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From iris.algebra Require Export cmra.

(** * Local updates *)
(** The idea is that lemams taking this class will usually have L explicit,
    and leave Lv implicit - it will be inferred by the typeclass machinery. *)
Class LocalUpdate {A : cmraT} (Lv : A  Prop) (L : A  A) := {
  local_update_ne n :> Proper (dist n ==> dist n) L;
  local_updateN n x y : Lv x  {n} (x  y)  L (x  y) {n} L x  y
}.
Arguments local_updateN {_ _} _ {_} _ _ _ _ _.

(** * Frame preserving updates *)
Definition cmra_updateP {A : cmraT} (x : A) (P : A  Prop) :=  n mz,
  {n} (x ? mz)   y, P y  {n} (y ? mz).
Instance: Params (@cmra_updateP) 1.
Infix "~~>:" := cmra_updateP (at level 70).

Definition cmra_update {A : cmraT} (x y : A) :=  n mz,
  {n} (x ? mz)  {n} (y ? mz).
Infix "~~>" := cmra_update (at level 70).
Instance: Params (@cmra_update) 1.

(** ** CMRAs *)
Section cmra.
Context {A : cmraT}.
Implicit Types x y : A.

Global Instance cmra_updateP_proper :
  Proper (() ==> pointwise_relation _ iff ==> iff) (@cmra_updateP A).
Proof.
  rewrite /pointwise_relation /cmra_updateP=> x x' Hx P P' HP;
    split=> ? n mz; setoid_subst; naive_solver.
Qed.
Global Instance cmra_update_proper :
  Proper (() ==> () ==> iff) (@cmra_update A).
Proof.
  rewrite /cmra_update=> x x' Hx y y' Hy; split=> ? n mz ?; setoid_subst; auto.
Qed.

(** ** Local updates *)
Global Instance local_update_proper (L : A  A) Lv :
  LocalUpdate Lv L  Proper (() ==> ()) L.
Proof. intros; apply (ne_proper _). Qed.

Lemma local_update (L : A  A) `{!LocalUpdate Lv L} x y :
  Lv x   (x  y)  L (x  y)  L x  y.
Proof.
  by rewrite cmra_valid_validN equiv_dist=>?? n; apply (local_updateN L).
Qed.

Global Instance op_local_update x : LocalUpdate (λ _, True) (op x).
Proof. split. apply _. by intros n y1 y2 _ _; rewrite assoc. Qed.

Global Instance id_local_update : LocalUpdate (λ _, True) (@id A).
Proof. split; auto with typeclass_instances. Qed.

