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omo.v
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omo.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 gpfsl.base_logic Require Import meta_data.
From gpfsl.examples.algebra Require Import mono_list_list.
From gpfsl.logic Require Import logatom atomics invariants.
From gpfsl.logic Require Import repeat_loop new_delete.
From gpfsl.examples Require Import map_seq loc_helper.
From stdpp Require Import namespaces.
From gpfsl.logic Require Import logatom.
From gpfsl.examples.omo Require Export omo_event omo_history.
Require Import iris.prelude.options.
Notation omoT := (list (event_id * list event_id))%type.
Notation lhb := hb_ord.
(** Pure definitions and lemmas for OMO **)
Section omo.
Context {eventT absStateT : Type}.
Implicit Types
(omo : omoT)
(st : absStateT).
Local Open Scope nat_scope.
(* Typeclass for state transition system *)
Class Interpretable eventT absStateT : Type :=
{
(* Initial State *)
init : absStateT;
(* Given an event and previous abstract state, interpreted result is new abstract state *)
step : event_id → omo_event eventT → absStateT → absStateT → Prop
}.
Inductive interp `{Interpretable eventT absStateT} : list (event_id * omo_event eventT) → absStateT → absStateT → Prop :=
| interp_nil : ∀ (st : absStateT), interp [] st st
| interp_snoc : ∀ e eV eeVs st1 st2 st3, interp eeVs st1 st2 → step e eV st2 st3 → interp (eeVs ++ [(e, eV)]) st1 st3
.
Notation history := (history eventT).
(** Index to an event in [omo]. [eidx] *)
Inductive event_idx : Set :=
| ro_event (gen i' : nat) (* [i']-th read-only operation of generation [gen] *)
| wr_event (gen : nat) (* the write event of generation [gen] *)
.
Definition gen_of eidx :=
match eidx with
| ro_event gen _ => gen
| wr_event gen => gen
end.
Definition lookup_omo omo eidx : option event_id :=
match eidx with
| ro_event gen i => omo.*2 !! gen ≫= (!!) i
| wr_event gen => omo.*1 !! gen
end.
(** "Lexicographic" order of [eidx] *)
Inductive eidx_le : ∀ (eidx1 eidx2 : event_idx), Prop :=
| eidx_le_r_r_1 gen1 i1' gen2 i2'
(LTgen : gen1 < gen2)
: eidx_le (ro_event gen1 i1') (ro_event gen2 i2')
| eidx_le_r_r_2 gen i1' i2'
(LEi' : i1' ≤ i2')
: eidx_le (ro_event gen i1') (ro_event gen i2')
| eidx_le_r_w gen1 i1' gen2
(LTgen : gen1 < gen2)
: eidx_le (ro_event gen1 i1') (wr_event gen2)
| eidx_le_w_r gen1 gen2 i2'
(LTgen : gen1 ≤ gen2)
: eidx_le (wr_event gen1) (ro_event gen2 i2')
| eidx_le_w_w gen1 gen2
(LEgen : gen1 ≤ gen2)
: eidx_le (wr_event gen1) (wr_event gen2)
.
(** local happens-before implies omo-before. *)
Definition lhb_omo (E : history) omo :=
∀ eidx1 eidx2 e1 e2 eV2
(omo_eidx1 : lookup_omo omo eidx1 = Some e1)
(omo_eidx2 : lookup_omo omo eidx2 = Some e2)
(EV2 : E !! e2 = Some eV2)
(IN_EVIEW : e1 ∈ eV2.(eview)),
eidx_le eidx1 eidx2.
(** Linearization of [omo]. Defined as flattening of omo into an array of event id *)
Definition lin_of_omo omo :=
concat ((λ '(e, es), e :: es) <$> omo).
(** Insert (append) read-only operation to the generation [gen] of [omo]. *)
Definition omo_insert_r omo gen e := alter (λ '(e', es'), (e', es'++[e])) gen omo.
(** Insert a new write operation at [gen]th generation *)
Definition omo_insert_w omo gen e es := take gen omo ++ (e, es) :: drop gen omo.
(** Insert a new write operation at the back *)
Definition omo_append_w omo e es := omo ++ [(e, es)].
(** Collects only write events of OMO.
This is widely used in the invariants of libraries. *)
Definition omo_write_op omo := omo.*1.
(** Collects only read-only events of OMO *)
Definition omo_read_op omo := omo.*2.
(* No [Inj] instance because it's partial *)
Definition omo_inj omo :=
∀ eidx1 eidx2 e
(OMO_LOOKUP1 : lookup_omo omo eidx1 = Some e)
(OMO_LOOKUP2 : lookup_omo omo eidx2 = Some e),
eidx1 = eidx2.
(* Prefix relation of two `OMO`'s *)
Definition omo_prefix omo1 omo2 :=
omo_read_op omo1 `prefixes_of` omo_read_op omo2 ∧
omo_write_op omo1 ⊑ omo_write_op omo2.
(** Convert an [eidx] into an index in [lin]. *)
Definition eidx_to_lin_idx omo eidx :=
match eidx with
| ro_event gen i' => length (concat ((λ '(e, es), e :: es) <$> take gen omo)) + S i'
| wr_event gen => length (concat ((λ '(e, es), e :: es) <$> take gen omo))
end.
Inductive interp_omo `{Interpretable eventT absStateT} : history → omoT → absStateT → list absStateT → Prop :=
| interp_omo_nil : ∀ E st, interp_omo E [] st []
| interp_omo_snoc :
∀ E e es eV omo st1 st2 st3 stlist,
E !! e = Some eV ∧
interp_omo E omo st1 stlist ∧
last (st1 :: stlist) = Some st2 ∧
step e eV st2 st3 ∧
Forall (λ e', ∃ eV', E !! e' = Some eV' ∧ step e' eV' st3 st3 ∧ e < e') es →
interp_omo E (omo ++ [(e, es)]) st1 (stlist ++ [st3])
.
Definition perm_omo (E : history) omo : Prop :=
lin_of_omo omo ≡ₚ seq 0 (length E).
(* Definition of linearizability in OMO version *)
Inductive Linearizability_omo (E : history) omo stlist `{Interpretable eventT absStateT} : Prop :=
| Linearizability_omo_intro
(INTERP_OMO : interp_omo E omo init stlist)
(LHB_OMO : lhb_omo E omo)
(PERM_OMO : perm_omo E omo)
.
Definition eid_to_event_valid (E : history) (eeVs : list (event_id * omo_event eventT)) : Prop :=
Forall (λ '(e, eV), E !! e = Some eV) eeVs.
(* Definition of linearizability (in linearizability history specification) *)
Inductive Linearizability E `{Interpretable eventT absStateT} : Prop :=
| Linearizability_intro xo lin st
(XO : xo.*1 = seq 0 (length E)) (* xo (execution order): list (event_id * omo_event) *)
(EEVS_VALID_XO : eid_to_event_valid E xo) (* Checks whether pairing between event_id and omo_event is valid *)
(LIN_PERM : lin ≡ₚ xo) (* lin: permutation of xo *)
(LHB_LIN : lhb E lin.*1) (* linearization order satisfies happens-before order *)
(INTERP_LIN : interp lin init st) (* linearization order is interpretable *)
.
