Commit 9bf43488 authored by Zhen Zhang's avatar Zhen Zhang

iter

parent 68445e22
......@@ -19,25 +19,35 @@ Definition pop: val :=
rec: "pop" "s" :=
let: "hd" := !"s" in
match: !"hd" with
SOME "pair" =>
if: CAS "s" "hd" (Snd "pair")
then SOME (Fst "pair")
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.
Definition iter: val := λ: "f" "s", iter' (!"s") "f".
Global Opaque new_stack push pop iter' iter.
Section proof.
Context `{!heapG Σ} (N: namespace) (R: val iProp Σ).
Context `{!heapG Σ} (N: namespace).
Fixpoint is_stack' (hd: loc) (xs: list val) : iProp Σ :=
Fixpoint is_stack' (R: val iProp Σ) (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') R x is_stack' hd' xs)%I
| x :: xs => ( (hd': loc) q, hd { q } SOMEV (x, #hd') R x is_stack' R hd' xs)%I
end.
(* how can we prove that it is persistent? *)
Lemma dup_is_stack': xs hd,
heap_ctx is_stack' hd xs is_stack' hd xs is_stack' hd xs.
Lemma dup_is_stack' R `{ v, PersistentP (R v)} : xs hd,
heap_ctx is_stack' R hd xs is_stack' R hd xs is_stack' R hd xs.
Proof.
induction xs as [|y xs' IHxs'].
- iIntros (hd) "(#? & Hs)".
......@@ -48,14 +58,14 @@ Section proof.
iSplitL "Hhd Hs1"; iExists hd', (q / 2)%Qp; by iFrame.
Qed.
Lemma uniq_is_stack':
xs ys hd, heap_ctx is_stack' hd xs is_stack' hd ys xs = ys.
Lemma uniq_is_stack' R:
xs ys hd, heap_ctx is_stack' R hd xs is_stack' R 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 & #Hy & Hys')".
simpl. iDestruct "Hys" as (hd' ?) "(Hhd & Hy & Hys')".
iExFalso. iDestruct "Hxs" as (?) "Hhd'".
iDestruct (heap_mapsto_op_1 with "[Hhd Hhd']") as "[% _]".
{ iSplitL "Hhd"; done. }
......@@ -78,12 +88,12 @@ Section proof.
by subst.
Qed.
Definition is_stack (s: loc) xs: iProp Σ := ( hd: loc, s #hd is_stack' hd xs)%I.
Definition is_stack R (s: loc) xs: iProp Σ := ( hd: loc, s #hd is_stack' R hd xs)%I.
Lemma new_stack_spec:
Lemma new_stack_spec R:
(Φ: val iProp Σ),
heapN N
heap_ctx ( s, is_stack s [] - Φ #s) WP new_stack #() {{ Φ }}.
heap_ctx ( s, is_stack R 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".
......@@ -92,15 +102,15 @@ Section proof.
iFrame. by iExists 1%Qp.
Qed.
Definition push_triple (s: loc) (x: val) :=
atomic_triple (fun xs => R x is_stack s xs)%I
(fun xs _ => is_stack s (x :: xs))
Definition push_triple R (s: loc) (x: val) :=
atomic_triple (fun xs => R x is_stack R s xs)%I
(fun xs _ => is_stack R s (x :: xs))
(nclose heapN)
(push #s x).
Lemma push_atomic_spec (s: loc) (x: val) :
heapN N heap_ctx push_triple s x.
Lemma push_atomic_spec R (s: loc) (x: val) :
heapN N heap_ctx push_triple R s x.
Proof.
iIntros (HN) "#?". rewrite /push_triple /atomic_triple.
iIntros (P Q) "#Hvs".
......@@ -126,19 +136,27 @@ Section proof.
iVsIntro. wp_if. by iApply "IH".
Qed.
Definition pop_triple (s: loc) :=
atomic_triple (fun xs => is_stack s xs)%I
(fun xs ret => (ret = NONEV xs = [] is_stack s [])
( x xs', ret = SOMEV x R x xs = x :: xs' is_stack s xs'))%I
Definition pop_triple_strong R (s: loc) :=
atomic_triple (fun xs => is_stack R s xs)%I
(fun xs ret => (ret = NONEV xs = [] is_stack R s [])
( x xs', ret = SOMEV x R x xs = x :: xs' is_stack R s xs'))%I
(nclose heapN)
(pop #s).
Definition pop_triple_weak R (s: loc) :=
atomic_triple (fun xs => is_stack R s xs)%I
(fun xs ret => (ret = NONEV xs = [] is_stack R s [])
( x, ret = SOMEV x R x))%I
(nclose heapN)
(pop #s).
Lemma pop_atomic_spec (s: loc) (x: val) :
heapN N heap_ctx pop_triple s.
Lemma pop_atomic_spec_strong R `{ v, PersistentP (R v)} (s: loc) (x: val) :
heapN N heap_ctx pop_triple_strong R s.
Proof.
iIntros (HN) "#?".
rewrite /pop_triple /atomic_triple.
rewrite /pop_triple_strong /atomic_triple.
iIntros (P Q) "#Hvs".
iLöb as "IH". iIntros "!# HP". wp_rec.
wp_bind (! _)%E.
......@@ -172,8 +190,8 @@ Section proof.
done.
- simpl. iDestruct "Hhd''" as (hd''' ?) "(Hhd'' & _ & Hxs'')".
iDestruct (heap_mapsto_op_1 with "[Hhd Hhd'']") as "[% _]".
