Move per-item stack spec in

b0f7bc5b · Zhen Zhang · ff73fa5b · b0f7bc5b · b0f7bc5b · b0f7bc5b
Commit b0f7bc5b authored Oct 10, 2016 by Zhen Zhang
--- a/Makefile.coq
+++ b/Makefile.coq
@@ -100,7 +100,8 @@ VFILES:=simple_sync.v\
  pair_cas.v\
  flat.v\
  atomic_pair.v\
-  atomic_sync.v
+  atomic_sync.v\
+  treiber.v

 ifneq ($(filter-out archclean clean cleanall printenv,$(MAKECMDGOALS)),)
 -include $(addsuffix .d,$(VFILES))

--- a/_CoqProject
+++ b/_CoqProject
@@ -4,3 +4,5 @@ pair_cas.v
 flat.v
 atomic_pair.v
 atomic_sync.v
+treiber.v
+
--- a/treiber.v
+++ b/treiber.v
+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 upred gmap dec_agree upred_big_op.
+From iris.program_logic Require Import auth.
+From iris.tests Require Import treiber_stack atomic misc.
+
+Section defs.
+  Context `{heapG Σ, !evidenceG loc val Σ} (N: namespace).
+  Context (R: val → iProp Σ) (γ: gname) `{∀ x, TimelessP (R x)}.
+
+  Definition allocated hd := (∃ q v, hd ↦{q} v)%I.
+
+  Definition evs := ev loc val γ.
+
+  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') ★ evs hd x ★ is_list' hd' xs)%I
+    end.
+
+  Lemma in_list' x xs:
+    ∀ hd, x ∈ xs →
+          is_list' hd xs
+          ⊢ ∃ (hd' hd'': loc) q, hd' ↦{q} SOMEV (x, #hd'') ★ evs hd' x.
+  Proof.
+    induction xs as [|x' xs' IHxs'].
+    - intros ? Hin. inversion Hin.
+    - intros hd Hin. destruct (decide (x = x')) as [->|Hneq].
+      + iIntros "Hls". simpl.
+        iDestruct "Hls" as (hd' q) "(? & ? & ?)".
+        iExists hd, hd', q. iFrame.
+      + assert (x ∈ xs') as Hin'; first set_solver.
+        iIntros "Hls". simpl.
+        iDestruct "Hls" as (hd' q) "(? & ? & ?)".
+        iApply IHxs'=>//.
+  Qed.
+
+  Definition perR' hd v v' := (v = (1%Qp, DecAgree v') ★ R v' ★ allocated hd)%I.
+  Definition perR  hd v := (∃ v', perR' hd v v')%I.
+  
+  Definition allR := (∃ m, own γ (● m) ★ [★ map] hd ↦ v ∈ m, perR hd v)%I.
+
+  Definition is_stack' xs s := (∃ hd: loc, s ↦ #hd ★ is_list' hd xs ★ allR)%I.
+
+  Global Instance is_list'_timeless hd xs: TimelessP (is_list' hd xs).
+  Proof. generalize hd. induction xs; apply _. Qed.
+
+  Global Instance is_stack'_timeless xs s: TimelessP (is_stack' xs s).
+  Proof. apply _. Qed.
+
+  Lemma dup_is_list': ∀ xs hd,
+    heap_ctx ★ is_list' hd xs ⊢ |=r=> 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'] & Hev & Hs')".
+      iDestruct (IHxs' with "[Hs']") as "==>[Hs1 Hs2]"; first by iFrame.
+      iDestruct (dup_ev with "Hev") as "==>[Hev1 Hev2]".
+      iVsIntro. iSplitL "Hhd Hs1 Hev1"; iExists hd', (q / 2)%Qp; by iFrame.
+  Qed.
+
+  Lemma extract_is_list: ∀ xs hd,
+    heap_ctx ★ is_list' hd xs ⊢ |=r=> 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'] & Hev & Hs')".
+      iDestruct (IHxs' with "[Hs']") as "==>[Hs1 Hs2]"; first by iFrame.
+      iVsIntro. iSplitL "Hhd Hs1 Hev"; iExists hd', (q / 2)%Qp; by iFrame.
+  Qed.
+
+  Definition f_spec (xs: list val) (s: loc) (f: val) (Rf RI: iProp Σ) := (* Rf, RI is some frame *)
+    ∀ Φ (x: val),
+      heapN ⊥ N → x ∈ xs →
+      heap_ctx ★ inv N ((∃ xs, is_stack' xs s) ★ RI) ★  Rf ★ (Rf -★ Φ #())
+      ⊢ WP f x {{ Φ }}.
+End defs.
+
+Section proofs.
+  Context `{heapG Σ, !evidenceG loc val Σ} (N: namespace).