Section omo_lemmas.
#[global] Instance eidx_eqdec : EqDecision event_idx.
Proof. solve_decision. Qed.
Lemma interp_omo_length E omo st stlist `{Interpretable eventT absStateT}
(INTERP_OMO : interp_omo E omo st stlist) :
length omo = length stlist.
Proof.
generalize dependent stlist. induction omo using rev_ind; intros.
- inversion INTERP_OMO; [done|]. apply (f_equal length) in H0. rewrite app_length /= in H0. lia.
- inversion INTERP_OMO; [done|]. destruct H2 as [_ [H2 _]].
apply app_inj_2 in H0 as [EQ _]; [|done]. subst omo0.
specialize (IHomo _ H2). rewrite !app_length /=. lia.
Qed.
Lemma interp_app eeVs1 eeVs2 st1 st2 `{Interpretable eventT absStateT} :
interp (eeVs1 ++ eeVs2) st1 st2 ↔
∃ st3, interp eeVs1 st1 st3 ∧ interp eeVs2 st3 st2.
Proof.
split; intros.
- revert st2 H0. induction eeVs2 using rev_ind; intros.
+ exists st2. simplify_list_eq. split; [done|]. apply interp_nil.
+ replace (eeVs1 ++ eeVs2 ++ [x]) with ((eeVs1 ++ eeVs2) ++ [x]) in H0; [|by simplify_list_eq].
inversion H0.
{ apply (f_equal length) in H2. rewrite app_length /= in H2. lia. }
apply app_inj_2 in H1 as [-> EQ]; [|done]. inversion EQ. subst x st0 st4. clear EQ.
specialize (IHeeVs2 _ H2) as [st4 [INTERP1 INTERP2]].
exists st4. split; [done|]. eapply interp_snoc; try done.
- revert st2 H0. induction eeVs2 using rev_ind; intros.
+ destruct H0 as [st3 [INTERP1 INTERP2]].
inversion INTERP2; last first.
{ apply (f_equal length) in H0. rewrite app_length /= in H0. lia. }
subst st st2. simplify_list_eq. done.
+ destruct H0 as [st3 [INTERP1 INTERP2]].
inversion INTERP2.
{ apply (f_equal length) in H1. rewrite app_length /= in H1. lia. }
apply app_inj_2 in H0 as [-> EQ]; [|done]. inversion EQ. subst st0 st5 x. clear EQ.
replace (eeVs1 ++ eeVs2 ++ [(e, eV)]) with ((eeVs1 ++ eeVs2) ++ [(e, eV)]); [|by simplify_list_eq].
eapply interp_snoc; try done. apply IHeeVs2. by eexists.
Qed.
Lemma interp_omo_app E omo1 omo2 st1 stlist1 stlist2 `{Interpretable eventT absStateT}
(EQlen : length omo1 = length stlist1) :
interp_omo E (omo1 ++ omo2) st1 (stlist1 ++ stlist2) ↔
∃ st2, interp_omo E omo1 st1 stlist1 ∧ interp_omo E omo2 st2 stlist2 ∧ last (st1 :: stlist1) = Some st2.
Proof.
split; intros.
- generalize dependent stlist2. induction omo2 using rev_ind; intros.
+ apply interp_omo_length in H0 as EQlen'.
rewrite !app_length EQlen /= Nat.add_0_r in EQlen'. destruct stlist2; last first.
{ simpl in EQlen'. clear -EQlen'. lia. }
simplify_list_eq.
have [st3 Hst3] : is_Some (last (st1 :: stlist1)) by rewrite last_is_Some.
exists st3. split_and!; try done. apply interp_omo_nil.
+ apply interp_omo_length in H0 as EQlen'.
rewrite !app_length EQlen /= in EQlen'.
destruct stlist2 using rev_ind; [simpl in *;lia|]. clear IHstlist2.
have EQlen2 : length omo2 = length stlist2.
{ rewrite app_length /= in EQlen'. lia. }
replace (omo1 ++ omo2 ++ [x]) with ((omo1 ++ omo2) ++ [x]) in H0; [|by simplify_list_eq].
replace (stlist1 ++ stlist2 ++ [x0]) with ((stlist1 ++ stlist2) ++ [x0]) in H0; [|by simplify_list_eq].
inversion H0.
{ apply (f_equal length) in H3. rewrite app_length /= in H3. lia. }
apply app_inj_2 in H1 as [-> EQ1]; [|done]. apply app_inj_2 in H3 as [-> EQ2]; [|done].
inversion EQ1. inversion EQ2. subst E0 st0 x x0. clear EQ1 EQ2.
destruct H5 as (H1 & H2 & H3 & H4 & H5).
specialize (IHomo2 _ H2) as (st5 & INTERP_OMO1 & INTERP_OMO2 & LAST).
exists st5. split_and!; try done. eapply interp_omo_snoc. split_and!; try done.
rewrite last_cons. rewrite last_cons last_app in H3.
destruct (last stlist2) eqn:Heq; [done|].
rewrite last_cons in LAST. destruct (last stlist1) eqn:Heq';
by rewrite -H3 -LAST.
- generalize dependent stlist2. induction omo2 using rev_ind; intros.
+ destruct H0 as (st3 & INTERP_OMO1 & INTERP_OMO2 & LAST).
inversion INTERP_OMO2; last first.
{ apply (f_equal length) in H0. rewrite app_length /= in H0. lia. }
subst E0 st stlist2. simplify_list_eq. done.
+ destruct H0 as (st3 & INTERP_OMO1 & INTERP_OMO2 & LAST).
inversion INTERP_OMO2.
{ apply (f_equal length) in H2. rewrite app_length /= in H2. lia. }
apply app_inj_2 in H0 as [-> EQ]; [|done]. inversion EQ. subst E0 st0 stlist2 x. rename stlist into stlist2. clear EQ.
destruct H2 as (H1 & H2 & H3 & H4 & H5).
have COND : ∃ st', interp_omo E omo1 st1 stlist1 ∧ interp_omo E omo2 st' stlist2 ∧ last (st1 :: stlist1) = Some st'.
{ exists st3. split_and!; try done. }
specialize (IHomo2 _ COND).
replace (omo1 ++ omo2 ++ [(e, es)]) with ((omo1 ++ omo2) ++ [(e, es)]); [|by simplify_list_eq].
replace (stlist1 ++ stlist2 ++ [st4]) with ((stlist1 ++ stlist2) ++ [st4]); [|by simplify_list_eq].
eapply interp_omo_snoc. split_and!; try done.
rewrite last_cons last_app. rewrite !last_cons in H3,LAST.
destruct (last stlist2) eqn:Heq1; [done|].
destruct (last stlist1) eqn:Heq2;
by rewrite LAST H3.
Qed.
Lemma interp_app_inv eeVs1 eeVs2 st1 st2 `{Interpretable eventT absStateT} :
interp (eeVs1 ++ eeVs2) st1 st2 →
∃ st3, interp eeVs1 st1 st3 ∧ interp eeVs2 st3 st2.
