Nuprl Lemma : pv11_p1_A5_C2
∀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). ∀i,l:Id. ∀b,b':pv11_p1_Ballot_Num(). ∀s:ℤ. ∀p,p':Cmd. ∀as,cs:bag(Id).
(pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ Inj(Id;ℤ;ldrs_uid)
⇒ (accpts = (as + cs) ∈ bag(Id))
⇒ #(as) < pv11_p1_threshold(accpts)
⇒ (∀a:Id
(a ↓∈ cs ⇒ (↓∃e':E. ((a = loc(e') ∈ Id) ∧ (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))))))
⇒ ((<b', s, p'> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)))
∨ pv11_p1_p2a'send(Cmd;f) i <l, b', s, p'> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e))
⇒ (↑(b < b'))
⇒ (p = p' ∈ Cmd))
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_threshold: pv11_p1_threshold(accpts),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_lt_bnum: pv11_p1_lt_bnum(ldrs_uid),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
msg-interface: Interface,
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-loc: loc(e),
es-E: E,
Id: Id,
l_member: (x ∈ l),
inject: Inj(A;B;f),
vatype: ValueAllType,
assert: ↑b,
less_than: a < b,
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag-size: #(bs),
bag-append: as + bs,
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),
or: P ∨ Q,
exists: ∃x:A. B[x],
nat: ℕ,
strongwellfounded: SWellFounded(R[x; y]),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
decidable: Dec(P),
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),
pv11_p1_p2a'base: pv11_p1_p2a'base(Cmd;mf),
uiff: uiff(P;Q),
es-header: header(e),
pv11_p1_headers_no_inputs: pv11_p1_headers_no_inputs(),
pv11_p1_p2a'send: pv11_p1_p2a'send(Cmd;f),
mk-msg-interface: mk-msg-interface(l;m),
make-msg-interface: make-msg-interface(i;l;m),
sq_type: SQType(T),
pv11_p1_init_active: pv11_p1_init_active(),
make-Msg: make-Msg(hdr;val),
mk-msg: mk-msg(auth;hdr;val),
has-es-info-type: has-es-info-type(es;e;f;T),
bag-deq-member: bag-deq-member(eq;x;b),
deq-member: x ∈b L,
reduce: reduce(f;k;as),
map: map(f;as),
bag-map: bag-map(f;bs),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
Id: Id,
cand: A c∧ B,
sq_stable: SqStable(P),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
pi1: fst(t),
assert: ↑b,
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
pi2: snd(t),
encodes-msg-type: hdr encodes T,
equal-info-body: v = body(e)
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{}i,l:Id. \mforall{}b,b':pv11\_\000Cp1\_Ballot\_Num(). \mforall{}s:\mBbbZ{}.
\mforall{}p,p':Cmd. \mforall{}as,cs: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}$)
{}\mRightarrow{} (accpts = (as + cs))
{}\mRightarrow{} \#(as) < pv11\_p1\_threshold(accpts)
{}\mRightarrow{} (\mforall{}a:Id
(a \mdownarrow{}\mmember{} cs
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((a = loc(e')) \mwedge{} (<b, s, p> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid\mbackslash{}\000Cff7d$;f;es;e')))))))
{}\mRightarrow{} ((<b', s, p'> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e)))
\mvee{} pv11\_p1\_p2a'send(Cmd;f) i <l, b', s, p'> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{ui\000Cd}$;reps;f)(e))
{}\mRightarrow{} (\muparrow{}(b < b'))
{}\mRightarrow{} (p = p'))
Date html generated:
2016_05_17-PM-04_09_54
Last ObjectModification:
2016_01_18-AM-11_19_30
Theory : paxos!synod
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