Nuprl Lemma : pv11_p1_commander_state_from_p2bs
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀start,e,e0:E. ∀accpts,ldrs:bag(Id).
∀ldrs_uid:Id ─→ ℤ. ∀reps:bag(Id). ∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀c:Cmd. ∀waitfor:bag(Id). ∀i,l:Id.
(start ≤loc e
⇒ Inj(Id;ℤ;ldrs_uid)
⇒ pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ pv11_p1_p2a'send(Cmd;f) i <l, b, s, c> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e0)
⇒ (waitfor = pv11_p1_CommanderStateFun(Cmd;accpts;f;b;s;es.start;e) ∈ bag(Id))
⇒ (↓∃L:E List
((waitfor = [i∈accpts|¬bi ∈b map(λe.loc(e);L))] ∈ bag(Id))
∧ (∀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_p2a'send: pv11_p1_p2a'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_CommanderStateFun: pv11_p1_CommanderStateFun(Cmd;accpts;mf;x;zzs;es;e),
pv11_p1_AcceptorStateFun: pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;mf;es;e),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
msg-interface: Interface,
Message: Message(f),
classrel: v ∈ X(e),
eo-forward: eo.e,
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
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,
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],
int: ℤ,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag-filter: [x∈b|p[x]],
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 :
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,
pv11_p1_CommanderStateFun_wf,
eo-forward_wf,
member-eo-forward-E,
es-loc_wf,
classrel_wf,
msg-interface_wf,
pv11_p1_main_wf,
pv11_p1_p2a'send_wf,
pv11_p1_message-constraint_wf,
inject_wf,
es-le_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,
es-causl-swellfnd,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_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_commander_state_fun_eq,
squash_wf,
exists_wf,
bag-filter_wf,
bnot_wf,
deq-member_wf,
id-deq_wf,
map_wf,
subtype_rel_bag,
assert_wf,
all_wf,
bag-member_wf,
pv11_p1_AcceptorStateFun_wf,
eo-forward-member-eclass,
event-ordering+_cumulative,
member-eclass_wf,
pv11_p1_p2b'base_wf,
bool_wf,
eqtt_to_assert,
es-first_wf2,
member-base-class,
eo-forward-base-classfun-res-sq,
subtype_rel_weakening,
ext-eq_weakening,
classfun-res-base,
subtype_base_sq,
list_subtype_base,
atom_subtype_base,
subtype_rel_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
decidable__lt,
not-equal-2,
le-add-cancel-alt,
not-le-2,
sq_stable__le,
add-mul-special,
zero-mul,
pair-eta,
es-info-body_wf,
subtype_rel_product,
top_wf,
subtype_top,
subtype_rel_transitivity,
es-info-type_wf,
pv11_p1_eq_bnums-assert,
assert_of_eq_int,
pv11_p1_eq_bnums_wf,
eq_int_wf,
neg_assert_of_eq_int,
decidable__assert,
bag-deq-member_wf,
assert-bag-deq-member,
pi1_wf_top,
assert-deq-member,
pv11_p1_headers_no_inputs_wf,
es-info-mk-msg,
es-info-auth_wf,
make-msg-interface_wf,
mk-msg_wf,
pv11_p1-p2b,
int_subtype_base,
pv11_p1-p2a,
pv11_p1_A4_C1_funC,
pv11_p1_acc_rcv_p2a2,
member_singleton,
map_cons_lemma,
map_nil_lemma,
deq_member_cons_lemma,
deq_member_nil_lemma,
bag-filter-equal,
bor_wf,
bfalse_wf,
assert_functionality_wrt_uiff,
true_wf,
bor_ff_simp,
deq_wf,
and_wf,
pair_eta_rw,
pi2_wf,
bag-filter-trivial,
btrue_wf,
assert_of_tt,
bag-remove-trivial,
null_nil_lemma,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
event_ordering_wf,
eo-forward-trivial,
es-le-first,
assert_elim,
ifthenelse_wf,
eo-forward-first,
es-le-self,
eq_id_wf,
assert-eq-id,
assert-es-eq-E-2,
eo-forward-loc,
es-pred_wf,
es-pred-locl,
eo-forward-locl,
es-causl_weakening,
es-le-pred,
eo-forward-pred,
equal-eo-forward-E,
eo-forward-E-subtype,
pv11_p1_on_p2b_wf,
bag-remove_wf,
bag-filter-filter,
iff_transitivity,
not_wf,
or_wf,
iff_weakening_uiff,
assert_of_bnot,
assert_of_bor,
assert_of_band,
es-causl_transitivity1,
es-causle_weakening_eq,
es-causle_wf,
es-causl_transitivity2,
es-causle_weakening_locl,
es-le_weakening
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}start,e,e0: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{}waitfor:bag(Id). \mforall{}i,l:Id.
(start \mleq{}loc e
{}\mRightarrow{} 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{} pv11\_p1\_p2a'send(Cmd;f) i <l, b, s, c> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid\mbackslash{}ff\000C7d$;reps;f)(e0)
{}\mRightarrow{} (waitfor = pv11\_p1\_CommanderStateFun(Cmd;accpts;f;b;s;es.start;e))
{}\mRightarrow{} (\mdownarrow{}\mexists{}L:E List
((waitfor = [i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))])
\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-04_59_07
Last ObjectModification:
2015_02_04-AM-08_05_38
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