Proof. by apply interp_app. Qed.
Lemma lookup_omo_ro_event omo gen i :
lookup_omo omo (ro_event gen i) = omo_read_op omo !! gen ≫= (!!) i.
Proof. done. Qed.
Lemma lookup_omo_inv_r omo gen i e
(LOOKUP : lookup_omo omo (ro_event gen i) = Some e) :
∃ es,
omo_read_op omo !! gen = Some es ∧
es !! i = Some e.
Proof.
rewrite lookup_omo_ro_event in LOOKUP.
destruct (omo_read_op omo !! gen) eqn:Heq; [|done].
simpl in LOOKUP. by exists l.
Qed.
Lemma lookup_omo_wr_event omo gen :
lookup_omo omo (wr_event gen) = omo_write_op omo !! gen.
Proof. done. Qed.
Lemma interp_omo_take E omo st stlist i `{Interpretable eventT absStateT}
(INTERP_OMO : interp_omo E omo st stlist) :
interp_omo E (take i omo) st (take i stlist).
Proof.
destruct (le_lt_dec (length omo) i) as [LE|LT].
{ have EQlen : length omo = length stlist by eapply interp_omo_length.
rewrite take_ge; [|done]. rewrite take_ge; [|lia]. done. }
generalize dependent stlist. induction omo using rev_ind; intros; [simpl in *; lia|].
inversion INTERP_OMO.
{ apply (f_equal length) in H2. rewrite app_length /= in H2. lia. }
apply app_inj_2 in H0 as [EQ1 EQ2]; [|done].
inversion EQ2. subst x omo0 st1 stlist E0. clear EQ2. rename stlist0 into stlist.
destruct (decide (i = length omo)) as [->|NEQ].
{ have EQlen : length (omo ++ [(e, es)]) = length (stlist ++ [st3]) by eapply interp_omo_length.
rewrite take_app. rewrite !app_length /= in EQlen. replace (length omo) with (length stlist) by lia.
rewrite take_app. by destruct H2 as [_ [? _]]. }
rewrite app_length /= in LT.
have LT' : i < length omo by lia.
destruct H2 as [_ [H2 _]]. specialize (IHomo LT' _ H2).
rewrite take_app_le; [|lia].
have EQlen : length (omo ++ [(e, es)]) = length (stlist ++ [st3]) by eapply interp_omo_length.
rewrite !app_length /= in EQlen.
rewrite take_app_le; [|lia]. done.
Qed.
Lemma lookup_omo_event_valid E omo stlist eidx e `{Interpretable eventT absStateT}
(OMO_GOOD : Linearizability_omo E omo stlist)
(LOOKUP : lookup_omo omo eidx = Some e) :
is_Some (E !! e).
Proof.
destruct OMO_GOOD. destruct eidx.
- specialize (interp_omo_take _ _ _ _ (S gen) INTERP_OMO) as INTERP_OMO'.
rewrite lookup_omo_ro_event /omo_read_op list_lookup_fmap in LOOKUP.
destruct (omo !! gen) eqn:Heq; [|done]. destruct p as [e' es']. simpl in LOOKUP.
rewrite (take_S_r _ _ (e', es')) in INTERP_OMO'; [|done].
inversion INTERP_OMO'.
{ apply (f_equal length) in H2. rewrite app_length /= in H2. lia. }
apply app_inj_2 in H0 as [-> EQ]; [|done]. inversion EQ. subst E0 st1 e0 es. clear EQ H2.
destruct H4 as (_ & _ & _ & _ & H1).
rewrite Forall_lookup in H1. specialize (H1 _ _ LOOKUP) as [eV' [H1 _]]. done.
- specialize (interp_omo_take _ _ _ _ (S gen) INTERP_OMO) as INTERP_OMO'.
rewrite lookup_omo_wr_event /omo_write_op list_lookup_fmap in LOOKUP.
destruct (omo !! gen) eqn:Heq; [|done]. destruct p as [e' es']. simpl in LOOKUP. inversion LOOKUP. subst e'.
rewrite (take_S_r _ _ (e, es')) in INTERP_OMO'; [|done].
inversion INTERP_OMO'.
{ apply (f_equal length) in H2. rewrite app_length /= in H2. lia. }
apply app_inj_2 in H0 as [-> EQ]; [|done]. inversion EQ. subst E0 st1 e0 es. clear EQ H2.
by destruct H4 as [H1 _].
Qed.
Lemma omo_same omo omo' :
omo = omo' ↔
omo_write_op omo = omo_write_op omo' ∧
omo_read_op omo = omo_read_op omo'.
Proof.
split; intros; [subst omo; done|].
destruct H as [H1 H2]. generalize dependent omo'. induction omo; intros.
- by destruct omo'.
- destruct omo'; try done.
inversion H1. inversion H2.
specialize (IHomo _ H3 H5). subst omo. destruct a, p. simpl in H0, H4.
subst e l. done.
Qed.
Lemma list_lookup_omo_destruct omo gen info :
omo !! gen = Some info ↔
omo_write_op omo !! gen = Some info.1 ∧
omo_read_op omo !! gen = Some info.2.
Proof.
split; intros.
- by split; rewrite list_lookup_fmap H.
- destruct H as [H1 H2]. generalize dependent info. generalize dependent gen. induction omo; intros.
+ done.
+ destruct gen.
* simpl in H1, H2. simpl. destruct a, info. inversion H1. inversion H2. done.
* simpl in H1, H2. simpl. by specialize (IHomo _ _ H1 H2).
Qed.
Lemma list_lookup_omo_destruct' omo gen e es :
omo !! gen = Some (e, es) ↔
omo_write_op omo !! gen = Some e ∧
omo_read_op omo !! gen = Some es.
Proof. apply list_lookup_omo_destruct. Qed.
Lemma list_lookup_omo_from_write_op omo gen e :
omo_write_op omo !! gen = Some e →
∃ es, omo !! gen = Some (e, es).
Proof.
rewrite /omo_write_op. intros.
have [x Hx] : is_Some (omo !! gen).
{ rewrite lookup_lt_is_Some. apply lookup_lt_Some in H. by rewrite !fmap_length in H. }
destruct x as [e' es].
rewrite !list_lookup_fmap Hx in H. inversion H. subst e'.
by exists es.
Qed.
Lemma list_lookup_omo_from_read_op omo gen es :
omo_read_op omo !! gen = Some es →
∃ e, omo !! gen = Some (e, es).
Proof.
rewrite /omo_read_op. intros. apply lookup_lt_Some in H as H'.
rewrite fmap_length in H'. apply lookup_lt_is_Some in H' as [ees Hees].
destruct ees as [e es'].
rewrite list_lookup_fmap Hees in H. inversion H. subst es'.
by eexists.
Qed.
Lemma list_lookup_omo_write_op_agree omo gen e e' es
(LOOKUP1 : omo !! gen = Some (e, es))
(LOOKUP2 : omo_write_op omo !! gen = Some e') :
e = e'.
