remove failing part of coin_flip; remove atomic_snapshot

Snapshot will re-appear in iris-examples eventually

_CoqProject

@@ -98,8 +98,6 @@ theories/heap_lang/lib/coin_flip.v
 theories/heap_lang/lib/counter.v
 theories/heap_lang/lib/atomic_heap.v
 theories/heap_lang/lib/increment.v
-theories/heap_lang/lib/atomic_snapshot.v
-theories/heap_lang/lib/atomic_snapshot_spec.v
 theories/proofmode/base.v
 theories/proofmode/tokens.v
 theories/proofmode/coq_tactics.v

theories/heap_lang/lib/atomic_snapshot.v

-From iris.algebra Require Import excl auth gmap agree.
-From iris.heap_lang Require Export lifting notation.
-From iris.base_logic.lib Require Export invariants.
-From iris.program_logic Require Export atomic.
-From iris.proofmode Require Import tactics.
-From iris.heap_lang Require Import proofmode notation par.
-From iris.bi.lib Require Import fractional.
-Set Default Proof Using "Type".
-
-(** Specifying snapshots with histories
-    Implementing atomic pair snapshot data structure from Sergey et al. (ESOP 2015) *)
-
-(** The CMRA & functor we need. *)
-
-Definition timestampUR := gmapUR Z $ agreeR valC.
-
-Class atomic_snapshotG Σ := AtomicSnapshotG {
-                                atomic_snapshot_stateG :> inG Σ $ authR $ optionUR $ exclR $ prodC valC valC;
-                                atomic_snapshot_timestampG :> inG Σ $ authR $ timestampUR
-}.
-Definition atomic_snapshotΣ : gFunctors :=
-  #[GFunctor (authR $ optionUR $ exclR $ prodC valC valC); GFunctor (authR timestampUR)].
-
-Instance subG_atomic_snapshotΣ {Σ} : subG atomic_snapshotΣ Σ → atomic_snapshotG Σ.
-Proof. solve_inG. Qed.
-
-Section atomic_snapshot.
-
-  Context {Σ} `{!heapG Σ, !atomic_snapshotG Σ}.
-
-  (*
-     newPair x y :=
-       (ref (ref (x, 0)), ref y)
-   *)
-  Definition newPair : val :=
-    λ: "args",
-      let: "x" := Fst "args" in
-      let: "y" := Snd "args" in
-      (ref (ref ("x", #0)), ref "y").
-
-  (*
-    writeY (xp, yp) y :=
-      yp <- y
-  *)
-  Definition writeY : val :=
-    λ: "args",
-      let: "p" := Fst "args" in
-      Snd "p" <- Snd "args".
-
-  (*
-    writeX (xp, yp) x :=
-      let xp1 = !xp in
-      let v   = (!xp1).2
-      let xp2 = ref (x, v + 1)
-      if CAS xp xp1 xp2
-        then ()
-        else writeX (xp, yp) x
-  *)
-  Definition writeX : val :=
-    rec: "writeX" "args" :=
-      let: "p"    := Fst "args"             in
-      let: "x"    := Snd "args"             in
-      let: "xp"   := Fst "p"                in
-      let: "xp1"  := !"xp"                  in
-      let: "v"    := Snd (!"xp1")           in
-      let: "xp2"  := ref ("x", "v" + #1)    in
-      if: CAS "xp" "xp1" "xp2"
-        then #()
-      else "writeX" "args".
-
-  (*
-    readX (xp, yp) :=
-      !!xp
-   *)
-  Definition readX : val :=
-    λ: "p",
-      let: "xp" := Fst "p" in
-      !(!"xp").
-
-   Definition readY : val :=
-    λ: "p",
-      let: "yp" := Snd "p" in
-      !"yp".
-
-  (*
-    readPair l :=
-      let (x, v)  = readX l in
-      let y       = readY l in
-      let (_, v') = readX l in
-      if v = v'
-        then (x, y)
-        else readPair l
-  *)
-  Definition readPair : val :=
-    rec: "readPair" "l" :=
-      let: "x"  := readX "l"  in
-      let: "y"  := readY "l"  in
-      let: "x'" := readX "l"  in
-      if: Snd "x" = Snd "x'"
-        then (Fst "x", Fst "y")
-        else  "readPair" "l".
