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proof_elim_history.v
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proof_elim_history.v
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From gpfsl.examples Require Import sflib.
From iris.algebra Require Import auth gset gmap agree.
From iris.algebra Require Import lib.mono_list.
From iris.proofmode Require Import tactics.
From iris.base_logic.lib Require Import ghost_var ghost_map.
From gpfsl.base_logic Require Import meta_data.
From gpfsl.examples.algebra Require Import mono_list_list saved_vprop.
From gpfsl.logic Require Import logatom atomics invariants.
From gpfsl.logic Require Import repeat_loop new_delete.
From gpfsl.examples Require Import uniq_token map_seq loc_helper.
From gpfsl.examples.omo Require Import omo omo_preds append_only_loc.
From gpfsl.examples.stack Require Import spec_history spec_treiber_composition code_elim.
From gpfsl.examples.exchanger Require Import spec_composition code_split.
Require Import iris.prelude.options.
Class elimG Σ := ElimG {
elim_omoGeneralG :> omoGeneralG Σ;
elim_stackG :> omoSpecificG Σ sevent_hist stack_state;
elim_exG :> omoSpecificG Σ xchg_event xchg_state;
elim_helpingG :> ghost_mapG Σ event_id (gname * view);
elim_uniqTokG :> uniqTokG Σ;
elim_savedvPropG :> savedvPropG Σ;
}.
Definition elimΣ : gFunctors := #[omoGeneralΣ; omoSpecificΣ sevent_hist stack_state; omoSpecificΣ xchg_event xchg_state; ghost_mapΣ event_id (gname * view); uniqTokΣ; savedvPropΣ].
Global Instance subG_elimΣ {Σ} : subG elimΣ Σ → elimG Σ.
Proof. solve_inG. Qed.
Section Elimstack.
Context `{!noprolG Σ, !atomicG Σ, !elimG Σ}.
Local Notation iProp := (iProp Σ).
Local Notation vProp := (vProp Σ).
Local Open Scope nat_scope.
Implicit Types
(γg γs γgt γst γgx γsx γm γh γi γu : gname)
(omo omot : omoT)
(E Et : history sevent_hist) (Ex : history xchg_event).
Hypothesis (ts : stack_spec Σ).
Hypothesis (xc : xchg_spec Σ).
Definition estackN (N : namespace) := N .@ "estack".
Definition tstackN (N : namespace) := N .@ "tstack".
Definition xchgN (N : namespace) := N .@ "xchg".
Definition helpN (N : namespace) := N .@ "helping".
Notation new_stack' := (spec_treiber_composition.new_stack ts).
Notation try_push' := (spec_treiber_composition.try_push ts).
Notation try_pop' := (spec_treiber_composition.try_pop ts).
Notation new_exchanger := (spec_composition.new_exchanger xc).
Notation new_offer := (spec_composition.new_offer xc).
Notation check_offer := (spec_composition.check_offer xc).
Notation take_offer := (spec_composition.take_offer xc).
Definition field_info γg (s lt lx : loc) : vProp :=
∃ (eV0 : omo_event sevent_hist) (q : Qp),
OmoEinfo γg 0%nat eV0 ∗
@{eV0.(sync)} (s ↦{q} #lt ∗ (s >> 1) ↦{q} #lx).
Definition last_state γg γgt (stlist tstlist : list stack_state) : vProp :=
∃ stf tstf,
⌜ last stlist = Some stf ∧ last tstlist = Some tstf ⌝ ∗
[∗ list] st; tst ∈ stf; tstf,
∃ (eV teV : omo_event sevent_hist),
OmoEinfo γg st.1.1.1 eV ∗ OmoEinfo γgt tst.1.1.1 teV ∗
⌜ st.1.1.2 = tst.1.1.2 ∧ st.1.2 = tst.1.2
∧ eV.(sync) = st.1.2 ∧ teV.(sync) = tst.1.2 ∧ eV.(eview) = st.2 ⌝.
Definition EStackInv γg (s : loc) E : vProp := ∃ γs omo stlist, OmoAuth γg γs (1/2)%Qp E omo stlist _.
Definition EStackLocal_inner N γg (s : loc) E M : vProp :=
∃ γgt γst γgx γsx γm γh (lt lx : loc) Mt Mx (eV0 : omo_event sevent_hist),
meta s nroot (γgt,γgx,γst,γsx,γm,γh,lt,lx) ∗
OmoSnapHistory γg E ∗
OmoEview γg M ∗
StackLocal ts (tstackN N) γgt lt Mt ∗
ExchangerLocal xc (xchgN N) γgx lx Mx ∗
OmoEinfo γg 0%nat eV0 ∗ ⊒(eV0.(sync)).
Definition atomic_update_push (N : namespace) γg s (v : Z) (V : view) (M : eView) (Q : bool → vProp) : vProp :=
atomic_update (⊤ ∖ ↑N) (↑histN)
(tele_app (TT:= [tele _]) (λ E, ▷ EStackInv γg s E)%I)
(tele_app (TT:= [tele _])
(λ E, tele_app (TT:= [tele _ _ _ _ _])
(λ (b: bool) V' E' ps M',
(* PUBLIC POST *)
▷ EStackInv γg s E' ∗
⌜ if b then (
(* successful case *)
E' = E ++ [mkOmoEvent ps V' M'] ∧
V ⊑ V' ∧
is_push ps v ∧
M ⊆ M')
else (
(* FAIL_RACE case *)
E' = E ∧ M' = M
) ⌝ ∗
⊒V' ∗ @{V'} EStackLocal_inner N γg s E' M')%I))
(tele_app (TT:= [tele _])
(λ E, tele_app (TT:= [tele _ _ _ _ _])
(λ b V' E' ps M', Q b)%I)).
Definition last_state_xchg N γg s γgx γh (xstlist : list xchg_state) (hmap : gmap event_id (gname * view)) : vProp :=
∃ xstl xstr, ⌜ last xstlist = Some (xstl, xstr) ⌝ ∗
([∗ map] xe↦xinfo ∈ xstl,
∃ (γi γpt γpf γu : gname) (Vi : view) M Q E,
⌜ γi = encode (γpt,γpf,γu) ∧ Vi ⊑ xinfo.1.2 ∧ (0 < xinfo.1.1)%Z ⌝ ∗
@{xinfo.1.2} (atomic_update_push N γg s xinfo.1.1 Vi M Q ∗ EStackLocal_inner N γg s E M) ∗
OmoTokenW γgx xe ∗
saved_vProp_own γpt DfracDiscarded (Q true) ∗ saved_vProp_own γpf DfracDiscarded (Q false) ∗
⎡xe ↪[γh]□ (γi,xinfo.1.2)⎤) ∗
([∗ map] xe↦xinfo ∈ xstr,
∃ (γi γpt γpf γu : gname) (Vb : view) P,
⌜ γi = encode (γpt,γpf,γu) ∧ xinfo.1.1.2 = 0%Z ⌝ ∗
⎡xe ↪[γh]□ (γi,Vb)⎤ ∗ saved_vProp_own γpt DfracDiscarded P ∗
(@{Vb ⊔ xinfo.1.2} P ∨ UTok γu)).
Definition AllEvents γg γs γgt γgx γm E ezfs ezfl : vProp :=
OmoSnap γg γs ezfl [] ∗
[∗ list] e↦eV ∈ E,
OmoLe γg e ezfl ∨
(∃ e' te', OmoLt γg ezfs e' ∗ OmoLe γg e' e ∗ OmoHb γg γg e e' ∗
OmoHb γg γgt e' te' ∗ OmoCW γg γgt e' te' ∗ CWMonoValid γm e').
Definition last_empty_t γg γgt γm ezfs (tes : list event_id) (tstlist : list stack_state) : vProp :=
∃ tezf tgenzf,
OmoCW γg γgt ezfs tezf ∗ CWMonoValid γm ezfs ∗
⌜ tes !! tgenzf = Some tezf ∧ tstlist !! tgenzf = Some [] ∧ (∀ tgen' st', tstlist !! tgen' = Some st' → (tgenzf < tgen')%nat → st' ≠ []) ⌝.
Definition last_empty ezfl (es : list event_id) (stlist : list stack_state) : Prop :=
∃ genzfl,
es !! genzfl = Some ezfl ∧ (∀ i st, stlist !! i = Some st → (genzfl < i)%nat → st ≠ []).
Definition hmap_valid Ex (hmap : gmap event_id (gname * view)) : Prop :=
∀ e, is_Some (hmap !! e) → e < length Ex.
Definition EStackInternalInv (N : namespace) γg s E : vProp :=
∃ (γs γgt γst γgx γsx γm γh : gname) (lt lx : loc) Et Ex omo omot omox stlist tstlist xstlist Mono hmap ezfs ezfl,
meta s nroot (γgt,γgx,γst,γsx,γm,γh,lt,lx) ∗
OmoAuth γg γs (1/2)%Qp E omo stlist _ ∗
StackInv ts γgt γst lt Et omot tstlist ∗
ExchangerInv xc γgx γsx lx Ex omox xstlist ∗
CWMono γg γgt γm Mono ∗
⎡ghost_map_auth γh 1 hmap⎤ ∗
field_info γg s lt lx ∗
last_state γg γgt stlist tstlist ∗
last_state_xchg N γg s γgx γh xstlist hmap ∗
AllEvents γg γs γgt γgx γm E ezfs ezfl ∗
last_empty_t γg γgt γm ezfs (omo_write_op omot) tstlist ∗
⌜ hmap_valid Ex hmap ∧ last_empty ezfl (omo_write_op omo) stlist ⌝.
Definition InternalInv_inner N γg s : vProp := ∃ E, EStackInternalInv N γg s E.
Definition InternInv N γg s := inv (estackN N) (InternalInv_inner N γg s).
Definition EStackLocal N γg s E M : vProp :=
EStackLocal_inner N γg s E M ∗ InternInv N γg s.
Local Tactic Notation "simplify_meta" "with" constr(Hs) :=
iDestruct (meta_agree with Hs) as %[[[[[[[-> ->]%pair_inj ->]%pair_inj ->]%pair_inj ->]%pair_inj ->]%pair_inj ->]%pair_inj ->]%pair_inj.
Lemma field_info_dup γg s lt lx :
field_info γg s lt lx -∗ field_info γg s lt lx ∗ field_info γg s lt lx.
Proof.
iDestruct 1 as (??) "[#ELEM [[s0↦ s0↦'] [s1↦ s1↦']]]".
iSplitL "s0↦ s1↦"; repeat iExists _; iFrame "∗#".
Qed.
Lemma EStackInv_Linearizable_instance :
∀ γg s E, EStackInv γg s E ⊢ ⌜ Linearizability E ⌝.
Proof.
intros. iDestruct 1 as (???) "M●".
iDestruct (OmoAuth_Linearizable with "M●") as %H. by apply omo_compatible in H.
