Library zoo_std.lazy

Require Import zoo.prelude.
Require Import zoo.common.countable.
Require Import zoo.iris.base_logic.lib.oneshot.
Require Import zoo.iris.base_logic.lib.subpreds.
Require Import zoo.base.
Require Export zoo_std.lazy__code.
Require Import zoo_std.lazy__types.
Require Import zoo.options.

Implicit Type b : bool.
Implicit Type v fn mtx : val.

Class LazyG Σ `{zoo۰G : !ZooG Σ} :=
  { #[local] lazy۰G۰mutex۰G :: MutexG Σ
  ; #[local] lazy۰G۰lstate۰G :: OneshotG Σ unit val
  ; #[local] lazy۰G۰consumer۰G :: SubpredsG Σ val
  }.

Definition lazy۰Σ :=
  #[mutex۰Σ
  ; oneshot۰Σ unit val
  ; subpreds۰Σ val
  ].
#[global] Instance subGｰlazy۰Σ Σ `{zoo۰G : !ZooG Σ} :
  subG lazy۰Σ Σ →
  LazyG Σ .

Module base.
  Section lazy۰G.
    Context `{lazy۰G : LazyG Σ}.

    Implicit Type t : location.
    Implicit Type Ψ Χ Ξ : val → iProp Σ.

    Record lazy۰name :=
      { lazy۰name۰thunk : val
      ; lazy۰name۰lstate : gname
      ; lazy۰name۰consumer : gname
      }.
    Implicit Type γ : lazy۰name.

    #[global] Instance lazy۰nameｰeq_dec : EqDecision lazy۰name :=
      ltac:(solve_decision).
    #[global] Instance lazy۰nameｰcountable :
      Countable lazy۰name.

    Variant state :=
      | Unset
      | Setting mtx
      | Set_ v.
    Implicit Type state : state.

    #[local] Instance stateｰinhabited : Inhabited state :=
      populate Unset.

    #[local] Definition state۰to_bool state :=
      match state with
      | Set_ _ ⇒
          true
      | _ ⇒
          false
      end.
    #[local] Definition state۰to_option state :=
      match state with
      | Set_ v ⇒
          Some v
      | _ ⇒
          None
      end.
    #[local] Definition state۰to_val γ state :=
      match state with
      | Unset ⇒
          ‘Unset( γ.(lazy۰name۰thunk) )
      | Setting mtx ⇒
          ‘Setting( mtx )
      | Set_ v ⇒
          ‘Set( v )
      end%V.

    #[local] Definition lstate۰unset₁' γ_lstate :=
      oneshot۰pending γ_lstate (DfracOwn (1/3)) ().
    #[local] Definition lstate۰unset₁ γ :=
      lstate۰unset₁' γ.(lazy۰name۰lstate).
    #[local] Definition lstate۰unset₂' γ_lstate :=
      oneshot۰pending γ_lstate (DfracOwn (2/3)) ().
    #[local] Definition lstate۰unset₂ γ :=
      lstate۰unset₂' γ.(lazy۰name۰lstate).
    #[local] Definition lstate۰set' γ_lstate :=
      oneshot۰shot γ_lstate.
    #[local] Definition lstate۰set γ :=
      lstate۰set' γ.(lazy۰name۰lstate).

    #[local] Definition consumer۰auth' :=
      subpreds۰auth.
    #[local] Definition consumer۰auth γ :=
      consumer۰auth' γ.(lazy۰name۰consumer).
    #[local] Definition consumer۰frag' :=
      subpreds۰frag.
    #[local] Definition consumer۰frag γ :=
      consumer۰frag' γ.(lazy۰name۰consumer).

    Definition lazy۰result :=
      lstate۰set.
    #[local] Instance : CustomIpat "result" :=
      " #Hlstate_set{_{}} ".
    Definition lazy۰resolved γ : iProp Σ :=
      ∃ v,
      lazy۰result γ v.

