws_deque_1۰name۰data) (hist ++ vs) priv ∗
history۰auth γ lhist ∗
prophet_multi۰model prophet_identifier γ.(inf_ws_deque_1۰name۰prophet) γ.(inf_ws_deque_1۰name۰prophet_name) pasts prophss ∗
⌜∀ i, front ≤ i → pasts i = []⌝ ∗
inv۰state γ state stable front back hist lhist vs (prophss front).
#[local] Instance : CustomIpat "inv۰inner" :=
" ( %state{} & %stable{} & %front{} & %back{} & %hist{} & %lhist{} & %vs{} & %priv{} & %pasts{} & %prophss{} & >Ht_front & >Ht_back & >Howner₂ & >Hfront_auth & >%Hfront{} & >Hmodel₂ & >%Hvs{} & >Hdata_model & >Hhistory_auth & >Hprophet_model & >%Hpasts{} & Hstate ) ".
#[local] Definition inv' t γ : iProp Σ :=
t.[data] ↦□ γ.(inf_ws_deque_1۰name۰data) ∗
t.[proph] ↦□ #γ.(inf_ws_deque_1۰name۰prophet) ∗
inf_array۰inv γ.(inf_ws_deque_1۰name۰data) ∗
inv γ.(inf_ws_deque_1۰name۰inv) (inv۰inner t γ).
#[local] Instance : CustomIpat "inv'" :=
" ( #Ht_data & #Ht_proph & #Hdata_inv & #Hinv ) ".
Definition inf_ws_deque_1۰inv t γ ι : iProp Σ :=
⌜ι = γ.(inf_ws_deque_1۰name۰inv)⌝ ∗
inv' t γ.
#[local] Instance : CustomIpat "inv" :=
" ( -> & (:inv') ) ".
Definition inf_ws_deque_1۰model :=
model₁.
#[local] Instance : CustomIpat "model" :=
" Hmodel₁{_{}} ".
Definition inf_ws_deque_1۰owner γ ws : iProp Σ :=
∃ back priv,
owner₁ γ Stable back priv ws.
#[local] Instance : CustomIpat "owner" :=
" ( %back{} & %priv{} & Howner₁{_{}} ) ".
#[global] Instance inf_ws_deque_1۰modelｰtimeless γ vs :
Timeless (inf_ws_deque_1۰model γ vs).
#[global] Instance inf_ws_deque_1۰ownerｰtimeless γ ws :
Timeless (inf_ws_deque_1۰owner γ ws).
#[global] Instance inf_ws_deque_1۰invｰpersistent t γ ι :
Persistent (inf_ws_deque_1۰inv t γ ι).
#[local] Lemma modelｰownerｰalloc :
⊢ |==>
∃ γ_model γ_owner,
model₁' γ_model [] ∗
model₂' γ_model [] ∗
owner₁' γ_owner γ_model Stable 1 (λ _, ()%V) [] ∗
owner₂' γ_owner Stable 1 (λ _, ()%V).
#[local] Lemma model₁ｰvalid γ stable back priv ws vs :
owner₁ γ stable back priv ws -∗
model₁ γ vs -∗
⌜vs `suffix_of` ws⌝.
#[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 model۰owner₁ｰagree γ stable back priv ws vs1 vs2 :
owner₁ γ stable back priv ws -∗
model₁ γ vs1 -∗
model₂ γ vs2 -∗
⌜vs1 `suffix_of` ws⌝ ∗
⌜vs1 = vs2⌝.
#[local] Lemma modelｰempty {γ stable back priv ws vs1 vs2} :
owner₁ γ stable back priv ws -∗
model₁ γ vs1 -∗
model₂ γ vs2 ==∗
owner₁ γ stable back priv [] ∗
model₁ γ [] ∗
model₂ γ [].
#[local] Lemma modelｰpush {γ stable back priv ws vs1 vs2} v :
owner₁ γ stable back priv ws -∗
model₁ γ vs1 -∗
model₂ γ vs2 ==∗
owner₁ γ stable back priv (vs1 ++ [v]) ∗
model₁ γ (vs1 ++ [v]) ∗
model₂ γ (vs1 ++ [v]).
