Nuprl Lemma : pv11_p1_acc_rcv_p2a
∀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.
(Inj(Id;ℤ;ldrs_uid)
⇒ pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ (header(e) = ``pv11_p1 p2a`` ∈ Name)
⇒ has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
⇒ (∃e':E. ∃i,l:Id. pv11_p1_p2a'send(Cmd;f) i <l, snd(msgval(e))> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e'))
⇒ ((fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))) = (fst(snd(msgval(e)))) ∈ pv11_p1_Ballot_Num())
⇒ (↓(snd(msgval(e)) ∈ 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_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,
es-info-body: msgval(e),
has-es-info-type: has-es-info-type(es;e;f;T),
es-header: header(e),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
name: Name,
l_member: (x ∈ l),
cons: [a / b],
nil: [],
inject: Inj(A;B;f),
vatype: ValueAllType,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
apply: f a,
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
token: "$token",
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 :
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,
pv11_p1_AcceptorStateFun_wf,
list_wf,
es-info-body_wf,
exists_wf,
classrel_wf,
msg-interface_wf,
pv11_p1_main_wf,
pv11_p1_p2a'send_wf,
has-es-info-type_wf,
equal-wf-T-base,
es-header_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,
member-base-class_iff,
member-eclass_wf,
pv11_p1_p1a'base_wf,
bool_wf,
eqtt_to_assert,
es-first_wf2,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
pv11_p1_p2a'base_wf,
pi1_wf_top,
squash_wf,
true_wf,
top_wf,
pv11_p1_acc_state_fun_eq2,
subtype_rel_product,
subtype_top,
pi2_wf,
member-base-class,
assert-name_eq,
classfun-res_wf,
base-headers-msg-val-single-val,
subtype_rel_weakening,
ext-eq_weakening,
es-pred_wf,
pair-eta,
subtype_rel_transitivity,
es-info-type_wf,
pv11_p1_leq_bnum_wf,
pv11_p1_dummy_ballot_wf,
pv11_p1_leq_bnum_dummy,
classfun-res-base,
reduce_nil_lemma,
pv11_p1_same_pvalue_wf,
or_wf,
bl-exists_wf,
assert-bl-exists,
false_wf,
l_exists_wf_nil,
assert_wf,
member_wf,
pv11_p1_add_if_new_iff2,
l_exists_iff,
pv11_p1_acc_fun_p2a_pv,
l_exists_wf,
base-noloc-classrel,
list_subtype_base,
atom_subtype_base,
assert-deq-member,
pv11_p1_headers_no_inputs_wf,
es-info-mk-msg,
event-ordering+_cumulative,
es-info-auth_wf,
make-msg-interface_wf,
es-loc_wf,
mk-msg_wf,
int_subtype_base,
pv11_p1-p2a,
pv11_p1_A4_C1,
eclass_wf,
pv11_p1_AcceptorState_wf,
pv11_p1_eq_bnums_wf,
pv11_p1_eq_bnums-assert,
pv11_p1_same_proposal_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
eq_int_wf,
assert_of_eq_int,
and_wf,
pv11_p1_bnum_p2a,
assert-union-deq,
unit_wf2,
product-deq_wf,
int-deq_wf,
id-deq_wf,
unit-deq_wf,
pv11_p1_mk_bnum_wf,
assert_of_band2,
union_subtype_base,
product_subtype_base,
atom2_subtype_base,
unit_subtype_base,
outl_wf,
isl_wf,
assert_functionality_wrt_uiff,
pv11_p1_max_bnum_wf,
pv11_p1_leq_bnum_max
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}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}reps:bag(Id). \mforall{}b:pv11\_p1\_Ballot\_Nu\000Cm(). \mforall{}s:\mBbbZ{}. \mforall{}c:Cmd.
(Inj(Id;\mBbbZ{};ldrs$_{uid}$)
{}\mRightarrow{} pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid\mbackslash{}ff7\000Cd$; reps; f; es)
{}\mRightarrow{} (header(e) = ``pv11\_p1 p2a``)
{}\mRightarrow{} has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)
{}\mRightarrow{} (\mexists{}e':E. \mexists{}i,l:Id. pv11\_p1\_p2a'send(Cmd;f) i <l, snd(msgval(e))> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;l\000Cdrs$_{uid}$;reps;f)(e'))
{}\mRightarrow{} ((fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e))) = (fst(snd(msgval\000C(e)))))
{}\mRightarrow{} (\mdownarrow{}(snd(msgval(e)) \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e)))\000C))
Date html generated:
2015_07_23-PM-04_57_53
Last ObjectModification:
2015_02_04-AM-08_05_58
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