### New atomic updates: defined as a fixed point with existential quantifier;...

`New atomic updates: defined as a fixed point with existential quantifier; intro lemma using class of Laterable assertions`
parent 278515b8
 ... ... @@ -40,6 +40,7 @@ theories/bi/embedding.v theories/bi/lib/counter_examples.v theories/bi/lib/fixpoint.v theories/bi/lib/fractional.v theories/bi/lib/laterable.v theories/bi/lib/atomic.v theories/bi/lib/core.v theories/base_logic/upred.v ... ...
 From iris.bi Require Export bi updates. From iris.bi.lib Require Import fixpoint laterable. From stdpp Require Import coPset. From iris.proofmode Require Import tactics. Set Default Proof Using "Type". Definition atomic_shift `{BiFUpd PROP} {A B : Type} (α : A → PROP) (* atomic pre-condition *) (β : A → B → PROP) (* atomic post-condition *) (Eo Em : coPset) (* outside/module masks *) (P : PROP) (* pre-condition *) (Q : A → B → PROP) (* post-condition *) : PROP := (□ (∀ E, ⌜Eo ⊆ E⌝ -∗ P ={E, E∖Em}=∗ ∃ x, α x ∗ ((α x ={E∖Em, E}=∗ P) ∧ (∀ y, β x y ={E∖Em, E}=∗ Q x y))) )%I. (** atomic_update as a fixed-point of the equation AU = ∃ P. ▷ P ∗ □ (▷ P ==∗ α ∗ (α ==∗ AU) ∧ (β ==∗ Q)) *) Section definition. Context `{BiFUpd PROP} {A B : Type} (α : A → PROP) (* atomic pre-condition *) (β : A → B → PROP) (* atomic post-condition *) (Eo Em : coPset) (* outside/module masks *) (Φ : A → B → PROP) (* post-condition *) . Definition atomic_update `{BiFUpd PROP} {A B : Type} (α : A → PROP) (* atomic pre-condition *) (β : A → B → PROP) (* atomic post-condition *) (Eo Em : coPset) (* outside/module masks *) (Q : A → B → PROP) (* post-condition *) : PROP := tc_opaque ( ∃ (F P : PROP), F ∗ ▷ P ∗ atomic_shift α β Eo Em (▷ P) (λ x y, F -∗ Q x y) Definition atomic_shift (P P' : PROP) : PROP := (□ (∀ E, ⌜Eo ⊆ E⌝ → P ={E, E∖Em}=∗ ∃ x, α x ∗ ((α x ={E∖Em, E}=∗ P') ∧ (∀ y, β x y ={E∖Em, E}=∗ Φ x y))) )%I. Lemma atomic_shift_mono P P1 P2: □ (P1 -∗ P2) -∗ □ (atomic_shift P P1 -∗ atomic_shift P P2). Proof. iIntros "#HP12 !# #AS !# * % HP". iMod ("AS" with "[% //] HP") as (x) "[Hα Hclose]". iModIntro. iExists x. iFrame "Hα". iSplit. - iIntros "Hα". iDestruct "Hclose" as "[Hclose _]". iApply "HP12". iApply "Hclose". done. - iIntros (y) "Hβ". iDestruct "Hclose" as "[_ Hclose]". iApply "Hclose". done. Qed. Definition atomic_update_pre (Ψ : () → PROP) (_ : ()) : PROP := (∃ (P : PROP), ▷ P ∗ atomic_shift (▷ P) (Ψ ()))%I. Local Instance atomic_update_pre_mono : BiMonoPred atomic_update_pre. Proof. constructor. - iIntros (P1 P2) "#HP12". iIntros ([]) "AU". iDestruct "AU" as (P) "[HP AS]". iExists P. iFrame. iApply (atomic_shift_mono with "[HP12]"); last done. iAlways. iApply "HP12". - intros ??. solve_proper. Qed. Definition atomic_update_def := bi_greatest_fixpoint atomic_update_pre (). End definition. (** Seal it *) Definition atomic_update_aux : seal (@atomic_update_def). by eexists. Qed. Definition atomic_update `{BiFUpd PROP} {A B : Type} := atomic_update_aux.(unseal) PROP _ A B. Definition atomic_update_eq : @atomic_update = @atomic_update_def := atomic_update_aux.(seal_eq). (** Lemmas about AU *) Section lemmas. Context `{BiFUpd PROP} {A B : Type}. Implicit Types (α : A → PROP) (β: A → B → PROP) (P : PROP) (Q : A → B → PROP). Implicit Types (α : A → PROP) (β: A → B → PROP) (P : PROP) (Φ : A → B → PROP). Local Existing Instance atomic_update_pre_mono. Lemma aupd_acc α β Eo Em Q E : (** The ellimination form: an accessor *) Lemma aupd_acc α β Eo Em Φ E : Eo ⊆ E → atomic_update α β Eo Em Q -∗ atomic_update α β Eo Em Φ -∗ |={E, E∖Em}=> ∃ x, α x ∗ ((α x ={E∖Em, E}=∗ atomic_update α β Eo Em Q) ∧ (∀ y, β x y ={E∖Em, E}=∗ Q x y)). ((α x ={E∖Em, E}=∗ atomic_update α β Eo Em Φ) ∧ (∀ y, β x y ={E∖Em, E}=∗ Φ x y)). Proof using Type*. rewrite {1}/atomic_update /=. iIntros (HE) "HUpd". iDestruct "HUpd" as (F P) "(HF & HP & #Hshift)". rewrite atomic_update_eq {1}/atomic_update_def /=. iIntros (HE) "HUpd". iPoseProof (greatest_fixpoint_unfold_1 with "HUpd") as "HUpd". iDestruct "HUpd" as (P) "(HP & Hshift)". iMod ("Hshift" with "[% //] HP") as (x) "[Hα Hclose]". iModIntro. iExists x. iFrame "Hα". iSplit. - iIntros "Hα". iDestruct "Hclose" as "[Hclose _]". iMod ("Hclose" with "Hα"). iModIntro. iExists F, P. by iFrame. - iIntros (y) "Hβ". iDestruct "Hclose" as "[_ Hclose]". iMod ("Hclose" with "Hβ") as "HQ". iModIntro. by iApply "HQ". iModIntro. iExists x. iFrame. Qed. Global Instance aupd_laterable α β Eo Em Φ : Laterable (atomic_update α β Eo Em Φ). Proof. rewrite /Laterable atomic_update_eq {1}/atomic_update_def /=. iIntros "AU". iPoseProof (greatest_fixpoint_unfold_1 with "AU") as (P) "[HP #AS]". iExists P. iFrame. iIntros "!# HP !>". iApply greatest_fixpoint_unfold_2. iExists P. iFrame "#∗". Qed. Lemma aupd_intro P α β Eo Em Φ : Em ⊆ Eo → Laterable P → P -∗ □ (P -∗ |={Eo, Eo∖Em}=> ∃ x, α x ∗ ((α x ={Eo∖Em, Eo}=∗ P) ∧ (∀ y, β x y ={Eo∖Em, Eo}=∗ Φ x y))) -∗ atomic_update α β Eo Em Φ. Proof. rewrite atomic_update_eq {1}/atomic_update_def /=. iIntros (??) "HP #AU". iApply (greatest_fixpoint_coind _ (λ _, P)); last done. iIntros "!#" ([]) "HP". iDestruct (laterable with "HP") as (P') "[HP' #HPi]". iExists P'. iFrame. iIntros "!#" (E HE) "HP'". iDestruct ("HPi" with "HP'") as ">HP {HPi}". iApply fupd_mask_frame_diff_open; last iMod ("AU" with "HP") as (x) "[Hα Hclose] {AU}"; [done..|]. iModIntro. iExists x. iFrame. iSplit. - iDestruct "Hclose" as "[Hclose _]". iIntros "Hα". iApply fupd_mask_frame_diff_close; last (by iApply "Hclose"); done. - iDestruct "Hclose" as "[_ Hclose]". iIntros (y) "Hβ". iApply fupd_mask_frame_diff_close; last (by iApply "Hclose"); done. Qed. End lemmas.
 From iris.bi Require Export bi. From iris.proofmode Require Import tactics. Set Default Proof Using "Type". (** The class of laterable assertions *) Class Laterable {PROP : sbi} (P : PROP) := laterable : P -∗ ∃ Q, ▷ Q ∗ □ (▷ Q -∗ ◇ P). Arguments Laterable {_} _%I : simpl never. Arguments laterable {_} _%I {_}. Hint Mode Laterable + ! : typeclass_instances. Section instances. Context {PROP : sbi}. Implicit Type (P : PROP). Global Instance later_laterable P : Laterable (▷ P). Proof. rewrite /Laterable. iIntros "HP". iExists P. iFrame. iIntros "!# HP !>". done. Qed. Global Instance timeless_laterable P : Timeless P → Laterable P. Proof. rewrite /Laterable. iIntros (?) "HP". iExists P%I. iFrame. iSplitR; first by iNext. iIntros "!# >HP !>". done. Qed. (** This lemma is not very useful: It needs a strange assumption about emp, and most of the time intuitionistic propositions can be just kept around anyway and don't need to be "latered". The lemma exists because the fact that it needs the side-condition is interesting; it is not an instance because it won't usually get used. *) Lemma intuitionistic_laterable P : Timeless (PROP:=PROP) emp → Affine P → Persistent P → Laterable P. Proof. rewrite /Laterable. iIntros (???) "#HP". iExists emp%I. iSplitL; first by iNext. iIntros "!# >_". done. Qed. Global Instance sep_laterable P Q : Laterable P → Laterable Q → Laterable (P ∗ Q). Proof. rewrite /Laterable. iIntros (LP LQ) "[HP HQ]". iDestruct (LP with "HP") as (P') "[HP' #HP]". iDestruct (LQ with "HQ") as (Q') "[HQ' #HQ]". iExists (P' ∗ Q')%I. iSplitL; first by iFrame. iIntros "!# [HP' HQ']". iSplitL "HP'". - iApply "HP". done. - iApply "HQ". done. Qed. End instances.