-
-  Definition readPair_proph : val :=
-    rec: "readPair" "l" :=
-      let: "xv1"    := readX "l"           in
-      let: "proph"  := new prophecy        in
-      let: "y"      := readY "l"           in
-      let: "xv2"    := readX "l"           in
-      let: "v2"     := Snd "xv2"           in
-      let: "v_eq"   := Snd "xv1" = "v2"    in
-      resolve "proph" to "v_eq" ;;
-      if: "v_eq"
-        then (Fst "xv1", "y")
-      else "readPair" "l".
-
-  Variable N: namespace.
-
-  Definition gmap_to_UR T : timestampUR := to_agree <$> T.
-
-  Definition pair_inv γ1 γ2 l1 l2 : iProp Σ :=
-    (∃ q (l1':loc) (T : gmap Z val) x y (t : Z),
-      (* we add the q to make the l1' map fractional. that way,
-         we can take a fraction of the l1' map out of the invariant
-         and do a case analysis on whether the pointer is the same
-         throughout invariant openings *)
-      l1 ↦ #l1' ∗ l1' ↦{q} (x, #t) ∗ l2 ↦ y ∗
-         own γ1 (● Excl' (x, y)) ∗ own γ2 (● gmap_to_UR T) ∗
-         ⌜T !! t = Some x⌝ ∗
-         ⌜forall (t' : Z), t' ∈ dom (gset Z) T -> (t' <= t)%Z⌝)%I.
-
-  Definition is_pair (γs: gname * gname) (p : val) :=
-    (∃ (l1 l2 : loc), ⌜p = (#l1, #l2)%V⌝ ∗ inv N (pair_inv γs.1 γs.2 l1 l2))%I.
-
-  Global Instance is_pair_persistent γs p : Persistent (is_pair γs p) := _.
-
-  Definition pair_content (γs : gname * gname) (a : val * val) :=
-    (own γs.1 (◯ Excl' a))%I.
-
-  Global Instance pair_content_timeless γs a : Timeless (pair_content γs a) := _.
-
-  Lemma pair_content_exclusive γs a1 a2 :
-    pair_content γs a1 -∗ pair_content γs a2 -∗ False.
-  Proof.
-    iIntros "H1 H2". iDestruct (own_valid_2 with "H1 H2") as %[].
-  Qed.
-
-  Definition new_timestamp t v : gmap Z val := {[ t := v ]}.
-
-  Lemma newPair_spec (e : expr) (v1 v2 : val) :
-      IntoVal e (v1, v2) ->
-      {{{ True }}}
-        newPair e
-      {{{ γs p, RET p; is_pair γs p ∗ pair_content γs (v1, v2) }}}.
-  Proof.
-    iIntros (<- Φ _) "Hp". rewrite /newPair. wp_lam.
-    repeat (wp_proj; wp_let).
-    iApply wp_fupd.
-    wp_alloc lx' as "Hlx'".
-    wp_alloc lx as "Hlx".
-    wp_alloc ly as "Hly".
-    set (Excl' (v1, v2)) as p.
-    iMod (own_alloc (● p ⋅ ◯ p)) as (γ1) "[Hp⚫ Hp◯]". {
-      rewrite /p. apply auth_valid_discrete_2. split; done.
-    }
-    set (new_timestamp 0 v1) as t.
-    iMod (own_alloc (● gmap_to_UR t ⋅ ◯ gmap_to_UR t)) as (γ2) "[Ht⚫ Ht◯]". {
-      rewrite /t /new_timestamp. apply auth_valid_discrete_2.
-      split; first done. rewrite /gmap_to_UR map_fmap_singleton. apply singleton_valid. done.
-    }
-    iApply ("Hp" $! (γ1, γ2)).
-    iMod (inv_alloc N _ (pair_inv γ1 γ2 lx ly) with "[-Hp◯ Ht◯]") as "#Hinv". {
-      iNext. rewrite /pair_inv. iExists _, _, _, _, _. iExists 0. iFrame.
-      iPureIntro. split; first done. intros ?. subst t. rewrite /new_timestamp dom_singleton.
-      rewrite elem_of_singleton. lia.