Global Instance exclusive_local_update y :
  LocalUpdate Exclusive (λ _, y) | 1000.
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Proof. split. apply _. by intros ?????%exclusiveN_l. Qed.
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(** ** Frame preserving updates *)
Lemma cmra_update_updateP x y : x ~~> y  x ~~>: (y =).
Proof. split=> Hup n z ?; eauto. destruct (Hup n z) as (?&<-&?); auto. Qed.
Lemma cmra_updateP_id (P : A  Prop) x : P x  x ~~>: P.
Proof. intros ? n mz ?; eauto. Qed.
Lemma cmra_updateP_compose (P Q : A  Prop) x :
  x ~~>: P  ( y, P y  y ~~>: Q)  x ~~>: Q.
Proof. intros Hx Hy n mz ?. destruct (Hx n mz) as (y&?&?); naive_solver. Qed.
Lemma cmra_updateP_compose_l (Q : A  Prop) x y : x ~~> y  y ~~>: Q  x ~~>: Q.
Proof.
  rewrite cmra_update_updateP.
  intros; apply cmra_updateP_compose with (y =); naive_solver.
Qed.
Lemma cmra_updateP_weaken (P Q : A  Prop) x :
  x ~~>: P  ( y, P y  Q y)  x ~~>: Q.
Proof. eauto using cmra_updateP_compose, cmra_updateP_id. Qed.
Global Instance cmra_update_preorder : PreOrder (@cmra_update A).
Proof.
  split.
  - intros x. by apply cmra_update_updateP, cmra_updateP_id.
  - intros x y z. rewrite !cmra_update_updateP.
    eauto using cmra_updateP_compose with subst.
Qed.
Lemma cmra_update_exclusive `{!Exclusive x} y:
   y  x ~~> y.
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Proof. move=>??[z|]=>[/exclusiveN_l[]|_]. by apply cmra_valid_validN. Qed.
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Lemma cmra_updateP_op (P1 P2 Q : A  Prop) x1 x2 :
  x1 ~~>: P1  x2 ~~>: P2  ( y1 y2, P1 y1  P2 y2  Q (y1  y2)) 
  x1  x2 ~~>: Q.
Proof.
  intros Hx1 Hx2 Hy n mz ?.
  destruct (Hx1 n (Some (x2 ? mz))) as (y1&?&?).
  { by rewrite /= -cmra_opM_assoc. }
  destruct (Hx2 n (Some (y1 ? mz))) as (y2&?&?).
  { by rewrite /= -cmra_opM_assoc (comm _ x2) cmra_opM_assoc. }
  exists (y1  y2); split; last rewrite (comm _ y1) cmra_opM_assoc; auto.
Qed.
Lemma cmra_updateP_op' (P1 P2 : A  Prop) x1 x2 :
  x1 ~~>: P1  x2 ~~>: P2 
  x1  x2 ~~>: λ y,  y1 y2, y = y1  y2  P1 y1  P2 y2.
Proof. eauto 10 using cmra_updateP_op. Qed.
Lemma cmra_update_op x1 x2 y1 y2 : x1 ~~> y1  x2 ~~> y2  x1  x2 ~~> y1  y2.
Proof.
  rewrite !cmra_update_updateP; eauto using cmra_updateP_op with congruence.
Qed.

(** ** Frame preserving updates for total CMRAs *)
Section total_updates.
  Context `{CMRATotal A}.

  Lemma cmra_total_updateP x (P : A  Prop) :
    x ~~>: P   n z, {n} (x  z)   y, P y  {n} (y  z).
  Proof.
    split=> Hup; [intros n z; apply (Hup n (Some z))|].
    intros n [z|] ?; simpl; [by apply Hup|].
    destruct (Hup n (core x)) as (y&?&?); first by rewrite cmra_core_r.
    eauto using cmra_validN_op_l.
  Qed.
  Lemma cmra_total_update x y : x ~~> y   n z, {n} (x  z)  {n} (y  z).
  Proof. rewrite cmra_update_updateP cmra_total_updateP. naive_solver. Qed.

  Context `{CMRADiscrete A}.

  Lemma cmra_discrete_updateP (x : A) (P : A  Prop) :
    x ~~>: P   z,  (x  z)   y, P y   (y  z).
  Proof.
    rewrite cmra_total_updateP; setoid_rewrite <-cmra_discrete_valid_iff.
    naive_solver eauto using 0.
  Qed.
  Lemma cmra_discrete_update `{CMRADiscrete A} (x y : A) :
    x ~~> y   z,  (x  z)   (y  z).
  Proof.
    rewrite cmra_total_update; setoid_rewrite <-cmra_discrete_valid_iff.
    naive_solver eauto using 0.
  Qed.
End total_updates.
End cmra.

(** ** CMRAs with a unit *)
Section ucmra.
  Context {A : ucmraT}.
  Implicit Types x y : A.

  Lemma ucmra_update_unit x : x ~~> .
  Proof.
    apply cmra_total_update=> n z. rewrite left_id; apply cmra_validN_op_r.
  Qed.
  Lemma ucmra_update_unit_alt y :  ~~> y   x, x ~~> y.
  Proof. split; [intros; trans |]; auto using ucmra_update_unit. Qed.
End ucmra.

