From iris.program_logic Require Export weakestpre.
From iris.heap_lang Require Export lang.
From iris.heap_lang Require Import proofmode notation.
From iris.algebra Require Import frac auth gmap dec_agree csum.
From iris.base_logic Require Import big_op.
From iris_atomic Require Import atomic misc.
Definition new_stack: val := λ: <>, ref (ref NONE).
Definition push: val :=
rec: "push" "s" "x" :=
let: "hd" := !"s" in
let: "s'" := ref SOME ("x", "hd") in
if: CAS "s" "hd" "s'"
then #()
else "push" "s" "x".
Definition pop: val :=
rec: "pop" "s" :=
let: "hd" := !"s" in
match: !"hd" with
SOME "cell" =>
if: CAS "s" "hd" (Snd "cell")
then SOME (Fst "cell")
else "pop" "s"
| NONE => NONE
end.
Definition iter: val :=
rec: "iter" "hd" "f" :=
match: !"hd" with
NONE => #()
| SOME "cell" => "f" (Fst "cell") ;; "iter" (Snd "cell") "f"
end.
Global Opaque new_stack push pop iter.
Section proof.
Context `{!heapG Σ} (N: namespace).
Fixpoint is_list (hd: loc) (xs: list val) : iProp Σ :=
match xs with
| [] => (∃ q, hd ↦{ q } NONEV)%I
| x :: xs => (∃ (hd': loc) q, hd ↦{ q } SOMEV (x, #hd') ★ is_list hd' xs)%I
end.
Lemma dup_is_list : ∀ xs hd,
heap_ctx ★ is_list hd xs ⊢ is_list hd xs ★ is_list hd xs.
Proof.
induction xs as [|y xs' IHxs'].
- iIntros (hd) "(#? & Hs)".
simpl. iDestruct "Hs" as (q) "[Hhd Hhd']". iSplitL "Hhd"; eauto.
- iIntros (hd) "(#? & Hs)". simpl.
iDestruct "Hs" as (hd' q) "([Hhd Hhd'] & Hs')".
iDestruct (IHxs' with "[Hs']") as "[Hs1 Hs2]"; first by iFrame.
iSplitL "Hhd Hs1"; iExists hd', (q / 2)%Qp; by iFrame.
Qed.
Lemma uniq_is_list:
∀ xs ys hd, heap_ctx ★ is_list hd xs ★ is_list hd ys ⊢ xs = ys.
Proof.
induction xs as [|x xs' IHxs'].
- induction ys as [|y ys' IHys'].
+ auto.
+ iIntros (hd) "(#? & Hxs & Hys)".
simpl. iDestruct "Hys" as (hd' ?) "(Hhd & Hys')".
iExFalso. iDestruct "Hxs" as (?) "Hhd'".
iDestruct (heap_mapsto_op_1 with "[Hhd Hhd']") as "[% _]".
{ iSplitL "Hhd"; done. }
done.
- induction ys as [|y ys' IHys'].
+ iIntros (hd) "(#? & Hxs & Hys)".
simpl.
iExFalso. iDestruct "Hxs" as (? ?) "(Hhd & _)".
iDestruct "Hys" as (?) "Hhd'".
iDestruct (heap_mapsto_op_1 with "[Hhd Hhd']") as "[% _]".
{ iSplitL "Hhd"; done. }
done.
+ iIntros (hd) "(#? & Hxs & Hys)".
simpl. iDestruct "Hxs" as (? ?) "(Hhd & Hxs')".
iDestruct "Hys" as (? ?) "(Hhd' & Hys')".
iDestruct (heap_mapsto_op_1 with "[Hhd Hhd']") as "[% _]".
{ iSplitL "Hhd"; done. }
inversion H3. (* FIXME: name *)
subst. iDestruct (IHxs' with "[Hxs' Hys']") as "%"; first by iFrame.
by subst.
Qed.
Definition is_stack (s: loc) xs: iProp Σ := (∃ hd: loc, s ↦ #hd ★ is_list hd xs)%I.
Global Instance is_list_timeless xs hd: TimelessP (is_list hd xs).
Proof. generalize hd. induction xs; apply _. Qed.
Global Instance is_stack_timeless xs hd: TimelessP (is_stack hd xs).
Proof. generalize hd. induction xs; apply _. Qed.
Lemma new_stack_spec:
∀ (Φ: val → iProp Σ),
heapN ⊥ N →
heap_ctx ★ (∀ s, is_stack s [] -★ Φ #s) ⊢ WP new_stack #() {{ Φ }}.
Proof.
iIntros (Φ HN) "[#Hh HΦ]". wp_seq.
wp_bind (ref NONE)%E. wp_alloc l as "Hl".
wp_alloc l' as "Hl'".
iApply "HΦ". rewrite /is_stack. iExists l.
iFrame. by iExists 1%Qp.
Qed.
Definition push_triple (s: loc) (x: val) :=
atomic_triple _ (fun xs_hd: list val * loc =>
let '(xs, hd) := xs_hd in s ↦ #hd ★ is_list hd xs)%I
(fun xs_hd ret =>
let '(xs, hd) := xs_hd in
∃ hd': loc,
ret = #() ★ s ↦ #hd' ★ hd' ↦ SOMEV (x, #hd) ★ is_list hd xs)%I
(nclose heapN)
⊤
(push #s x).
