ghost_ownership.v 5.03 KB
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Require Export algebra.iprod program_logic.pviewshifts.
Require Import program_logic.ownership.
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Import uPred.

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(** Index of a CMRA in the product of global CMRAs. *)
Definition gid := nat.
(** Name of one instance of a particular CMRA in the ghost state. *)
Definition gname := positive.
(** The global CMRA: Indexed product over a gid i to (gname --fin--> Σ i) *)
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Definition globalC (Σ : gid  iFunctor) : iFunctor :=
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  iprodF (λ i, mapF gname (Σ i)).
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Class InG (Λ : language) (Σ : gid  iFunctor) (i : gid) (A : cmraT) :=
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  inG : A = Σ i (laterC (iPreProp Λ (globalC Σ))).
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Definition to_globalC {Λ Σ A}
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    (i : gid) `{!InG Λ Σ i A} (γ : gname) (a : A) : iGst Λ (globalC Σ) :=
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  iprod_singleton i {[ γ  cmra_transport inG a ]}.
Definition own {Λ Σ A}
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    (i : gid) `{!InG Λ Σ i A} (γ : gname) (a : A) : iProp Λ (globalC Σ) :=
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  ownG (to_globalC i γ a).
Instance: Params (@to_globalC) 6.
Instance: Params (@own) 6.
Typeclasses Opaque to_globalC own.

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Section global.
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Context {Λ : language} {Σ : gid  iFunctor} (i : gid) `{!InG Λ Σ i A}.
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Implicit Types a : A.

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(** * Properties of to_globalC *)
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Instance to_globalC_ne γ n : Proper (dist n ==> dist n) (to_globalC i γ).
Proof. by intros a a' Ha; apply iprod_singleton_ne; rewrite Ha. Qed.
Lemma to_globalC_validN n γ a : {n} (to_globalC i γ a)  {n} a.
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Proof.
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  by rewrite /to_globalC
    iprod_singleton_validN map_singleton_validN cmra_transport_validN.
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Qed.
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Lemma to_globalC_op γ a1 a2 :
  to_globalC i γ (a1  a2)  to_globalC i γ a1  to_globalC i γ a2.
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Proof.
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  by rewrite /to_globalC iprod_op_singleton map_op_singleton cmra_transport_op.
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Qed.
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Lemma to_globalC_unit γ a : unit (to_globalC i γ a)  to_globalC i γ (unit a).
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Proof.
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  by rewrite /to_globalC
    iprod_unit_singleton map_unit_singleton cmra_transport_unit.
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Qed.
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Instance to_globalC_timeless γ m : Timeless m  Timeless (to_globalC i γ m).
Proof. rewrite /to_globalC; apply _. Qed.
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(** * Transport empty *)
Instance inG_empty `{Empty A} : Empty (Σ i (laterC (iPreProp Λ (globalC Σ)))) :=
  cmra_transport inG .
Instance inG_empty_spec `{Empty A} :
  CMRAIdentity A  CMRAIdentity (Σ i (laterC (iPreProp Λ (globalC Σ)))).
Proof.
  split.
  * apply cmra_transport_valid, cmra_empty_valid.
  * intros x; rewrite /empty /inG_empty; destruct inG. by rewrite left_id.
  * apply _.
Qed.

(** * Properties of own *)
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Global Instance own_ne γ n : Proper (dist n ==> dist n) (own i γ).
Proof. by intros m m' Hm; rewrite /own Hm. Qed.
Global Instance own_proper γ : Proper (() ==> ()) (own i γ) := ne_proper _.

Lemma own_op γ a1 a2 : own i γ (a1  a2)  (own i γ a1  own i γ a2)%I.
Proof. by rewrite /own -ownG_op to_globalC_op. Qed.
Lemma always_own_unit γ a : ( own i γ (unit a))%I  own i γ (unit a).
Proof. by rewrite /own -to_globalC_unit always_ownG_unit. Qed.
Lemma own_valid γ a : own i γ a   a.
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Proof.
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  rewrite /own ownG_valid; apply valid_mono=> ?; apply to_globalC_validN.
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Qed.
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Lemma own_valid_r γ a : own i γ a  (own i γ a   a).
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Proof. apply (uPred.always_entails_r' _ _), own_valid. Qed.
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Global Instance own_timeless γ a : Timeless a  TimelessP (own i γ a).
Proof. unfold own; apply _. Qed.
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(* TODO: This also holds if we just have ✓ a at the current step-idx, as Iris
   assertion. However, the map_updateP_alloc does not suffice to show this. *)
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Lemma own_alloc a E :  a  True  pvs E E ( γ, own i γ a).
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Proof.
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  intros Ha.
  rewrite -(pvs_mono _ _ ( m,  ( γ, m = to_globalC i γ a)  ownG m)%I).
  * eapply pvs_ownG_updateP_empty, (iprod_singleton_updateP_empty i);
      first (eapply map_updateP_alloc', cmra_transport_valid, Ha); naive_solver.
  * apply exist_elim=>m; apply const_elim_l=>-[γ ->].
    by rewrite -(exist_intro γ).
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Qed.
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Lemma own_updateP γ a P E :
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  a ~~>: P  own i γ a  pvs E E ( a',  P a'  own i γ a').
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Proof.
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  intros Ha.
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  rewrite -(pvs_mono _ _ ( m,  ( a', m = to_globalC i γ a'  P a')  ownG m)%I).
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  * eapply pvs_ownG_updateP, iprod_singleton_updateP;
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      first by (eapply map_singleton_updateP', cmra_transport_updateP', Ha).
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    naive_solver.
  * apply exist_elim=>m; apply const_elim_l=>-[a' [-> HP]].
    rewrite -(exist_intro a'). by apply and_intro; [apply const_intro|].
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Qed.

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Lemma own_updateP_empty `{Empty A, !CMRAIdentity A} γ a P E :
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   ~~>: P  True  pvs E E ( a,  P a  own i γ a).
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Proof.
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  intros Hemp.
  rewrite -(pvs_mono _ _ ( m,  ( a', m = to_globalC i γ a'  P a')  ownG m)%I).
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  * eapply pvs_ownG_updateP_empty, iprod_singleton_updateP_empty;
      first eapply map_singleton_updateP_empty', cmra_transport_updateP', Hemp.
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    naive_solver.
  * apply exist_elim=>m; apply const_elim_l=>-[a' [-> HP]].
    rewrite -(exist_intro a'). by apply and_intro; [apply const_intro|].
Qed.

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Lemma own_update γ a a' E : a ~~> a'  own i γ a  pvs E E (own i γ a').
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Proof.
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  intros; rewrite (own_updateP _ _ (a' =)); last by apply cmra_update_updateP.
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  by apply pvs_mono, uPred.exist_elim=> m''; apply uPred.const_elim_l=> ->.
Qed.
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End global.