Section cmra_transport.
  Context {A B : cmraT} (H : A = B).
  Notation T := (cmra_transport H).
  Lemma cmra_transport_updateP (P : A  Prop) (Q : B  Prop) x :
    x ~~>: P  ( y, P y  Q (T y))  T x ~~>: Q.
  Proof. destruct H; eauto using cmra_updateP_weaken. Qed.
  Lemma cmra_transport_updateP' (P : A  Prop) x :
    x ~~>: P  T x ~~>: λ y,  y', y = cmra_transport H y'  P y'.
  Proof. eauto using cmra_transport_updateP. Qed.
End cmra_transport.

(** * Product *)
Section prod.
  Context {A B : cmraT}.
  Implicit Types x : A * B.

  Lemma prod_updateP P1 P2 (Q : A * B  Prop) x :
    x.1 ~~>: P1  x.2 ~~>: P2  ( a b, P1 a  P2 b  Q (a,b))  x ~~>: Q.
  Proof.
    intros Hx1 Hx2 HP n mz [??]; simpl in *.
    destruct (Hx1 n (fst <$> mz)) as (a&?&?); first by destruct mz.
    destruct (Hx2 n (snd <$> mz)) as (b&?&?); first by destruct mz.
    exists (a,b); repeat split; destruct mz; auto.
  Qed.
  Lemma prod_updateP' P1 P2 x :
    x.1 ~~>: P1  x.2 ~~>: P2  x ~~>: λ y, P1 (y.1)  P2 (y.2).
  Proof. eauto using prod_updateP. Qed.
  Lemma prod_update x y : x.1 ~~> y.1  x.2 ~~> y.2  x ~~> y.
  Proof.
    rewrite !cmra_update_updateP.
    destruct x, y; eauto using prod_updateP with subst.
  Qed.

  Global Instance prod_local_update
      (LA : A  A) `{!LocalUpdate LvA LA} (LB : B  B) `{!LocalUpdate LvB LB} :
    LocalUpdate (λ x, LvA (x.1)  LvB (x.2)) (prod_map LA LB).
  Proof.
    constructor.
    - intros n x y [??]; constructor; simpl; by apply local_update_ne.
    - intros n ?? [??] [??];
        constructor; simpl in *; eapply local_updateN; eauto.
  Qed.
End prod.

(** * Option *)
Section option.
  Context {A : cmraT}.
  Implicit Types x y : A.

  Global Instance option_fmap_local_update (L : A  A) Lv :
    LocalUpdate Lv L 
    LocalUpdate (λ mx, if mx is Some x then Lv x else False) (fmap L).
  Proof.
    split; first apply _.
    intros n [x|] [z|]; constructor; by eauto using (local_updateN L).
  Qed.
  Global Instance option_const_local_update Lv y :
    LocalUpdate Lv (λ _, y) 
    LocalUpdate (λ mx, if mx is Some x then Lv x else False) (λ _, Some y).
  Proof.
    split; first apply _.
    intros n [x|] [z|]; constructor; by eauto using (local_updateN (λ _, y)).
  Qed.

  Lemma option_updateP (P : A  Prop) (Q : option A  Prop) x :
    x ~~>: P  ( y, P y  Q (Some y))  Some x ~~>: Q.
  Proof.
    intros Hx Hy; apply cmra_total_updateP=> n [y|] ?.
    { destruct (Hx n (Some y)) as (y'&?&?); auto. exists (Some y'); auto. }
    destruct (Hx n None) as (y'&?&?); rewrite ?cmra_core_r; auto.
    by exists (Some y'); auto.
  Qed.
  Lemma option_updateP' (P : A  Prop) x :
    x ~~>: P  Some x ~~>: from_option P False.
  Proof. eauto using option_updateP. Qed.
  Lemma option_update x y : x ~~> y  Some x ~~> Some y.
  Proof.
    rewrite !cmra_update_updateP; eauto using option_updateP with congruence.
  Qed.
End option.