 ... ... @@ -69,9 +69,9 @@ Section proof. ⊤ ⊤ (λ '(v, q) _, v). Proof. iIntros (Φ) "Aupd". wp_let. iMod (aupd_acc with "Aupd") as ((v, q)) "[H↦ [_ Hclose]]"; first solve_ndisj. wp_load. iMod ("Hclose" \$! () with "H↦"). done. iIntros (Q Φ) "? AU". wp_let. iMod (aupd_acc with "AU") as ((v, q)) "[H↦ [_ Hclose]]"; first solve_ndisj. wp_load. iMod ("Hclose" \$! () with "H↦") as "HΦ". by iApply "HΦ". Qed. Lemma primitive_store_spec (l : loc) (e : expr) (w : val) : ... ... @@ -82,9 +82,9 @@ Section proof. ⊤ ⊤ (λ _ _, #()%V). Proof. iIntros (<-%of_to_val Φ) "Aupd". wp_let. wp_proj. wp_proj. iMod (aupd_acc with "Aupd") as (v) "[H↦ [_ Hclose]]"; first solve_ndisj. wp_store. iMod ("Hclose" \$! () with "H↦"). done. iIntros (<-%of_to_val Q Φ) "? AU". wp_let. wp_proj. wp_proj. iMod (aupd_acc with "AU") as (v) "[H↦ [_ Hclose]]"; first solve_ndisj. wp_store. iMod ("Hclose" \$! () with "H↦") as "HΦ". by iApply "HΦ". Qed. Lemma primitive_cas_spec (l : loc) e1 e2 (w1 w2 : val) : ... ... @@ -95,10 +95,10 @@ Section proof. ⊤ ⊤ (λ v _, #(if decide (v = w1) then true else false)%V). Proof. iIntros (<-%of_to_val <-%of_to_val Φ) "Aupd". wp_let. repeat wp_proj. iMod (aupd_acc with "Aupd") as (v) "[H↦ [_ Hclose]]"; first solve_ndisj. iIntros (<-%of_to_val <-%of_to_val Q Φ) "? AU". wp_let. repeat wp_proj. iMod (aupd_acc with "AU") as (v) "[H↦ [_ Hclose]]"; first solve_ndisj. destruct (decide (v = w1)) as [Hv|Hv]; [wp_cas_suc|wp_cas_fail]; iMod ("Hclose" \$! () with "H↦"); done. iMod ("Hclose" \$! () with "H↦") as "HΦ"; by iApply "HΦ". Qed. Definition primitive_atomic_heap : atomic_heap Σ := ... ...