-    }
-    iModIntro. iSplitR. rewrite /is_pair. repeat (iExists _). iSplitL; eauto.
-    rewrite /pair_content. rewrite /p. iApply "Hp◯".
-  Qed.
-
- Lemma excl_update γ n' n m :
-    own γ (● (Excl' n)) -∗ own γ (◯ (Excl' m)) ==∗
-      own γ (● (Excl' n')) ∗ own γ (◯ (Excl' n')).
-  Proof.
-    iIntros "Hγ● Hγ◯".
-    iMod (own_update_2 _ _ _ (● Excl' n' ⋅ ◯ Excl' n') with "Hγ● Hγ◯") as "[$$]".
-    { by apply auth_update, option_local_update, exclusive_local_update. }
-    done.
-  Qed.
-
-  Lemma excl_sync γ n m :
-    own γ (● (Excl' n)) -∗ own γ (◯ (Excl' m)) -∗ ⌜m = n⌝.
-  Proof.
-    iIntros "Hγ● Hγ◯".
-    iDestruct (own_valid_2 with "Hγ● Hγ◯") as
-        %[H%Excl_included%leibniz_equiv _]%auth_valid_discrete_2.
-    done.
-  Qed.
-
-  Lemma timestamp_dupl γ T:
-    own γ (● T) ==∗ own γ (● T) ∗ own γ (◯ T).
-  Proof.
-    iIntros "Ht". iApply own_op. iApply (own_update with "Ht").
-    apply auth_update_alloc. apply local_update_unital_discrete => f Hv. rewrite left_id => <-.
-    split; first done. apply core_id_dup. apply _.
-  Qed.
-
-  Lemma timestamp_update γ (T : gmap Z val) (t : Z) x :
-    T !! t = None ->
-    own γ (● gmap_to_UR T) ==∗ own γ (● gmap_to_UR (<[ t := x ]> T)).
-  Proof.
-    iIntros (HT) "Ht".
-    set (<[ t := x ]> T) as T'.
-    iDestruct (own_update _ _ (● gmap_to_UR T' ⋅ ◯ gmap_to_UR {[ t := x ]}) with "Ht") as "Ht". {
-      apply auth_update_alloc. rewrite /T' /gmap_to_UR map_fmap_singleton. rewrite fmap_insert.
-      apply alloc_local_update; last done. rewrite lookup_fmap HT. done.
-    }
-    iMod (own_op with "Ht") as "[Ht● Ht◯]". iModIntro. iFrame.
-  Qed.
-
-  Lemma timestamp_sub γ (T1 T2 : gmap Z val):
-    own γ (● gmap_to_UR T1) ∗ own γ (◯ gmap_to_UR T2) -∗
-    ⌜forall t x, T2 !! t = Some x -> T1 !! t = Some x⌝.
-  Proof.
-    iIntros "[Hγ⚫ Hγ◯]".
-    iDestruct (own_valid_2 with "Hγ⚫ Hγ◯") as
-        %[H Hv]%auth_valid_discrete_2. iPureIntro. intros t x HT2.
-    pose proof (iffLR (lookup_included (gmap_to_UR T2) (gmap_to_UR T1)) H t) as Ht.
-    rewrite !lookup_fmap HT2 /= in Ht.
-    destruct (is_Some_included _ _ Ht) as [? [t2 [Ht2 ->]]%fmap_Some_1]; first by eauto.
-    revert Ht.
-    rewrite Ht2 Some_included_total to_agree_included. fold_leibniz.
-    by intros ->.
-  Qed.
-
-  Lemma writeY_spec e (y2: val) γ p :
-      IntoVal e y2 ->
-      is_pair γ p -∗
-      <<< ∀ x y : val, pair_content γ (x, y) >>>
-        writeY (p, e)
-        @ ⊤∖↑N
-      <<< pair_content γ (x, y2), RET #() >>>.
-  Proof.
-    iIntros (<-) "Hp". iApply wp_atomic_intro. iIntros (Φ) "AU".
-    iDestruct "Hp" as (l1 l2 ->) "#Hinv".
-    repeat (wp_lam; wp_proj). wp_proj.
-    iApply wp_fupd.
-    iInv N as (q l1' T x) "Hinvp".