Qed.
Lemma EStackInv_history_wf_instance :
∀ γg s E, EStackInv γg s E ⊢ ⌜ history_wf E ⌝.
Proof.
intros. iDestruct 1 as (???) "M●".
by iDestruct (OmoAuth_wf with "M●") as "[_ %H2]".
Qed.
Lemma EStackInv_StackLocal_instance :
∀ N γg s E E' M',
EStackInv γg s E -∗ EStackLocal N γg s E' M' -∗ ⌜ E' ⊑ E ⌝.
Proof.
intros. iDestruct 1 as (???) "M●".
iDestruct 1 as "[(%&%&%&%&%&%&%&%&%&%&%& _ & E◯ & _) _]".
by iDestruct (OmoAuth_OmoSnapHistory with "M● E◯") as %?.
Qed.
Lemma EStackLocal_lookup_instance :
∀ N γg s E M e V,
sync <$> E !! e = Some V → e ∈ M →
EStackLocal N γg s E M -∗ ⊒V.
Proof.
intros. iDestruct 1 as "[(%&%&%&%&%&%&%&%&%&%&%& _ & E◯ & M◯ & _) _]".
by iApply (OmoSnapHistory_OmoEview with "E◯ M◯").
Qed.
Lemma new_stack_spec :
spec_history.new_stack_spec' (new_estack new_stack' new_exchanger) EStackLocal EStackInv.
Proof.
iIntros (N tid Φ) "_ Post". wp_lam. wp_apply wp_new; [done..|].
iIntros (s) "([s†|%] & s↦ & HM & _)"; [|done]. rewrite own_loc_na_vec_cons own_loc_na_vec_singleton.
iDestruct "s↦" as "[s0↦ s1↦]".
wp_pures. rewrite -[Z.to_nat 0]/(0%nat) shift_0.
iDestruct (view_at_intro emp with "[]") as (V0) "[#⊒V0 _]"; [done|].
wp_apply (new_stack_spec ts (tstackN N) _ V0 with "⊒V0").
iIntros (γgt γst lt Mt V1) "(#⊒V1 & Mt● & #⊒Mt@V1 & WCOMMIT & %LeV0V1)".
wp_write. wp_pures. rewrite -[Z.to_nat 1]/(1%nat).
wp_apply (new_exchanger_spec xc (xchgN N) _ V1 with "⊒V1").
iIntros (γgx γsx lx V2 Mx) "(#⊒V2 & Mx● & #⊒Mx@V2 & _ & %LeV1V2)".
wp_apply wp_fupd. wp_write.
iCombine "s0↦ s1↦" as "s↦".
iDestruct (view_at_intro_incl with "s↦ ⊒V2") as (V3) "(#⊒V3 & %LeV2V3 & s↦@V3)".
set Minit : eView := {[0%nat]}.
set eVinit := mkOmoEvent (stack_event_omo.Init) V3 Minit.
set initst : stack_state := [].
iMod (@OmoAuth_alloc _ _ _ _ _ eVinit initst _ _ stack_interpretable with "WCOMMIT") as (γg γs) "(M● & #⊒M@V2 & #e0↦te0 & #einit✓eVinit & #einit↪emp & _)".
{ eapply stack_step_Init; try done. } { done. }
iDestruct (OmoHbToken_finish with "M●") as "[M● M●']".
iDestruct (OmoSnapHistory_get with "M●") as "#E◯".
iMod (ghost_map_alloc_empty) as (γh) "Hγh".
iMod (CWMono_new γg γgt) as (γm) "MONO".
iMod (meta_set _ _ (γgt,γgx,γst,γsx,γm,γh,lt,lx) nroot with "HM") as "#HM"; [done|].
iDestruct (StackInv_OmoAuth_acc with "Mt●") as "[Mt● Close]".
iMod (CWMono_insert_last_last (wr_event 0) with "MONO M● Mt● e0↦te0") as "(MONO & #MONO✓e0 & M● & Mt●)"; [done|done|].
iDestruct ("Close" with "Mt●") as "Mt●".
iMod (inv_alloc (estackN N) _ (InternalInv_inner N γg s) with "[-Post M●]") as "#SInv".
{ iModIntro. do 20 iExists _. iExists 0,0. iFrame "HM M●' Hγh MONO Mt● Mx●". rewrite !omo_append_w_write_op /=.
iSplitL; last iSplit; last iSplit; last iSplit; last iSplit.
- iExists _,1%Qp. iFrame "einit✓eVinit". subst eVinit. done.
- iExists initst,initst. iSplit; [|done]. done.
- iExists ∅,∅. iSplit; try done. rewrite !big_sepM_empty. done.
- iFrame "einit↪emp". rewrite big_sepL_singleton. iLeft. iApply OmoLe_get_from_Eq. iApply (OmoEq_get_from_Einfo with "einit✓eVinit").
- iExists 0,0. iFrame "e0↦te0 MONO✓e0". iPureIntro. split_and!; try done. intros. destruct tgen'; try done. lia.
- iPureIntro. split.
+ unfold hmap_valid. intros. rewrite lookup_empty in H. destruct H. done.
+ exists 0. split; [done|]. intros. destruct i; try done. lia. }
iDestruct ("Post" $! γg s V3 _ Minit with "[M●]") as "HΦ".
{ iFrame "⊒V3". iSplitL; [repeat iExists _; iFrame|]. iSplit; [done|]. iFrame "SInv".
repeat iExists _. iFrame "HM ⊒M@V2 ⊒Mt@V1 ⊒Mx@V2 einit✓eVinit E◯". subst eVinit. simpl. done. }
iModIntro. by iApply "HΦ".
Qed.
Lemma try_push_spec :
spec_history.try_push_spec' (try_push try_push' new_offer check_offer) EStackLocal EStackInv.
Proof.
intros N DISJ s tid γg E0 M V0 v NZ. iIntros "#⊒V0 #EStackLocal" (Φ) "AU".
iDestruct "EStackLocal" as "[(%&%&%&%&%&%&%&%&%&%&%& HM & #E◯0 & ⊒M & ⊒Mt & ⊒Mx & e0✓eV0 & ⊒eV0) SInv]".
wp_lam. wp_pures. rewrite -[Z.to_nat 0]/(0%nat) shift_0. wp_bind (! _)%E.
(* Inv access 1 *)
iInv "SInv" as (E1) "H" "Hclose".
iDestruct "H" as (γs1 ???????? Et1) "(%Ex1 & %omo1 & %omot1 & %omox1 & %stlist1 & %tstlist1 & %xstlist1 & %Mono1 & %hmap1 & %ezfs1 & %ezfl1 &
>HM' & >M●1 & Mt●1 & Mx●1 & MONO1 & Hγh1 & >Field1 & >LASTst1 & LASTxchg1 & #AllEvents1 & >#LASTemp1 & >[%VALhmap1 %LASTemp1])".
simplify_meta with "HM' HM". iClear "HM'".
iDestruct (field_info_dup with "Field1") as "[Field1 (%&%& #e0✓eV0' & s↦)]".
iDestruct (OmoEinfo_agree with "e0✓eV0 e0✓eV0'") as %<-. iClear "e0✓eV0'".
iDestruct (view_at_elim with "⊒eV0 s↦") as "[s0↦ s1↦]".
iCombine "⊒M ⊒Mt ⊒Mx ⊒eV0" as "⊒INFO".
iDestruct (view_at_intro_incl with "⊒INFO ⊒V0") as (V0') "(⊒V0' & %LeV0V0' & (⊒M@V0' & ⊒Mt@V0' & ⊒Mx@V0' & ⊒eV0@V0'))".
iMod ("Hclose" with "[-AU s0↦ s1↦]") as "_".
{ repeat iExists _. iFrame. iFrame "HM AllEvents1 LASTemp1". done. }
wp_read. iApply fupd_mask_intro_subseteq; [done|]. wp_pures.
(* Inv access 2 *)
have NDISJ' : (tstackN N) ## histN by solve_ndisj.
awp_apply (try_push_spec ts _ NDISJ' lt _ γgt _ V0' v with "⊒V0' ⊒Mt"); [done|].
iInv "SInv" as (E2) "H".
iDestruct "H" as (γs2 ???????? Et2) "(%Ex2 & %omo2 & %omot2 & %omox2 & %stlist2 & %tstlist2 & %xstlist2 & %Mono2 & %hmap2 & %ezfs2 & %ezfl2 &
>HM' & >M●2 & >Mt●2 & Mx●2 & >MONO2 & >Hγh2 & >Field2 & >LASTst2 & LASTxchg2 & #AllEvents2 & >#LASTemp2 & >[%VALhmap2 %LASTemp2])".
simplify_meta with "HM' HM". iClear "HM'".
iDestruct (StackInv_OmoAuth_acc with "Mt●2") as "[Mt●2 Close]".
iDestruct (OmoAuth_wf with "Mt●2") as %[OMO_GOODt2 _].
iDestruct (OmoAuth_omo_nonempty with "Mt●2") as %Nomot2.
have NZomot2 : length omot2 ≠ 0 by destruct omot2.
eapply omo_stlist_length in OMO_GOODt2 as EQlentST2.
iDestruct ("Close" with "Mt●2") as "Mt●2".
iAaccIntro with "Mt●2".
{ (* Peeking case *)
iIntros "Mt●2". iModIntro. iFrame "AU s0↦ s1↦". repeat iExists _. iFrame. iFrame "HM AllEvents2 LASTemp2". done. }
iIntros (b2 V2 Et2' omot2' tstlist2' Mt2') "(#⊒V2 & >Mt●2 & #⊒Mt2'@V2 & CASE)".
destruct b2.
{ (* Push on tstack success: commit here *)
iDestruct "CASE" as "((%HEt2' & %Homot2' & [%tst2 %Htstlist2'] & %LeV0'V2 & %SubMtMt2') & #MAXgenMt & WCOMMIT)".
iMod "AU" as (?) "[>(%&%&%& M●2') [_ Commit]]".
iDestruct (OmoAuth_agree with "M●2 M●2'") as %(<- & <- & <- & <-).
iCombine "M●2 M●2'" as "M●2".
iDestruct "LASTemp2" as (tezf2 tgenzf2) "(#ezfs2↦tezf2 & #MONO✓ezfs2 & (%Htezf2 & %Htgenzf2st & %LASTemp))".
iDestruct (OmoAuth_OmoCW_l with "M●2 ezfs2↦tezf2") as %[eVzfs2 Hezfs2].
iDestruct "LASTst2" as (stf2 tstf2) "[[%LASTstf2 %LASTtstf2] BIG2]".
iDestruct (StackInv_OmoAuth_acc with "Mt●2") as "[Mt●2 Close]".
iDestruct (OmoAuth_wf with "Mt●2") as %[OMO_GOODt2' _].
iDestruct (OmoEinfo_get _ _ _ _ _ _ (length Et2) with "Mt●2") as "#ten2✓teV2"; [by rewrite HEt2' lookup_app_1_eq|].
iDestruct ("Close" with "Mt●2") as "Mt●2".
eapply (omo_write_op_step _ _ _ (length omot2 - 1) (length Et2)) in OMO_GOODt2' as STEP; last first.
