Nuprl Lemma : consensus-rcv-crosses-threshold
∀[V:Type]
∀A:Id List. ∀t:ℕ+. ∀n:ℤ. ∀L:consensus-rcv(V;A) List. ∀r:consensus-rcv(V;A).
(∃a:{a:Id| (a ∈ A)}
∃v:V. ((r = Vote[a;n;v] ∈ consensus-rcv(V;A)) ∧ (¬↑null(filter(λr.n - 1 <z inning(r);L))) ∧ (0 ≤ n))) supposing
((((2 * t) + 1) ≤ ||values-for-distinct(IdDeq;votes-from-inning(n;L @ [r]))||) and
(||values-for-distinct(IdDeq;votes-from-inning(n;L))|| ≤ (2 * t)))
Proof
Definitions occuring in Statement :
votes-from-inning: votes-from-inning(i;L),
rcvd-inning-gt: i <z inning(r),
cs-rcv-vote: Vote[a;i;v],
consensus-rcv: consensus-rcv(V;A),
id-deq: IdDeq,
Id: Id,
values-for-distinct: values-for-distinct(eq;L),
l_member: (x ∈ l),
length: ||as||,
null: null(as),
filter: filter(P;l),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
nat_plus: ℕ+,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat_plus: ℕ+,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
values-for-distinct: values-for-distinct(eq;L),
top: Top,
pi1: fst(t),
votes-from-inning: votes-from-inning(i;L),
rcvd-inning-eq: inning(r) =z i,
rcvd-vote: rcvd-vote(x),
mapfilter: mapfilter(f;P;L),
rcv-vote?: rcv-vote?(x),
consensus-rcv: consensus-rcv(V;A),
outr: outr(x),
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
spreadn: spread3,
btrue: tt,
nat: ℕ,
ge: i ≥ j ,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cs-rcv-vote: Vote[a;i;v],
cand: A c∧ B,
sq_type: SQType(T),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
Id: Id,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Latex:
\mforall{}[V:Type]
\mforall{}A:Id List. \mforall{}t:\mBbbN{}\msupplus{}. \mforall{}n:\mBbbZ{}. \mforall{}L:consensus-rcv(V;A) List. \mforall{}r:consensus-rcv(V;A).
(\mexists{}a:\{a:Id| (a \mmember{} A)\}
\mexists{}v:V. ((r = Vote[a;n;v]) \mwedge{} (\mneg{}\muparrow{}null(filter(\mlambda{}r.n - 1 <z inning(r);L))) \mwedge{} (0 \mleq{} n))) supposing
((((2 * t) + 1) \mleq{} ||values-for-distinct(IdDeq;votes-from-inning(n;L @ [r]))||) and
(||values-for-distinct(IdDeq;votes-from-inning(n;L))|| \mleq{} (2 * t)))
Date html generated:
2016_05_16-PM-00_39_02
Last ObjectModification:
2016_01_17-PM-08_01_04
Theory : event-ordering
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