    #[local] Definition inv۰state۰unset γ Ψ Ξ : iProp Σ :=
      lstate۰unset₁ γ ∗
      lstate۰unset₂ γ ∗
      WP γ.(lazy۰name۰thunk) () {{ v,
        ▷ Ψ v ∗
        ▷ □ Ξ v
      }}.
    #[local] Instance : CustomIpat "inv۰state۰unset" :=
      " ( {>;}Hlstate_unset₁{_{}} & {>;}Hlstate_unset₂{_{}} & Hthunk ) ".
    #[local] Definition inv۰state۰setting γ mtx : iProp Σ :=
      lstate۰unset₁ γ ∗
      mutex۰inv mtx (lazy۰resolved γ).
    #[local] Instance : CustomIpat "inv۰state۰setting" :=
      " ( {>;}Hlstate_unset₁{_{}} & #Hmtx_inv{_{}} ) ".
    #[local] Definition inv۰state۰set γ Ξ v : iProp Σ :=
      lstate۰set γ v ∗
      □ Ξ v.
    #[local] Instance : CustomIpat "inv۰state۰set" :=
      " ( {>;}#Hlstate_set{_{}} & #HΞ{_{}} ) ".
    #[local] Definition inv۰state γ Ψ Ξ state :=
      match state with
      | Unset ⇒
          inv۰state۰unset γ Ψ Ξ
      | Setting mtx ⇒
          inv۰state۰setting γ mtx
      | Set_ v ⇒
          inv۰state۰set γ Ξ v
      end.

    #[local] Definition inv۰inner t γ Ψ Ξ : iProp Σ :=
      ∃ state,
      t ↦ᵣ state۰to_val γ state ∗
      consumer۰auth γ Ψ (state۰to_option state) ∗
      inv۰state γ Ψ Ξ state.
    #[local] Instance : CustomIpat "inv۰inner" :=
      " ( %state & Ht & Hconsumer_auth & Hstate ) ".
    Definition lazy۰inv t γ Ψ Ξ : iProp Σ :=
      inv nroot (inv۰inner t γ Ψ Ξ).
    #[local] Instance : CustomIpat "inv" :=
      " #Hinv ".

    Definition lazy۰consumer :=
      consumer۰frag.
    #[local] Instance : CustomIpat "consumer" :=
      " Hconsumer{}_frag ".

    #[global] Instance lazy۰invｰcontractive t γ n :
      Proper (
        (pointwise_relation _ (dist_later n)) ==>
        (pointwise_relation _ (dist_later n)) ==>
        (≡{n}≡)
      ) (lazy۰inv t γ).
    #[global] Instance lazy۰invｰproper t γ :
      Proper (
        (pointwise_relation _ (≡)) ==>
        (pointwise_relation _ (≡)) ==>
        (≡)
      ) (lazy۰inv t γ).
    #[global] Instance lazy۰consumerｰcontractive γ n :
      Proper (
        (pointwise_relation _ (dist_later n)) ==>
        (≡{n}≡)
      ) (lazy۰consumer γ).
    #[global] Instance lazy۰consumerｰproper γ :
      Proper (
        (pointwise_relation _ (≡)) ==>
        (≡)
      ) (lazy۰consumer γ).

    #[global] Instance lazy۰resultｰtimeless γ v :
      Timeless (lazy۰result γ v).

    #[global] Instance lazy۰invｰpersistent t γ Ψ Ξ :
      Persistent (lazy۰inv t γ Ψ Ξ).
    #[global] Instance lazy۰resultｰpersistent γ v :
      Persistent (lazy۰result γ v).

    #[local] Lemma lstateｰalloc :
      ⊢ |==>
        ∃ γ_lstate,
        lstate۰unset₁' γ_lstate ∗
        lstate۰unset₂' γ_lstate.
    #[local] Lemma lstate۰unset₂ｰexclusive γ :
      lstate۰unset₂ γ -∗
      lstate۰unset₂ γ -∗
      False.
    #[local] Lemma lstate۰setｰagree γ v1 v2 :
      lstate۰set γ v1 -∗
      lstate۰set γ v2 -∗
      ⌜v1 = v2⌝.
    #[local] Lemma lstateｰunset₁ｰset γ v :
      lstate۰unset₁ γ -∗
      lstate۰set γ v -∗
      False.
    #[local] Lemma lstateｰunset₂ｰset γ v :
      lstate۰unset₂ γ -∗
      lstate۰set γ v -∗
      False.
    #[local] Lemma lstateｰupdate {γ} v :
      lstate۰unset₁ γ -∗
      lstate۰unset₂ γ ==∗
      lstate۰set γ v.