Proof.
rewrite /omo_write_op list_lookup_fmap LOOKUP1 in LOOKUP2.
inversion LOOKUP2. done.
Qed.
Lemma list_lookup_omo_read_op_agree omo gen e es es'
(LOOKUP1 : omo !! gen = Some (e, es))
(LOOKUP2 : omo_read_op omo !! gen = Some es') :
es = es'.
Proof.
rewrite /omo_read_op list_lookup_fmap LOOKUP1 in LOOKUP2.
inversion LOOKUP2. done.
Qed.
Lemma omo_write_op_lt_is_Some omo gen :
is_Some (omo_write_op omo !! gen) ↔ gen < length omo.
Proof. by rewrite /omo_write_op !list_lookup_fmap !fmap_is_Some lookup_lt_is_Some. Qed.
Lemma omo_write_op_lt_Some omo gen e :
omo_write_op omo !! gen = Some e → gen < length omo.
Proof. by rewrite -omo_write_op_lt_is_Some. Qed.
Lemma omo_read_op_lt_is_Some omo gen :
is_Some (omo_read_op omo !! gen) ↔ gen < length omo.
Proof. by rewrite /omo_read_op !list_lookup_fmap !fmap_is_Some lookup_lt_is_Some. Qed.
Lemma omo_read_op_lt_Some omo gen es :
omo_read_op omo !! gen = Some es → gen < length omo.
Proof. by rewrite -omo_read_op_lt_is_Some. Qed.
Lemma lookup_omo_lt_is_Some omo eidx :
is_Some (lookup_omo omo eidx) → gen_of eidx < length omo.
Proof.
destruct eidx; intros.
- simpl in H. destruct (omo !! gen) eqn:Heq.
+ apply lookup_lt_Some in Heq. done.
+ rewrite list_lookup_fmap Heq /= in H. by inversion H.
- rewrite /= lookup_lt_is_Some fmap_length in H. done.
Qed.
Lemma lookup_omo_lt_Some omo eidx e :
lookup_omo omo eidx = Some e → gen_of eidx < length omo.
Proof. intros. by apply lookup_omo_lt_is_Some. Qed.
Lemma omo_write_op_length omo :
length omo = length (omo_write_op omo).
Proof. by rewrite /omo_write_op !fmap_length. Qed.
Lemma omo_read_op_length omo :
length omo = length (omo_read_op omo).
Proof. by rewrite /omo_read_op !fmap_length. Qed.
Lemma omo_stlist_length E omo stlist `{Interpretable eventT absStateT}
(OMO_GOOD : Linearizability_omo E omo stlist) :
length omo = length stlist.
Proof.
destruct OMO_GOOD. clear LHB_OMO PERM_OMO.
generalize dependent stlist. induction omo using rev_ind; intros.
- inversion INTERP_OMO.
+ done.
+ apply (f_equal length) in H0. rewrite app_length /= in H0. lia.
- inversion INTERP_OMO.
+ done.
+ rewrite !app_length /=.
destruct H2 as [_ [H2 _]].
apply app_inj_2 in H0 as [-> _]; [|done].
specialize (IHomo _ H2). rewrite IHomo. done.
Qed.
Lemma lookup_omo_mono omo1 omo2 eidx e
(LOOKUP : lookup_omo omo1 eidx = Some e)
(PREFIX : omo_prefix omo1 omo2) :
lookup_omo omo2 eidx = Some e.
Proof.
destruct eidx.
- destruct PREFIX as [H1 H2].
destruct (omo_read_op omo1 !! gen) eqn:Heq1; destruct (omo_read_op omo2 !! gen) eqn:Heq2;
specialize (H1 gen) as H3; rewrite Heq1 Heq2 /= in H3; try done.
+ rewrite lookup_omo_ro_event Heq1 /= in LOOKUP.
rewrite lookup_omo_ro_event Heq2 /=.
by eapply prefix_lookup_Some.
+ rewrite lookup_omo_ro_event Heq1 /= in LOOKUP. done.
+ rewrite lookup_omo_ro_event Heq1 /= in LOOKUP. done.
- destruct PREFIX as [H1 H2].
rewrite lookup_omo_wr_event in LOOKUP. rewrite lookup_omo_wr_event.
by eapply prefix_lookup_Some.
Qed.
Lemma omo_prefix_refl omo :
omo_prefix omo omo.
Proof. done. Qed.
Lemma omo_prefix_trans omo1 omo2 omo3
(PREFIX1 : omo_prefix omo1 omo2)
(PREFIX2 : omo_prefix omo2 omo3) :
omo_prefix omo1 omo3.
Proof.
destruct PREFIX1 as [SubR1 SubW1].
destruct PREFIX2 as [SubR2 SubW2].
split.
- by transitivity (omo_read_op omo2).
- by transitivity (omo_write_op omo2).
Qed.
Global Instance omo_prefix_Reflexive : Reflexive omo_prefix := omo_prefix_refl.
Global Instance omo_prefix_Transitive : Transitive omo_prefix := omo_prefix_trans.
Lemma omo_write_op_take omo gen :
omo_write_op (take gen omo) = take gen (omo_write_op omo).
Proof. by rewrite /omo_write_op !fmap_take. Qed.
Lemma omo_write_op_drop omo gen :
omo_write_op (drop gen omo) = drop gen (omo_write_op omo).
Proof. by rewrite /omo_write_op !fmap_drop. Qed.
Lemma omo_read_op_take omo gen :
omo_read_op (take gen omo) = take gen (omo_read_op omo).
Proof. by rewrite /omo_read_op !fmap_take. Qed.
Lemma omo_read_op_drop omo gen :
omo_read_op (drop gen omo) = drop gen (omo_read_op omo).
Proof. by rewrite /omo_read_op !fmap_drop. Qed.
Lemma lookup_omo_take omo gen eidx
(LT : gen_of eidx < gen) :
lookup_omo (take gen omo) eidx = lookup_omo omo eidx.
Proof. by destruct eidx; simpl; rewrite fmap_take lookup_take. Qed.
Lemma lookup_omo_drop omo gen eidx :
lookup_omo (drop gen omo) eidx
= match eidx with
| wr_event gen' => lookup_omo omo (wr_event (gen + gen'))
| ro_event gen' idx => lookup_omo omo (ro_event (gen + gen') idx)
end.
Proof. by destruct eidx; simpl; rewrite fmap_drop lookup_drop. Qed.
Lemma omo_prefix_take omo gen :
omo_prefix (take gen omo) omo.
Proof.
split.
- rewrite /prefixes /map_included /map_relation. intros.
destruct (omo_read_op (take gen omo) !! i) eqn:Heq.
+ apply lookup_lt_Some in Heq as LT. rewrite -omo_read_op_length take_length in LT.
rewrite /omo_read_op fmap_take lookup_take in Heq; [|lia]. rewrite Heq /=. done.
+ simpl. by destruct (omo_read_op omo !! i).
- rewrite /omo_write_op fmap_take. apply prefix_take.
Qed.
