From 67abfa937b0314b2205e587d56b7cdcc7ee38105 Mon Sep 17 00:00:00 2001
From: Marianna Rapoport <mrapoport@uwaterloo.ca>
Date: Thu, 28 Jun 2018 14:41:55 +0200
Subject: [PATCH] Implementing atomic-pair-snapshot example from Sergey et al.
(ESOP'15)
---
_CoqProject | 4 +
theories/heap_lang/lib/atomic_snapshot.v | 503 ++++++++++++++++++
theories/heap_lang/lib/atomic_snapshot_spec.v | 56 ++
.../lib/{coin-flip.v => coin_flip.v} | 0
4 files changed, 563 insertions(+)
create mode 100644 theories/heap_lang/lib/atomic_snapshot.v
create mode 100644 theories/heap_lang/lib/atomic_snapshot_spec.v
rename theories/heap_lang/lib/{coin-flip.v => coin_flip.v} (100%)
diff --git a/_CoqProject b/_CoqProject
index 92fc6bd86..bdd1e49ce 100644
--- a/_CoqProject
+++ b/_CoqProject
@@ -87,15 +87,19 @@ theories/heap_lang/notation.v
theories/heap_lang/proofmode.v
theories/heap_lang/adequacy.v
theories/heap_lang/total_adequacy.v
+theories/heap_lang/prophecy.v
theories/heap_lang/lib/spawn.v
theories/heap_lang/lib/par.v
theories/heap_lang/lib/assert.v
theories/heap_lang/lib/lock.v
theories/heap_lang/lib/spin_lock.v
theories/heap_lang/lib/ticket_lock.v
+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
diff --git a/theories/heap_lang/lib/atomic_snapshot.v b/theories/heap_lang/lib/atomic_snapshot.v
new file mode 100644
index 000000000..f8575dcdd
--- /dev/null
+++ b/theories/heap_lang/lib/atomic_snapshot.v
@@ -0,0 +1,503 @@
+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 prophecy.
+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 pr : val :=
+ rec: "readPair" "l" :=
+ let: "xv1" := readX "l" in
+ let: "proph" := new_prophecy pr #() in
+ let: "y" := readY "l" in
+ let: "xv2" := readX "l" in
+ let: "v2" := Snd "xv2" in
+ let: "v_eq" := Snd "xv1" = "v2" in
+ resolve_prophecy pr "proph" "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)%ZâŒ)%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 "Hpâ—¯".
+ 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 (â— (Excl' n)) -∗ own γ (â—¯ (Excl' m)) -∗ ⌜m = nâŒ.
+ 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 _ _ (â— gmap_to_UR T' â‹… â—¯ 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 "[Htâ— Htâ—¯]". iModIntro. iFrame.
+ Qed.
+
+ Lemma fmap_undo {A B} (f : A -> B) (m : gmap Z A) k v :
+ f <$> m !! k = Some v ->
+ exists v', m !! k = Some v' /\ v = f v'.
+ Proof.
+ intros Hl. destruct (m !! k); inversion Hl. subst. eauto.
+ 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 xâŒ.
+ Proof.
+ iIntros "[Hγ⚫ Hγ◯]".
+ iDestruct (own_valid_2 with "Hγ⚫ Hγ◯") as
+ %[H Hv]%auth_valid_discrete_2. iPureIntro. intros t x Ht.
+ pose proof (iffLR (lookup_included (gmap_to_UR T2) (gmap_to_UR T1)) H t) as Hsub.
+ repeat rewrite lookup_fmap in Hsub.
+ rewrite Ht in Hsub. simpl in Hsub.
+ pose proof (mk_is_Some (Some (to_agree x)) _ eq_refl) as Hsome.
+ pose proof (is_Some_included _ _ Hsub Hsome) as Hsome'; clear Hsome.
+ destruct Hsome' as [c Heqx]. rewrite Heqx in Hsub.
+ apply (iffLR (Some_included_total _ _)) in Hsub.
+ destruct (fmap_undo to_agree _ _ _ Heqx) as [c' [Heq1 Heq2]]. subst.