#[local] Lemma modelｰsteal γ vs1 vs2 :
model₁ γ vs1 -∗
model₂ γ vs2 ==∗
model₁ γ (tail vs1) ∗
model₂ γ (tail vs1).
#[local] Lemma modelｰpop γ stable back priv ws vs1 vs2 :
owner₁ γ stable back priv ws -∗
model₁ γ vs1 -∗
model₂ γ vs2 ==∗
owner₁ γ stable back priv (removelast vs1) ∗
model₁ γ (removelast vs1) ∗
model₂ γ (removelast vs1).
#[local] Lemma modelｰpop' γ stable back priv ws vs1 v vs2 :
owner₁ γ stable back priv ws -∗
model₁ γ (vs1 ++ [v]) -∗
model₂ γ vs2 ==∗
owner₁ γ stable back priv vs1 ∗
model₁ γ vs1 ∗
model₂ γ vs1.
#[local] Lemma owner₁ｰexclusive γ stable1 back1 priv1 ws1 stable2 back2 priv2 ws2 :
owner₁ γ stable1 back1 priv1 ws1 -∗
owner₁ γ stable2 back2 priv2 ws2 -∗
False.
#[local] Lemma ownerｰagree γ stable1 back1 priv1 ws stable2 back2 priv2 :
owner₁ γ stable1 back1 priv1 ws -∗
owner₂ γ stable2 back2 priv2 -∗
⌜stable1 = stable2⌝ ∗
⌜back1 = back2⌝ ∗
⌜priv1 = priv2⌝.
#[local] Lemma owner₁ｰupdate γ stable back priv ws vs :
owner₁ γ stable back priv ws -∗
model₁ γ vs -∗
model₂ γ vs ==∗
owner₁ γ stable back priv vs ∗
model₁ γ vs ∗
model₂ γ vs.
#[local] Lemma ownerｰupdate {γ stable1 back1 priv1 ws stable2 back2 priv2} stable back priv :
owner₁ γ stable1 back1 priv1 ws -∗
owner₂ γ stable2 back2 priv2 ==∗
owner₁ γ stable back priv ws ∗
owner₂ γ stable back priv.
#[local] Lemma frontｰalloc :
⊢ |==>
∃ γ_front,
front۰auth' γ_front 1.
#[local] Lemma front۰lbｰget γ front :
front۰auth γ front ⊢
front۰lb γ front.
#[local] Lemma front۰lbｰle {γ front} front' :
front' ≤ front →
front۰lb γ front ⊢
front۰lb γ front'.
#[local] Lemma front۰lbｰvalid γ front1 front2 :
front۰auth γ front1 -∗
front۰lb γ front2 -∗
⌜front2 ≤ front1⌝.
#[local] Lemma frontｰupdate γ front :
front۰auth γ front ⊢ |==>
front۰auth γ ˖front.
#[local] Lemma historyｰalloc :
⊢ |==>
∃ γ_hist,
history۰auth' γ_hist [()%V].
#[local] Lemma history۰atｰget {γ hist v} i :
i = length hist →
history۰auth γ (hist ++ [v]) ⊢
history۰at γ i v.
#[local] Lemma history۰atｰlookup γ hist i v :
history۰auth γ hist -∗
history۰at γ i v -∗
⌜hist !! i = Some v⌝.
#[local] Lemma history۰atｰagree γ i v1 v2 :
history۰at γ i v1 -∗
history۰at γ i v2 -∗
⌜v1 = v2⌝.
#[local] Lemma historyｰupdate {γ hist} i v :
i = length hist →
history۰auth γ hist ⊢ |==>
history۰auth γ (hist ++ [v]) ∗
history۰at γ i v.
#[local] Lemma winnerｰalloc :
⊢ |==>
∃ γ_winner,
winner۰pop' γ_winner 1 True ∗
winner۰steal' γ_winner 1 True.