-    iDestruct "Hinvp" as (y v') "[Hl1 [Hl1' [Hl2 [Hp⚫ Htime]]]]".
-    wp_store.
-    iMod "AU" as (xv yv) "[Hpair [_ Hclose]]".
-    rewrite /pair_content.
-    iDestruct (excl_sync with "Hp⚫ Hpair") as %[= -> ->].
-    iMod (excl_update _ (x, y2) with "Hp⚫ Hpair") as "[Hp⚫ Hp◯]".
-    iMod ("Hclose" with "Hp◯") as "HΦ".
-    iModIntro. iSplitR "HΦ"; last done.
-    iNext. unfold pair_inv. eauto 7 with iFrame.
-  Qed.
-
-  Lemma writeX_spec e (x2: val) γ p :
-      IntoVal e x2 ->
-      is_pair γ p  -∗
-      <<< ∀ x y : val, pair_content γ (x, y) >>>
-        writeX (p, e)
-        @ ⊤∖↑N
-      <<< pair_content γ (x2, y), RET #() >>>.
-  Proof.
-    iIntros (<-) "Hp". iApply wp_atomic_intro. iIntros (Φ) "AU". iLöb as "IH".
-    iDestruct "Hp" as (l1 l2 ->) "#Hinv".
-    repeat (wp_lam; wp_proj). wp_lam. wp_bind (!_)%E.
-    (* first read *)
-    (* open invariant *)
-    iInv N as (q l1' T x) "Hinvp".
-      iDestruct "Hinvp" as (y v') "[Hl1 [Hl1' [Hl2 [Hp⚫ Htime]]]]".
-      wp_load.
-      iDestruct "Hl1'" as "[Hl1'frac1 Hl1'frac2]".
-      iModIntro. iSplitR "AU Hl1'frac2".
-      (* close invariant *)
-      { iNext. rewrite /pair_inv. eauto 10 with iFrame. }
-    wp_let. wp_bind (!_)%E. clear T.
-    wp_load. wp_proj. wp_let. wp_op. wp_alloc l1'new as "Hl1'new".
-    wp_let.
-    (* CAS *)
-    wp_bind (CAS _ _ _)%E.
-    (* open invariant *)
-    iInv N as (q' l1'' T x') ">Hinvp".
-    iDestruct  "Hinvp" as (y' v'') "[Hl1 [Hl1'' [Hl2 [Hp⚫ [Ht● Ht]]]]]".
-    iDestruct "Ht" as %[Ht Hvt].
-    destruct (decide (l1'' = l1')) as [-> | Hn].
-    - wp_cas_suc.
-      iDestruct (mapsto_agree with "Hl1'frac2 Hl1''") as %[= -> ->]. iClear "Hl1'frac2".
-      (* open AU *)
-      iMod "AU" as (xv yv) "[Hpair [_ Hclose]]".
-        (* update pair ghost state to (x2, y') *)
-        iDestruct (excl_sync with "Hp⚫ Hpair") as %[= -> ->].
-        iMod (excl_update _ (x2, y') with "Hp⚫ Hpair") as "[Hp⚫ Hp◯]".
-        (* close AU *)
-        iMod ("Hclose" with "Hp◯") as "HΦ".
-      (* update timestamp *)
-      iMod (timestamp_update _ T (v'' + 1)%Z x2 with "[Ht●]") as "Ht".
-      { eapply (not_elem_of_dom (D:=gset Z) T). intros Hd. specialize (Hvt _ Hd). omega. }
-      { done. }
-      (* close invariant *)
-      iModIntro. iSplitR "HΦ".
-      + iNext. rewrite /pair_inv.
-        set (<[ v'' + 1 := x2]> T) as T'.
-        iExists 1%Qp, l1'new, T', x2, y', (v'' + 1).
-        iFrame.
-        iPureIntro. split.
-        * apply: lookup_insert.
-        * intros ? Hv. destruct (decide (t' = (v'' + 1)%Z)) as [-> | Hn]; first done.
-          assert (dom (gset Z) T' = {[(v'' + 1)%Z]} ∪ dom (gset Z) T) as Hd. {
-            apply leibniz_equiv. rewrite dom_insert. done.