+ apply to_agree_included in Hsub. apply leibniz_equiv in Hsub. subst. done.
+ 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 [Hpâš« Htime]]]]".
+ wp_store.
+ iMod "AU" as (xv yv) "[Hpair [_ Hclose]]".
+ rewrite /pair_content.
+ iDestruct (excl_sync with "Hpâš« Hpair") as %[= -> ->].
+ iMod (excl_update _ (x, y2) with "Hpâš« Hpair") as "[Hpâš« Hpâ—¯]".
+ 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 [Hpâš« 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 [Hpâš« [Htâ— 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 "Hpâš« Hpair") as %[= -> ->].
+ iMod (excl_update _ (x2, y') with "Hpâš« Hpair") as "[Hpâš« Hpâ—¯]".
+ (* close AU *)
+ iMod ("Hclose" with "Hp◯") as "HΦ".
+ (* update timestamp *)
+ iMod (timestamp_update _ T (v'' + 1)%Z x2 with "[Htâ—]") 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 [Hpâš« Htime]]]]".
+ wp_load.
+ iMod "AU" as (xv yv) "[Hpair [_ Hclose]]".
+ rewrite /pair_content.
+ iDestruct (excl_sync with "Hpâš« 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 [Hpâš« Htime]]]]".
+ wp_load.
+ iDestruct "Hl1'" as "[Hl1' Hl1'frac]".
+ iMod "AU" as (xv yv) "[Hpair [_ Hclose]]".
+ iDestruct (excl_sync with "Hpâš« 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:val) : bool :=
+ match v with
+ | LitV (LitBool b) => b
+ | _ => false
+ end.
+
+ Variable pr: prophecy.
+
+ Lemma readPair_spec γ p :
+ is_pair γ p -∗
+ <<< ∀ v1 v2 : val, pair_content γ (v1, v2) >>>
+ readPair_proph pr 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 [Hpâš« [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 "Hpâš« Hpair") as %[= -> ->].
+ iMod ("Hclose" with "Hpair") as "AU".
+ (* duplicate timestamp T_x1 *)
+ iMod (timestamp_dupl with "Ht_x") as "[Ht_x1âš« Ht_x1â—¯]".
+ (* close invariant *)
+ iModIntro. iSplitR "AU Hl1' Ht_x1â—¯". {
+ iNext. unfold pair_inv. eauto 8 with iFrame.
+ }
+ wp_load. wp_let.
+ (* ************ new prophecy ********** *)
+ wp_apply new_prophecy_spec; 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 [Hpâš« [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 "Hpâš« 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_yâ— Ht_yâ—¯]".
+ (* show that T_x <= T_y *)
+ iDestruct (timestamp_sub with "[Ht_yâ— Ht_x1â—¯]") 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 [Hpâš« [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_yâ—¯]") 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 (resolve_prophecy_spec 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 [Hpâš« [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 (resolve_prophecy_spec 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.
diff --git a/theories/heap_lang/lib/atomic_snapshot_spec.v b/theories/heap_lang/lib/atomic_snapshot_spec.v
new file mode 100644
index 000000000..679975031
--- /dev/null
+++ b/theories/heap_lang/lib/atomic_snapshot_spec.v
@@ -0,0 +1,56 @@
+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 prophecy.
+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 -∗
+ <<< ∀ v1 v2 : val, pair_content γ (v1, v2) >>>
+ writeY (p, e)
+ @ ⊤∖↑N
+ <<< pair_content γ (v1, v), RET #() >>>;
+ readPair_spec N γ p :
+ is_pair N γ p -∗
+ <<< ∀ v1 v2 : val, pair_content γ (v1, v2) >>>
+ readPair p
+ @ ⊤∖↑N
+ <<< pair_content γ (v1, v2), RET (v1, v2) >>>;
+ }.
+
+ End atomic_snapshot_spec.
diff --git a/theories/heap_lang/lib/coin-flip.v b/theories/heap_lang/lib/coin_flip.v
similarity index 100%
rename from theories/heap_lang/lib/coin-flip.v
rename to theories/heap_lang/lib/coin_flip.v
--
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