{ rewrite Nat.sub_add; [|lia]. rewrite EQlentST2 Htstlist2' lookup_app_1_eq. done. }
{ rewrite last_lookup in LASTtstf2. replace (Init.Nat.pred (length tstlist2)) with (length tstlist2 - 1) in LASTtstf2 by lia.
rewrite Htstlist2' EQlentST2 lookup_app_1_ne; [done|]. lia. }
{ by rewrite HEt2' lookup_app_1_eq. }
{ rewrite Nat.sub_add; [|lia]. rewrite Homot2' omo_append_w_write_op omo_write_op_length lookup_app_1_eq. done. }
inversion STEP; [|inversion POP|inversion EMPPOP|inversion INIT]. inversion PUSH. subst v0. clear PUSH. subst u uV stk.
have LeV0V2 : V0 ⊑ V2 by solve_lat.
set psId := length E2.
set M' := {[psId]} ∪ M.
set eVpush := mkOmoEvent (Push v) V2 M'.
set E2' := E2 ++ [eVpush].
set nst : stack_state := (psId, v, V2, M') :: stf2.
iDestruct (view_at_mono_2 _ V2 with "⊒M@V0'") as "⊒M@V2"; [done|].
iMod (OmoAuth_insert_last with "M●2 ⊒M@V2 WCOMMIT []") as "(M●2 & #⊒M'@V2 & #psId↦ten2 & #psId✓eVpush & _)".
{ have ? : step psId eVpush stf2 nst; [|done].
apply stack_step_Push; try done. set_solver. }
iMod (OmoHb_new_1 with "M●2 psId✓eVpush ten2✓teV2") as "[M●2 #psId⊒ten2]"; [done|].
iMod (OmoHb_new_1 with "M●2 psId✓eVpush psId✓eVpush") as "[M●2 #psId⊒psId]"; [done|].
iDestruct (OmoHbToken_finish with "M●2") as "[M●2 M●2']".
iDestruct (OmoSnapHistory_get with "M●2") as "#E◯2".
iDestruct (StackInv_OmoAuth_acc with "Mt●2") as "[Mt●2 Close]". rewrite Homot2'.
iMod (CWMono_insert_last_last (wr_event (length omo2)) with "MONO2 M●2 Mt●2 psId↦ten2") as "(MONO2 & #MONO✓psId & M●2 & Mt●2)".
{ rewrite lookup_omo_wr_event omo_append_w_write_op omo_write_op_length lookup_app_1_eq. done. } { rewrite omo_append_w_length /=. lia. }
iDestruct ("Close" with "Mt●2") as "Mt●2".
iMod ("Commit" $! true V2 E2' (Push v) M' with "[M●2']") as "HΦ".
{ iFrame "⊒V2". iSplitL "M●2'"; [by repeat iExists _|]. iSplit.
- iPureIntro. split_and!; try done. set_solver.
- iSplit; [|by iFrame "SInv"]. repeat iExists _. iFrame "HM E◯2 ⊒eV0@V0' ⊒M'@V2 ⊒Mt@V0' ⊒Mx@V0' e0✓eV0". done. }
iDestruct (OmoEinfo_get _ _ _ _ _ _ ezfs2 with "M●2") as "#ezfs2✓eVzfs2".
{ rewrite lookup_app_1_ne; [done|]. apply lookup_lt_Some in Hezfs2. lia. }
iDestruct (OmoLt_get_append_w with "M●2 ezfs2✓eVzfs2") as "#ezfs2<psId"; [by apply lookup_lt_Some in Hezfs2; lia|].
iModIntro. iSplitR "HΦ".
{ repeat iExists _. iFrame. iFrame "HM". iSplitL "BIG2"; last iSplit; last iSplit.
- iModIntro. iExists nst,tst2. rewrite Htstlist2'. iSplit.
+ rewrite !last_app /=. done.
+ subst nst tst2. rewrite big_sepL2_cons. iFrame "BIG2". repeat iExists _. iFrame "psId✓eVpush ten2✓teV2". done.
- iModIntro. rewrite /AllEvents big_sepL_snoc. iDestruct "AllEvents2" as "[ezfl2↪emp BIG]".
iFrame "ezfl2↪emp BIG". subst eVpush. simpl. iRight.
iExists psId,(length Et2). iDestruct (OmoEq_get_from_Einfo with "psId✓eVpush") as "#psId=psId".
iDestruct (OmoLe_get_from_Eq with "psId=psId") as "psId≤psId".
iFrame "ezfs2<psId psId≤psId psId⊒psId psId⊒ten2 psId↦ten2 MONO✓psId".
- iModIntro. iExists tezf2,tgenzf2. iFrame "ezfs2↦tezf2 MONO✓ezfs2". iPureIntro. split_and!.
+ rewrite omo_append_w_write_op lookup_app_1_ne; [done|]. apply lookup_lt_Some in Htezf2. lia.
+ rewrite Htstlist2' lookup_app_1_ne; [done|]. apply lookup_lt_Some in Htgenzf2st. lia.
+ intros. subst tstlist2'. destruct (decide (tgen' = length tstlist2)) as [->|NEQ].
* rewrite lookup_app_1_eq in H. inversion H. subst st'. subst tst2. done.
* rewrite lookup_app_1_ne in H; [|done]. by eapply LASTemp.
- iPureIntro. split; [done|]. rewrite omo_append_w_write_op. destruct LASTemp2 as (genzfl2 & Hgenzfl2 & H).
exists genzfl2. split.
+ rewrite lookup_app_1_ne; [done|]. apply lookup_lt_Some in Hgenzfl2. lia.
+ intros. destruct (decide (i = length stlist2)) as [->|NEQ].
* rewrite lookup_app_1_eq in H0. inversion H0. subst nst. done.
* rewrite lookup_app_1_ne in H0; [|done]. eapply H; try done. }
iIntros "_". wp_pures. by iApply "HΦ". }
(* Push on tstack failed, use exchanger *)
iDestruct "CASE" as %(-> & -> & -> & ->).
iModIntro. iSplitR "AU s0↦ s1↦". { repeat iExists _. iFrame. iFrame "HM AllEvents2 LASTemp2". done. }
iIntros "_". wp_pures. rewrite -[Z.to_nat 1]/(1%nat). wp_read. wp_pures.
iDestruct (view_at_intro_incl with "AU ⊒V0'") as (V0'') "(#⊒V0'' & %LeV0'V0'' & AU@V0'')".
(* Inv access 3 *)
have NDISJ'' : (xchgN N) ## histN by solve_ndisj.
awp_apply (new_offer_spec xc _ NDISJ'' lx _ γgx _ V0'' v with "⊒V0'' ⊒Mx"); [lia|].
iInv "SInv" as (E3) "H".
iDestruct "H" as (γs3 ???????? Et3) "(%Ex3 & %omo3 & %omot3 & %omox3 & %stlist3 & %tstlist3 & %xstlist3 & %Mono3 & %hmap3 & %ezfs3 & %ezfl3 &
>HM' & >M●3 & >Mt●3 & >Mx●3 & >MONO3 & >Hγh3 & >Field3 & >LASTst3 & LASTxchg3 & #AllEvents3 & >#LASTemp3 & >[%VALhmap3 %LASTemp3])".
simplify_meta with "HM' HM". iClear "HM'".
iDestruct (ExchangerInv_OmoAuth_acc with "Mx●3") as "[Mx●3 Close]".
iDestruct (OmoAuth_wf with "Mx●3") as %[OMO_GOODx3 _].
iDestruct (OmoAuth_omo_nonempty with "Mx●3") as %Nomox3.
have NZomox3 : length omox3 ≠ 0 by destruct omox3.
eapply omo_stlist_length in OMO_GOODx3 as EQlenxST3.
iDestruct ("Close" with "Mx●3") as "Mx●3".
iAaccIntro with "Mx●3".
{ (* Peeking case *)
iIntros "Mx●3". iModIntro. iFrame "AU@V0'' s0↦ s1↦ ". repeat iExists _. iFrame. iFrame "HM AllEvents3 LASTemp3". done. }
iIntros (v3 V3 Ex3' omox3' xstlist3' Mx3) "(#⊒V3 & >Mx●3 & #⊒Mx3@V3 & CASE)".
destruct (decide (v3 = 0%Z)) as [->|NEQ].
{ (* [new_offer] failed, commit nothing *)
iDestruct "CASE" as %(-> & -> & -> & ->). iDestruct (view_at_elim with "⊒V0'' AU@V0''") as "AU".
iMod "AU" as (E3'') "[M●3' [_ Commit]]". iDestruct "M●3'" as (???) ">M●3'".
iDestruct (OmoSnapHistory_get with "M●3'") as "#E◯3'".
iMod ("Commit" $! false V0' E3'' dummy_sevt_hist M with "[M●3']") as "HΦ".
{ iFrame "⊒V0'". iSplitL "M●3'"; [repeat iExists _; iFrame "M●3'"|]. iSplit; [done|].
iSplit; [|by iFrame "SInv"]. repeat iExists _. iFrame "HM E◯3' ⊒M@V0' ⊒Mt@V0' ⊒Mx@V0' e0✓eV0 ⊒eV0@V0'". }
iModIntro. iSplitR "HΦ". { repeat iExists _. iFrame. iFrame "HM AllEvents3 LASTemp3". done. }
iIntros "_". wp_pures. by iApply "HΦ". }
(* [new_offer] succeeded, register helping *)
iDestruct "CASE" as "[([%l3 %EQv3] & %LeV2'V3 & %Homox3' & [%xst3 %Hxstlist3'] & %HEx3' & %SubMxMx3) WCOMMIT]".
iAssert (@{V3} atomic_update_push N γg s v V0 M (λ b, Φ #b))%I with "[AU@V0'']" as "AU@V3".
{ iDestruct (view_at_objective_iff _ V0'' with "SInv") as "SInv@V0''".
iCombine "SInv@V0'' AU@V0''" as "RES".
iApply (view_at_mono with "RES"); [done|]. iIntros "[#SInv AU]".
rewrite /atomic_update_push. iAuIntro. iApply (aacc_aupd with "AU"); [done|].
iIntros (E) "H". iAaccIntro with "H".