Lemma lin_of_omo_empty :
lin_of_omo [] = [].
Proof. done. Qed.
Lemma length_concat_fmap_take_plus_S (f : _ → list event_id) gen1' omo x N
(GEN1 : omo !! gen1' = Some x) :
length (concat (f <$> take (gen1' + S N) omo))
= length (concat (f <$> take gen1' omo)) + (length (f x) + length (concat (f <$> take N (drop (S gen1') omo)))).
Proof.
apply lookup_lt_Some in GEN1 as LEgen1'.
rewrite -{1}(take_drop gen1' omo).
rewrite take_add_app; last first. { rewrite take_length. lia. }
rewrite fmap_app concat_app app_length.
rewrite (drop_S _ _ _ GEN1). rewrite firstn_cons fmap_cons /=.
rewrite app_length. done.
Qed.
Lemma eidx_to_lin_idx_inj omo eidx1 eidx2 e
(OMO_LOOKUP1 : lookup_omo omo eidx1 = Some e)
(OMO_LOOKUP2 : lookup_omo omo eidx2 = Some e)
(EQlin : eidx_to_lin_idx omo eidx1 = eidx_to_lin_idx omo eidx2) :
eidx1 = eidx2.
Proof.
unfold eidx_to_lin_idx in *.
set (f := λ '(e0, es), e0 :: es) in *.
destruct eidx1 as [gen1 i1|gen1], eidx2 as [gen2 i2|gen2].
- case (decide (gen1 = gen2)) as [->|NEgen].
{ have EQ : i1 = i2 by lia. subst i1. done. }
exfalso.
wlog: gen1 gen2 i1 i2 OMO_LOOKUP1 OMO_LOOKUP2 EQlin NEgen / gen1 < gen2.
{ destruct (le_lt_dec gen1 gen2) as [LE|LT].
- intros. apply (H gen1 gen2 i1 i2); try done. lia.
- intros. apply (H gen2 gen1 i2 i1); try done. }
intros LTgen.
have [N EQ] : ∃ N, gen2 = gen1 + (S N) by exists (gen2 - gen1 - 1); lia. subst gen2.
have {}EQlin : length (concat (f <$> take gen1 omo)) + i1 = length (concat (f <$> take (gen1 + S N) omo)) + i2 by lia.
destruct (omo !! gen1) eqn:Heq; last first.
{ rewrite lookup_omo_ro_event /omo_read_op list_lookup_fmap Heq /= in OMO_LOOKUP1. done. }
destruct p as [e1 es1].
erewrite length_concat_fmap_take_plus_S in EQlin; [|done]. simpl in EQlin.
assert (i1 = S (length es1 + length (concat (f <$> take N (drop (S gen1) omo)))) + i2) as ->; try lia.
rewrite lookup_omo_ro_event /omo_read_op list_lookup_fmap Heq /= in OMO_LOOKUP1.
apply lookup_lt_Some in OMO_LOOKUP1. lia.
- exfalso. destruct (le_lt_dec gen1 gen2) as [LE|LT].
+ have LT : gen1 < gen2 by destruct (decide (gen1 = gen2)) as [<-|NE]; lia.
destruct (omo !! gen1) eqn:Heq; [|by rewrite lookup_omo_ro_event /omo_read_op list_lookup_fmap Heq /= in OMO_LOOKUP1].
destruct p as [e1 es1].
have [N EQ] : ∃ N, gen2 = gen1 + (S N) by exists (gen2 - gen1 - 1); lia. subst gen2.
erewrite length_concat_fmap_take_plus_S in EQlin; [|done]. simpl in EQlin.
have EQ : i1 = length es1 + length (concat (f <$> take N (drop (S gen1) omo))) by lia.
rewrite EQ lookup_omo_ro_event /omo_read_op list_lookup_fmap Heq /= in OMO_LOOKUP1.
apply lookup_lt_Some in OMO_LOOKUP1. lia.
+ have [N EQ] : ∃ N, gen1 = gen2 + (S N) by exists (gen1 - gen2 - 1); lia. subst gen1.
destruct (omo !! gen2) eqn:Heq; last first.
{ rewrite lookup_omo_wr_event /omo_write_op list_lookup_fmap Heq /= in OMO_LOOKUP2. done. }
erewrite length_concat_fmap_take_plus_S in EQlin; [|done]. simpl in EQlin. lia.
- exfalso. destruct (le_lt_dec gen1 gen2) as [LE|LT].
+ have LT : gen1 < gen2 by destruct (decide (gen1 = gen2)) as [<-|NE]; lia.
have [N EQ] : ∃ N, gen2 = gen1 + (S N) by exists (gen2 - gen1 - 1); lia. subst gen2.
destruct (omo !! gen1) eqn:Heq; last first.
{ rewrite lookup_omo_wr_event /omo_write_op list_lookup_fmap Heq /= in OMO_LOOKUP1. done. }
erewrite length_concat_fmap_take_plus_S in EQlin; [|done]. simpl in EQlin. lia.
+ have [N EQ] : ∃ N, gen1 = gen2 + (S N) by exists (gen1 - gen2 - 1); lia. subst gen1.
destruct (omo !! gen2) eqn:Heq; last first.
{ rewrite lookup_omo_ro_event /omo_read_op list_lookup_fmap Heq /= in OMO_LOOKUP2. done. }
destruct p as [e2 es2].
erewrite length_concat_fmap_take_plus_S in EQlin; [|done]. simpl in EQlin.
have EQ : i2 = length es2 + length (concat (f <$> take N (drop (S gen2) omo))) by lia.
rewrite EQ lookup_omo_ro_event /omo_read_op list_lookup_fmap Heq /= in OMO_LOOKUP2.
apply lookup_lt_Some in OMO_LOOKUP2. lia.
- destruct (decide (gen1 = gen2)) as [->|NE]; [done|]. exfalso.
wlog: gen1 gen2 OMO_LOOKUP1 OMO_LOOKUP2 EQlin NE / gen1 < gen2.
{ destruct (le_lt_dec gen1 gen2) as [LE|LT].
- have LT : gen1 < gen2 by lia. intros. by apply (H gen1 gen2).
- intros. by apply (H gen2 gen1). }
intros LT.
have [N EQ] : ∃ N, gen2 = gen1 + (S N) by exists (gen2 - gen1 - 1); lia. subst gen2.
destruct (omo !! gen1) eqn:Heq; last first.
{ rewrite lookup_omo_wr_event /omo_write_op list_lookup_fmap Heq /= in OMO_LOOKUP1. done. }
destruct p as [e1 es1].
erewrite length_concat_fmap_take_plus_S in EQlin; [|done]. simpl in EQlin. lia.
Qed.
Lemma lin_of_omo_take_drop omo gen :
lin_of_omo omo = lin_of_omo (take gen omo) ++ lin_of_omo (drop gen omo).
Proof.
destruct (le_lt_dec (length omo) gen) as [Hgen|Hgen].
- rewrite take_ge; [|done].
rewrite drop_ge; [|done].
by rewrite lin_of_omo_empty app_nil_r.
