Fix merge with master

theories/heap_lang/proph_map.v

-From iris.algebra Require Import excl auth gmap.
-From iris.base_logic.lib Require Export own.
-From iris.proofmode Require Import tactics.
-Set Default Proof Using "Type".
-Import uPred.
-
-Definition proph_map (P V : Type) `{Countable P} := gmap P (option V).
-Definition proph_val_list (P V : Type) := list (P * V).
-
-Definition proph_mapUR (P V : Type) `{Countable P} : ucmraT :=
-  gmapUR P $ exclR $ optionC $ leibnizC V.
-
-Definition to_proph_map {P V} `{Countable P} (pvs : proph_map P V) : proph_mapUR P V :=
-  fmap (λ v, Excl (v : option (leibnizC V))) pvs.
-
-(** The CMRA we need. *)
-Class proph_mapG (P V : Type) (Σ : gFunctors) `{Countable P} := ProphMapG {
-  proph_map_inG :> inG Σ (authR (proph_mapUR P V));
-  proph_map_name : gname
-}.
-Arguments proph_map_name {_ _ _ _ _} _ : assert.
-
-Class proph_mapPreG (P V : Type) (Σ : gFunctors) `{Countable P} :=
-  { proph_map_preG_inG :> inG Σ (authR (proph_mapUR P V)) }.
-
-Definition proph_mapΣ (P V : Type) `{Countable P} : gFunctors :=
-  #[GFunctor (authR (proph_mapUR P V))].
-
-Instance subG_proph_mapPreG {Σ P V} `{Countable P} :
-  subG (proph_mapΣ P V) Σ → proph_mapPreG P V Σ.
-Proof. solve_inG. Qed.
-
-Section definitions.
-  Context `{Countable P, pG : !proph_mapG P V Σ}.
-
-  (** The first resolve for [p] in [pvs] *)
-  Definition first_resolve (pvs : proph_val_list P V) (p : P) : option V :=
-    (list_to_map pvs : gmap P V) !! p.
-
-  Definition first_resolve_in_list (R : proph_map P V) (pvs : proph_val_list P V) :=
-    ∀ p v, p ∈ dom (gset _) R →
-           first_resolve pvs p = Some v →
-           R !! p = Some (Some v).
-
-  Definition proph_map_auth (R : proph_map P V) : iProp Σ :=
-    own (proph_map_name pG) (● (to_proph_map R)).
-
-  Definition proph_map_ctx (pvs : proph_val_list P V) (ps : gset P) : iProp Σ :=
-    (∃ R, ⌜first_resolve_in_list R pvs ∧
-          dom (gset _) R ⊆ ps⌝ ∗
-          proph_map_auth R)%I.
-
-  Definition proph_def (p : P) (v: option V) : iProp Σ :=
-    own (proph_map_name pG) (◯ {[ p := Excl (v : option $ leibnizC V) ]}).
-  Definition proph_aux : seal (@proph_def). by eexists. Qed.
-  Definition proph := proph_aux.(unseal).
-  Definition proph_eq :
-    @proph = @proph_def := proph_aux.(seal_eq).
-End definitions.
-
-Section first_resolve.
-  Context {P V : Type} `{Countable P}.
-  Implicit Type pvs : proph_val_list P V.
-  Implicit Type p : P.
-  Implicit Type v : V.
-  Implicit Type R : proph_map P V.
-
-  Lemma first_resolve_insert pvs p R :
-    first_resolve_in_list R pvs →
-    p ∉ dom (gset _) R →
-    first_resolve_in_list (<[p := first_resolve pvs p]> R) pvs.
-  Proof.
-    intros Hf Hnotin p' v' Hp'. rewrite (dom_insert_L R p) in Hp'.
-    erewrite elem_of_union in Hp'. destruct Hp' as [->%elem_of_singleton | Hin].
-    - intros ->. by rewrite lookup_insert.
-    - intros <-%Hf; last done. rewrite lookup_insert_ne; first done.
-      intros ?. subst. done.
-  Qed.
-
-  Lemma first_resolve_delete pvs p v R :
-    first_resolve_in_list R ((p, v) :: pvs) →
-    first_resolve_in_list (delete p R) pvs.
-  Proof.
-    intros Hfr p' v' Hpin Heq. rewrite dom_delete_L in Hpin. rewrite /first_resolve in Heq.
-    apply elem_of_difference in Hpin as [Hpin Hne%not_elem_of_singleton].
-    erewrite <- lookup_insert_ne in Heq; last done. rewrite lookup_delete_ne; eauto.
-  Qed.