-          }
-          rewrite Hd in Hv. clear Hd. apply elem_of_union in Hv.
-          destruct Hv as [Hv%elem_of_singleton_1 | Hv]; first done.
-          specialize (Hvt _ Hv). lia.
-      + wp_if. done.
-    - wp_cas_fail. iModIntro. iSplitR "AU".
-      + iNext. rewrite /pair_inv. eauto 10 with iFrame.
-      + wp_if. iApply "IH"; last eauto. rewrite /is_pair. eauto.
-  Qed.
-
-  Lemma readY_spec γ p :
-    is_pair γ p -∗
-    <<< ∀ v1 v2 : val, pair_content γ (v1, v2) >>>
-      readY p
-      @ ⊤∖↑N
-    <<< pair_content γ (v1, v2), RET v2 >>>.
-  Proof.
-    iIntros "Hp". iApply wp_atomic_intro. iIntros (Φ) "AU".
-    iDestruct "Hp" as (l1 l2 ->) "#Hinv".
-    repeat (wp_lam; wp_proj). wp_let.
-    iApply wp_fupd.
-    iInv N as (q l1' T x) "Hinvp".
-    iDestruct "Hinvp" as (y v') "[Hl1 [Hl1' [Hl2 [Hp⚫ Htime]]]]".
-    wp_load.
-    iMod "AU" as (xv yv) "[Hpair [_ Hclose]]".
-    rewrite /pair_content.
-    iDestruct (excl_sync with "Hp⚫ Hpair") as %[= -> ->].
-    iMod ("Hclose" with "Hpair") as "HΦ".
-    iModIntro. iSplitR "HΦ"; last done.
-    iNext. unfold pair_inv. eauto 7 with iFrame.
-  Qed.
-
-  Lemma readX_spec γ p :
-    is_pair γ p -∗
-    <<< ∀ v1 v2 : val, pair_content γ (v1, v2) >>>
-      readX p
-      @ ⊤∖↑N
-    <<< ∃ (t: Z), pair_content γ (v1, v2), RET (v1, #t) >>>.
-  Proof.
-    iIntros "Hp". iApply wp_atomic_intro. iIntros (Φ) "AU".
-    iDestruct "Hp" as (l1 l2 ->) "#Hinv".
-    repeat (wp_lam; wp_proj). wp_let. wp_bind (! #l1)%E.
-    (* open invariant for 1st read *)
-    iInv N as (q l1' T x) ">Hinvp".
-      iDestruct "Hinvp" as (y v') "[Hl1 [Hl1' [Hl2 [Hp⚫ Htime]]]]".
-      wp_load.
-      iDestruct "Hl1'" as "[Hl1' Hl1'frac]".
-      iMod "AU" as (xv yv) "[Hpair [_ Hclose]]".
-      iDestruct (excl_sync with "Hp⚫ Hpair") as %[= -> ->].
-      iMod ("Hclose" with "Hpair") as "HΦ".
-      (* close invariant *)
-      iModIntro. iSplitR "HΦ Hl1'". {
-        iNext. unfold pair_inv. eauto 7 with iFrame.
-      }
-    iApply wp_fupd. clear T y.
-    wp_load. eauto.
-  Qed.
-
-  Definition val_to_bool (v : option val) : bool :=
-    match v with
-    | Some (LitV (LitBool b)) => b
-    | _                       => false
-    end.
-
-  Lemma readPair_spec γ p :
-    is_pair γ p -∗
-    <<< ∀ v1 v2 : val, pair_content γ (v1, v2) >>>
-       readPair_proph p
-       @ ⊤∖↑N
-    <<< pair_content γ (v1, v2), RET (v1, v2) >>>.
-  Proof.
-    iIntros "Hp". iApply wp_atomic_intro. iIntros (Φ) "AU". iLöb as "IH".
-    wp_let.
-    (* ************ 1st readX ********** *)
-    iDestruct "Hp" as (l1 l2 ->) "#Hinv".
-    repeat (wp_lam; wp_proj). wp_let. wp_bind (! #l1)%E.
-    (* open invariant for 1st read *)
-    iInv N as (q_x1 l1' T_x x1) ">Hinvp".