+  Context (R: val → iProp Σ).
+
+Lemma new_stack_spec' Φ RI:
+    heapN ⊥ N →
+    heap_ctx ★ RI ★ (∀ γ s : loc, inv N ((∃ xs, is_stack' R γ xs s) ★ RI) -★ Φ #s)
+    ⊢ WP new_stack #() {{ Φ }}.
+  Proof.
+    iIntros (HN) "(#Hh & HR & HΦ)".
+    iVs (own_alloc (● (∅: evmapR loc val) ⋅ ◯ ∅)) as (γ) "[Hm Hm']".
+    { apply auth_valid_discrete_2. done. }
+    wp_seq. wp_bind (ref NONE)%E. wp_alloc l as "Hl".
+    wp_alloc s as "Hs".
+    iAssert ((∃ xs : list val, is_stack' R γ xs s) ★ RI)%I with "[-HΦ Hm']" as "Hinv".
+    { iFrame. iExists [], l. iFrame. simpl. iSplitL "Hl".
+      - eauto.
+      - iExists ∅. iFrame. by iApply (big_sepM_empty (fun hd v => perR R hd v)). }
+    iVs (inv_alloc N _ ((∃ xs : list val, is_stack' R γ xs s) ★ RI)%I with "[-HΦ Hm']") as "#?"; first eauto.
+    by iApply "HΦ".
+  Qed.
+    
+  Lemma iter_spec Φ γ s (Rf RI: iProp Σ):
+    ∀ xs hd (f: expr) (f': val),
+      heapN ⊥ N → f_spec N R γ xs s f' Rf RI → to_val f = Some f' →
+      heap_ctx ★ inv N ((∃ xs, is_stack' R γ xs s) ★ RI) ★
+      is_list' γ hd xs ★ Rf ★ (Rf -★ Φ #())
+      ⊢ WP (iter #hd) f {{ v, Φ v }}.
+  Proof.
+    induction xs as [|x xs' IHxs'].
+    - simpl. iIntros (hd f f' HN ? ?) "(#Hh & #? & Hxs1 & HRf & HΦ)".
+      iDestruct "Hxs1" as (q) "Hhd".
+      wp_rec. wp_value. iVsIntro. wp_let. wp_load. wp_match. by iApply "HΦ".
+    - simpl. iIntros (hd f f' HN Hf ?) "(#Hh & #? & Hxs1 & HRf & HΦ)".
+      iDestruct "Hxs1" as (hd2 q) "(Hhd & Hev & Hhd2)".
+      wp_rec. wp_value. iVsIntro. wp_let. wp_load. wp_match. wp_proj.
+      wp_bind (f' _). iApply Hf=>//; first set_solver. iFrame "#". iFrame. iIntros "HRf".
+      wp_seq. wp_proj. iApply (IHxs' with "[-]")=>//.
+      + rewrite /f_spec. iIntros (? ? ? ?) "(#? & #? & ? & ?)".
+        iApply Hf=>//.
+        * set_solver.
+        * iFrame "#". by iFrame.
+      + apply to_of_val.
+      + iFrame "#". by iFrame.
+  Qed.
+  
+  Lemma push_spec Φ γ (s: loc) (x: val) RI:
+    heapN ⊥ N →
+    heap_ctx ★ R x ★ inv N ((∃ xs, is_stack' R γ xs s) ★ RI) ★ ((∃ hd, evs γ hd x) -★ Φ #())
+    ⊢ WP push #s x {{ Φ }}.
+  Proof.
+    iIntros (HN) "(#Hh & HRx & #? & HΦ)".
+    iDestruct (push_atomic_spec N s x with "Hh") as "Hpush"=>//.
+    rewrite /push_triple /atomic_triple.
+    iSpecialize ("Hpush" $! (R x) (fun _ ret => (∃ hd, evs γ hd x) ★ ret = #())%I with "[]").
+    - iIntros "!# Rx".
+      (* open the invariant *)
+      iInv N as "[IH1 ?]" "Hclose".
+      iDestruct "IH1" as (xs hd) "[>Hs [>Hxs HR]]".
+      iDestruct (extract_is_list with "[Hxs]") as "==>[Hxs Hxs']"; first by iFrame.
+      iDestruct (dup_is_list with "[Hxs']") as "[Hxs'1 Hxs'2]"; first by iFrame.
+      (* mask magic *)
+      iApply pvs_intro'.
+      { apply ndisj_subseteq_difference; auto. }
+      iIntros "Hvs".
+      iExists (xs, hd).
+      iFrame "Hs Hxs'1".
+      iSplit.
+      + (* provide a way to rollback *)
+        iIntros "[Hs Hl']".
+        iVs "Hvs". iVs ("Hclose" with "[-Rx]"); last done.
