Nuprl Lemma : pv11_p1_consistency_lemma
∀Cmd:{T:Type| valueall-type(T)} . ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀accpts,ldrs:bag(Id).
∀ldrs_uid:Id ─→ ℤ. ∀reps:bag(Id).
(pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ Inj(Id;ℤ;ldrs_uid)
⇒ (∀e:E. ∀b:pv11_p1_Ballot_Num(). ∀k:ℤ. ∀v:Cmd.
(pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e;<b, k, v>;f)
⇒ (∀e':E. ∀w:Cmd.
(pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e';k;w)
⇒ has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) (snd(snd(snd(msgval(e'))))) b))
⇒ (w = v ∈ Cmd))))))
Proof
Definitions occuring in Statement :
pv11_p1_decision: pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e;k;v),
pv11_p1_pvalue: pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e;x;f),
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_leq_bnum: pv11_p1_leq_bnum(ldrs_uid),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
es-info-body: msgval(e),
has-es-info-type: has-es-info-type(es;e;f;T),
Message: Message(f),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
inject: Inj(A;B;f),
valueall-type: valueall-type(T),
assert: ↑b,
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T,
bag: bag(T)
Definitions :
assert: ↑b,
ifthenelse: if b then t else f fi ,
deq-member: x ∈b L),
reduce: reduce(f;k;as),
pv11_p1_headers_no_inputs: pv11_p1_headers_no_inputs(),
cons: [a / b],
bor: p ∨bq,
list-deq: list-deq(eq),
band: p ∧b q,
atom-deq: AtomDeq,
eq_atom: x =a y,
btrue: tt,
bfalse: ff,
nil: [],
it: ⋅,
null: null(as),
true: True,
member: t ∈ T
Lemmas :
es-causl-swellfnd,
event-ordering+_subtype,
Message_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
es-info-body_wf,
assert_wf,
pv11_p1_leq_bnum_wf,
has-es-info-type_wf,
pv11_p1_decision_wf,
es-E_wf,
pv11_p1_pvalue_wf,
int_seg_wf,
int_seg_subtype-nat,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__equal_int,
subtype_rel-int_seg,
le_weakening,
int_seg_properties,
le_wf,
nat_wf,
zero-le-nat,
lelt_wf,
es-causl_wf,
pv11_p1_pvalue_from_p2a,
equal_wf,
decidable__lt,
not-equal-2,
le-add-cancel-alt,
not-le-2,
sq_stable__le,
add-mul-special,
zero-mul,
equal-wf-T-base,
name_wf,
es-header_wf,
subtype_rel-equal,
iff_weakening_equal,
l_member_wf,
exists_wf,
es-le_wf,
pv11_p1_ScoutStateFun_wf,
eo-forward_wf,
member-eo-forward-E,
Id_wf,
es-loc_wf,
bag_wf,
list_wf,
pv11_p1_AcceptorStateFun_wf,
pv11_p1_from-p2a_wf,
subtype_base_sq,
list_subtype_base,
atom_subtype_base,
subtype_rel_wf,
assert-deq-member,
pv11_p1_headers_no_inputs_wf,
cons_wf,
nil_wf,
es-info-mk-msg,
event-ordering+_cumulative,
es-info-auth_wf,
subtype_rel_transitivity,
es-info-type_wf,
subtype_rel_weakening,
ext-eq_weakening,
classrel_wf,
msg-interface_wf,
pv11_p1_main_wf,
make-msg-interface_wf,
pair-eta,
subtype_rel_product,
top_wf,
subtype_top,
mk-msg_wf,
pi1_wf_top,
int_subtype_base,
bool_wf,
bool_subtype_base,
pv11_p1-p2a,
pv11_p1_leq_bnum_implies_eq_or_lt,
pv11_p1_decision_from_p2a,
make-Msg-as-mk-msg,
and_wf,
pv11_p1_Ballot_Num_wf,
pv11_p1_p2a'send_wf,
pi2_wf,
bfalse_wf,
pv11_p1_A4_C1_funC,
bag-size-bag-member,
bag-member_wf,
pv11_p1_decision_from_maj_acceptors,
make-Msg_wf,
cons_wf_listp,
listp_wf,
pv11_p1-decision,
pv11_p1_A5_C2,
bag-filter_wf,
bnot_wf,
deq-member_wf,
id-deq_wf,
map_wf,
subtype_rel_bag,
bag-append-comm,
bag-split,
bag-member-filter,
member_map
Latex:
\mforall{}Cmd:\{T:Type| valueall-type(T)\} . \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)).
\mforall{}accpts,ldrs:bag(Id). \mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}reps:bag(Id).
(pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\mbackslash{}\000Cff24; reps; f; es)
{}\mRightarrow{} Inj(Id;\mBbbZ{};ldrs$_{uid}$)
{}\mRightarrow{} (\mforall{}e:E. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}k:\mBbbZ{}. \mforall{}v:Cmd.
(pv11\_p1\_pvalue\{i:l\}(Cmd;ldrs$_{uid}$;accpts;es;e;<b, k, v>f)
{}\mRightarrow{} (\mforall{}e':E. \mforall{}w:Cmd.
(pv11\_p1\_decision\{i:l\}(Cmd;f;accpts;ldrs;ldrs$_{uid}$;reps;es;e';k\000C;w)
{}\mRightarrow{} has-es-info-type(es;e';f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())
{}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (snd(snd(snd(msgval(e'))))) b\000C))
{}\mRightarrow{} (w = v))))))
Date html generated:
2015_07_23-PM-05_05_55
Last ObjectModification:
2015_02_04-AM-07_51_55
Home
Index