-      iDestruct "Hinvp" as (y_x1 v_x1) "[Hl1 [Hl1' [Hl2 [Hp⚫ [Ht_x Htime_x]]]]]".
-      iDestruct "Htime_x" as %[Hlookup_x Hdom_x].
-      wp_load.
-      iDestruct "Hl1'" as "[Hl1' Hl1'frac]".
-      iMod "AU" as (xv yv) "[Hpair [Hclose _]]".
-      iDestruct (excl_sync with "Hp⚫ Hpair") as %[= -> ->].
-      iMod ("Hclose" with "Hpair") as "AU".
-      (* duplicate timestamp T_x1 *)
-      iMod (timestamp_dupl with "Ht_x") as "[Ht_x1⚫ Ht_x1◯]".
-      (* close invariant *)
-      iModIntro. iSplitR "AU Hl1' Ht_x1◯". {
-        iNext. unfold pair_inv. eauto 8 with iFrame.
-      }
-    wp_load. wp_let.
-    (* ************ new prophecy ********** *)
-    wp_apply wp_new_proph; first done.
-    iIntros (proph) "Hpr". iDestruct "Hpr" as (proph_val) "Hpr".
-    wp_let.
-    (* ************ readY ********** *)
-    repeat (wp_lam; wp_proj). wp_let. wp_bind (!_)%E.
-    iInv N as (q_y l1'_y T_y x_y) ">Hinvp".
-    iDestruct "Hinvp" as (y_y v_y) "[Hl1 [Hl1'_y [Hl2 [Hp⚫ [Ht_y Htime_y]]]]]".
-    iDestruct "Htime_y" as %[Hlookup_y Hdom_y].
-    wp_load.
-    (* linearization point *)
-    iMod "AU" as (xv yv) "[Hpair Hclose]".
-    rewrite /pair_content.
-    iDestruct (excl_sync with "Hp⚫ Hpair") as %[= -> ->].
-    destruct (val_to_bool proph_val) eqn:Hproph.
-    - (* prophecy value is predicting that timestamp has not changed, so we commit *)
-      (* committing AU *)
-      iMod ("Hclose" with "Hpair") as "HΦ".
-      (* duplicate timestamp T_y *)
-      iMod (timestamp_dupl with "Ht_y") as "[Ht_y● Ht_y◯]".
-      (* show that T_x <= T_y *)
-      iDestruct (timestamp_sub with "[Ht_y● Ht_x1◯]") as "#Hs"; first by iFrame.
-      iDestruct "Hs" as %Hs.
-      iModIntro.
-      (* closing invariant *)
-      iSplitR "HΦ Hl1' Ht_x1◯ Ht_y◯ Hpr".
-      { iNext. unfold pair_inv. eauto 10 with iFrame. }
-      wp_let.
-      (* ************ 2nd readX ********** *)
-      repeat (wp_lam; wp_proj). wp_let. wp_bind (! #l1)%E.
-      (* open invariant *)
-      iInv N as (q_x2 l1'_x2 T_x2 x2) ">Hinvp".
-      iDestruct "Hinvp" as (y_x2 v_x2) "[Hl1 [Hl1'_x2 [Hl2 [Hp⚫ [Ht_x2 Htime_x2]]]]]".
-      iDestruct "Htime_x2" as %[Hlookup_x2 Hdom_x2].
-      iDestruct "Hl1'_x2" as "[Hl1'_x2 Hl1'_x2_frag]".
-      wp_load.
-      (* show that T_y <= T_x2 *)
-      iDestruct (timestamp_sub with "[Ht_x2 Ht_y◯]") as "#Hs'"; first by iFrame.
-      iDestruct "Hs'" as %Hs'.
-      iModIntro. iSplitR "HΦ Hl1'_x2_frag Hpr". {
-        iNext. unfold pair_inv. eauto 8 with iFrame.
-      }
-      wp_load. wp_let. wp_proj. wp_let. wp_proj. wp_op.
-      case_bool_decide; wp_let; wp_apply (wp_resolve_proph with "Hpr");
-        iIntros (->); wp_seq; wp_if.
-      + inversion H; subst; clear H. wp_proj.
-        assert (v_x2 = v_y) as ->. {
-          assert (v_x2 <= v_y) as vneq. {
-            apply Hdom_y.