#[local] Lemma winner۰popｰexclusive γ front1 P1 front2 P2 :
winner۰pop γ front1 P1 -∗
winner۰pop γ front2 P2 -∗
False.
#[local] Lemma winner۰popｰexclusive' γ front P :
winner۰pop γ front P -∗
winner γ -∗
False.
#[local] Lemma winner۰stealｰexclusive γ front1 P1 front2 P2 :
winner۰steal γ front1 P1 -∗
winner۰steal γ front2 P2 -∗
False.
#[local] Lemma winner۰stealｰexclusive' γ front P :
winner۰steal γ front P -∗
winner γ -∗
False.
#[local] Lemma winnerｰagree γ front1 P1 front2 P2 :
winner۰pop γ front1 P1 -∗
winner۰steal γ front2 P2 -∗
⌜front1 = front2⌝ ∗
▷ (P1 ≡ P2).
#[local] Lemma winnerｰupdate {γ front1 P1 front2 P2} front P :
winner۰pop γ front1 P1 -∗
winner۰steal γ front2 P2 ==∗
winner۰pop γ front P ∗
winner۰steal γ front P.
Opaque owner₁'.
Lemma inf_ws_deque_1۰modelｰexclusive γ vs1 vs2 :
inf_ws_deque_1۰model γ vs1 -∗
inf_ws_deque_1۰model γ vs2 -∗
False.
Lemma inf_ws_deque_1۰ownerｰexclusive γ ws1 ws2 :
inf_ws_deque_1۰owner γ ws1 -∗
inf_ws_deque_1۰owner γ ws2 -∗
False.
Lemma inf_ws_deque_1۰ownerｰmodel γ ws vs :
inf_ws_deque_1۰owner γ ws -∗
inf_ws_deque_1۰model γ vs -∗
⌜vs `suffix_of` ws⌝.
#[local] Lemma inv۰stateｰStable γ state front back hist lhist vs prophs :
length vs = back - front →
inv۰state γ state Stable front back hist lhist vs prophs ⊢
⌜state = Empty ∨ state = Nonempty⌝ ∗
⌜front ≤ back⌝ ∗
⌜length (hist ++ vs) = back⌝.
#[local] Lemma inv۰stateｰUnstable γ state front back hist lhist vs prophs :
inv۰state γ state Unstable front back hist lhist vs prophs ⊢
⌜state = Emptyish ∨ state = Superempty⌝ ∗
⌜front = back ∨ front = ˖back⌝.
#[local] Lemma inv۰stateｰNonempty γ state stable front back hist lhist vs prophs :
front < back →
inv۰state γ state stable front back hist lhist vs prophs ⊢
⌜state = Nonempty⌝.
#[local] Lemma inv۰stateｰSuperempty γ state front back hist lhist vs prophs :
back < front →
inv۰state γ state Unstable front back hist lhist vs prophs -∗
⌜state = Superempty⌝.
#[local] Lemma inv۰stateｰwinner۰pop γ state stable front1 back hist lhist vs prophs front2 P :
inv۰state γ state stable front1 back hist lhist vs prophs -∗
winner۰pop γ front2 P -∗
∃ P_,
⌜front1 = front2⌝ ∗
▷ (P ≡ P_) ∗
( inv۰state۰nonempty۰steal γ state stable front2 back hist lhist vs prophs P_
∨ inv۰state۰emptyish۰steal γ state stable front2 back hist lhist P_
) ∗
winner۰pop γ front2 P.
#[local] Lemma inv۰stateｰwinner۰steal γ state stable front1 back hist lhist vs prophs front2 P :
inv۰state γ state stable front1 back hist lhist vs prophs -∗
winner۰steal γ front2 P -∗
∃ P_,
⌜front1 = front2⌝ ∗
▷ (P_ ≡ P) ∗
inv۰state۰emptyish۰pop γ state stable front2 back hist lhist P_ ∗
winner۰steal γ front2 P.