{ iIntros "H !>". iFrame. iIntros "AU !>". iFrame. }
iIntros (b V' E' ps M') "(H & %CASE & ⊒V' & LocalInner)". iRight. iModIntro.
iExists b,V',E',ps,M'. iFrame. iFrame "SInv". iSplit; [destruct b; des; done|]. iIntros "HΦ". iModIntro. by iApply "HΦ". }
(* Obtain latest state and corresponding information of exchanger *)
iDestruct (ExchangerInv_OmoAuth_acc with "Mx●3") as "[Mx●3 Close]".
iDestruct (OmoAuth_wf with "Mx●3") as %[OMO_GOODx3' _].
iDestruct ("Close" with "Mx●3") as "Mx●3".
iDestruct "LASTxchg3" as (xstl3 xstr3) "(>%LASTxst3 & BIGxstl3 & BIGxstr3)".
eapply (omo_write_op_step _ _ _ (length omox3 - 1) (length Ex3)) in OMO_GOODx3' as STEP; last first.
{ rewrite Nat.sub_add; [|lia]. rewrite EQlenxST3 Hxstlist3' lookup_app_1_eq. done. }
{ rewrite last_lookup in LASTxst3. replace (Init.Nat.pred (length xstlist3)) with (length xstlist3 - 1) in LASTxst3 by lia.
rewrite Hxstlist3' EQlenxST3 lookup_app_1_ne; [done|]. lia. }
{ by rewrite HEx3' lookup_app_1_eq. }
{ rewrite Nat.sub_add; [|lia]. rewrite Homox3' omo_append_w_write_op omo_write_op_length lookup_app_1_eq. done. }
inversion STEP; [inversion INIT|..|inversion ACK|inversion RVK|inversion ACP].
inversion REG. subst e eV stl str v0. clear STEP REG. rename H1 into Hxst3.
iMod UTok_alloc as (γu) "Hγu".
iMod (saved_vProp_alloc DfracDiscarded (Φ #true)) as (γpt) "#Hγpt"; [done|].
iMod (saved_vProp_alloc DfracDiscarded (Φ #false)) as (γpf) "#Hγpf"; [done|].
remember (encode (γpt,γpf,γu)) as γi eqn:Hγi.
iMod (ghost_map_insert_persist (length Ex3) (γi,V3) with "Hγh3") as "[Hγh3 #KEY]".
{ destruct (hmap3 !! length Ex3) eqn:Heq; try done.
have LOOKUP : is_Some (hmap3 !! length Ex3) by done. apply VALhmap3 in LOOKUP. lia. }
iModIntro. iSplitR "Hγu".
{ repeat iExists _. iFrame. iFrame "HM AllEvents3 LASTemp3". iModIntro. iSplit.
- repeat iExists _. iSplit; [by rewrite Hxstlist3' -Hxst3 last_app|].
rewrite big_sepM_insert; [|done]. iFrame "BIGxstr3 BIGxstl3".
iExists γi,γpt,γpf,γu,V0,M,(λ b, Φ #b),E0. simpl. iFrame "AU@V3 WCOMMIT Hγpt Hγpf KEY". iSplit.
+ iPureIntro. split_and!; try done. solve_lat.
+ repeat iExists _. iFrame "HM E◯0 ⊒M@V0' ⊒Mt@V0' ⊒Mx@V0' e0✓eV0 ⊒eV0@V0'".
- iPureIntro. split; [|done]. unfold hmap_valid in *. intros. destruct H as [γ He]. destruct (decide (e = length Ex3)) as [->|NEe].
+ subst Ex3'. rewrite app_length /=. lia.
+ have LOOKUP : is_Some (hmap3 !! e) by rewrite lookup_insert_ne in He.
apply VALhmap3 in LOOKUP. subst Ex3'. rewrite app_length /=. lia. }
subst v3. iIntros "(%&%& Token)". inversion H. subst l'. clear H. wp_pures.
(* Inv access 4 *)
awp_apply (check_offer_spec xc _ NDISJ'' lx l3 _ γgx _ V3 with "⊒V3 ⊒Mx Token").
iInv "SInv" as (E4) "H".
iDestruct "H" as (γs4 ???????? Et4) "(%Ex4 & %omo4 & %omot4 & %omox4 & %stlist4 & %tstlist4 & %xstlist4 & %Mono4 & %hmap4 & %ezfs4 & %ezfl4 &
>HM' & >M●4 & >Mt●4 & >Mx●4 & >MONO4 & >Hγh4 & >Field4 & >LASTst4 & LASTxchg4 & #AllEvents4 & >#LASTemp4 & >[%VALhmap4 %LASTemp4])".
simplify_meta with "HM' HM". iClear "HM'".
iDestruct (ExchangerInv_OmoAuth_acc with "Mx●4") as "[Mx●4 Close]".
iDestruct (OmoAuth_wf with "Mx●4") as %[OMO_GOODx4 _].
iDestruct (OmoAuth_omo_nonempty with "Mx●4") as %Nomox4.
have NZomox4 : length omox4 ≠ 0 by destruct omox4.
eapply omo_stlist_length in OMO_GOODx4 as EQlenxST4.
iDestruct ("Close" with "Mx●4") as "Mx●4".
iAaccIntro with "[Mx●4 LASTxchg4]".
{ instantiate (1 := [tele_arg γsx; Ex4; omox4; xstlist4; (last_state_xchg N γg s γgx γh xstlist4 hmap4)]). iFrame "Mx●4 LASTxchg4". }
{ (* Peeking case *)
iIntros "[Mx●4 LASTxchg4]". iModIntro. iFrame "Hγu". repeat iExists _. iFrame. iFrame "HM AllEvents4 LASTemp4". done. }
iIntros (v4 V4 xev4 omox4' xstlist4' Mx4) "(LASTxchg4 & #⊒V4 & >Mx●4 & #⊒Mx4@V4 & [%LeV3V4 %SubMxMx4] & CASE)".
iDestruct (ExchangerInv_OmoAuth_acc with "Mx●4") as "[Mx●4 Close]".
iDestruct (OmoAuth_wf with "Mx●4") as %[OMO_GOODx4' _].
iDestruct ("Close" with "Mx●4") as "Mx●4".
iDestruct "LASTxchg4" as (xstl4 xstr4) "(%LASTxst4 & BIGxstl4 & BIGxstr4)".
destruct (decide (v4 = (-1)%Z)) as [->|NEQv4].
{ (* revoked my offer, commit nothing *)
iDestruct "CASE" as "[(-> & %Homox4' & [%xst4 %Hxstlist4']) _]".
eapply (omo_write_op_step _ _ _ (length omox4 - 1) (length Ex4)) in OMO_GOODx4' as STEP; last first.
{ rewrite Nat.sub_add; [|lia]. rewrite EQlenxST4 Hxstlist4' lookup_app_1_eq. done. }
{ rewrite last_lookup in LASTxst4. replace (Init.Nat.pred (length xstlist4)) with (length xstlist4 - 1) in LASTxst4 by lia.
rewrite Hxstlist4' EQlenxST4 lookup_app_1_ne; [done|]. lia. }
{ by rewrite lookup_app_1_eq. }
{ rewrite Nat.sub_add; [|lia]. rewrite Homox4' omo_append_w_write_op omo_write_op_length lookup_app_1_eq. done. }
inversion STEP; [inversion INIT|inversion REG|inversion ACK|inversion RVK|inversion ACP].
subst e eV stl str e'. rename H1 into Hxst4.
destruct ACKS as [xinfo4 Hxinfo4].
iDestruct (big_sepM_delete with "BIGxstl4") as "[H BIGxstl4]"; [done|].
iDestruct "H" as (????????) "((%Hγi' & %LeVixinfo4 & %GZ) & [AU #⊒M0] & WCOMMIT & #Hγpt' & #Hγpf' & #KEY')".
iDestruct (ghost_map_elem_agree with "KEY KEY'") as %H. inversion H. rewrite -H1 in Hγi'. encode_agree Hγi. subst γi0 V3. clear H.
iDestruct (view_at_elim with "⊒V3 AU") as "AU".
iMod "AU" as (?) "[>M● [_ Commit]]". iDestruct "M●" as (???) "M●".
iDestruct (OmoSnapHistory_get with "M●") as "#E◯".
iMod ("Commit" $! false V4 x dummy_sevt_hist M0 with "[M●]") as "HΦ".
{ iFrame "⊒V4". iSplitL; [repeat iExists _; iFrame|]. iSplit; [done|].
iDestruct (view_at_objective_iff _ xinfo4.1.2 with "E◯") as "E◯'".
iCombine "⊒M0 E◯'" as "RES". iApply (view_at_mono with "RES"); [done|]. iIntros "[#⊒M #E◯]".
iDestruct "⊒M" as (??????????) "(%& H1 & H2 & H3 & H4 & H5 & H6 & H7)". repeat iExists _. iFrame "E◯". iFrame "#". }
iModIntro. iSplitR "HΦ".
{ repeat iExists _. iFrame. iFrame "HM AllEvents4 LASTemp4". iModIntro. iSplit.
- repeat iExists _. iFrame "BIGxstr4 BIGxstl4". rewrite Hxstlist4' Hxst4 last_app /=. done.
- iPureIntro. split; [|done]. unfold hmap_valid in *. intros. apply VALhmap4 in H. rewrite app_length /=. lia. }
iIntros "_". iDestruct (saved_vProp_agree with "Hγpf Hγpf'") as "EQ". wp_pures. iRewrite "EQ". done. }
(* Exchange success, already committed by Pop thread *)
iDestruct "CASE" as "[(_ & -> & -> & %Homox4') _]".
have [[xe xes] Hxgen4] : is_Some (omox4 !! (length omox4 - 1)) by rewrite lookup_lt_is_Some; lia.
eapply (omo_read_op_step _ _ _ (length omox4 - 1) (length xes) (length Ex4)) in OMO_GOODx4' as STEP; last first.
{ rewrite last_lookup -EQlenxST4 in LASTxst4. replace (Init.Nat.pred (length omox4)) with (length omox4 - 1) in LASTxst4 by lia. done. }
{ rewrite lookup_app_1_eq. done. }
{ rewrite lookup_omo_ro_event Homox4' omo_insert_r_read_op list_lookup_alter /omo_read_op list_lookup_fmap Hxgen4 /= lookup_app_1_eq. done. }
inversion STEP; [inversion INIT|inversion REG|inversion ACK|inversion RVK|inversion ACP].
subst e eV stl str e' v0.
iDestruct (big_sepM_lookup_acc with "BIGxstr4") as "[Inner BIGxstr4]"; [exact ACKS|].
iDestruct "Inner" as (??????) "([%Hγi' %EQv4] & #KEY' & #Hγpt' & CASE)". simpl in EQv4. subst v4.
iDestruct (ghost_map_elem_agree with "KEY KEY'") as %H. inversion H. subst γi0 Vb. encode_agree Hγi. clear H.
iDestruct "CASE" as "[HΦ|Hγu']"; [|by iDestruct (UTok_unique with "Hγu Hγu'") as %?]. simpl.
iDestruct ("BIGxstr4" with "[Hγu]") as "BIGxstr4".
