Nuprl Lemma : pv11_p1_decision_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). ∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀c:Cmd. ∀i:Id.
(Inj(Id;ℤ;ldrs_uid)
⇒ pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ 0 < #(accpts)
⇒ has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
⇒ (b = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num())
⇒ pv11_p1_decision'send(Cmd;f) i <s, c> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e)
⇒ (↓∃L:E List
(#([i∈accpts|¬bi ∈b map(λe.loc(e);L))]) < pv11_p1_threshold(accpts)
∧ (∀e':E
((e' ∈ L)
⇒ ((e' < e)
∧ loc(e') ↓∈ accpts
∧ (b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())
∧ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))))))))
Proof
Definitions occuring in Statement :
pv11_p1_decision'send: pv11_p1_decision'send(Cmd;f),
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
pv11_p1_main: pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;mf),
pv11_p1_AcceptorStateFun: pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;mf;es;e),
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(),
msg-interface: Interface,
es-info-body: msgval(e),
has-es-info-type: has-es-info-type(es;e;f;T),
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),
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],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag-size: #(bs),
bag-filter: [x∈b|p[x]],
bag: bag(T)
Lemmas :
int_seg_wf,
length_wf,
name_wf,
pv11_p1_headers_wf,
l_all_iff,
l_member_wf,
equal_wf,
pv11_p1_headers_fun_wf,
cons_wf_listp,
nil_wf,
listp_wf,
cons_member,
cons_wf,
equal-wf-base,
iff_weakening_equal,
list_wf,
classrel_wf,
msg-interface_wf,
pv11_p1_main_wf,
pv11_p1_decision'send_wf,
es-info-body_wf,
has-es-info-type_wf,
less_than_wf,
bag-size_wf,
nat_wf,
pv11_p1_message-constraint_wf,
inject_wf,
pv11_p1_Ballot_Num_wf,
bag_wf,
Id_wf,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
Message_wf,
subtype_rel_dep_function,
vatype_wf,
pv11_p1_headers_type_wf,
set_wf,
valueall-type_wf,
pv11_p1-decision,
bfalse_wf,
bag-size-bag-member,
es-loc_wf,
pv11_p1_p2a'send_wf,
exists_wf,
es-le-loc,
and_wf,
bag-member_wf,
pv11_p1-p2a,
subtype_base_sq,
int_subtype_base,
union_subtype_base,
unit_wf2,
product_subtype_base,
atom2_subtype_base,
unit_subtype_base,
pv11_p1_commander_state_from_p2bs,
pv11_p1_CommanderStateFun_wf,
eo-forward_wf,
member-eo-forward-E,
pv11_p1_threshold_wf,
assert_wf,
bnot_wf,
deq-member_wf,
id-deq_wf,
map_wf,
bag-filter_wf,
all_wf,
es-causl_wf,
pv11_p1_AcceptorStateFun_wf
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e:E.
\mforall{}accpts,ldrs:bag(Id). \mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}reps:bag(Id). \mforall{}b:pv11\_p1\_Ballot\_Nu\000Cm(). \mforall{}s:\mBbbZ{}. \mforall{}c:Cmd. \mforall{}i:Id.
(Inj(Id;\mBbbZ{};ldrs$_{uid}$)
{}\mRightarrow{} pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid\mbackslash{}ff7\000Cd$; reps; f; es)
{}\mRightarrow{} 0 < \#(accpts)
{}\mRightarrow{} has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())
{}\mRightarrow{} (b = (snd(snd(snd(msgval(e))))))
{}\mRightarrow{} pv11\_p1\_decision'send(Cmd;f) i <s, c> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid\mbackslash{}ff7\000Cd$;reps;f)(e)
{}\mRightarrow{} (\mdownarrow{}\mexists{}L:E List
(\#([i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))]) < pv11\_p1\_threshold(accpts)
\mwedge{} (\mforall{}e':E
((e' \mmember{} L)
{}\mRightarrow{} ((e' < e)
\mwedge{} loc(e') \mdownarrow{}\mmember{} accpts
\mwedge{} (b = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e')))\000C)
\mwedge{} (<b, s, c> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;e\000Cs;e')))))))))
Date html generated:
2015_07_23-PM-05_00_09
Last ObjectModification:
2015_02_04-AM-07_52_38
Home
Index