Nuprl Lemma : pv11_p1_adopted_from_maj_acceptors
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e:E. ∀accpts,ldrs:bag(Id).
∀ldrs_uid:Id ─→ ℤ. ∀reps:bag(Id). ∀bnum:pv11_p1_Ballot_Num(). ∀pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.
(Inj(Id;ℤ;ldrs_uid)
⇒ pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e)
⇒ (↓∃L:E List
(#([i∈accpts|¬bi ∈b map(λe.loc(e);L))]) < pv11_p1_threshold(accpts)
∧ (∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
((pv ∈ pvals) ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L)))))
∧ (∀e':E
((e' ∈ L)
⇒ ((e' < e)
∧ loc(e') ↓∈ accpts
∧ (bnum = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())))))))
Proof
Definitions occuring in Statement :
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
pv11_p1_AcceptorStateFun: pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;mf;es;e),
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
pv11_p1_threshold: pv11_p1_threshold(accpts),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-causl: (e < e'),
es-loc: loc(e),
es-E: E,
id-deq: IdDeq,
Id: Id,
deq-member: x ∈b L),
l_member: (x ∈ l),
concat: concat(ll),
map: map(f;as),
list: T List,
inject: Inj(A;B;f),
vatype: ValueAllType,
bnot: ¬bb,
less_than: a < b,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
lambda: λx.A[x],
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag-size: #(bs),
bag-filter: [x∈b|p[x]],
bag: bag(T)
Lemmas :
int_seg_wf,
length_wf,
name_wf,
pv11_p1_headers_wf,
l_all_iff,
l_member_wf,
equal_wf,
pv11_p1_headers_fun_wf,
cons_wf_listp,
nil_wf,
listp_wf,
cons_member,
cons_wf,
equal-wf-base,
iff_weakening_equal,
classrel_wf,
pv11_p1_adopted'base_wf,
pv11_p1_message-constraint_wf,
inject_wf,
list_wf,
pv11_p1_Ballot_Num_wf,
bag_wf,
Id_wf,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
Message_wf,
subtype_rel_dep_function,
vatype_wf,
pv11_p1_headers_type_wf,
set_wf,
valueall-type_wf,
base-noloc-classrel-make-Msg2,
hdrmkmsg_lemma,
msg-header_wf,
pv11_p1_headers_no_inputs_wf,
squash_wf,
exists_wf,
es-causl_wf,
msg-interface_wf,
pv11_p1_main_wf,
make-msg-interface_wf,
es-loc_wf,
msg-authentic_wf,
es-info_wf,
pv11_p1-adopted,
es-init-le,
no-classrel-in-interval_wf,
true_wf,
eclass_wf,
eclass2_wf,
eclass1_wf,
pv11_p1_p1b'base_wf,
pv11_p1_scout_output_wf,
pv11_p1_ScoutState_wf,
es-info-body_wf,
pv11_p1_upd_bnum_wf,
subtype_base_sq,
union_subtype_base,
unit_wf2,
product_subtype_base,
int_subtype_base,
atom2_subtype_base,
unit_subtype_base,
pv11_p1_ScoutStateFun_wf,
eo-forward_wf,
member-eo-forward-E,
and_wf,
pi2_wf,
less_than_wf,
bag-size_wf,
nat_wf,
pv11_p1_threshold_wf,
es-init_wf,
es-le_wf,
es-first-init,
assert_of_tt,
pi1_wf_top,
top_wf,
eo-forward-trivial,
subtype_rel_product,
subtype_top,
es-locl_wf,
not_wf,
pv11_p1_scout_state_from_p1bs,
length_wf_nat,
pair-eta,
es-causl_transitivity2,
es-causle_weakening,
assert_wf,
bnot_wf,
deq-member_wf,
id-deq_wf,
map_wf,
bag-filter_wf,
all_wf,
iff_wf,
concat_wf,
pv11_p1_AcceptorStateFun_wf,
bag-member_wf
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e:E.
\mforall{}accpts,ldrs:bag(Id). \mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}reps:bag(Id). \mforall{}bnum:pv11\_p1\_Ballot\000C\_Num().
\mforall{}pvals:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List.
(Inj(Id;\mBbbZ{};ldrs$_{uid}$)
{}\mRightarrow{} pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid\mbackslash{}ff7\000Cd$; reps; f; es)
{}\mRightarrow{} <bnum, pvals> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e)
{}\mRightarrow{} (\mdownarrow{}\mexists{}L:E List
(\#([i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))]) < pv11\_p1\_threshold(accpts)
\mwedge{} (\mforall{}pv:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd
((pv \mmember{} pvals)
\mLeftarrow{}{}\mRightarrow{} (pv \mmember{} concat(map(\mlambda{}e.(snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}\mbackslash{}f\000Cf24;f;es;e)));L)))))
\mwedge{} (\mforall{}e':E
((e' \mmember{} L)
{}\mRightarrow{} ((e' < e)
\mwedge{} loc(e') \mdownarrow{}\mmember{} accpts
\mwedge{} (bnum = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e'\000C))))))))))
Date html generated:
2015_07_23-PM-04_59_16
Last ObjectModification:
2015_02_04-AM-07_52_54
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