    #[local] Lemma consumerｰalloc Ψ :
      ⊢ |==>
        ∃ γ_consumer,
        consumer۰auth' γ_consumer Ψ None ∗
        consumer۰frag' γ_consumer Ψ.
    #[local] Lemma consumerｰwand {γ Ψ} {state : option val} {Χ1} Χ2 E :
      ▷ consumer۰auth γ Ψ state -∗
      consumer۰frag γ Χ1 -∗
      (∀ v, Χ1 v -∗ Χ2 v) ={E}=∗
        ▷ consumer۰auth γ Ψ state ∗
        consumer۰frag γ Χ2.
    #[local] Lemma consumerｰdivide {γ Ψ} {state : option val} Χs E :
      ▷ consumer۰auth γ Ψ state -∗
      consumer۰frag γ (λ v, [∗ list] Χ ∈ Χs, Χ v) ={E}=∗
        ▷ consumer۰auth γ Ψ state ∗
        [∗ list] Χ ∈ Χs, consumer۰frag γ Χ.
    #[local] Lemma consumerｰproduce {γ Ψ} v :
      consumer۰auth γ Ψ None -∗
      Ψ v -∗
      consumer۰auth γ Ψ (Some v).
    #[local] Lemma consumerｰconsume γ Ψ v Χ E :
      ▷ consumer۰auth γ Ψ (Some v) -∗
      consumer۰frag γ Χ ={E}=∗
        ▷ consumer۰auth γ Ψ (Some v) ∗
        ▷^2 Χ v.

    #[local] Lemma inv۰stateｰlstate۰set γ Ψ Ξ state v :
      ▷ inv۰state γ Ψ Ξ state -∗
      lstate۰set γ v -∗
      ◇ (
        ⌜state = Set_ v⌝ ∗
        ▷ inv۰state۰set γ Ξ v
      ).

    Lemma lazy۰consumerｰwand {t γ Ψ Ξ Χ1} Χ2 :
      lazy۰inv t γ Ψ Ξ -∗
      lazy۰consumer γ Χ1 -∗
      (∀ v, Χ1 v -∗ Χ2 v) ={⊤}=∗
      lazy۰consumer γ Χ2.
    Lemma lazy۰consumerｰdivide {t γ Ψ Ξ} Χs :
      lazy۰inv t γ Ψ Ξ -∗
      lazy۰consumer γ (λ v, [∗ list] Χ ∈ Χs, Χ v) ={⊤}=∗
      [∗ list] Χ ∈ Χs, lazy۰consumer γ Χ.

    Lemma lazy۰resultｰagree γ v1 v2 :
      lazy۰result γ v1 -∗
      lazy۰result γ v2 -∗
      ⌜v1 = v2⌝.

    Lemma lazyｰinvｰresult t γ Ψ Ξ v :
      lazy۰inv t γ Ψ Ξ -∗
      lazy۰result γ v ={⊤}=∗
      ▷ □ Ξ v.
    Lemma lazyｰinvｰresultｰconsumer t γ Ψ Ξ v Χ :
      lazy۰inv t γ Ψ Ξ -∗
      lazy۰result γ v -∗
      lazy۰consumer γ Χ ={⊤}=∗
        ▷^2 Χ v ∗
        ▷ □ Ξ v.

    Lemma lazy٠makeｰspec Ψ Ξ fn :
      {{{
        WP fn () {{ v,
          ▷ Ψ v ∗
          ▷ □ Ξ v
        }}
      }}}
        lazy٠make fn
      {{{
        t γ
      , RET #t;
        meta_token t ⊤ ∗
        lazy۰inv t γ Ψ Ξ ∗
        lazy۰consumer γ Ψ
      }}}.

    Lemma lazy٠returnｰspec Ψ Ξ v :
      {{{
        ▷ Ψ v ∗
        ▷ □ Ξ v
      }}}
        lazy٠return v
      {{{
        t γ
      , RET #t;
        meta_token t ⊤ ∗
        lazy۰inv t γ Ψ Ξ ∗
        lazy۰result γ v ∗
        lazy۰consumer γ Ψ
      }}}.

    Lemma lazy٠is_setｰspec t γ Ψ Ξ :
      {{{
        lazy۰inv t γ Ψ Ξ
      }}}
        lazy٠is_set #t
      {{{
        b
      , RET #b;
        if b then
          £ 2 ∗
          lazy۰resolved γ
        else
          True
      }}}.
    Lemma lazy٠is_setｰspecｰresult t γ Ψ Ξ v :
      {{{
        lazy۰inv t γ Ψ Ξ ∗
        lazy۰result γ v
      }}}
        lazy٠is_set #t
      {{{
        RET true;
        £ 2
      }}}.

