Nuprl Lemma : vote-crosses-threshold
∀[V:Type]. ∀[A:Id List]. ∀[t:ℕ+]. ∀[n1:ℕ]. ∀[n2:ℤ]. ∀[v1:V]. ∀[L1:consensus-rcv(V;A) List]. ∀[a:{a:Id| (a ∈ A)} ].
({(n1 = n2 ∈ ℤ) ∧ (¬↑null(filter(λr.n1 - 1 <z inning(r);L1)))}) supposing
((((2 * t) + 1) ≤ ||values-for-distinct(IdDeq;votes-from-inning(n2;L1 @ [Vote[a;n1;v1]]))||) and
(||values-for-distinct(IdDeq;votes-from-inning(n2;L1))|| ≤ (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+,
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
le: A ≤ B,
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],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
guard: {T},
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
le: A ≤ B,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
cs-rcv-vote: Vote[a;i;v],
consensus-rcv: consensus-rcv(V;A),
pi1: fst(t),
pi2: snd(t),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
squash: ↓T,
true: True
Latex:
\mforall{}[V:Type]. \mforall{}[A:Id List]. \mforall{}[t:\mBbbN{}\msupplus{}]. \mforall{}[n1:\mBbbN{}]. \mforall{}[n2:\mBbbZ{}]. \mforall{}[v1:V]. \mforall{}[L1:consensus-rcv(V;A) List].
\mforall{}[a:\{a:Id| (a \mmember{} A)\} ].
(\{(n1 = n2) \mwedge{} (\mneg{}\muparrow{}null(filter(\mlambda{}r.n1 - 1 <z inning(r);L1)))\}) supposing
((((2 * t) + 1) \mleq{} ||values-for-distinct(IdDeq;votes-from-inning(n2;L1
@ [Vote[a;n1;v1]]))||) and
(||values-for-distinct(IdDeq;votes-from-inning(n2;L1))|| \mleq{} (2 * t)))
Date html generated:
2016_05_16-PM-00_39_23
Last ObjectModification:
2016_01_17-PM-08_00_21
Theory : event-ordering
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