Lemma inf_ws_deque_1٠createｰspec ι :
{{{
True
}}}
inf_ws_deque_1٠create ()
{{{
t γ
, RET #t;
meta_token t ⊤ ∗
inf_ws_deque_1۰inv t γ ι ∗
inf_ws_deque_1۰model γ [] ∗
inf_ws_deque_1۰owner γ []
}}}.
#[local] Lemma frontｰspec t γ :
{{{
inv' t γ
}}}
(#t).{front}
{{{
front
, RET #front;
front۰lb γ front
}}}.
#[local] Lemma frontｰspecｰownerｰStable t γ back priv ws :
{{{
inv' t γ ∗
owner₁ γ Stable back priv ws
}}}
(#t).{front}
{{{
front
, RET #front;
owner₁ γ Stable back priv ws ∗
front۰lb γ front ∗
⌜front ≤ back⌝
}}}.
#[local] Lemma frontｰspecｰownerｰUnstable t γ back priv ws :
{{{
inv' t γ ∗
owner₁ γ Unstable back priv ws
}}}
(#t).{front}
{{{
front
, RET #front;
owner₁ γ Unstable back priv ws ∗
front۰lb γ front ∗
⌜front = back ∨ front = ˖back⌝
}}}.
#[local] Lemma frontｰspecｰSuperempty t γ back priv ws front :
back < front →
{{{
inv' t γ ∗
owner₁ γ Unstable back priv ws ∗
front۰lb γ front
}}}
(#t).{front}
{{{
RET #front;
owner₁ γ Unstable back priv ws
}}}.
#[local] Lemma frontｰspecｰwinner۰steal t γ front P :
{{{
inv' t γ ∗
winner۰steal γ front P
}}}
(#t).{front}
{{{
RET #front;
winner۰steal γ front P
}}}.
#[local] Lemma backｰspec t γ stable back priv ws :
{{{
inv' t γ ∗
owner₁ γ stable back priv ws
}}}
(#t).{back}
{{{
RET #back;
owner₁ γ stable back priv ws
}}}.
#[local] Lemma set_backｰspecｰSuperempty t γ back priv ws front (back' : Z) :
back < front →
back' = ˖back →
{{{
inv' t γ ∗
owner₁ γ Unstable back priv ws ∗
front۰lb γ front
}}}
#t <-{back} #back'
{{{
RET ();
owner₁ γ Stable ˖back priv ws
}}}.
#[local] Lemma inf_array٠getｰspecｰhistory t γ (i : nat) (i_ : Z) v :
i_ = i →
{{{
inv' t γ ∗
history۰at γ i v
}}}
inf_array٠get γ.(inf_ws_deque_1۰name۰data) #i_
{{{
RET v;
True
}}}.
#[local] Lemma inf_array٠getｰspecｰowner t γ back (back_ : Z) priv ws v :
back_ = back →
priv 0 = v →
{{{
inv' t γ ∗
owner₁ γ Stable back priv ws
}}}
inf_array٠get γ.(inf_ws_deque_1۰name۰data) #back_
{{{
RET v;
owner₁ γ Stable back priv ws
}}}.
#[local] Lemma inf_array٠setｰspecｰowner t γ back priv ws v :
{{{
inv' t γ ∗
owner₁ γ Stable back priv ws
}}}
inf_array٠set γ.(inf_ws_deque_1۰name۰data) #back v
{{{
RET ();
owner₁ γ Stable back (<[0 := v]> priv) ws
}}}.
#[local] Lemma resolveｰspecｰloser₁ t γ front1 front2 id :
front1 < front2 →
{{{
inv' t γ ∗
front۰lb γ front2
}}}
Resolve (CAS (#t).[front]%V #front1 #(front1 + 1)) #γ.(inf_ws_deque_1۰name۰prophet) (#front1, #id)%V
{{{
RET false;
True
}}}.
