Nuprl Lemma : pv11_p1_acc_rcv_p2a2
∀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)
⇒ ((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_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_headers_type: pv11_p1_headers_type{i:l}(Cmd),
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),
es-header: header(e),
Message: Message(f),
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],
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
int: ℤ,
token: "$token",
equal: s = t ∈ T,
bag: bag(T)
Definitions unfolded in proof :
vatype: ValueAllType,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
l_all: (∀x∈L.P[x]),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
listp: A List+,
name: Name,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
length: ||as||,
list_ind: list_ind,
pv11_p1_headers: pv11_p1_headers(),
cons: [a / b],
nil: [],
it: ⋅,
true: True,
select: L[n],
subtract: n - m,
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
pv11_p1_headers_fun: pv11_p1_headers_fun(Cmd),
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
atom-deq: AtomDeq,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
null: null(as),
has-es-info-type: has-es-info-type(es;e;f;T),
pi1: fst(t),
pi2: snd(t),
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
eo-msg-interface-constraint: eo-msg-interface-constraint(es;X;hdrs;f),
sq_type: SQType(T),
pv11_p1_headers_no_inputs: pv11_p1_headers_no_inputs(),
exists: ∃x:A. B[x],
es-info-body: msgval(e),
es-info-auth: es-info-auth(es;e),
top: Top,
make-msg-interface: make-msg-interface(i;l;m),
pv11_p1_p2a'send: pv11_p1_p2a'send(Cmd;f),
mk-msg-interface: mk-msg-interface(l;m),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
pv11_p1_init_active: pv11_p1_init_active()
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e:E.
\mforall{}accpts,ldrs:bag(Id). \mforall{}ldrs$_{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$; reps; f; es)
{}\mRightarrow{} (header(e) = ``pv11\_p1 p2a``)
{}\mRightarrow{} has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)
{}\mRightarrow{} ((fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{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:
2016_05_17-PM-04_06_57
Last ObjectModification:
2016_01_18-AM-11_16_41
Theory : paxos!synod
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