 ... ... @@ -5,7 +5,7 @@ From iris.proofmode Require Import tactics. From iris.heap_lang Require Import proofmode notation atomic_heap par. Set Default Proof Using "Type". (** Show taht implementing fetch-and-add on top of CAS preserves logical (** Show that implementing fetch-and-add on top of CAS preserves logical atomicity. *) (* TODO: Move this to iris-examples once gen_proofmode is merged. *) ... ... @@ -26,32 +26,28 @@ Section increment. ⊤ ⊤ (λ v _, #v). Proof. iIntros (Φ) "AUpd". iLöb as "IH". wp_let. wp_apply load_spec. iIntros (Q Φ) "HQ AU". iLöb as "IH". wp_let. wp_apply (load_spec with "HQ"). (* Prove the atomic shift for load *) iDestruct "AUpd" as (F P) "(HF & HP & #AShft)". iExists F, P. iFrame. iIntros "!#" (E ?) "HP". iMod ("AShft" with "[%] HP") as (x) "[H↦ Hclose]"; first done. iModIntro. iExists (#x, 1%Qp). iFrame. iSplit. { iDestruct "Hclose" as "[Hclose _]". iApply "Hclose". } iIntros (_) "H↦". iDestruct "Hclose" as "[Hclose _]". iMod ("Hclose" with "H↦") as "HP". iIntros "!> HF". clear dependent E. iApply (aupd_intro with "AU"); first done. iIntros "!# AU". iMod (aupd_acc with "AU") as (x) "[H↦ [Hclose _]]"; first solve_ndisj. iModIntro. iExists (#x, 1%Qp). iFrame "H↦". iSplit; first done. iIntros ([]) "H↦". iMod ("Hclose" with "H↦") as "AU". iIntros "!> HQ". (* Now go on *) wp_let. wp_op. wp_bind (aheap.(cas) _)%I. wp_apply cas_spec. wp_apply (cas_spec with "HQ"). (* Prove the atomic shift for CAS *) iExists F, P. iFrame. iIntros "!# * % HP". iMod ("AShft" with "[%] HP") as (x') "[H↦ Hclose]"; first done. iApply (aupd_intro with "AU"); first done. iIntros "!# AU". iMod (aupd_acc with "AU") as (x') "[H↦ Hclose]"; first solve_ndisj. iModIntro. iExists #x'. iFrame. iSplit. { iDestruct "Hclose" as "[Hclose _]". iApply "Hclose". } iIntros (_). destruct (decide (#x' = #x)) as [[= Hx]|Hx]. iIntros ([]). destruct (decide (#x' = #x)) as [[= Hx]|Hx]. - iIntros "H↦". iDestruct "Hclose" as "[_ Hclose]". subst. iMod ("Hclose" \$! () with "H↦") as "HΦ". iIntros "!> HF". iMod ("Hclose" \$! () with "H↦") as "HΦ". iIntros "!> HQ". wp_if. by iApply "HΦ". - iDestruct "Hclose" as "[Hclose _]". iIntros "H↦". iMod ("Hclose" with "H↦") as "HP". iIntros "!> HF". wp_if. iApply "IH". iExists F, P. iFrame. done. iMod ("Hclose" with "H↦") as "AU". iIntros "!> HQ". wp_if. iApply ("IH" with "HQ"). done. Qed. End increment. ... ... @@ -71,13 +67,9 @@ Section increment_client. iMod (inv_alloc nroot _ (∃x':Z, l ↦ #x')%I with "[Hl]") as "#?"; first eauto. (* FIXME: I am only usign persistent stuff, so I should be allowed to move this to the persisten context even without the additional □. *) iAssert (□ atomic_update (λ (v: Z), l ↦ #v) (λ v (_:()), l ↦ #(v + 1)) ⊤ ⊤ (λ _ _, True))%I as "#Aupd". { iAlways. iExists True%I, True%I. repeat (iSplit; first done). clear x. iIntros "!#" (E) "% _". assert (E = ⊤) as -> by set_solver. iAssert (□ WP incr primitive_atomic_heap #l {{ _, True }})%I as "#Aupd". { iAlways. wp_apply (incr_spec with "[]"); first iEmpIntro. clear x. iApply (aupd_intro with "[]"); try iEmpIntro; [done|apply _|]. iIntros "!# _". iInv nroot as (x) ">H↦" "Hclose". iMod fupd_intro_mask' as "Hclose2"; last iModIntro; first set_solver. iExists _. iFrame. iSplit. ... ... @@ -85,10 +77,9 @@ Section increment_client. iNext. iExists _. iFrame. } iIntros (_) "H↦". iMod "Hclose2" as "_". iMod ("Hclose" with "[-]"); last done. iNext. iExists _. iFrame. } wp_apply (wp_par (λ _, True)%I (λ _, True)%I). - wp_apply incr_spec. iAssumption. (* FIXME: without giving the postconditions above, this diverges. *) - wp_apply incr_spec. iAssumption. wp_apply wp_par. - iAssumption. - iAssumption. - iIntros (??) "_ !>". done. Qed. ... ...
 ... ... @@ -10,7 +10,7 @@ Definition atomic_wp `{irisG Λ Σ} {A B : Type} (Eo Em : coPset) (* outside/module masks *) (f: A → B → val Λ) (* Turn the return data into the return value *) : iProp Σ := (∀ Φ, atomic_update α β Eo Em (λ x y, Φ (f x y)) -∗ WP e {{ Φ }})%I. (∀ Q Φ, Q -∗ atomic_update α β Eo Em (λ x y, Q -∗ Φ (f x y)) -∗ WP e {{ Φ }})%I. (* Note: To add a private postcondition, use atomic_update α β Eo Em (λ x y, POST x y -∗ Φ (f x y)) *)
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