- by rewrite /lin_of_omo -concat_app -fmap_app take_drop.
Qed.
Lemma omo_take_drop_lin_singleton omo gen e es
(LOOKUP : omo !! gen = Some (e, es)) :
lin_of_omo (drop gen (take (S gen) omo)) = e :: es.
Proof.
erewrite take_S_r; [|done].
have Hgen : gen = length (take gen omo).
{ rewrite take_length. apply lookup_lt_Some in LOOKUP. lia. }
by rewrite {1}Hgen drop_app /lin_of_omo /= app_nil_r.
Qed.
Lemma lin_of_omo_take_S_r omo gen e es
(LOOKUP : omo !! gen = Some (e, es)) :
lin_of_omo (take (S gen) omo) = lin_of_omo (take gen omo) ++ (e :: es).
Proof.
rewrite /lin_of_omo. erewrite take_S_r; [|done]. simplify_list_eq.
by rewrite concat_app /= app_nil_r.
Qed.
Lemma lin_of_omo_append_w omo e es :
lin_of_omo (omo_append_w omo e es) = lin_of_omo omo ++ e :: es.
Proof.
rewrite /lin_of_omo /omo_append_w. simplify_list_eq. by rewrite concat_app /= app_nil_r.
Qed.
Lemma lin_of_omo_insert_w omo gen e es :
lin_of_omo (omo_insert_w omo gen e es) = lin_of_omo (take gen omo) ++ e :: es ++ lin_of_omo (drop gen omo).
Proof.
rewrite /lin_of_omo /omo_insert_w /=.
list_simplifier.
rewrite concat_app /=.
list_simplifier.
done.
Qed.
Lemma lookup_omo_app omo omo' eidx e
(OMO_LOOKUP : lookup_omo omo eidx = Some e) :
lookup_omo (omo ++ omo') eidx = Some e.
Proof.
destruct eidx.
- rewrite lookup_omo_ro_event /omo_read_op fmap_app. rewrite lookup_omo_ro_event in OMO_LOOKUP.
destruct (omo_read_op omo !! gen) eqn:Heq; [|done].
rewrite lookup_app_l; [|by apply lookup_lt_Some in Heq].
rewrite Heq. done.
- rewrite lookup_omo_wr_event /omo_write_op fmap_app. rewrite lookup_omo_wr_event in OMO_LOOKUP.
destruct (omo_write_op omo !! gen) eqn:Heq; [|done].
rewrite lookup_app_l; [|by apply lookup_lt_Some in Heq].
rewrite Heq. done.
Qed.
Lemma eidx_to_lin_idx_omo_app omo eidx e e' es'
(OMO_LOOKUP : lookup_omo omo eidx = Some e) :
eidx_to_lin_idx omo eidx = eidx_to_lin_idx (omo_append_w omo e' es') eidx.
Proof.
unfold eidx_to_lin_idx. destruct eidx.
- rewrite /omo_append_w take_app_le; [done|].
rewrite lookup_omo_ro_event in OMO_LOOKUP. destruct (omo_read_op omo !! gen) eqn:Heq; [|done].
apply lookup_lt_Some in Heq. rewrite -omo_read_op_length in Heq. lia.
- rewrite /omo_append_w take_app_le; [done|].
rewrite lookup_omo_wr_event in OMO_LOOKUP. destruct (omo_write_op omo !! gen) eqn:Heq; [|done].
apply lookup_lt_Some in Heq. rewrite -omo_write_op_length in Heq. lia.
Qed.
Lemma lin_of_omo_lookup_lookup_omo omo i e
(NODUP : NoDup (lin_of_omo omo))
(LIN_I : lin_of_omo omo !! i = Some e) :
∃ eidx,
i = eidx_to_lin_idx omo eidx ∧
lookup_omo omo eidx = Some e.
Proof.
revert i e NODUP LIN_I.
induction omo using rev_ind; intros; [done|].
destruct x as [e' es'].
have H : omo_append_w omo e' es' = omo ++ [(e', es')] by done.
rewrite -H in NODUP, LIN_I.
rewrite ->lin_of_omo_append_w,NoDup_app in NODUP. des.
rewrite lin_of_omo_append_w in LIN_I.
set len := length (lin_of_omo omo).
have Hi : i < len ∨ i = len ∨ i > len by lia. des.
- (* use [i] in IH to get a [eidx] and use it *)
rewrite lookup_app_l in LIN_I; [|done].
specialize (IHomo _ _ NODUP LIN_I). des.
exists eidx. split.
+ erewrite eidx_to_lin_idx_omo_app in IHomo; last done.
rewrite -H. done.
+ by eapply lookup_omo_app in IHomo0.
- rewrite list_lookup_middle in LIN_I; [|done]. simplify_eq.
exists (wr_event (length omo)). list_simplifier. split.
+ subst len. done.
+ by rewrite -!(fmap_length fst) lookup_app_1_eq.
- rewrite ->lookup_app_r,lookup_cons in LIN_I; [|lia].
case Esub: (i - len) => [|i']; [lia|]. rewrite -/len Esub in LIN_I.
have {Esub}Hi : i = len + S i' by lia. subst.
subst len. unfold lin_of_omo in *. exists (ro_event (length omo) i'). split.
+ rewrite /eidx_to_lin_idx. rewrite take_app_le; [|lia]. rewrite take_ge; [|lia]. done.
+ rewrite -H lookup_omo_ro_event /omo_append_w /omo_read_op fmap_app.
replace (length omo) with (length omo.*2); [|by rewrite fmap_length].
rewrite lookup_app_1_eq. done.
Qed.
Lemma lookup_omo_lin_of_omo_lookup omo eidx e
(OMO_LOOKUP : lookup_omo omo eidx = Some e) :
lin_of_omo omo !! (eidx_to_lin_idx omo eidx) = Some e.
Proof.
unfold lin_of_omo, eidx_to_lin_idx.
set (f := λ '(e0, es), e0 :: es) in *.
destruct eidx as [gen' i'|gen']; simpl in *.
- rewrite -{2}(take_drop gen' omo) fmap_app concat_app.
rewrite lookup_app_r; [|lia].
replace (length (concat (f <$> take gen' omo)) + S i' - length (concat (f <$> take gen' omo))) with (S i') by lia.
destruct (omo !! gen') eqn:Heq; last first.
{ rewrite list_lookup_fmap Heq /= in OMO_LOOKUP. done. }
destruct p as [e' es'].
rewrite (drop_S _ (e', es')); [|done]. simpl. rewrite lookup_app_l.
+ by rewrite list_lookup_fmap Heq /= in OMO_LOOKUP.
+ rewrite list_lookup_fmap Heq /= in OMO_LOOKUP. by apply lookup_lt_Some in OMO_LOOKUP.
- rewrite -{2}(take_drop gen' omo) fmap_app concat_app.
rewrite lookup_app_r; [|lia].
replace (length (concat (f <$> take gen' omo)) - length (concat (f <$> take gen' omo))) with 0 by lia.
destruct (omo !! gen') eqn:Heq; last first.
