Add the lazy_coin example

_CoqProject

@@ -110,7 +110,8 @@ 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/nondet_bool.v
+theories/heap_lang/lib/lazy_coin.v
 theories/heap_lang/lib/counter.v
 theories/heap_lang/lib/atomic_heap.v
 theories/heap_lang/lib/increment.v

theories/heap_lang/lib/coin_flip.v

-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.
-Set Default Proof Using "Type".
-
-(** Nondeterminism and Speculation:
-    Implementing "Late choice versus early choice" example from
-    Logical Relations for Fine-Grained Concurrency by Turon et al. (POPL'13) *)
-
-Definition rand: val :=
-  λ: "_",
-    let: "y" := ref #false in
-    Fork ("y" <- #true) ;;
-    !"y".
-
-Definition earlyChoice: val :=
-  λ: "x",
-    let: "r" := rand #() in
-    "x" <- #0 ;;
-    "r".
-
-Definition lateChoice: val :=
-  λ: "x",
-    let: "p" := NewProph in
-    "x" <- #0 ;;
-    let: "r" := rand #() in
-    resolve_proph: "p" to: "r" ;;
-    "r".
-
-Section coinflip.
-  Context `{!heapG Σ}.
-
-  Local Definition N := nroot .@ "coin".
-
-  Lemma rand_spec :
-    {{{ True }}} rand #() {{{ (b : bool), RET #b; True }}}.
-  Proof.
-    iIntros (Φ) "_ HP". wp_lam. wp_alloc l as "Hl".
-    iMod (inv_alloc N _ (∃ (b: bool), l ↦ #b)%I with "[Hl]") as "#Hinv"; first by eauto.
-    wp_apply wp_fork.
-    - iInv N as (b) ">Hl". wp_store. iModIntro. iSplitL; eauto.
-    - wp_pures. iInv N as (b) ">Hl". wp_load. iModIntro. iSplitL "Hl"; first by eauto.
-      iApply "HP". done.
-  Qed.
-
-  Lemma earlyChoice_spec (x: loc) :
-    <<< x ↦ - >>>
-        earlyChoice #x
-        @ ⊤
-    <<< ∃ (b: bool), x ↦ #0, RET #b >>>.
-  Proof.
-    iIntros (Φ) "AU". wp_lam.
-    wp_apply rand_spec; first done.
-    iIntros (b) "_". wp_let.
-    wp_bind (_ <- _)%E.
-    iMod "AU" as "[Hl [_ Hclose]]".
-    iDestruct "Hl" as (v) "Hl".
-    wp_store.
-    iMod ("Hclose" with "[Hl]") as "HΦ"; first by eauto.
-    iModIntro. wp_seq. done.
-  Qed.
-
-  Definition extract_bool (l : list (val * val)) : bool :=
-    match l with
-    | (_, LitV (LitBool b)) :: _ => b
-    | _                          => true
-    end.
-
-  Lemma lateChoice_spec (x: loc) :
-    <<< x ↦ - >>>
-        lateChoice #x
-        @ ⊤
-    <<< ∃ (b: bool), x ↦ #0, RET #b >>>.
-  Proof.
-    iIntros (Φ) "AU". wp_lam.
-    wp_apply wp_new_proph; first done.
-    iIntros (v p) "Hp".
-    wp_let.
-    wp_bind (_ <- _)%E.
-    iMod "AU" as "[Hl [_ Hclose]]".
-    iDestruct "Hl" as (v') "Hl".
-    wp_store.
-    iMod ("Hclose" $! (extract_bool v) with "[Hl]") as "HΦ"; first by eauto.
-    iModIntro. wp_apply rand_spec; try done.
-    iIntros (b') "_".
-    wp_apply (wp_resolve_proph with "Hp").
-    iIntros (vs) "[HEq _]". iDestruct "HEq" as "->". wp_seq. done.
-  Qed.
-
-End coinflip.