+        { iNext. iFrame. iExists xs. iExists hd. by iFrame. }
+      + (* provide a way to commit *)
+        iIntros (v) "Hs".
+        iDestruct "Hs" as (hd') "[% [Hs [[Hhd'1 Hhd'2] Hxs']]]". subst.
+        iVs "Hvs".
+        iDestruct "HR" as (m) "[>Hom HRm]".
+        destruct (m !! hd') eqn:Heqn.
+        * iDestruct (big_sepM_delete_later (perR R) m with "HRm") as "[Hx ?]"=>//.
+          iDestruct "Hx" as (?) "(_ & _ & >Hhd'')".
+          iDestruct (heap_mapsto_op_1 with "[Hhd'1 Hhd'2]") as "[_ Hhd]"; first iFrame.
+          rewrite Qp_div_2.
+          iDestruct "Hhd''" as (q v) "Hhd''". iExFalso.
+          iApply (bogus_heap hd' 1%Qp q); first apply Qp_not_plus_q_ge_1.
+          iFrame "#". iFrame.
+        * iAssert (evs γ hd' x ★ ▷ (allR R γ))%I
+                  with "==>[Rx Hom HRm Hhd'1]" as "[Hox ?]".
+          {
+            iDestruct (evmap_alloc _ _ _ m with "[Hom]") as "==>[Hom Hox]"=>//.
+            iDestruct (dup_ev with "[Hox]") as "==>[Hox1 Hox2]".
+            { rewrite /ev. eauto. }
+            iFrame.
+            iDestruct (big_sepM_insert_later (perR R) m) as "H"=>//.
+            rewrite (insert_singleton_op m)=>//.
+            iExists ({[hd' := (1%Qp, DecAgree x)]} ⋅ m).
+            iFrame. iApply "H". iFrame. iExists x.
+            iFrame. rewrite /allocated. iSplitR "Hhd'1"; auto.
+          }
+          iDestruct (dup_ev with "[Hox]") as "==>[Hox1 Hox2]"=>//.
+          iVs ("Hclose" with "[-Hox2]").
+          { iNext. iFrame. iExists (x::xs).
+            iExists hd'. iFrame.
+            iExists hd, (1/2)%Qp. iFrame.
+          }
+        iVsIntro. iSplitL; last auto. by iExists hd'.
+    - iApply wp_wand_r. iSplitL "HRx Hpush".
+      + by iApply "Hpush".
+      + iIntros (?) "H". iDestruct "H" as (_) "[? %]". subst.
+        by iApply "HΦ".
+  Qed.
+
+  (* some helpers *)
+  Lemma access (γ: gname) (x: val) (xs: list val) m:
+    ∀ hd: loc,
+    x ∈ xs  →
+    ▷ ([★ map] hd↦v ∈ m, perR R hd v) ★ own γ (● m) ★
+    is_list' γ hd xs
+    ⊢ ∃ hd' q, ▷ ([★ map] hd↦v ∈ delete hd' m, perR R hd v) ★
+               ▷ perR R hd' (q, DecAgree x) ★ m !! hd' = Some (q, DecAgree x) ★ own γ (● m).
+  Proof.
+    induction xs as [|x' xs' IHxs'].
+    - iIntros (? Habsurd). inversion Habsurd.
+    - destruct (decide (x = x')) as [->|Hneq].
+      + iIntros (hd _) "(HR & Hom & Hxs)".
+        simpl. iDestruct "Hxs" as (hd' q) "[Hhd [Hev Hxs']]".
+        rewrite /ev. destruct (m !! hd) as [[q' [x|]]|] eqn:Heqn.
+        * iDestruct (big_sepM_delete_later (perR R) m with "HR") as "[Hp HRm]"=>//.
+          iDestruct (map_agree_eq' _ _ _ m with "[Hom Hev]") as "(Hom & Hev & %)"=>//; first iFrame.
+          subst. iExists hd, q'. inversion H0. subst. by iFrame.
+        * iDestruct (big_sepM_delete_later (perR R) m with "HR") as "[Hp HRm]"=>//.
+          iDestruct (map_agree_eq' _ _ _ m with "[Hom Hev]") as "(Hom & Hev & %)"=>//; first iFrame.
+        * iExFalso. iApply (map_agree_none' _ _ _ m)=>//. iFrame.
+      + iIntros (hd ?).
+        assert (x ∈ xs'); first set_solver.
+        iIntros "(HRs & Hom & Hxs')".
+        simpl. iDestruct "Hxs'" as (hd' q) "[Hhd [Hev Hxs']]".
+        iDestruct (IHxs' hd' with "[HRs Hxs' Hom]") as "?"=>//.
+        iFrame.
+  Qed.
+
+End proofs.