#[local] Lemma resolveｰspecｰloser₂ t γ front id prophs0 :
head prophs0 ≠ Some id →
{{{
inv' t γ ∗
front۰lb γ front ∗
prophet_multi۰full prophet_identifier γ.(inf_ws_deque_1۰name۰prophet_name) front prophs0
}}}
Resolve (CAS (#t).[front]%V #front #(front + 1)) #γ.(inf_ws_deque_1۰name۰prophet) (#front, #id)%V
{{{
RET false;
front۰lb γ ˖front
}}}.
#[local] Lemma resolveｰspecｰwinnerｰpop t γ front P id :
{{{
inv' t γ ∗
winner۰pop γ front P
}}}
Resolve (CAS (#t).[front]%V #front #(front + 1)) #γ.(inf_ws_deque_1۰name۰prophet) (#front, #id)%V
{{{
RET true;
▷ P
}}}.
#[local] Lemma resolveｰspecｰwinnerｰsteal t γ front P id :
{{{
inv' t γ ∗
winner۰steal γ front P
}}}
Resolve (CAS (#t).[front]%V #front #(front + 1)) #γ.(inf_ws_deque_1۰name۰prophet) (#front, #id)%V
{{{
RET true;
front۰lb γ ˖front
}}}.
#[local] Lemma resolveｰspecｰEmpty t γ back priv ws id :
{{{
inv' t γ ∗
owner₁ γ Stable back priv ws ∗
front۰lb γ back
}}}
Resolve (CAS (#t).[front]%V #back #(back + 1)) #γ.(inf_ws_deque_1۰name۰prophet) (#back, #id)%V
{{{
RET true;
owner₁ γ Unstable back (priv ∘ S) ws ∗
front۰lb γ ˖back ∗
history۰at γ back (priv 0)
}}}.
Lemma inf_ws_deque_1٠sizeｰspec t γ ι ws :
<<<
inf_ws_deque_1۰inv t γ ι ∗
inf_ws_deque_1۰owner γ ws
| ∀∀ vs,
inf_ws_deque_1۰model γ vs
>>>
inf_ws_deque_1٠size #t @ ↑ι
<<<
⌜vs `suffix_of` ws⌝ ∗
inf_ws_deque_1۰model γ vs
| RET #(length vs);
inf_ws_deque_1۰owner γ vs
>>>.
Lemma inf_ws_deque_1٠is_emptyｰspec t γ ι ws :
<<<
inf_ws_deque_1۰inv t γ ι ∗
inf_ws_deque_1۰owner γ ws
| ∀∀ vs,
inf_ws_deque_1۰model γ vs
>>>
inf_ws_deque_1٠is_empty #t @ ↑ι
<<<
⌜vs `suffix_of` ws⌝ ∗
inf_ws_deque_1۰model γ vs
| RET #(bool_decide (vs = []%list));
inf_ws_deque_1۰owner γ vs
>>>.
Lemma inf_ws_deque_1٠pushｰspec t γ ι ws v :
<<<
inf_ws_deque_1۰inv t γ ι ∗
inf_ws_deque_1۰owner γ ws
| ∀∀ vs,
inf_ws_deque_1۰model γ vs
>>>
inf_ws_deque_1٠push #t v @ ↑ι
<<<
⌜vs `suffix_of` ws⌝ ∗
inf_ws_deque_1۰model γ (vs ++ [v])
| RET ();
inf_ws_deque_1۰owner γ (vs ++ [v])
>>>.
Lemma inf_ws_deque_1٠stealｰspec t γ ι :
<<<
inf_ws_deque_1۰inv t γ ι
| ∀∀ vs,
inf_ws_deque_1۰model γ vs
>>>
inf_ws_deque_1٠steal #t @ ↑ι
<<<
inf_ws_deque_1۰model γ (tail vs)
| RET head vs;
True
>>>.
Variant pop۰state :=
| PopNonempty v
| PopEmptyishWinner v
| PopEmptyishLoser
| PopSuperempty.