{ repeat iExists _. iFrame "KEY Hγpt'". iSplit; [done|]. iRight. done. }
iModIntro. iSplitR "HΦ".
{ repeat iExists _. iFrame. iFrame "HM AllEvents4 LASTemp4". iSplit.
- repeat iExists _. iFrame. done.
- iPureIntro. split; [|done]. unfold hmap_valid in *. intros. apply VALhmap4 in H. rewrite app_length /=. lia. }
iDestruct (saved_vProp_agree with "Hγpt Hγpt'") as "EQ". iIntros "_". wp_pures.
simpl in *.
iDestruct (view_at_mono_2 _ V4 with "HΦ") as "HΦ"; [solve_lat|].
iDestruct (view_at_elim with "⊒V4 HΦ") as "HΦ". iRewrite "EQ". done.
Qed.
Lemma push_spec :
spec_history.push_spec' (push try_push' new_offer check_offer) EStackLocal EStackInv.
Proof.
intros N DISJ s tid γg E0 M V0 v NZ. iIntros "#⊒V0 #EStackLocal" (Φ) "AU".
iLöb as "IH" forall "#". wp_rec. awp_apply (try_push_spec with "⊒V0 EStackLocal"); [done|done|].
iApply (aacc_aupd with "AU"); [done|].
iIntros (E) "M●". iAaccIntro with "M●"; first by eauto with iFrame.
iIntros (b V1 E1 ps M1) "(M● & %CASE & #⊒V1 & #⊒M1@V1)".
iModIntro. destruct b.
- iRight. destruct CASE as (EQE1 & PS & LeV0V1 & SubMM1).
iExists V1,E1,ps,M1. iFrame "⊒V1 M● ⊒M1@V1". iSplit; [done|]. iIntros "HΦ !> _". wp_pures. by iApply "HΦ".
- iLeft. destruct CASE as [-> ->]. iFrame "M●". iIntros "AU !> _". wp_if. iApply ("IH" with "AU"); try done.
Qed.
Lemma try_pop_spec :
spec_history.try_pop_spec' (try_pop try_pop' take_offer) EStackLocal EStackInv.
Proof.
intros N DISJ s tid γg E0 M V0. iIntros "#⊒V0 #EStackLocal" (Φ) "AU".
iDestruct "EStackLocal" as "[(%&%&%&%&%&%&%&%&%&%&%& HM & #E◯0 & ⊒M & ⊒Mt & ⊒Mx & e0✓eV0 & ⊒eV0) SInv]".
wp_lam. wp_pures. rewrite -[Z.to_nat 0]/(0%nat) shift_0. wp_bind (! _)%E.
(* Inv access 1 *)
iInv "SInv" as (E1) "H" "Hclose".
iDestruct "H" as (γs1 ???????? Et1) "(%Ex1 & %omo1 & %omot1 & %omox1 & %stlist1 & %tstlist1 & %xstlist1 & %Mono1 & %hmap1 & %ezfs1 & %ezfl1 &
>HM' & >M●1 & Mt●1 & Mx●1 & MONO1 & Hγh1 & >Field1 & >LASTst1 & LASTxchg1 & #AllEvents1 & >#LASTemp1 & >[%VALhmap1 %LASTemp1])".
simplify_meta with "HM' HM". iClear "HM'".
iDestruct (field_info_dup with "Field1") as "[Field1 (%&%& #e0✓eV0' & s↦)]".
iDestruct (OmoEinfo_agree with "e0✓eV0 e0✓eV0'") as %<-. iClear "e0✓eV0'".
iDestruct (view_at_elim with "⊒eV0 s↦") as "[s0↦ s1↦]".
iDestruct (StackLocal_OmoEview with "⊒Mt") as "⊒Mt'".
iDestruct (OmoEview_update with "M●1 ⊒M ⊒M") as (M') "(M●1 & #⊒M' & M⊢M' & %LeMM')".
iDestruct (OmoEview_update with "M●1 ⊒M' ⊒Mt'") as (Mt1) "(M●1 & #⊒Mt1 & M'⊢Mt1 & %LEMh0Mh1)".
iDestruct (StackLocal_Eview_update with "⊒Mt ⊒Mt1") as "⊒Mt1'".
replace (Mt ∪ Mt1) with Mt1; [|set_solver].
iCombine "⊒M ⊒Mt1' ⊒Mx ⊒eV0" as "⊒INFO".
iDestruct (view_at_intro_incl with "⊒INFO ⊒V0") as (V0') "(⊒V0' & %LeV0V0' & (⊒M@V0' & ⊒Mt1@V0' & ⊒Mx@V0' & ⊒eV0@V0'))".
iMod ("Hclose" with "[-AU s0↦ s1↦ M⊢M' M'⊢Mt1]") as "_".
{ repeat iExists _. iFrame. iFrame "HM AllEvents1 LASTemp1". done. }
wp_read. iApply fupd_mask_intro_subseteq; [done|]. wp_pures.
(* Inv access 2 *)
have NDISJ' : (tstackN N) ## histN by solve_ndisj.
awp_apply (try_pop_spec ts _ NDISJ' lt _ γgt _ V0' with "⊒V0' ⊒Mt1'").
iInv "SInv" as (E2) "H".
iDestruct "H" as (γs2 ???????? Et2) "(%Ex2 & %omo2 & %omot2 & %omox2 & %stlist2 & %tstlist2 & %xstlist2 & %Mono2 & %hmap2 & %ezfs2 & %ezfl2 &
>HM' & >M●2 & >Mt●2 & Mx●2 & >MONO2 & >Hγh2 & >Field2 & >LASTst2 & LASTxchg2 & >#AllEvents2 & >#LASTemp2 & >[%VALhmap2 %LASTemp2])".
simplify_meta with "HM' HM". iClear "HM'".
iDestruct (StackInv_OmoAuth_acc with "Mt●2") as "[Mt●2 Close]".
iDestruct (OmoAuth_wf with "Mt●2") as %[OMO_GOODt2 _].
iDestruct (OmoAuth_omo_nonempty with "Mt●2") as %Nomot2.
have NZomot2 : length omot2 ≠ 0 by destruct omot2.
eapply omo_stlist_length in OMO_GOODt2 as EQlentST2.
iDestruct ("Close" with "Mt●2") as "Mt●2".
iAaccIntro with "Mt●2".
{ (* Peeking case *)
iIntros "Mt●2". iModIntro. iFrame "AU s0↦ s1↦ M⊢M' M'⊢Mt1". repeat iExists _. iFrame. iFrame "HM AllEvents2 LASTemp2". done. }
iIntros (v2 V2 Et2' omot2' tstlist2' Mt2) "(#⊒V2 & >Mt●2 & #⊒Mt2@V2 & CASE)".
destruct (decide (v2 = (-1)%Z)) as [->|NEv2]; last first.
{ (* If there was no race, then commit here, either Pop or EmpPop *)
iDestruct "CASE" as "([%LeV0'V2 %SubMt1Mt2] & #MAXgenMt1 & CASE)".
destruct (decide (v2 = 0%Z)) as [->|NEv2'].
- (* Commit EmpPop *)
iDestruct "CASE" as "[(%HEt2' & -> & (%tgen2 & %Homot2' & %LTtgen2)) RCOMMIT]".
iAssert (OmoUB γg M ezfl2 ∗ OmoSnap γg γs2 ezfl2 [])%I with "[-]" as "[#MAXgenM #ezf2↪emp]".
{ iDestruct "LASTemp2" as (tezf2 tgenzf2) "(#ezfs2↦tezf2 & #MONO✓ezfs2 & (%Htezf2 & %Htgenzf2st & %LASTemp))".
iDestruct "AllEvents2" as "[ezfl2↪emp AllEvents2]". iFrame "ezfl2↪emp".
rewrite {2}/OmoUB. iApply big_sepS_forall. iIntros "%e %IN".
iDestruct (OmoAuth_OmoEview with "M●2 ⊒M") as %CLOSED.
apply CLOSED in IN as He. destruct He as [eV He].
iDestruct (big_sepL_lookup with "AllEvents2") as "[CASE|CASE]"; [exact He|iFrame "CASE"|].
(* Only one contradiction case remains *)
iDestruct "CASE" as (??) "(#ezfs2<e' & #e'≤e & #e⊒e' & #e'⊒te' & #e'↦te' & #MONO✓e')".
iDestruct (OmoHb_HbIncluded with "e⊒e' M⊢M'") as %IN'; [done|].
iDestruct (OmoHb_HbIncluded with "e'⊒te' M'⊢Mt1") as %tIN; [done|].
iDestruct (big_sepS_elem_of with "MAXgenMt1") as "#te'≤ten2"; [exact tIN|].
iDestruct (StackInv_OmoAuth_acc with "Mt●2") as "[Mt●2 Close]".
iDestruct (OmoAuth_wf with "Mt●2") as %[OMO_GOODt2' _].
have [[te tes] Htgen2] : is_Some (omot2 !! tgen2) by rewrite lookup_lt_is_Some.
have Htgen2st : tstlist2 !! tgen2 = Some [].
{ have [st Hst] : is_Some (tstlist2 !! tgen2) by rewrite lookup_lt_is_Some -EQlentST2.
eapply (omo_read_op_step _ _ _ tgen2 (length tes) (length Et2)) in OMO_GOODt2'; last first; [done|..].
- rewrite HEt2' lookup_app_1_eq. done.
- rewrite lookup_omo_ro_event Homot2' omo_insert_r_read_op list_lookup_alter /omo_read_op list_lookup_fmap Htgen2 /= lookup_app_1_eq. done.
- inversion OMO_GOODt2'; try done. subst st. done. }
destruct (le_lt_dec tgen2 tgenzf2) as [LE|LT]; [|by specialize (LASTemp _ _ Htgen2st LT)].
iDestruct (OmoLe_get _ _ _ _ _ _ (ro_event tgen2 (length tes)) (wr_event tgenzf2) with "Mt●2") as "#ten2≤tezf2"; [..|done|].
{ rewrite lookup_omo_ro_event Homot2' omo_insert_r_read_op list_lookup_alter /omo_read_op list_lookup_fmap Htgen2 /= lookup_app_1_eq. done. }
{ rewrite lookup_omo_wr_event Homot2' omo_insert_r_write_op. done. }
iDestruct (OmoLe_trans with "te'≤ten2 ten2≤tezf2") as "te'≤tezf2".
iDestruct (CWMono_acc with "MONO2 MONO✓e' MONO✓ezfs2 e'↦te' ezfs2↦tezf2 te'≤tezf2") as "#e'≤ezfs2".
iDestruct (OmoLe_Lt_contra with "e'≤ezfs2 ezfs2<e'") as %?. done. }
have LeV0V2 : V0 ⊑ V2 by solve_lat.
set ppId := length E2.
set M2 := {[ppId]} ∪ M.
set eVpop := mkOmoEvent EmpPop V2 M2.
set E2' := E2 ++ [eVpop].
iDestruct (view_at_mono_2 _ V2 with "⊒M@V0'") as "⊒M@V2"; [done|].
iMod "AU" as (?) "[M●2' [_ Commit]]". iDestruct "M●2'" as (???) ">M●2'".
iDestruct (OmoAuth_agree with "M●2 M●2'") as %(<- & <- & <- & <-).
iCombine "M●2 M●2'" as "M●2".
iMod (OmoAuth_insert_ro with "M●2 ⊒M@V2 RCOMMIT ezf2↪emp MAXgenM []") as (gen2) "(M●2 & #⊒M2@V2 & #ppId↦ten2 & #ezf2=ppId & _)".