Lemma push_atomic_spec (s: loc) (x: val) :
heapN ⊥ N → heap_ctx ⊢ push_triple s x.
Proof.
iIntros (HN) "#?". rewrite /push_triple /atomic_triple.
iIntros (P Q) "#Hvs".
iLöb as "IH". iIntros "!# HP". wp_rec.
wp_let. wp_bind (! _)%E.
iMod ("Hvs" with "HP") as ([xs hd]) "[[Hs Hhd] [Hvs' _]]".
wp_load. iMod ("Hvs'" with "[Hs Hhd]") as "HP"; first by iFrame.
iModIntro. wp_let. wp_alloc l as "Hl". wp_let.
wp_bind (CAS _ _ _)%E.
iMod ("Hvs" with "HP") as ([xs' hd']) "[[Hs Hhd'] Hvs']".
destruct (decide (hd = hd')) as [->|Hneq].
* wp_cas_suc. iDestruct "Hvs'" as "[_ Hvs']".
iMod ("Hvs'" $! #() with "[-]") as "HQ".
{ iExists l. iSplitR; first done. by iFrame. }
iModIntro. wp_if. iModIntro. eauto.
* wp_cas_fail.
iDestruct "Hvs'" as "[Hvs' _]".
iMod ("Hvs'" with "[-]") as "HP"; first by iFrame.
iModIntro. wp_if. by iApply "IH".
Qed.
Definition pop_triple (s: loc) :=
atomic_triple _ (fun xs_hd: list val * loc =>
let '(xs, hd) := xs_hd in s ↦ #hd ★ is_list hd xs)%I
(fun xs_hd ret =>
let '(xs, hd) := xs_hd in
(ret = NONEV ★ xs = [] ★ s ↦ #hd ★ is_list hd []) ∨
(∃ x q (hd': loc) xs', hd ↦{q} SOMEV (x, #hd') ★ ret = SOMEV x ★
xs = x :: xs' ★ s ↦ #hd' ★ is_list hd' xs'))%I
(nclose heapN)
⊤
(pop #s).
Lemma pop_atomic_spec (s: loc):
heapN ⊥ N → heap_ctx ⊢ pop_triple s.
Proof.
iIntros (HN) "#?".
rewrite /pop_triple /atomic_triple.
iIntros (P Q) "#Hvs".
iLöb as "IH". iIntros "!# HP". wp_rec.
wp_bind (! _)%E.
iMod ("Hvs" with "HP") as ([xs hd]) "[[Hs Hhd] Hvs']".
destruct xs as [|y' xs'].
- simpl. wp_load. iDestruct "Hvs'" as "[_ Hvs']".
iDestruct "Hhd" as (q) "[Hhd Hhd']".
iMod ("Hvs'" $! NONEV with "[-Hhd]") as "HQ".
{ iLeft. iSplit=>//. iSplit=>//. iFrame. eauto. }
iModIntro. wp_let. wp_load. wp_match.
iModIntro. eauto.
- simpl. iDestruct "Hhd" as (hd' q) "([[Hhd1 Hhd2] Hhd'] & Hxs')".
iDestruct (dup_is_list with "[Hxs']") as "[Hxs1 Hxs2]"; first by iFrame.
wp_load. iDestruct "Hvs'" as "[Hvs' _]".
iMod ("Hvs'" with "[-Hhd1 Hhd2 Hxs1]") as "HP".
{ iFrame. iExists hd', (q / 2)%Qp. by iFrame. }
iModIntro. wp_let. wp_load. wp_match. wp_proj.
wp_bind (CAS _ _ _). iMod ("Hvs" with "HP") as ([xs hd'']) "[[Hs Hhd''] Hvs']".
destruct (decide (hd = hd'')) as [->|Hneq].
+ wp_cas_suc. iDestruct "Hvs'" as "[_ Hvs']".
iMod ("Hvs'" $! (SOMEV y') with "[-]") as "HQ".
{ iRight. iExists y', (q / 2 / 2)%Qp, hd', xs'.
destruct xs as [|x' xs''].
- simpl. iDestruct "Hhd''" as (?) "H".
iExFalso. iDestruct (heap_mapsto_op_1 with "[Hhd1 H]") as "[% _]".
{ iSplitL "Hhd1"; done. }
done.
- simpl. iDestruct "Hhd''" as (hd''' ?) "(Hhd'' & Hxs'')".
iDestruct (heap_mapsto_op_1 with "[Hhd1 Hhd'']") as "[% _]".
{ iSplitL "Hhd1"; done. }
inversion H0. (* FIXME: bad naming *) subst.
iDestruct (uniq_is_list with "[Hxs1 Hxs'']") as "%"; first by iFrame. subst.
repeat (iSplitR "Hxs1 Hs"; first done).
iFrame. }
iModIntro. wp_if. wp_proj. eauto.
+ wp_cas_fail. iDestruct "Hvs'" as "[Hvs' _]".
iMod ("Hvs'" with "[-]") as "HP"; first by iFrame.
iModIntro. wp_if. by iApply "IH".
Qed.
End proof.