#[local] Lemma inf_ws_deque_1٠pop₁ｰspec {t γ} (state : pop۰state) stable back (back_ : Z) priv ws id :
back_ = back →
{{{
inv' t γ ∗
owner₁ γ stable back priv ws ∗
match state with
| PopNonempty v ⇒
⌜stable = Stable⌝ ∗
⌜priv 0 = v⌝
| PopEmptyishWinner v ⇒
⌜stable = Unstable⌝ ∗
history۰at γ back v ∗
winner۰steal γ back inhabitant
| PopEmptyishLoser ⇒
∃ id_winner prophs,
⌜stable = Unstable⌝ ∗
prophet_multi۰full prophet_identifier γ.(inf_ws_deque_1۰name۰prophet_name) back (id_winner :: prophs) ∗
⌜head (id_winner :: prophs) ≠ Some id⌝
| PopSuperempty ⇒
∃ front,
⌜stable = Unstable⌝ ∗
front۰lb γ front ∗
⌜front = ˖back⌝
end
}}}
inf_ws_deque_1٠pop₁ #t #id #back_
{{{
o back priv
, RET o;
owner₁ γ Stable back priv ws ∗
match state with
| PopNonempty v ⇒
⌜o = Some v⌝
| PopEmptyishWinner v ⇒
⌜o = Some v⌝
| PopEmptyishLoser ⇒
⌜o = None⌝
| PopSuperempty ⇒
⌜o = None⌝
end
}}}.
Lemma inf_ws_deque_1٠popｰspec t γ ι ws :
<<<
inf_ws_deque_1۰inv t γ ι ∗
inf_ws_deque_1۰owner γ ws
| ∀∀ vs,
inf_ws_deque_1۰model γ vs
>>>
inf_ws_deque_1٠pop #t @ ↑ι
<<<
∃∃ o ws',
⌜vs `suffix_of` ws⌝ ∗
match o with
| None ⇒
⌜vs = []⌝ ∗
⌜ws' = []⌝ ∗
inf_ws_deque_1۰model γ []
| Some v ⇒
∃ vs',
⌜vs = vs' ++ [v]⌝ ∗
⌜ws' = vs'⌝ ∗
inf_ws_deque_1۰model γ vs'
end
| RET o;
inf_ws_deque_1۰owner γ ws'
>>>.
End inf_ws_deque_1۰G.
#[global] Opaque inf_ws_deque_1۰inv.
#[global] Opaque inf_ws_deque_1۰model.
#[global] Opaque inf_ws_deque_1۰owner.
End base.
Require zoo_saturn.inf_ws_deque_1__opaque.
Section inf_ws_deque_1۰G.
Context `{inf_ws_deque_1۰G : InfWsDeque1G Σ}.
Implicit Type 𝑡 : location.
Implicit Type t : val.
Definition inf_ws_deque_1۰inv t ι : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
𝑡 ↪ γ ∗
base.inf_ws_deque_1۰inv 𝑡 γ ι.
#[local] Instance : CustomIpat "inv" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hinv{_{}} ) ".
Definition inf_ws_deque_1۰model t vs : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
𝑡 ↪ γ ∗
base.inf_ws_deque_1۰model γ vs.
#[local] Instance : CustomIpat "model" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hmodel{_{}} ) ".
Definition inf_ws_deque_1۰owner t ws : iProp Σ :=
∃ 𝑡 γ,
⌜t = #𝑡⌝ ∗
𝑡 ↪ γ ∗
base.inf_ws_deque_1۰owner γ ws.
#[local] Instance : CustomIpat "owner" :=
" ( %𝑡{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Howner{_{}} ) ".
#[global] Instance inf_ws_deque_1۰modelｰtimeless γ vs :
Timeless (inf_ws_deque_1۰model γ vs).
#[global] Instance inf_ws_deque_1۰ownerｰtimeless γ ws :
Timeless (inf_ws_deque_1۰owner γ ws).
#[global] Instance inf_ws_deque_1۰invｰpersistent t ι :
Persistent (inf_ws_deque_1۰inv t ι).