theories/heap_lang/lib/lazy_coin.v

+From iris.base_logic Require Export invariants.
+From iris.program_logic Require Export weakestpre.
+From iris.heap_lang Require Export lang proofmode notation.
+From iris.heap_lang.lib Require Export nondet_bool.
+
+Definition new_coin: val := λ: <>, (ref NONE, NewProph).
+
+Definition read_coin : val :=
+  λ: "cp",
+  let: "c" := Fst "cp" in
+  let: "p" := Snd "cp" in
+  match: !"c" with
+    NONE => let: "r" := nondet_bool #() in
+           "c" <- SOME "r";; resolve_proph: "p" to: "r";; "r"
+  | SOME "b" => "b"
+  end.
+
+Section proof.
+  Context `{!heapG Σ}.
+
+  Definition val_to_bool (v : val) : bool := bool_decide (v = #true).
+
+  Definition prophecy_to_bool (vs : list (val * val)) : bool :=
+    default false (val_to_bool ∘ snd <$> head vs).
+
+  Lemma prophecy_to_bool_of_bool (b : bool) v vs :
+    prophecy_to_bool ((v, #b) :: vs) = b.
+  Proof. by destruct b. Qed.
+
+  Definition coin (cp : val) (b : bool) : iProp Σ :=
+    (∃ (c : loc) (p : proph_id) (vs : list (val * val)),
+        ⌜cp = (#c, #p)%V⌝ ∗ proph p vs ∗
+        (c ↦ SOMEV #b ∨ (c ↦ NONEV ∗ ⌜b = prophecy_to_bool vs⌝)))%I.
+
+  Lemma new_coin_spec : {{{ True }}} new_coin #() {{{ c b, RET c; coin c b }}}.
+  Proof.
+    iIntros (Φ) "_ HΦ".
+    wp_lam.
+    wp_apply wp_new_proph; first done.
+    iIntros (vs p) "Hp".
+    wp_alloc c as "Hc".
+    wp_pair.
+    iApply ("HΦ" $! (#c, #p)%V ).
+    rewrite /coin; eauto 10 with iFrame.
+  Qed.
+
+  Lemma read_coin_spec cp b :
+    {{{ coin cp b }}} read_coin cp {{{ RET #b; coin cp b }}}.
+  Proof.
+    iIntros (Φ) "Hc HΦ".
+    iDestruct "Hc" as (c p vs ->) "[Hp [Hc | [Hc ->]]]".
+    - wp_lam. wp_load. wp_match.
+      iApply "HΦ".
+      rewrite /coin; eauto 10 with iFrame.
+    - wp_lam. wp_load. wp_match.
+      wp_apply nondet_bool_spec; first done.
+      iIntros (r) "_".
+      wp_let.
+      wp_store.
+      wp_apply (wp_resolve_proph with "[Hp]"); first done.
+      iIntros (ws) "[-> Hws]".
+      rewrite !prophecy_to_bool_of_bool.
+      wp_seq.
+      iApply "HΦ".
+      rewrite /coin; eauto with iFrame.
+  Qed.
+
+End proof.

theories/heap_lang/lib/nondet_bool.v

+From iris.base_logic Require Export invariants.
+From iris.program_logic Require Export weakestpre.
+From iris.heap_lang Require Export lang proofmode notation.
+
+Definition nondet_bool : val :=
+  λ: <>, let: "l" := ref #true in Fork ("l" <- #false);; !"l".
+
+Section proof.
+  Context `{!heapG Σ}.
+
+  Lemma nondet_bool_spec : {{{ True }}} nondet_bool #() {{{ (b : bool), RET #b; True }}}.
+  Proof.
+    iIntros (Φ) "_ HΦ".
+    wp_lam. wp_alloc l as "Hl". wp_let.
+    pose proof (nroot .@ "rnd") as rndN.
+    iMod (inv_alloc rndN _ (∃ (b : bool), l ↦ #b)%I with "[Hl]") as "#Hinv";
+      first by eauto.
+    wp_apply wp_fork.
+    - iInv rndN as (?) "?". wp_store; eauto.
+    - wp_seq. iInv rndN as (?) "?". wp_load.
+      iSplitR "HΦ"; first by eauto.
+      by iApply "HΦ".
+  Qed.
+
+End proof.