{ have ? : step ppId eVpop [] []; [|done]. apply stack_step_EmpPop; try done. set_solver. }
iDestruct (OmoHbToken_finish with "M●2") as "[M●2 M●2']".
iDestruct (OmoSnapHistory_get with "M●2") as "#E◯2".
iMod ("Commit" $! 0%Z V2 E2' EmpPop M2 with "[M●2']") as "HΦ".
{ iFrame "⊒V2". iSplitL; [repeat iExists _; iFrame "M●2'"|]. iSplit; last iSplit; [|done|].
- iFrame "SInv". repeat iExists _. iFrame "HM E◯2 ⊒M2@V2 ⊒Mt1@V0' ⊒Mx@V0' e0✓eV0 ⊒eV0@V0'".
- iPureIntro. simpl. split_and!; try done. set_solver. }
iModIntro. iSplitR "HΦ".
{ repeat iExists _. iFrame. rewrite Homot2' !omo_insert_r_write_op. iFrame "HM LASTemp2". iSplit; [|done].
iModIntro. rewrite /AllEvents big_sepL_snoc. iDestruct "AllEvents2" as "[ezfl2↪emp AllEvents2]".
iFrame "ezfl2↪emp AllEvents2". iLeft. iApply OmoLe_get_from_Eq. by iApply OmoEq_sym. }
iIntros "_". wp_pures. by iApply "HΦ".
- (* Commit Pop *)
iDestruct "CASE" as "[(%HEt2' & %Homot2' & [%tst2 %Htstlist2']) WCOMMIT]".
iDestruct "LASTemp2" as (tezf2 tgenzf2) "(#ezfs2↦tezf2 & #MONO✓ezfs2 & (%Htezf2 & %Htgenzf2st & %LASTemp))".
iDestruct (OmoAuth_OmoCW_l with "M●2 ezfs2↦tezf2") as %[eVzfs2 Hezfs2].
iDestruct "LASTst2" as (stf2 tstf2) "[[%LASTstf2 %LASTtstf2] BIG2]".
iDestruct (StackInv_OmoAuth_acc with "Mt●2") as "[Mt●2 Close]".
iDestruct (OmoAuth_wf with "Mt●2") as %[OMO_GOODt2' _].
iDestruct (OmoEinfo_get _ _ _ _ _ _ (length Et2) with "Mt●2") as "#ten2✓teV2"; [by rewrite HEt2' lookup_app_1_eq|].
iDestruct ("Close" with "Mt●2") as "Mt●2".
iDestruct (OmoAuth_wf with "M●2") as %[OMO_GOOD2 _].
eapply omo_stlist_length in OMO_GOOD2 as EQlenST2.
eapply (omo_write_op_step _ _ _ (length omot2 - 1) (length Et2)) in OMO_GOODt2' as STEP; last first.
{ rewrite Nat.sub_add; [|lia]. rewrite EQlentST2 Htstlist2' lookup_app_1_eq. done. }
{ rewrite last_lookup in LASTtstf2. replace (Init.Nat.pred (length tstlist2)) with (length tstlist2 - 1) in LASTtstf2 by lia.
rewrite Htstlist2' EQlentST2 lookup_app_1_ne; [done|]. lia. }
{ by rewrite HEt2' lookup_app_1_eq. }
{ rewrite Nat.sub_add; [|lia]. rewrite Homot2' omo_append_w_write_op omo_write_op_length lookup_app_1_eq. done. }
inversion STEP; [inversion PUSH|inversion POP|inversion EMPPOP|inversion INIT]. subst o oV stk v tstf2. rename tst2 into ntstf2.
simpl in *.
destruct stf2 as [|stf2 nstf2].
{ iDestruct (big_sepL2_length with "BIG2") as %H. simpl in H. lia. }
iDestruct (big_sepL2_cons with "BIG2") as "[H BIG2]".
iDestruct (big_sepL2_length with "BIG2") as %EQlenSTs.
iDestruct "H" as (eVf2 teVf2) "(#ef2✓eVf2 & #tef2✓teVf2 & (%EQ1 & %EQ2 & %eVf2OUT & %teVf2OUT & %eVf2LVIEW))".
destruct stf2 as [[[ef2 zf2] Vf2] lVf2]. simpl in *. subst zf2 V lVf2.
iDestruct (view_at_mono_2 _ V2 with "⊒M@V0'") as "⊒M@V2"; [solve_lat|].
iDestruct (OmoEview_insert_new_obj with "⊒M@V2 ef2✓eVf2") as "⊒M2@V2"; [by rewrite eVf2OUT|].
have LeV0V2 : V0 ⊑ V2 by solve_lat.
set ppId := length E2.
set M2 := {[ef2]} ∪ eVf2.(eview) ∪ M.
set M2' := {[ppId]} ∪ M2.
set eVpop := mkOmoEvent (Pop v2) V2 M2'.
set E2' := E2 ++ [eVpop].
iMod "AU" as (?) "[M●2' [_ Commit]]". iDestruct "M●2'" as (???) ">M●2'".
iDestruct (OmoAuth_agree with "M●2 M●2'") as %(<- & <- & <- & <-).
iCombine "M●2 M●2'" as "M●2".
iMod (OmoAuth_insert_last with "M●2 ⊒M2@V2 WCOMMIT []") as "(M●2 & #⊒M2'@V2 & #ppId↦ten2 & #ppId✓eVpop & #ppId↪nstf2 & _)".
{ have ? : step ppId eVpop ((ef2, v2, Vf2, eVf2.(eview)) :: nstf2) nstf2; [|done].
apply stack_step_Pop; try done. set_solver. }
iMod (OmoHb_new_1 with "M●2 ppId✓eVpop ppId✓eVpop") as "[M●2 #ppId⊒ppId]"; [done|].
iMod (OmoHb_new_1 with "M●2 ppId✓eVpop ten2✓teV2") as "[M●2 #ppId⊒ten2]"; [done|].
iDestruct (OmoHbToken_finish with "M●2") as "[M●2 M●2']".
iDestruct (OmoSnapHistory_get with "M●2") as "#E◯2".
iDestruct (StackInv_OmoAuth_acc with "Mt●2") as "[Mt●2 Close]". rewrite Homot2'.
iMod (CWMono_insert_last_last (wr_event (length omo2)) with "MONO2 M●2 Mt●2 ppId↦ten2") as "(MONO2 & #MONO✓ppId & M●2 & Mt●2)".
{ rewrite lookup_omo_wr_event omo_append_w_write_op omo_write_op_length lookup_app_1_eq. done. } { rewrite omo_append_w_length /=. lia. }
iDestruct ("Close" with "Mt●2") as "Mt●2".
iDestruct (OmoEinfo_get _ _ _ _ _ _ ezfs2 with "M●2") as "#ezfs2✓eVzfs2".
{ rewrite lookup_app_1_ne; [done|]. apply lookup_lt_Some in Hezfs2. lia. }
iDestruct (OmoLt_get_append_w with "M●2 ezfs2✓eVzfs2") as "#ezfs2<ppId"; [by apply lookup_lt_Some in Hezfs2; lia|].
iMod ("Commit" $! v2 V2 E2' (Pop v2) M2' with "[M●2']") as "HΦ".
{ iFrame "⊒V2". iSplit; [repeat iExists _; iFrame "M●2'"|]. iSplit; last iSplit; [|done|].
- iFrame "SInv". repeat iExists _. iFrame "HM E◯2 ⊒M2'@V2 ⊒Mt2@V2 ⊒Mx@V0' e0✓eV0 ⊒eV0@V0'".
- iPureIntro. rewrite decide_False; [|lia]. rewrite decide_False; [|lia]. split_and!; try done. set_solver. }
set ezfs2' := match ntstf2 with | [] => ppId | _ => ezfs2 end.
set ezfl2' := match ntstf2 with | [] => ppId | _ => ezfl2 end.
iAssert (AllEvents γg γs2 γgt γgx γm E2' ezfs2' ezfl2' ∗ last_empty_t γg γgt γm ezfs2' (omo_write_op (omo_append_w omot2 (length Et2) [])) tstlist2'
∗ ⌜ last_empty ezfl2' (omo_write_op (omo_append_w omo2 ppId [])) (stlist2 ++ [nstf2]) ⌝)%I
with "[-]" as "(#AllEvents2' & #LASTemp2' & %LASTemp2')".
{ destruct ntstf2; subst ezfs2' ezfl2'; (iSplit; last iSplit).
- destruct nstf2; [|done]. iFrame "ppId↪nstf2".
iApply big_sepL_forall. iIntros "%e %eV %Lookup". destruct (decide (e = length E2)) as [->|NEQ].
+ iLeft. iDestruct (OmoEq_get_from_Einfo with "ppId✓eVpop") as "#ppId=ppId".
iApply OmoLe_get_from_Eq. done.
+ iLeft. iDestruct (OmoEinfo_get with "M●2") as "#ELEM"; [done|]. iApply OmoLe_get_from_Lt.
iApply (OmoLt_get_append_w with "M●2 ELEM"). done.
- iExists (length Et2),(length omot2). iFrame "ppId↦ten2 MONO✓ppId". iPureIntro. split_and!.
+ rewrite omo_append_w_write_op omo_write_op_length lookup_app_1_eq. done.
+ rewrite Htstlist2' EQlentST2 lookup_app_1_eq. done.
+ intros. apply lookup_lt_Some in H. subst tstlist2'. rewrite app_length -EQlentST2 /= in H. lia.
- iPureIntro. rewrite omo_append_w_write_op. destruct nstf2; [|done]. exists (length omo2). split.
+ rewrite omo_write_op_length lookup_app_1_eq. done.
+ intros. apply lookup_lt_Some in H. rewrite app_length -EQlenST2 /= in H. lia.
- subst E2'. iDestruct "AllEvents2" as "[ezfl2↪emp AllEvents2]". iFrame "ezfl2↪emp".
iApply big_sepL_snoc. iFrame "AllEvents2". iRight. iExists ppId,(length Et2).
iDestruct (OmoEq_get_from_Einfo with "ppId✓eVpop") as "ppId=ppId".
iDestruct (OmoLe_get_from_Eq with "ppId=ppId") as "ppId≤ppId".
iFrame "ezfs2<ppId ppId≤ppId ppId⊒ppId ppId⊒ten2 ppId↦ten2 MONO✓ppId".