Lemma inf_ws_deque_1۰modelｰexclusive t vs1 vs2 :
inf_ws_deque_1۰model t vs1 -∗
inf_ws_deque_1۰model t vs2 -∗
False.
Lemma inf_ws_deque_1۰ownerｰexclusive t ws1 ws2 :
inf_ws_deque_1۰owner t ws1 -∗
inf_ws_deque_1۰owner t ws2 -∗
False.
Lemma inf_ws_deque_1ｰownerｰmodel γ ws vs :
inf_ws_deque_1۰owner γ ws -∗
inf_ws_deque_1۰model γ vs -∗
⌜vs `suffix_of` ws⌝.
Lemma inf_ws_deque_1٠createｰspec ι :
{{{
True
}}}
inf_ws_deque_1٠create ()
{{{
t
, RET t;
inf_ws_deque_1۰inv t ι ∗
inf_ws_deque_1۰model t [] ∗
inf_ws_deque_1۰owner t []
}}}.
Lemma inf_ws_deque_1٠sizeｰspec t ι ws :
<<<
inf_ws_deque_1۰inv t ι ∗
inf_ws_deque_1۰owner t ws
| ∀∀ vs,
inf_ws_deque_1۰model t vs
>>>
inf_ws_deque_1٠size t @ ↑ι
<<<
⌜vs `suffix_of` ws⌝ ∗
inf_ws_deque_1۰model t vs
| RET #(length vs);
inf_ws_deque_1۰owner t vs
>>>.
Lemma inf_ws_deque_1٠is_emptyｰspec t ι ws :
<<<
inf_ws_deque_1۰inv t ι ∗
inf_ws_deque_1۰owner t ws
| ∀∀ vs,
inf_ws_deque_1۰model t vs
>>>
inf_ws_deque_1٠is_empty t @ ↑ι
<<<
⌜vs `suffix_of` ws⌝ ∗
inf_ws_deque_1۰model t vs
| RET #(bool_decide (vs = []%list));
inf_ws_deque_1۰owner t vs
>>>.
Lemma inf_ws_deque_1٠pushｰspec t ι ws v :
<<<
inf_ws_deque_1۰inv t ι ∗
inf_ws_deque_1۰owner t ws
| ∀∀ vs,
inf_ws_deque_1۰model t vs
>>>
inf_ws_deque_1٠push t v @ ↑ι
<<<
⌜vs `suffix_of` ws⌝ ∗
inf_ws_deque_1۰model t (vs ++ [v])
| RET ();
inf_ws_deque_1۰owner t (vs ++ [v])
>>>.
Lemma inf_ws_deque_1٠stealｰspec t ι :
<<<
inf_ws_deque_1۰inv t ι
| ∀∀ vs,
inf_ws_deque_1۰model t vs
>>>
inf_ws_deque_1٠steal t @ ↑ι
<<<
inf_ws_deque_1۰model t (tail vs)
| RET head vs;
True
>>>.
Lemma inf_ws_deque_1٠popｰspec t ι ws :
<<<
inf_ws_deque_1۰inv t ι ∗
inf_ws_deque_1۰owner t ws
| ∀∀ vs,
inf_ws_deque_1۰model t vs
>>>
inf_ws_deque_1٠pop t @ ↑ι
<<<
∃∃ o ws',
⌜vs `suffix_of` ws⌝ ∗
match o with
| None ⇒
⌜vs = []⌝ ∗
⌜ws' = []⌝ ∗
inf_ws_deque_1۰model t []
| Some v ⇒
∃ vs',
⌜vs = vs' ++ [v]⌝ ∗
⌜ws' = vs'⌝ ∗
inf_ws_deque_1۰model t vs'
end
| RET o;
inf_ws_deque_1۰owner t ws'
>>>.
End inf_ws_deque_1۰G.
#[global] Opaque inf_ws_deque_1۰inv.
#[global] Opaque inf_ws_deque_1۰model.
#[global] Opaque inf_ws_deque_1۰owner.