    Lemma lazy٠is_unsetｰspec t γ Ψ Ξ :
      {{{
        lazy۰inv t γ Ψ Ξ
      }}}
        lazy٠is_unset #t
      {{{
        b
      , RET #b;
        if b then
          True
        else
          £ 2 ∗
          lazy۰resolved γ
      }}}.
    Lemma lazy٠is_unsetｰspecｰresult t γ Ψ Ξ v :
      {{{
        lazy۰inv t γ Ψ Ξ ∗
        lazy۰result γ v
      }}}
        lazy٠is_unset #t
      {{{
        RET false;
        £ 2
      }}}.

    Lemma lazy٠getｰspec t γ Ψ Ξ :
      {{{
        lazy۰inv t γ Ψ Ξ
      }}}
        lazy٠get #t
      {{{
        v
      , RET v;
        £ 2 ∗
        lazy۰result γ v
      }}}.
    Lemma lazy٠getｰspecｰresult t γ Ψ Ξ v :
      {{{
        lazy۰inv t γ Ψ Ξ ∗
        lazy۰result γ v
      }}}
        lazy٠get #t
      {{{
        RET v;
        £ 2
      }}}.
  End lazy۰G.

  #[global] Opaque lazy۰inv.
  #[global] Opaque lazy۰consumer.
  #[global] Opaque lazy۰result.
End base.

Require zoo_std.lazy__opaque.

Section lazy۰G.
  Context `{lazy۰G : LazyG Σ}.

  Implicit Type 𝑡 : location.
  Implicit Type t : val.
  Implicit Type γ : base.lazy۰name.
  Implicit Type Ψ Χ Ξ : val → iProp Σ.

  Definition lazy۰inv t Ψ Ξ : iProp Σ :=
    ∃ 𝑡 γ,
    ⌜t = #𝑡⌝ ∗
    𝑡 ↪ γ ∗
    base.lazy۰inv 𝑡 γ Ψ Ξ.
  #[local] Instance : CustomIpat "inv" :=
    " ( %l{} & %γ{} & {%Heq{};->} & #Hmeta{_{}} & Hinv{_{}} ) ".

  Definition lazy۰consumer t Χ : iProp Σ :=
    ∃ 𝑡 γ,
    ⌜t = #𝑡⌝ ∗
    𝑡 ↪ γ ∗
    base.lazy۰consumer γ Χ.
  #[local] Instance : CustomIpat "consumer" :=
    " ( %l{;_} & %γ{;_} & {%Heq{};->} & #Hmeta{_{}} & Hconsumer{_{}} ) ".

  Definition lazy۰result t v : iProp Σ :=
    ∃ 𝑡 γ,
    ⌜t = #𝑡⌝ ∗
    𝑡 ↪ γ ∗
    base.lazy۰result γ v.
  #[local] Instance : CustomIpat "result" :=
    " ( %l{;_} & %γ{;_} & {%Heq{};->} & #Hmeta{_{}} & Hresult{_{}} ) ".
  Definition lazy۰resolved t : iProp Σ :=
    ∃ v,
    lazy۰result t v.

  #[global] Instance lazy۰invｰcontractive t n :
    Proper (
      (pointwise_relation _ (dist_later n)) ==>
      (pointwise_relation _ (dist_later n)) ==>
      (≡{n}≡)
    ) (lazy۰inv t).
  #[global] Instance lazy۰invｰproper t :
    Proper (
      (pointwise_relation _ (≡)) ==>
      (pointwise_relation _ (≡)) ==>
      (≡)
    ) (lazy۰inv t).
  #[global] Instance lazy۰consumerｰcontractive t n :
    Proper (
      (pointwise_relation _ (dist_later n)) ==>
      (≡{n}≡)
    ) (lazy۰consumer t).
  #[global] Instance lazy۰consumerｰproper t :
    Proper (
      (pointwise_relation _ (≡)) ==>
      (≡)
    ) (lazy۰consumer t).

  #[global] Instance lazy۰resultｰtimeless t v :
    Timeless (lazy۰result t v).

  #[global] Instance lazy۰invｰpersistent t Ψ Ξ :
    Persistent (lazy۰inv t Ψ Ξ).
  #[global] Instance lazy۰resultｰpersistent t v :
    Persistent (lazy۰result t v).