{ rewrite list_lookup_fmap Heq /= in OMO_LOOKUP. done. }
destruct p as [e' es'].
rewrite (drop_S _ (e', es')); [|done]. simpl.
by rewrite list_lookup_fmap Heq /= in OMO_LOOKUP.
Qed.
Lemma event_in_omo (E : history) omo e
(LIN_PERM : lin_of_omo omo ≡ₚ seq 0 (length E))
(IS : is_Some (E !! e)) :
∃ eidx, lookup_omo omo eidx = Some e.
Proof.
have NODUP : NoDup (lin_of_omo omo) by by eapply ord_nodup.
move: IS => /lookup_lt_is_Some /lookup_xo => XO_LOOKUP.
apply Permutation_sym,Permutation_inj in LIN_PERM as [_ (f & _ & LIN_PERM)].
specialize (LIN_PERM e). rewrite LIN_PERM in XO_LOOKUP.
specialize (lin_of_omo_lookup_lookup_omo _ _ _ NODUP XO_LOOKUP). i. des.
by exists eidx.
Qed.
Lemma logview_in_omo (E : history) omo (M : eView)
(LIN_PERM : lin_of_omo omo ≡ₚ seq 0 (length E))
(SubME : set_in_bound M E) :
set_Forall (λ e, ∃ eidx, lookup_omo omo eidx = Some e) M.
Proof. intros ??. eapply event_in_omo; [done|]. by apply SubME. Qed.
Lemma longer_take i1 i2 (lss : list (list event_id)) (LE: i1 ≤ i2):
length (concat (take i1 lss))
≤ length (concat (take i2 lss)).
Proof.
apply prefix_length.
revert lss i2 LE. induction i1; intros; simpl.
{ apply prefix_nil. }
destruct i2; first lia. destruct lss; first done; simpl.
apply prefix_app, IHi1; lia.
Qed.
Lemma lin_idx_r_i'_mono omo gen i1' i2' (LEi' : i1' ≤ i2') :
eidx_to_lin_idx omo (ro_event gen i1') ≤ eidx_to_lin_idx omo (ro_event gen i2').
Proof.
unfold eidx_to_lin_idx.
destruct gen; lia.
Qed.
Lemma lin_idx_w_gen_mono omo gen1' gen2' (LEgen : gen1' ≤ gen2') :
eidx_to_lin_idx omo (wr_event gen1') ≤ eidx_to_lin_idx omo (wr_event gen2').
Proof.
simpl. do 2 rewrite fmap_take.
by apply longer_take.
Qed.
Lemma lin_idx_r_w_gen_mono omo gen1 i1' gen2
(OMO_LOOKUP1 : is_Some (lookup_omo omo (ro_event gen1 i1')))
(LTgen : gen1 < gen2) :
eidx_to_lin_idx omo (ro_event gen1 i1') < eidx_to_lin_idx omo (wr_event gen2).
Proof.
inversion OMO_LOOKUP1 as [e LOOKUP].
apply lookup_omo_inv_r in LOOKUP as [es [LOOKUP1 LOOKUP2]].
apply lookup_lt_Some in LOOKUP2; clear e.
eapply Nat.lt_le_trans; last first.
{ apply lin_idx_w_gen_mono. instantiate (1 := S gen1); lia. }
unfold eidx_to_lin_idx.
destruct (omo !! gen1) eqn:Heq; last first.
{ rewrite /omo_read_op list_lookup_fmap Heq /= in LOOKUP1. done. }
destruct p as [e1 es1]. rewrite (take_S_r _ _ (e1,es1)); [|done].
rewrite fmap_app concat_app app_length. rewrite app_length /=.
rewrite list_lookup_fmap Heq /= in LOOKUP1. inversion LOOKUP1. lia.
Qed.
Lemma lin_idx_w_r_gen_mono omo gen1 gen2 i2' (LEgen : gen1 ≤ gen2) :
eidx_to_lin_idx omo (wr_event gen1) < eidx_to_lin_idx omo (ro_event gen2 i2').
Proof.
unfold eidx_to_lin_idx. do 2 rewrite fmap_take. eapply Nat.le_lt_trans.
- apply longer_take. done.
- lia.
Qed.
Lemma lin_idx_r_gen_mono omo gen1 gen2 i1' i2'
(OMO_LOOKUP1 : is_Some (lookup_omo omo (ro_event gen1 i1')))
(LTgen : gen1 < gen2) :
eidx_to_lin_idx omo (ro_event gen1 i1') < eidx_to_lin_idx omo (ro_event gen2 i2').
Proof.
eapply Nat.lt_le_trans.
- by apply lin_idx_r_w_gen_mono.
- simpl; lia.
Qed.
Lemma omo_extract_eeVs E omo stlist es `{Interpretable eventT absStateT}
(OMO_GOOD : Linearizability_omo E omo stlist)
(EIDS_VALID : Forall (λ e, e < length E) es) :
∃ eeVs,
eeVs.*1 = es ∧
eid_to_event_valid E eeVs.
Proof.
induction es.
- exists []. split; try done. unfold eid_to_event_valid. by rewrite Forall_nil.
- rewrite Forall_cons in EIDS_VALID. destruct EIDS_VALID as [H1 H2].
specialize (IHes H2) as [eeVs [EQ VALID]].
rewrite -lookup_lt_is_Some in H1. destruct H1 as [eV Ha].
exists ((a, eV) :: eeVs). split.
+ rewrite fmap_cons. rewrite EQ. done.
+ unfold eid_to_event_valid. rewrite Forall_cons. done.
Qed.
Lemma omo_write_op_to_eeVs E omo stlist `{Interpretable eventT absStateT}
(OMO_GOOD : Linearizability_omo E omo stlist) :
∃ eeVs,
eeVs.*1 = omo_write_op omo ∧
eid_to_event_valid E eeVs.
Proof.
eapply omo_extract_eeVs; [done|].
rewrite Forall_lookup. intros. rewrite -lookup_lt_is_Some.
eapply lookup_omo_event_valid; [done|].
rewrite -lookup_omo_wr_event in H0. done.
Qed.
Lemma lin_of_omo_to_eeVs E omo stlist `{Interpretable eventT absStateT}
(OMO_GOOD : Linearizability_omo E omo stlist) :
∃ eeVs,
eeVs.*1 = lin_of_omo omo ∧
eid_to_event_valid E eeVs.
Proof.
eapply omo_extract_eeVs; [done|].
rewrite Forall_lookup. intros. rewrite -lookup_lt_is_Some.
apply lin_of_omo_lookup_lookup_omo in H0 as [eidx [_ H']].
- by eapply lookup_omo_event_valid.
- destruct OMO_GOOD. by eapply ord_nodup.
Qed.
Lemma omo_xo_to_eeVs E omo stlist `{Interpretable eventT absStateT}
(OMO_GOOD : Linearizability_omo E omo stlist) :
∃ eeVs,
eeVs.*1 = seq 0 (length E) ∧
eid_to_event_valid E eeVs.
Proof.
eapply omo_extract_eeVs; [done|].
rewrite Forall_lookup. intros.
rewrite lookup_seq in H0. lia.
