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),
filter: filter(P;l),
length: ||as||,
append: as @ bs,
null: null(as),
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
Lemmas :
less_than_wf,
length_wf,
values-for-distinct_wf,
Id_wf,
l_member_wf,
id-deq_wf,
subtype_rel-deq,
sq_stable__l_member,
decidable__equal_Id,
equal_wf,
set_wf,
votes-from-inning_wf,
append_wf,
consensus-rcv_wf,
cons_wf,
nil_wf,
length-map,
remove-repeats_wf,
map_wf,
mapfilter-append,
map_append_sq,
le_wf,
list_wf,
nat_plus_wf,
filter_cons_lemma,
filter_nil_lemma,
map_nil_lemma,
append-nil,
subtype_rel_list,
top_wf,
condition-implies-le,
minus-add,
minus-one-mul,
mul-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
equal-wf-base-T,
int_subtype_base,
map_cons_lemma,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
subtype_base_sq,
cs-rcv-vote_wf,
sq_stable__le,
le_transitivity,
le_weakening,
null_wf3,
filter_wf5,
rcvd-inning-gt_wf,
subtract_wf,
exists_wf,
assert_of_null,
votes-from-inning-is-nil,
subtract-is-less,
list_subtype_base,
set_subtype_base,
atom2_subtype_base,
list_ind_nil_lemma,
remove_repeats_cons_lemma,
remove_repeats_nil_lemma,
length_of_cons_lemma,
length_of_nil_lemma
\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:
2015_07_17-AM-11_49_32
Last ObjectModification:
2015_01_28-AM-00_44_06
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