- iExists tezf2,tgenzf2. iFrame "ezfs2↦tezf2 MONO✓ezfs2". iPureIntro. split_and!.
+ rewrite omo_append_w_write_op lookup_app_1_ne; [done|]. apply lookup_lt_Some in Htezf2. lia.
+ rewrite Htstlist2' lookup_app_1_ne; [done|]. apply lookup_lt_Some in Htgenzf2st. lia.
+ intros. destruct (decide (tgen' = length tstlist2)) as [->|NEQ].
* rewrite Htstlist2' lookup_app_1_eq in H. inversion H. done.
* eapply LASTemp; try done. by rewrite Htstlist2' lookup_app_1_ne in H.
- iPureIntro. rewrite omo_append_w_write_op. destruct LASTemp2 as (genzfl2 & Hezfl2 & LASTemp2).
exists genzfl2. split.
+ rewrite lookup_app_1_ne; [done|]. apply lookup_lt_Some in Hezfl2. lia.
+ intros. destruct (decide (i = length stlist2)) as [->|NEQ].
* rewrite lookup_app_1_eq in H. inversion H. subst st. unfold not. intros. subst nstf2. done.
* rewrite lookup_app_1_ne in H; [|done]. eapply LASTemp2; try done. }
iModIntro. iSplitR "HΦ".
{ iModIntro. repeat iExists _. iFrame. iFrame "HM LASTemp2' AllEvents2'". iSplit; [|done].
repeat iExists _. iFrame "BIG2". subst tstlist2'. rewrite !last_app. done. }
iIntros "_". wp_pures. rewrite bool_decide_false; [|lia]. wp_pures. by iApply "HΦ". }
(* Race happened, try using exchanger *)
iDestruct "CASE" as %(-> & -> & -> & ->).
iModIntro. iSplitR "AU s0↦ s1↦ M⊢M' M'⊢Mt1".
{ repeat iExists _. iFrame. iFrame "HM AllEvents2 LASTemp2". done. }
iIntros "_". wp_pures. rewrite -[Z.to_nat 1]/(1%nat). wp_read. wp_pures.
(* Inv access 3 *)
iDestruct (view_at_intro_incl with "AU ⊒V0'") as (V0'') "(#⊒V0'' & %LeV0'V0'' & AU@V0'')".
have NDISJ'' : (xchgN N) ## histN by solve_ndisj.
awp_apply (take_offer_spec xc _ NDISJ'' lx _ γgx _ V0'' 0%Z with "⊒V0'' ⊒Mx"); [done|].
iInv "SInv" as (E3) "H".
iDestruct "H" as (γs3 ???????? Et3) "(%Ex3 & %omo3 & %omot3 & %omox3 & %stlist3 & %tstlist3 & %xstlist3 & %Mono3 & %hmap3 & %ezfs3 & %ezfl3 &
>HM' & >M●3 & >Mt●3 & >Mx●3 & >MONO3 & >Hγh3 & >Field3 & >#LASTst3 & LASTxchg3 & >#AllEvents3 & >#LASTemp3 & >[%VALhmap3 %LASTemp3])".
simplify_meta with "HM' HM". iClear "HM'".
iDestruct (ExchangerInv_OmoAuth_acc with "Mx●3") as "[Mx●3 Close]".
iDestruct (OmoAuth_wf with "Mx●3") as %[OMO_GOODx3 _].
iDestruct (OmoAuth_omo_nonempty with "Mx●3") as %Nomox3.
have NZomox3 : length omox3 ≠ 0 by destruct omox3.
eapply omo_stlist_length in OMO_GOODx3 as EQlenxST3.
iDestruct ("Close" with "Mx●3") as "Mx●3".
iAaccIntro with "[Mx●3 LASTxchg3]".
{ instantiate (1 := [tele_arg γsx; Ex3; omox3; xstlist3; (last_state_xchg N γg s γgx γh xstlist3 hmap3)]). iFrame "Mx●3 LASTxchg3". }
{ (* Peeking case *)
iIntros "[Mx●3 LASTxchg3]". iModIntro. iFrame "AU@V0'' s0↦ s1↦ M⊢M' M'⊢Mt1". repeat iExists _. iFrame. iFrame "HM AllEvents3 LASTemp3 LASTst3". done. }
iIntros (v3 V3 Ex3' omox3' xstlist3' Mx3) "(LASTxchg3 & #⊒V3 & >Mx●3 & #⊒Mx3@V3 & [%LeV0''V3 %SubMxMx3] & CASE)".
destruct (decide (v3 = (-1)%Z)) as [->|NEQv3].
{ (* [take_offer] failed. [try_pop] also fails. Commit nothing *)
iDestruct "CASE" as %(-> & -> & -> & ->).
iDestruct (view_at_elim with "⊒V0'' AU@V0''") as "AU".
iMod "AU" as (?) "[>M●3' [_ Commit]]". iDestruct "M●3'" as (???) "M●3'".
iDestruct (OmoSnapHistory_get with "M●3'") as "#E◯3'".
iMod ("Commit" $! (-1)%Z V0' x dummy_sevt_hist M with "[M●3']") as "HΦ".
{ iFrame "⊒V0'". iSplitL; [repeat iExists _; iFrame|]. iSplit; last iSplit; [|done|done].
iFrame "SInv". repeat iExists _. iFrame "HM ⊒M@V0' ⊒Mx@V0' ⊒Mt1@V0' E◯3' e0✓eV0 ⊒eV0@V0'". }
iModIntro. iSplitR "HΦ". { repeat iExists _. iFrame. iFrame "HM AllEvents3 LASTemp3 LASTst3". done. }
iIntros "_". wp_pures. by iApply "HΦ". }
(* [take_offer] succeeded. Commit Pop *)
iDestruct "CASE" as "[(%HEx3' & %Homox3' & [%xst3 %Hxstlist3']) WCOMMIT]".
iDestruct (ExchangerInv_OmoAuth_acc with "Mx●3") as "[Mx●3 Close]".
iDestruct (OmoAuth_wf with "Mx●3") as %[OMO_GOODx3' _].
iDestruct ("Close" with "Mx●3") as "Mx●3".
iDestruct "LASTxchg3" as (xstl3 xstr3) "(%LASTxst3 & BIGxstl3 & BIGxstr3)".
eapply (omo_write_op_step _ _ _ (length omox3 - 1) (length Ex3)) in OMO_GOODx3' as STEP; last first.
{ rewrite Nat.sub_add; [|lia]. rewrite EQlenxST3 Hxstlist3' lookup_app_1_eq. done. }
{ rewrite last_lookup in LASTxst3. replace (Init.Nat.pred (length xstlist3)) with (length xstlist3 - 1) in LASTxst3 by lia.
rewrite Hxstlist3' EQlenxST3 lookup_app_1_ne; [done|]. lia. }
{ by rewrite HEx3' lookup_app_1_eq. }
{ rewrite Nat.sub_add; [|lia]. rewrite Homox3' omo_append_w_write_op omo_write_op_length lookup_app_1_eq. done. }
inversion STEP; [inversion INIT|inversion REG|inversion ACK|inversion RVK|inversion ACP].
subst e eV stl str v' v. rename H1 into Hxst3.
iDestruct "LASTemp3" as (tezf3 tgenzf3) "(#ezfs3↦tezf3 & #MONO✓ezfs3 & (%Htezf3 & %Htgenzf3st & %LASTemp))".
iDestruct "AllEvents3" as "[ezfl3↪emp AllEvents3]".
iDestruct (big_sepM_delete with "BIGxstl3") as "[H BIGxstl3]"; [exact ACKS|].
iDestruct "H" as (????????) "((%Hγi & %LeViV' & %NZv3) & AU' & WCOMMIT' & #Hγpt & #Hγpf & #KEY)".
iAssert (|={⊤ ∖ ↑(xchgN N) ∖ ↑(estackN N)}=> @{V' ⊔ V3} (∃ E3' omo3' stlist3' γs3' ezfl3',
Q true ∗ (emp -∗ Φ #v3) ∗ OmoAuth γg γs3' (1/2)%Qp E3' omo3' stlist3' stack_interpretable ∗
AllEvents γg γs3' γgt γgx γm E3' ezfs3 ezfl3' ∗ last_state γg γgt stlist3' tstlist3 ∗ ⌜ last_empty ezfl3' (omo_write_op omo3') stlist3' ⌝))%I
with "[M●3 AU@V0'' AU' WCOMMIT WCOMMIT']" as ">(%&%&%&%&%& HQ & HΦ & M●3 & #AllEvents3' & #LASTst3' & %LASTemp3')".
{ iDestruct "AU'" as "[AU' ⊒M0]".
iCombine "⊒M@V0' MONO✓ezfs3 M●3 ⊒M0 WCOMMIT WCOMMIT' AllEvents3 ezfl3↪emp HM SInv LASTst3" as "RES".
iDestruct (view_at_objective_iff _ (V' ⊔ V3) with "RES") as "RES".
iDestruct (view_at_mono_2 _ (V' ⊔ V3) with "AU@V0''") as "AU@V"; [solve_lat|].
iDestruct (view_at_mono_2 _ (V' ⊔ V3) with "AU'") as "AU'@V"; [solve_lat|].
iAssert (@{V' ⊔ V3} ⊒(V' ⊔ V3))%I with "[]" as "#⊒V@V"; [done|].
iCombine "RES AU@V AU'@V ⊒V@V" as "RES".
rewrite -view_at_fupd. iApply (view_at_mono with "RES"); [done|].
iIntros "((#⊒M@V0' & #MONO✓ezfs3 & M●3 & #⊒M0@V' & WCOMMIT2 & WCOMMIT1 & #AllEvents3 & #ezfl3↪emp & #HM & #SInv & #LASTst3) & AU2 & AU1 & #⊒V3')".
set V3' := V' ⊔ V3. clear eV0.
iDestruct (view_at_mono_2 _ V3' with "⊒M@V0'") as "⊒M@V3'"; [solve_lat|].
iDestruct (view_at_mono_2 _ V3' with "⊒M0@V'") as "⊒M0@V3'"; [solve_lat|].
iDestruct "⊒M0@V3'" as (??????????) "(%eV0 & HM' & _ & ⊒M0@V3' & ⊒Mt@V3' & ⊒Mx@V3' & #e0✓eV0 & ⊒eV0@V3')".
rewrite !view_at_objective_iff. simplify_meta with "HM' HM". iClear "HM'".
iDestruct (OmoEview_union_obj with "⊒M@V3' ⊒M0@V3'") as "⊒M3@V3'".
set M3 := M ∪ M0.
