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From algebra Require Export cmra.
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From algebra Require Import functor upred.
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(* COFE *)
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Section cofe.
Context {A : cofeT}.
Inductive option_dist : Dist (option A) :=
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  | Some_dist n x y : x {n} y  Some x {n} Some y
  | None_dist n : None {n} None.
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Existing Instance option_dist.
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Program Definition option_chain
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    (c : chain (option A)) (x : A) (H : c 1 = Some x) : chain A :=
  {| chain_car n := from_option x (c n) |}.
Next Obligation.
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  intros c x ? n [|i] ?; [omega|]; simpl.
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  destruct (c 1) eqn:?; simplify_eq/=.
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  by feed inversion (chain_cauchy c n (S i)).
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Qed.
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Instance option_compl : Compl (option A) := λ c,
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  match Some_dec (c 1) with
  | inleft (exist x H) => Some (compl (option_chain c x H)) | inright _ => None
  end.
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Definition option_cofe_mixin : CofeMixin (option A).
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Proof.
  split.
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  - intros mx my; split; [by destruct 1; constructor; apply equiv_dist|].
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    intros Hxy; feed inversion (Hxy 1); subst; constructor; apply equiv_dist.
    by intros n; feed inversion (Hxy n).
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  - intros n; split.
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    + by intros [x|]; constructor.
    + by destruct 1; constructor.
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    + destruct 1; inversion_clear 1; constructor; etrans; eauto.
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  - by inversion_clear 1; constructor; apply dist_S.
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  - intros n c; unfold compl, option_compl.
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    destruct (Some_dec (c 1)) as [[x Hx]|].
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    { assert (is_Some (c (S n))) as [y Hy].
      { feed inversion (chain_cauchy c 0 (S n)); eauto with lia congruence. }
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      rewrite Hy; constructor.
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      by rewrite (conv_compl n (option_chain c x Hx)) /= Hy. }
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    feed inversion (chain_cauchy c 0 (S n)); eauto with lia congruence.
    constructor.
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Qed.
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Canonical Structure optionC := CofeT option_cofe_mixin.
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Global Instance option_discrete : Discrete A  Discrete optionC.
Proof. inversion_clear 2; constructor; by apply (timeless _). Qed.

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Global Instance Some_ne : Proper (dist n ==> dist n) (@Some A).
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Proof. by constructor. Qed.
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Global Instance is_Some_ne n : Proper (dist n ==> iff) (@is_Some A).
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Proof. inversion_clear 1; split; eauto. Qed.
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Global Instance Some_dist_inj : Inj (dist n) (dist n) (@Some A).
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Proof. by inversion_clear 1. Qed.
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Global Instance None_timeless : Timeless (@None A).
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Proof. inversion_clear 1; constructor. Qed.
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Global Instance Some_timeless x : Timeless x  Timeless (Some x).
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Proof. by intros ?; inversion_clear 1; constructor; apply timeless. Qed.
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End cofe.

Arguments optionC : clear implicits.

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(* CMRA *)
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Section cmra.
Context {A : cmraT}.

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Instance option_valid : Valid (option A) := λ mx,
  match mx with Some x =>  x | None => True end.
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Instance option_validN : ValidN (option A) := λ n mx,
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  match mx with Some x => {n} x | None => True end.
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Global Instance option_empty : Empty (option A) := None.
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Instance option_unit : Unit (option A) := fmap unit.
Instance option_op : Op (option A) := union_with (λ x y, Some (x  y)).
Instance option_minus : Minus (option A) :=
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  difference_with (λ x y, Some (x  y)).
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Lemma option_includedN n (mx my : option A) :
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  mx {n} my  mx = None   x y, mx = Some x  my = Some y  x {n} y.
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Proof.
  split.
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  - intros [mz Hmz].
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    destruct mx as [x|]; [right|by left].
    destruct my as [y|]; [exists x, y|destruct mz; inversion_clear Hmz].
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    destruct mz as [z|]; inversion_clear Hmz; split_and?; auto;
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      cofe_subst; eauto using cmra_includedN_l.
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  - intros [->|(x&y&->&->&z&Hz)]; try (by exists my; destruct my; constructor).
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    by exists (Some z); constructor.
Qed.
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Lemma None_includedN n (mx : option A) : None {n} mx.
Proof. rewrite option_includedN; auto. Qed.
Lemma Some_Some_includedN n (x y : A) : x {n} y  Some x {n} Some y.
Proof. rewrite option_includedN; eauto 10. Qed.
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Definition Some_op a b : Some (a  b) = Some a  Some b := eq_refl.
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Definition option_cmra_mixin  : CMRAMixin (option A).
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Proof.
  split.
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  - by intros n [x|]; destruct 1; constructor; cofe_subst.
  - by destruct 1; constructor; cofe_subst.
  - by destruct 1; rewrite /validN /option_validN //=; cofe_subst.
  - by destruct 1; inversion_clear 1; constructor; cofe_subst.
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  - intros [x|]; [apply cmra_valid_validN|done].
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  - intros n [x|]; unfold validN, option_validN; eauto using cmra_validN_S.
  - intros [x|] [y|] [z|]; constructor; rewrite ?assoc; auto.
  - intros [x|] [y|]; constructor; rewrite 1?comm; auto.
  - by intros [x|]; constructor; rewrite cmra_unit_l.
  - by intros [x|]; constructor; rewrite cmra_unit_idemp.
  - intros n mx my; rewrite !option_includedN;intros [->|(x&y&->&->&?)]; auto.
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    right; exists (unit x), (unit y); eauto using cmra_unit_preservingN.
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  - intros n [x|] [y|]; rewrite /validN /option_validN /=;
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      eauto using cmra_validN_op_l.
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  - intros n mx my; rewrite option_includedN.
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    intros [->|(x&y&->&->&?)]; [by destruct my|].
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    by constructor; apply cmra_op_minus.
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  - intros n mx my1 my2.
    destruct mx as [x|], my1 as [y1|], my2 as [y2|]; intros Hx Hx';
      try (by exfalso; inversion Hx'; auto).
    + destruct (cmra_extend n x y1 y2) as ([z1 z2]&?&?&?); auto.
      { by inversion_clear Hx'. }
      by exists (Some z1, Some z2); repeat constructor.
    + by exists (Some x,None); inversion Hx'; repeat constructor.
    + by exists (None,Some x); inversion Hx'; repeat constructor.
    + exists (None,None); repeat constructor.
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Qed.
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Canonical Structure optionRA := CMRAT option_cofe_mixin option_cmra_mixin.
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Global Instance option_cmra_identity : CMRAIdentity optionRA.
Proof. split. done. by intros []. by inversion_clear 1. Qed.
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Global Instance option_cmra_discrete : CMRADiscrete A  CMRADiscrete optionRA.
Proof. split; [apply _|]. by intros [x|]; [apply (cmra_discrete_valid x)|]. Qed.
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(** Misc *)
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Lemma op_is_Some mx my : is_Some (mx  my)  is_Some mx  is_Some my.
Proof.
  destruct mx, my; rewrite /op /option_op /= -!not_eq_None_Some; naive_solver.
Qed.
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Lemma option_op_positive_dist_l n mx my : mx  my {n} None  mx {n} None.
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Proof. by destruct mx, my; inversion_clear 1. Qed.
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Lemma option_op_positive_dist_r n mx my : mx  my {n} None  my {n} None.
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Proof. by destruct mx, my; inversion_clear 1. Qed.