-            eapply (iffRL (elem_of_dom T_y _)). eauto using mk_is_Some.
-          }
-          assert (v_y <= v_x2) as vneq'. {
-            apply Hdom_x2.
-            eapply (iffRL (elem_of_dom T_x2 _)). eauto using mk_is_Some.
-          }
-          apply Z.le_antisymm; auto.
-        }
-        assert (x1 = x_y) as ->. {
-          specialize (Hs _ _ Hlookup_x). rewrite Hs in Hlookup_y. inversion Hlookup_y. done.
-        }
-        done.
-      + inversion Hproph.
-    - iDestruct "Hclose" as "[Hclose _]". iMod ("Hclose" with "Hpair") as "AU".
-      iModIntro.
-      (* closing invariant *)
-      iSplitR "AU Hpr".
-      { iNext. unfold pair_inv. eauto 10 with iFrame. }
-      wp_let.
-      (* ************ 2nd readX ********** *)
-      repeat (wp_lam; wp_proj). wp_let. wp_bind (! #l1)%E.
-      (* open invariant *)
-      iInv N as (q_x2 l1'_x2 T_x2 x2) ">Hinvp".
-      iDestruct "Hinvp" as (y_x2 v_x2) "[Hl1 [Hl1'_x2 [Hl2 [Hp⚫ [Ht_x2 Htime_x2]]]]]".
-      iDestruct "Hl1'_x2" as "[Hl1'_x2 Hl1'_x2_frag]".
-      wp_load.
-      iModIntro. iSplitR "AU Hl1'_x2_frag Hpr". {
-        iNext. unfold pair_inv. eauto 8 with iFrame.
-      }
-      wp_load. repeat (wp_let; wp_proj). wp_op. wp_let.
-      wp_apply (wp_resolve_proph with "Hpr").
-      iIntros (Heq). subst.
-      case_bool_decide.
-      + inversion H; subst; clear H. inversion Hproph.
-      + wp_seq. wp_if. iApply "IH"; rewrite /is_pair; eauto.
-  Qed.
-
-End atomic_snapshot.

theories/heap_lang/lib/atomic_snapshot_spec.v

-From iris.algebra Require Import excl auth list.
-From iris.heap_lang Require Export lifting notation.
-From iris.base_logic.lib Require Export invariants.
-From iris.program_logic Require Export atomic.
-From iris.proofmode Require Import tactics.
-From iris.heap_lang Require Import proofmode notation par.
-From iris.bi.lib Require Import fractional.
-Set Default Proof Using "Type".
-
-(** Specifying snapshots with histories
-    Implementing atomic pair snapshot data structure from Sergey et al. (ESOP 2015) *)
-
-Section atomic_snapshot_spec.
-
-  Record atomic_snapshot {Σ} `{!heapG Σ} := AtomicSnapshot {
-    newPair : val;
-    writeX : val;
-    writeY : val;
-    readPair : val;
-    (* other data *)
-    name: Type;
-    (* predicates *)
-    is_pair (N : namespace) (γ : name) (p : val) : iProp Σ;
-    pair_content (γ : name) (a: val * val) : iProp Σ;
-    (* predicate properties *)
-    is_pair_persistent N γ p : Persistent (is_pair N γ p);
-    pair_content_timeless γ a : Timeless (pair_content γ a);
-    pair_content_exclusive γ a1 a2 :
-      pair_content γ a1 -∗ pair_content γ a2 -∗ False;
-    (* specs *)
-    newPair_spec N (e : expr) (v1 v2 : val) :
-      IntoVal e (v1, v2) ->
-      {{{ True }}} newPair e {{{ γ p, RET p; is_pair N γ p ∗ pair_content γ (v1, v2) }}};
-    writeX_spec N e (v: val) p γ :
-      IntoVal e v ->
-      is_pair N γ p -∗
-      <<< ∀ v1 v2 : val, pair_content γ (v1, v2) >>>
-        writeX (p, e)
-        @ ⊤∖↑N
-      <<< pair_content γ (v, v2), RET #() >>>;
-    writeY_spec N e (v: val) p γ:
-      IntoVal e v ->
-      is_pair N γ p -∗