(* Pick latest observed event (e3) *)
iDestruct (OmoEview_latest_elem_obj with "⊒M3@V3'") as (e3) "[MAXe3 %e3IN]".
iDestruct (OmoAuth_wf with "M●3") as %[OMO_GOOD3 _].
iDestruct (OmoAuth_OmoEview_obj with "M●3 ⊒M3@V3'") as %INCL.
apply INCL in e3IN as e3IN'. eapply lookup_omo_surjective in e3IN'; [|done]. destruct e3IN' as [eidx3 Heidx3].
eapply omo_stlist_length in OMO_GOOD3 as EQlenST3.
set gen3 := gen_of eidx3.
have LeViV3' : Vi ⊑ V3' by solve_lat.
set psId := length E3.
set M3' := {[psId]} ∪ M3.
set eVpush := mkOmoEvent (Push v3) V3' M3'.
set E3' := E3 ++ [eVpush].
have LeV0V3' : V0 ⊑ V3' by solve_lat.
set ppId := length E3'.
set M3'' := {[ppId]} ∪ M3'.
set eVpop := mkOmoEvent (Pop v3) V3' M3''.
set E3'' := E3' ++ [eVpop].
destruct LASTemp3 as (genzfl3 & Hgenzfl3 & LASTemp3).
destruct (le_lt_dec gen3 genzfl3) as [LEgen|LTgen].
- (* If current observation is ealier than [genzfl3], then commit Push and Pop consecutively at [genzfl3] *)
iDestruct (OmoAuth_OmoSnap with "M●3 ezfl3↪emp") as %Hst; [by rewrite -lookup_omo_wr_event in Hgenzfl3|]. simpl in Hst.
iAssert (∀ len, ∃ stnew, ⌜ interp_omo E3' ((psId, []) :: take len (drop (genzfl3 + 1)%nat omo3)) [] stnew ∧ (∀ i st1 st2,
stlist3 !! (genzfl3 + i)%nat = Some st1 → stnew !! i = Some st2 → st2 = st1 ++ [(psId, v3, V3', M3')]) ⌝)%I with "[M●3]" as "%Hgengood".
{ iIntros "%len". iInduction len as [|] "IH".
- rewrite take_0. iExists [[(psId,v3,V3',M3')]]. iPureIntro. split.
+ rewrite -(app_nil_l [(psId, [])]) -(app_nil_l [[(psId, v3, V3', M3')]]). eapply interp_omo_snoc. split_and!; try done.
* rewrite lookup_app_1_eq. done.
* apply interp_omo_nil.
* eapply stack_step_Push; try done. set_solver-.
+ intros. destruct i; [|done]. inversion H0. subst st2. rewrite Nat.add_0_r Hst in H. inversion H. done.
- iDestruct ("IH" with "M●3") as %(stnew & IH1 & IH2).
destruct (le_lt_dec (length (drop (genzfl3 + 1) omo3)) len) as [LE'|LT'].
{ rewrite take_ge in IH1; [|done]. do 2 (rewrite take_ge; [|lia]). iExists stnew. done. }
have [ees Hlast] : is_Some ((drop (genzfl3 + 1) omo3) !! len) by rewrite lookup_lt_is_Some.
destruct ees as [e es]. rewrite (take_S_r _ _ (e, es)); [|done]. rewrite lookup_drop in Hlast.
specialize (omo_gen_info _ _ _ _ _ _ OMO_GOOD3 Hlast) as (eV & eeVs & st' & EID_MATCH & EEVS_VALID & E3e & Lookup_stlist3 & Interp & ES).
eapply LASTemp3 in Lookup_stlist3 as Nempst'; [|lia].
destruct es; last first.
{ rewrite Forall_cons in ES. destruct ES as [(?&?&?&?) _].
inversion H0; try done; apply (f_equal length) in H5; rewrite cons_length in H5; lia. }
have [stp Hstp] : is_Some (stlist3 !! (genzfl3 + len)).
{ rewrite lookup_lt_is_Some. apply lookup_lt_Some in Lookup_stlist3. lia. }
have STEP' : step e eV stp st'.
{ eapply omo_write_op_step; try done; replace (genzfl3 + len + 1) with (genzfl3 + 1 + len) by lia; try done.
rewrite list_lookup_omo_destruct in Hlast. by destruct Hlast as [? _]. }
apply interp_omo_length in IH1 as EQlen. apply lookup_lt_Some in Hlast as LTlen.
rewrite /= take_length drop_length Nat.min_l in EQlen; [|lia].
have [stp' Hstp'] : is_Some (stnew !! len) by rewrite lookup_lt_is_Some; lia.
eapply IH2 in Hstp' as EQstp'; [|done].
replace ((psId, []) :: take len (drop (genzfl3 + 1)%nat omo3) ++ [(e, [])]) with
(((psId, []) :: take len (drop (genzfl3 + 1)%nat omo3)) ++ [(e, [])]); [|by simplify_list_eq].
inversion STEP'; try (by subst).
+ iExists (stnew ++ [(e, v, eV.(sync), eV.(eview)) :: stp']). iPureIntro. split.
* eapply interp_omo_snoc. split_and!; try done.
-- rewrite lookup_app_1_ne; [done|]. apply lookup_lt_Some in E3e. lia.
-- rewrite last_cons last_lookup -EQlen. replace (Init.Nat.pred (S len)) with len by lia. rewrite Hstp'. done.
-- eapply stack_step_Push; try done.
* intros. destruct (decide (i = length stnew)) as [->|NEQ].
-- rewrite lookup_app_1_eq in H4. inversion H4. rewrite -EQlen in H3.
replace (genzfl3 + S len) with (genzfl3 + 1 + len) in H3 by lia. rewrite Lookup_stlist3 in H3. inversion H3.
subst st1. rewrite EQstp' -H2. clear. simplify_list_eq. done.
-- eapply IH2; try done. by rewrite lookup_app_1_ne in H4.
+ iExists (stnew ++ [stk ++ [(psId, v3, V3', M3')]]). iPureIntro. split.
* eapply interp_omo_snoc. split_and!; try done.
-- rewrite lookup_app_1_ne; [done|]. apply lookup_lt_Some in E3e. lia.
-- rewrite last_cons last_lookup -EQlen. replace (Init.Nat.pred (S len)) with len by lia. rewrite Hstp'. done.
-- rewrite EQstp' -H1.
replace (((u, v, V, lV) :: stk) ++ [(psId, v3, V3', M3')]) with ((u, v, V, lV) :: (stk ++ [(psId, v3, V3', M3')])); [|by simplify_list_eq].
eapply stack_step_Pop; try done.
* intros. destruct (decide (i = length stnew)) as [->|NEQ].
-- rewrite lookup_app_1_eq in H4. inversion H4. rewrite -EQlen in H3.
replace (genzfl3 + S len) with (genzfl3 + 1 + len) in H3 by lia. rewrite Lookup_stlist3 in H3. inversion H3. subst. done.
-- eapply IH2; try done. by rewrite lookup_app_1_ne in H4. }
specialize (Hgengood (length (drop (genzfl3 + 1) omo3))) as (stnew & Hgengood & Hstnew).
rewrite take_ge in Hgengood; [|done].
iDestruct (OmoUB_into_gen with "M●3 MAXe3") as %MAXe3; [done|].
apply (OmoUBgen_mono _ _ _ genzfl3) in MAXe3 as MAXgenzfl3; [|done].
(* Commit Push *)
iMod "AU1" as (?) "[M●3' [_ Commit]]". iDestruct "M●3'" as (???) ">M●3'".
iDestruct (OmoAuth_agree with "M●3 M●3'") as %(<- & <- & <- & <-). iCombine "M●3 M●3'" as "M●3".
iMod (OmoAuth_insert with "M●3 ⊒M3@V3' WCOMMIT1 []") as (γs3') "(M●3 & #⊒M3'@V3' & _ & #psId✓eVpush & _)".
{ iPureIntro. split_and!; try done; by subst eVpush. }
iDestruct (OmoHbToken_finish with "M●3") as "[M●3 M●3']".
iDestruct (OmoSnapHistory_get with "M●3") as "#E◯3'".
iMod ("Commit" $! true V3' E3' (Push v3) M3' with "[M●3']") as "HQ".
{ iFrame "⊒V3'". iSplit; [repeat iExists _; iFrame|]. iSplit.
- iPureIntro. split_and!; try done. set_solver-.
- repeat iExists _. iFrame "HM E◯3' ⊒M3'@V3' ⊒Mt@V3' ⊒Mx@V3' e0✓eV0 ⊒eV0@V3'". }
(* Commit Pop *)
set omo3' := omo_insert_w omo3 (genzfl3 + 1) psId [].
set stlist3' := take (genzfl3 + 1) stlist3 ++ stnew.
set gen := (genzfl3 + 1)%nat.
iDestruct (OmoAuth_wf with "M●3") as %[OMO_GOOD3' _].
eapply omo_stlist_length in OMO_GOOD3' as EQlenST3'.
iAssert (∀ len, ∃ stnew', ⌜ interp_omo E3'' ((ppId, []) :: take len (drop (gen + 1)%nat omo3')) [(psId, v3, V3', M3')] stnew' ∧ (∀ i st1 st2,
stlist3 !! (genzfl3 + i)%nat = Some st1 → stnew' !! i = Some st2 → st1 = st2) ⌝)%I with "[M●3]" as "%Hgengood'".
{ iIntros "%len". iInduction len as [|] "IH".
- rewrite take_0. iExists [[]]. iPureIntro. split.
+ rewrite -(app_nil_l [(ppId, [])]) -(app_nil_l [[]]). eapply interp_omo_snoc. split_and!; try done.
* rewrite lookup_app_1_eq. done.
* apply interp_omo_nil.
* eapply stack_step_Pop; try done. set_solver-.
+ intros. destruct i; [|done]. inversion H0. subst st2. rewrite Nat.add_0_r Hst in H. inversion H. done.
- iDestruct ("IH" with "M●3") as %(stnew' & IH1 & IH2).
destruct (le_lt_dec (length (drop (gen + 1) omo3')) len) as [LE'|LT'].
{ rewrite take_ge in IH1; [|done]. do 2 (rewrite take_ge; [|lia]). iExists stnew'. done. }
have [ees Hlast] : is_Some ((drop (gen + 1) omo3') !! len) by rewrite lookup_lt_is_Some.
destruct ees as [e es]. rewrite (take_S_r _ _ (e, es)); [|done]. rewrite lookup_drop in Hlast.
specialize (omo_gen_info _ _ _ _ _ _ OMO_GOOD3' Hlast) as (eV & eeVs & st' & EID_MATCH & EEVS_VALID & E3e & Lookup_stlist3' & Interp & ES).
apply interp_omo_length in Hgengood as EQlenstnew. rewrite /= drop_length in EQlenstnew.
have Lookup_stnew : stnew !! (len + 1)%nat = Some st'.