{ iSplitL "Hhd"; done. }
inversion H0. (* FIXME: bad naming *) subst.
{ iSplitL "Hhd"; done. }
inversion H1. (* FIXME: bad naming *) subst.
iDestruct (uniq_is_stack' with "[Hxs1 Hxs'']") as "%"; first by iFrame. subst.
repeat (iSplitR "Hxs1 Hs"; first done).
iExists hd'''. by iFrame.
......@@ -184,5 +202,88 @@ Section proof.
{ iExists hd''. by iFrame. }
iVsIntro. wp_if. by iApply "IH".
Qed.
(* FIXME: Code dup with pop_atomic_spec_strong *)
Lemma pop_atomic_spec_weak R (s: loc) (x: val) :
heapN N heap_ctx pop_triple_weak R s.
Proof.
iIntros (HN) "#?".
rewrite /pop_triple_strong /atomic_triple.
iIntros (P Q) "#Hvs".
iLöb as "IH". iIntros "!# HP". wp_rec.
wp_bind (! _)%E.
iVs ("Hvs" with "HP") as (xs) "[Hxs Hvs']".
destruct xs as [|y' xs'].
- simpl. iDestruct "Hxs" as (hd) "[Hs Hhd]".
wp_load. iDestruct "Hvs'" as "[_ Hvs']".
iDestruct "Hhd" as (q) "[Hhd Hhd']".
iVs ("Hvs'" $! NONEV with "[-Hhd]") as "HQ".
{ iLeft. iSplit=>//. iSplit=>//.
iExists hd. iFrame. rewrite /is_stack'. eauto. }
iVsIntro. wp_let. wp_load. wp_match.
iVsIntro. by iExists [].
- simpl. iDestruct "Hxs" as (hd) "[Hs Hhd]".
simpl. iDestruct "Hhd" as (hd' q) "([Hhd Hhd'] & Hy' & Hxs')".
wp_load. iDestruct "Hvs'" as "[Hvs' _]".
iVs ("Hvs'" with "[-Hhd]") as "HP".
{ iExists hd. iFrame. iExists hd', (q / 2)%Qp. by iFrame. }
iVsIntro. wp_let. wp_load. wp_match. wp_proj.
wp_bind (CAS _ _ _). iVs ("Hvs" with "HP") as (xs) "[Hxs Hvs']".
iDestruct "Hxs" as (hd'') "[Hs Hhd'']".
destruct (decide (hd = hd'')) as [->|Hneq].
+ wp_cas_suc. iDestruct "Hvs'" as "[_ Hvs']".
iVs ("Hvs'" $! (SOMEV y') with "[-]") as "HQ".
{ iRight. rewrite /is_stack. iExists y'.
destruct xs as [|x' xs''].
- simpl. iDestruct "Hhd''" as (?) "H".
iExFalso. iDestruct (heap_mapsto_op_1 with "[Hhd H]") as "[% _]".
{ iSplitL "Hhd"; done. }
done.
- simpl. iDestruct "Hhd''" as (hd''' ?) "(Hhd'' & Hx' & Hxs'')".
iDestruct (heap_mapsto_op_1 with "[Hhd Hhd'']") as "[% _]".
{ iSplitL "Hhd"; done. }
inversion H0. (* FIXME: bad naming *) subst.
eauto.
}
iVsIntro. wp_if. wp_proj. eauto.
+ wp_cas_fail. iDestruct "Hvs'" as "[Hvs' _]".
iVs ("Hvs'" with "[-]") as "HP".
{ iExists hd''. by iFrame. }
iVsIntro. wp_if. by iApply "IH".
Qed.
Lemma iter_spec' R R' (f: val) :
xs (hd: loc),
heapN N ( x: val, {{ R x }} f x {{ _, R' x }} )
heap_ctx is_stack' R hd xs WP iter' #hd f {{ _, is_stack' R' hd xs }}.
Proof.
induction xs as [|x xs' IHxs'].
- iIntros (hd HN Hf) "[#? Hs]".
simpl. iDestruct "Hs" as (?) "Hhd".
wp_rec. wp_let. wp_load. wp_match. eauto.
- iIntros (hd HN Hf) "[#? Hs]".
simpl. iDestruct "Hs" as (hd' q) "(Hhd & Hx & Hxs')".
wp_rec. wp_let. wp_load. wp_match. wp_proj.
wp_bind (f _). iApply wp_wand_r.
iSplitL "Hx"; first by iApply Hf.
iIntros (?) "Hx". wp_seq.
iApply wp_wand_r.
iSplitL "Hxs'". wp_proj.
{ iApply (IHxs' with "[Hxs']")=>//. by iFrame. }
iIntros (?) "Hs".
iExists hd', q. by iFrame.
Qed.
Lemma iter_spec R R' (s: loc) xs (f: val) :
heapN N ( x: val, {{ R x }} f x {{ _, R' x }} )
heap_ctx is_stack R s xs WP iter f #s {{ _, is_stack R' s xs }}.
Proof.
iIntros (HN Hf) "[#? Hs]".
wp_seq. wp_let. iDestruct "Hs" as (hd) "[Hs Hhd]".
wp_load. iApply wp_wand_r. iSplitL "Hhd".
{ iApply (iter_spec' with "[Hhd]")=>//. by iFrame. }
iIntros (?) "Hhd'".
iExists hd. by iFrame.
Qed.
End proof.
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