Nuprl Lemma : consensus-accum-num-property1
∀[V:Type]
∀A:Id List. ∀t:ℕ+. ∀f:(V List) ─→ V. ∀v0:V. ∀L:consensus-rcv(V;A) List.
let b,i,as,vs,v = consensus-accum-num-state(t;f;v0;L) in (filter(λr.i - 1 <z inning(r);L)
= []
∈ (consensus-rcv(V;A) List))
∧ (||as|| = ||vs|| ∈ ℤ)
∧ (zip(as;vs) = votes-from-inning(i - 1;L) ∈ (({b:Id| (b ∈ A)} × V) List))
∧ (0 ≤ i)
∧ 1 ≤ i supposing ¬↑null(L)
∧ (||values-for-distinct(IdDeq;votes-from-inning(i - 1;L))|| ≤ (2 * t))
Proof
Definitions occuring in Statement :
consensus-accum-num-state: consensus-accum-num-state(t;f;v0;L),
votes-from-inning: votes-from-inning(i;L),
rcvd-inning-gt: i <z inning(r),
consensus-rcv: consensus-rcv(V;A),
id-deq: IdDeq,
Id: Id,
values-for-distinct: values-for-distinct(eq;L),
zip: zip(as;bs),
l_member: (x ∈ l),
filter: filter(P;l),
length: ||as||,
null: null(as),
nil: [],
list: T List,
nat_plus: ℕ+,
assert: ↑b,
spreadn: let a,b,c,d,e = u in v[a; b; c; d; e],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ─→ B[x],
product: x:A × B[x],
multiply: n * m,
subtract: n - m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Lemmas :
id-deq_wf,
subtype_rel-deq,
Id_wf,
l_member_wf,
sq_stable__l_member,
decidable__equal_Id,
equal_wf,
set_wf,
last_induction,
consensus-rcv_wf,
consensus-accum-num-state_wf,
nat_plus_subtype_nat,
bool_wf,
list_wf,
equal-wf-T-base,
filter_wf5,
rcvd-inning-gt_wf,
subtract_wf,
length_wf,
length_of_nil_lemma,
zip_wf,
votes-from-inning_wf,
le_wf,
not_wf,
assert_wf,
null_wf3,
subtype_rel_list,
top_wf,
values-for-distinct_wf,
filter_nil_lemma,
zip_nil_lemma,
null_nil_lemma,
mapfilter_nil_lemma,
map_nil_lemma,
remove_repeats_nil_lemma,
nil_wf,
false_wf,
true_wf,
mul_bounds_1a,
list_accum_append,
list_accum_cons_lemma,
list_accum_nil_lemma,
nat_plus_wf,
eq_int_wf,
equal-wf-base,
int_subtype_base,
mapfilter-append,
votes-from-inning-is-nil,
decidable__lt,
le_antisymmetry_iff,
condition-implies-le,
add-associates,
minus-zero,
add-zero,
add-commutes,
add_functionality_wrt_le,
zero-add,
le-add-cancel-alt,
list_ind_nil_lemma,
filter_cons_lemma,
less_than_wf,
append_wf,
cons_wf,
cs-initial-rcv_wf,
bnot_wf,
append-nil,
rcvd-innings-nil,
decidable__le,
not-le-2,
minus-add,
add-swap,
le-add-cancel2,
not-equal-2,
le-add-cancel,
minus-one-mul,
and_wf,
length_wf_nat,
nat_wf,
append_back_nil,
lt_int_wf,
sq_stable__le,
less-iff-le,
cs-rcv-vote_wf,
assert_of_lt_int,
less_than_transitivity2,
le_weakening2,
less_than_irreflexivity,
equal-wf-base-T,
map_cons_lemma,
less_than_transitivity1,
le_weakening,
le_int_wf,
length_of_cons_lemma,
zip_cons_cons_lemma,
minus-minus,
apply_alist_cons_lemma,
remove_repeats_cons_lemma,
add-mul-special,
zero-mul,
remove-repeats_wf,
length-append,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
filter_append_sq,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
zip-append,
le_antisymmetry,
unzip_zip,
length-map,
map_functionality,
iff_weakening_equal,
subtype_base_sq,
list_subtype_base,
set_subtype_base,
atom2_subtype_base,
map_wf,
map_append_sq,
remove-repeats-append-one
\mforall{}[V:Type]
\mforall{}A:Id List. \mforall{}t:\mBbbN{}\msupplus{}. \mforall{}f:(V List) {}\mrightarrow{} V. \mforall{}v0:V. \mforall{}L:consensus-rcv(V;A) List.
let b,i,as,vs,v = consensus-accum-num-state(t;f;v0;L) in (filter(\mlambda{}r.i - 1 <z inning(r);L) = [])
\mwedge{} (||as|| = ||vs||)
\mwedge{} (zip(as;vs) = votes-from-inning(i - 1;L))
\mwedge{} (0 \mleq{} i)
\mwedge{} 1 \mleq{} i supposing \mneg{}\muparrow{}null(L)
\mwedge{} (||values-for-distinct(IdDeq;votes-from-inning(i - 1;L))|| \mleq{} (2 * t))
Date html generated:
2015_07_17-AM-11_50_30
Last ObjectModification:
2015_02_04-PM-05_47_55
Home
Index