  Lemma lazy۰consumerｰwand {t Ψ Ξ Χ1} Χ2 :
    lazy۰inv t Ψ Ξ -∗
    lazy۰consumer t Χ1 -∗
    (∀ v, Χ1 v -∗ Χ2 v) ={⊤}=∗
    lazy۰consumer t Χ2.
  Lemma lazy۰consumerｰdivide {t Ψ Ξ} Χs :
    lazy۰inv t Ψ Ξ -∗
    lazy۰consumer t (λ v, [∗ list] Χ ∈ Χs, Χ v) ={⊤}=∗
    [∗ list] Χ ∈ Χs, lazy۰consumer t Χ.
  Lemma lazy۰consumerｰsplit {t Ψ Ξ} Χ1 Χ2 :
    lazy۰inv t Ψ Ξ -∗
    lazy۰consumer t (λ v, Χ1 v ∗ Χ2 v) ={⊤}=∗
      lazy۰consumer t Χ1 ∗
      lazy۰consumer t Χ2.
  Lemma lazy۰resultｰagree t v1 v2 :
    lazy۰result t v1 -∗
    lazy۰result t v2 -∗
    ⌜v1 = v2⌝.

  Lemma lazyｰinvｰresult t Ψ Ξ v :
    lazy۰inv t Ψ Ξ -∗
    lazy۰result t v ={⊤}=∗
    ▷ □ Ξ v.
  Lemma lazyｰinv_result' t Ψ Ξ v :
    £ 1 -∗
    lazy۰inv t Ψ Ξ -∗
    lazy۰result t v ={⊤}=∗
    □ Ξ v.
  Lemma lazyｰinvｰresultｰconsumer t Ψ Ξ v Χ :
    lazy۰inv t Ψ Ξ -∗
    lazy۰result t v -∗
    lazy۰consumer t Χ ={⊤}=∗
      ▷^2 Χ v ∗
      ▷ □ Ξ v.
  Lemma lazyｰinvｰresultｰconsumer' t Ψ Ξ v Χ :
    £ 2 -∗
    lazy۰inv t Ψ Ξ -∗
    lazy۰result t v -∗
    lazy۰consumer t Χ ={⊤}=∗
      Χ v ∗
      □ Ξ v.

  Lemma lazy٠makeｰspec Ψ Ξ fn :
    {{{
      WP fn () {{ v,
        ▷ Ψ v ∗
        ▷ □ Ξ v
      }}
    }}}
      lazy٠make fn
    {{{
      t
    , RET t;
      lazy۰inv t Ψ Ξ ∗
      lazy۰consumer t Ψ
    }}}.

  Lemma lazy٠returnｰspec Ψ Ξ v :
    {{{
      ▷ Ψ v ∗
      ▷ □ Ξ v
    }}}
      lazy٠return v
    {{{
      t
    , RET t;
      lazy۰inv t Ψ Ξ ∗
      lazy۰result t v ∗
      lazy۰consumer t Ψ
    }}}.

  Lemma lazy٠is_setｰspec t Ψ Ξ :
    {{{
      lazy۰inv t Ψ Ξ
    }}}
      lazy٠is_set t
    {{{
      b
    , RET #b;
      if b then
        £ 2 ∗
        lazy۰resolved t
      else
        True
    }}}.
  Lemma lazy٠is_setｰspecｰresult t Ψ Ξ v :
    {{{
      lazy۰inv t Ψ Ξ ∗
      lazy۰result t v
    }}}
      lazy٠is_set t
    {{{
      RET true;
      £ 2
    }}}.

  Lemma lazy٠is_unsetｰspec t Ψ Ξ :
    {{{
      lazy۰inv t Ψ Ξ
    }}}
      lazy٠is_unset t
    {{{
      b
    , RET #b;
      if b then
        True
      else
        £ 2 ∗
        lazy۰resolved t
    }}}.
  Lemma lazy٠is_unsetｰspecｰresult t Ψ Ξ v :
    {{{
      lazy۰inv t Ψ Ξ ∗
      lazy۰result t v
    }}}
      lazy٠is_unset t
    {{{
      RET false;
      £ 2
    }}}.

  Lemma lazy٠getｰspec t Ψ Ξ :
    {{{
      lazy۰inv t Ψ Ξ
    }}}
      lazy٠get t
    {{{
      v
    , RET v;
      £ 2 ∗
      lazy۰result t v
    }}}.
  Lemma lazy٠getｰspecｰresult t Ψ Ξ v :
    {{{
      lazy۰inv t Ψ Ξ ∗
      lazy۰result t v
    }}}
      lazy٠get t
    {{{
      RET v;
      £ 2
    }}}.
End lazy۰G.

#[global] Opaque lazy۰inv.
#[global] Opaque lazy۰consumer.
#[global] Opaque lazy۰result.