Qed.
Lemma eid_to_event_valid_agree_from_eid E eeVs eeVs'
(EEVS_VALID_1 : eid_to_event_valid E eeVs)
(EEVS_VALID_2 : eid_to_event_valid E eeVs')
(EID_MATCH : eeVs.*1 = eeVs'.*1) :
eeVs = eeVs'.
Proof.
generalize dependent eeVs'. induction eeVs; intros.
- by destruct eeVs'.
- destruct eeVs'; try done.
rewrite !fmap_cons in EID_MATCH. inversion EID_MATCH.
destruct a, p. simpl in H0. subst e0.
have [VALID VALID'] : eid_to_event_valid E eeVs ∧ eid_to_event_valid E eeVs'.
{ unfold eid_to_event_valid in *. rewrite !Forall_cons in EEVS_VALID_1, EEVS_VALID_2. by destruct EEVS_VALID_1, EEVS_VALID_2. }
rewrite /eid_to_event_valid !Forall_cons in EEVS_VALID_1, EEVS_VALID_2.
destruct EEVS_VALID_1 as [H2 _]. destruct EEVS_VALID_2 as [H3 _].
rewrite H2 in H3. inversion H3. subst o0.
specialize (IHeeVs VALID _ VALID' H1). subst eeVs'. done.
Qed.
Lemma eid_to_event_valid_mono_history E E' eeVs
(SubE : E ⊑ E')
(EEVS_VALID : eid_to_event_valid E eeVs) :
eid_to_event_valid E' eeVs.
Proof.
unfold eid_to_event_valid in *. rewrite Forall_lookup. intros. destruct x as [e eV].
rewrite Forall_lookup in EEVS_VALID. specialize (EEVS_VALID _ _ H). simpl in EEVS_VALID.
eapply prefix_lookup_Some; try done.
Qed.
Lemma eid_to_event_valid_mono_history_rev E E' eeVs
(SubE : E' ⊑ E)
(EEVS_VALID : eid_to_event_valid E eeVs)
(EID_VALID : Forall (λ e, is_Some (E' !! e)) eeVs.*1) :
eid_to_event_valid E' eeVs.
Proof.
unfold eid_to_event_valid in *. rewrite Forall_lookup. intros. destruct x as [e eV].
rewrite Forall_lookup in EID_VALID.
have LOOKUP : eeVs.*1 !! i = Some e by rewrite list_lookup_fmap H.
specialize (EID_VALID _ _ LOOKUP).
rewrite Forall_lookup in EEVS_VALID.
specialize (EEVS_VALID _ _ H). simpl in EEVS_VALID.
eapply prefix_lookup_inv; try done.
Qed.
Lemma eid_to_event_valid_mono_eeVs E eeVs eeVs'
(SubEEVS : eeVs ⊑ eeVs')
(EEVS_VALID : eid_to_event_valid E eeVs') :
eid_to_event_valid E eeVs.
Proof.
unfold eid_to_event_valid in *. rewrite Forall_lookup. intros. destruct x as [e eV].
apply (prefix_lookup_Some _ eeVs') in H; [|done].
rewrite Forall_lookup in EEVS_VALID. specialize (EEVS_VALID _ _ H). done.
Qed.
Lemma eid_to_event_valid_eeVs_app E eeVs eeVs'
(EEVS_VALID_1 : eid_to_event_valid E eeVs)
(EEVS_VALID_2 : eid_to_event_valid E eeVs') :
eid_to_event_valid E (eeVs ++ eeVs').
Proof. by unfold eid_to_event_valid in *; rewrite Forall_app. Qed.
Lemma eid_to_event_valid_perm E eeVs eeVs'
(PERM : eeVs ≡ₚ eeVs')
(EEVS_VALID : eid_to_event_valid E eeVs) :
eid_to_event_valid E eeVs'.
Proof.
unfold eid_to_event_valid in *. rewrite Forall_lookup. intros.
destruct x as [e eV]. symmetry in PERM.
apply elem_of_list_lookup_2 in H.
eapply Permutation_in in PERM; last first.
{ rewrite elem_of_list_In in H. done. }
rewrite -elem_of_list_In in PERM.
apply elem_of_list_lookup_1 in PERM as [i' LOOKUP].
rewrite Forall_lookup in EEVS_VALID. by specialize (EEVS_VALID _ _ LOOKUP).
Qed.
Lemma eid_to_event_valid_perm_from_eid E eeVs eeVs'
(EEVS_VALID_1 : eid_to_event_valid E eeVs)
(EEVS_VALID_2 : eid_to_event_valid E eeVs')
(PERM : eeVs.*1 ≡ₚ eeVs'.*1) :
eeVs ≡ₚ eeVs'.
Proof.
generalize dependent eeVs'. induction eeVs; intros.
- destruct eeVs'; try done. apply Permutation_length in PERM. simpl in PERM. lia.
- destruct a as [e eV]. rewrite fmap_cons in PERM.
apply (elem_of_Permutation_proper e) in PERM as ELEM.
have [eeVs'' PERM'] : ∃ eeVs'', eeVs' ≡ₚ (e, eV) :: eeVs''.
{ simpl in ELEM.
have ELEM' : e ∈ e :: eeVs.*1 by apply elem_of_list_here.
rewrite ELEM in ELEM'. apply elem_of_list_lookup_1 in ELEM' as [i Hi].
apply list_lookup_fmap_inv in Hi as [[e' eV'] [EQ Hi]]. simpl in EQ. subst e'.
have EQ : eV = eV'.
{ unfold eid_to_event_valid in *. rewrite !Forall_lookup in EEVS_VALID_1, EEVS_VALID_2.
have LOOKUP : ((e, eV) :: eeVs) !! 0 = Some (e, eV) by done.
specialize (EEVS_VALID_1 _ _ LOOKUP). specialize (EEVS_VALID_2 _ _ Hi). simpl in *.
rewrite EEVS_VALID_2 in EEVS_VALID_1. inversion EEVS_VALID_1. done. }
subst eV'. apply elem_of_list_lookup_2 in Hi. rewrite elem_of_Permutation in Hi. done. }
have PERM'' : eeVs'.*1 ≡ₚ ((e, eV) :: eeVs'').*1 by eapply fmap_Permutation. rewrite fmap_cons /= in PERM''.
have H1 : e :: eeVs.*1 ≡ₚ e :: eeVs''.*1 by eapply Permutation_trans.
apply cons_Permutation_inj_r in H1 as H2.
have EEVS_VALID_1' : eid_to_event_valid E eeVs.
{ unfold eid_to_event_valid in *. rewrite Forall_cons in EEVS_VALID_1. by destruct EEVS_VALID_1 as [_ ?]. }
have EEVS_VALID_2' : eid_to_event_valid E eeVs''.
{ unfold eid_to_event_valid in *. rewrite Forall_lookup. intros. destruct x as [e' eV'].
have IN : (e', eV') ∈ ((e, eV) :: eeVs'').
{ apply (elem_of_list_lookup_2 _ (S i)). simpl. done. }