-
-  Lemma first_resolve_eq R p v w pvs :
-    first_resolve_in_list R ((p, v) :: pvs) →
-    R !! p = Some w →
-    w = Some v.
-  Proof.
-    intros Hfr Hlookup. specialize (Hfr p v (elem_of_dom_2 _ _ _ Hlookup)).
-    rewrite /first_resolve lookup_insert in Hfr. rewrite Hfr in Hlookup; last done.
-    inversion Hlookup. done.
-  Qed.
-
-End first_resolve.
-
-Section to_proph_map.
-  Context (P V : Type) `{Countable P}.
-  Implicit Types p : P.
-  Implicit Types R : proph_map P V.
-
-  Lemma to_proph_map_valid R : ✓ to_proph_map R.
-  Proof. intros l. rewrite lookup_fmap. by case (R !! l). Qed.
-
-  Lemma to_proph_map_insert p v R :
-    to_proph_map (<[p := v]> R) = <[p := Excl (v: option (leibnizC V))]> (to_proph_map R).
-  Proof. by rewrite /to_proph_map fmap_insert. Qed.
-
-  Lemma to_proph_map_delete p R :
-    to_proph_map (delete p R) = delete p (to_proph_map R).
-  Proof. by rewrite /to_proph_map fmap_delete. Qed.
-
-  Lemma lookup_to_proph_map_None R p :
-    R !! p = None → to_proph_map R !! p = None.
-  Proof. by rewrite /to_proph_map lookup_fmap=> ->. Qed.
-
-  Lemma proph_map_singleton_included R p v :
-    {[p := Excl v]} ≼ to_proph_map R → R !! p = Some v.
-  Proof.
-    rewrite singleton_included_exclusive; last by apply to_proph_map_valid.
-    by rewrite leibniz_equiv_iff /to_proph_map lookup_fmap fmap_Some=> -[v' [-> [->]]].
-  Qed.
-End to_proph_map.
-
-
-Lemma proph_map_init `{Countable P, !proph_mapPreG P V PVS} pvs ps :
-  (|==> ∃ _ : proph_mapG P V PVS, proph_map_ctx pvs ps)%I.
-Proof.
-  iMod (own_alloc (● to_proph_map ∅)) as (γ) "Hh".
-  { rewrite -auth_auth_valid. exact: to_proph_map_valid. }
-  iModIntro. iExists (ProphMapG P V PVS _ _ _ γ), ∅. iSplit; last by iFrame.
-  iPureIntro. split =>//.
-Qed.
-
-Section proph_map.
-  Context `{Countable P, !proph_mapG P V Σ}.
-  Implicit Types p : P.
-  Implicit Types v : option V.
-  Implicit Types R : proph_map P V.
-
-  (** General properties of mapsto *)
-  Global Instance proph_timeless p v : Timeless (proph p v).
-  Proof. rewrite proph_eq /proph_def. apply _. Qed.
-
-  Lemma proph_map_alloc R p v :
-    p ∉ dom (gset _) R →
-    proph_map_auth R ==∗ proph_map_auth (<[p := v]> R) ∗ proph p v.
-  Proof.
-    iIntros (Hp) "HR". rewrite /proph_map_ctx proph_eq /proph_def.
-    iMod (own_update with "HR") as "[HR Hl]".
-    { eapply auth_update_alloc,
-        (alloc_singleton_local_update _ _ (Excl $ (v : option (leibnizC _))))=> //.
-      apply lookup_to_proph_map_None. apply (iffLR (not_elem_of_dom _ _) Hp). }
-    iModIntro. rewrite /proph_map_auth to_proph_map_insert. iFrame.
-  Qed.
-
-  Lemma proph_map_remove R p v :
-    proph_map_auth R -∗ proph p v ==∗ proph_map_auth (delete p R).
-  Proof.
-    iIntros "HR Hp". rewrite /proph_map_ctx proph_eq /proph_def.
-    rewrite /proph_map_auth to_proph_map_delete. iApply (own_update_2 with "HR Hp").
-    apply auth_update_dealloc, (delete_singleton_local_update _ _ _).
-  Qed.
-
-  Lemma proph_map_valid R p v : proph_map_auth R -∗ proph p v -∗ ⌜R !! p = Some v⌝.
-  Proof.
-    iIntros "HR Hp". rewrite /proph_map_ctx proph_eq /proph_def.
-    iDestruct (own_valid_2 with "HR Hp")
-      as %[HH%proph_map_singleton_included _]%auth_valid_discrete_2; auto.
-  Qed.
-End proph_map.