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(** Internalized properties *)
Lemma option_equivI {M} (x y : option A) :
  (x  y)%I  (match x, y with
               | Some a, Some b => a  b | None, None => True | _, _ => False
               end : uPred M)%I.
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Proof. do 2 split. by destruct 1. by destruct x, y; try constructor. Qed.
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Lemma option_validI {M} (x : option A) :
  ( x)%I  (match x with Some a =>  a | None => True end : uPred M)%I.
Proof. by destruct x. Qed.

(** Updates *)
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Lemma option_updateP (P : A  Prop) (Q : option A  Prop) x :
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  x ~~>: P  ( y, P y  Q (Some y))  Some x ~~>: Q.
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Proof.
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  intros Hx Hy n [y|] ?.
  { destruct (Hx n y) as (y'&?&?); auto. exists (Some y'); auto. }
  destruct (Hx n (unit x)) as (y'&?&?); rewrite ?cmra_unit_r; auto.
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  by exists (Some y'); split; [auto|apply cmra_validN_op_l with (unit x)].
Qed.
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Lemma option_updateP' (P : A  Prop) x :
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  x ~~>: P  Some x ~~>: λ y, default False y P.
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Proof. eauto using option_updateP. Qed.
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Lemma option_update x y : x ~~> y  Some x ~~> Some y.
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Proof.
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  rewrite !cmra_update_updateP; eauto using option_updateP with congruence.
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Qed.
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Lemma option_update_None `{Empty A, !CMRAIdentity A} :  ~~> Some .
Proof.
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  intros n [x|] ?; rewrite /op /cmra_op /validN /cmra_validN /= ?left_id;
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    auto using cmra_empty_validN.
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Qed.
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End cmra.
Arguments optionRA : clear implicits.

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(** Functor *)
Instance option_fmap_ne {A B : cofeT} (f : A  B) n:
  Proper (dist n ==> dist n) f  Proper (dist n==>dist n) (fmap (M:=option) f).
Proof. by intros Hf; destruct 1; constructor; apply Hf. Qed.
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Instance option_fmap_cmra_monotone {A B : cmraT} (f: A  B) `{!CMRAMonotone f} :
  CMRAMonotone (fmap f : option A  option B).
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Proof.
  split.
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  - intros n mx my; rewrite !option_includedN.
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    intros [->|(x&y&->&->&?)]; simpl; eauto 10 using @includedN_preserving.
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  - by intros n [x|] ?; rewrite /cmra_validN /=; try apply validN_preserving.
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Qed.
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Definition optionC_map {A B} (f : A -n> B) : optionC A -n> optionC B :=
  CofeMor (fmap f : optionC A  optionC B).
Instance optionC_map_ne A B n : Proper (dist n ==> dist n) (@optionC_map A B).
Proof. by intros f f' Hf []; constructor; apply Hf. Qed.
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Program Definition optionF (Σ : iFunctor) : iFunctor := {|
  ifunctor_car := optionRA  Σ; ifunctor_map A B := optionC_map  ifunctor_map Σ
|}.
Next Obligation.
  by intros Σ A B n f g Hfg; apply optionC_map_ne, ifunctor_map_ne.
Qed.
Next Obligation.
  intros Σ A x. rewrite /= -{2}(option_fmap_id x).
  apply option_fmap_setoid_ext=>y; apply ifunctor_map_id.
Qed.
Next Obligation.
  intros Σ A B C f g x. rewrite /= -option_fmap_compose.
  apply option_fmap_setoid_ext=>y; apply ifunctor_map_compose.
Qed.