Nuprl Lemma : three-cs-archive-invariant
∀[V:Type]
∀eq:EqDecider(V). ∀A:Id List. ∀t:ℕ+. ∀f:(V List) ─→ V.
((∀vs:V List. (f vs ∈ vs) supposing ||vs|| ≥ 1 )
⇒ (∀s:ts-reachable(three-consensus-ts(V;A;t;f)). ∀v:V. ∀a:{a:Id| (a ∈ A)} . ∀n:ℤ. ∀L:consensus-rcv(V;A) List.
(L ≤ s a
⇒ archive-condition(V;A;t;f;n;v;L)
⇒ (∃a∈A. (||s a|| ≥ 1 ) ∧ (hd(s a) = Init[v] ∈ consensus-rcv(V;A))))))
Proof
Definitions occuring in Statement :
three-consensus-ts: three-consensus-ts(V;A;t;f),
archive-condition: archive-condition(V;A;t;f;n;v;L),
cs-initial-rcv: Init[v],
consensus-rcv: consensus-rcv(V;A),
Id: Id,
deq: EqDecider(T),
iseg: l1 ≤ l2,
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
hd: hd(l),
length: ||as||,
list: T List,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ─→ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T,
ts-reachable: ts-reachable(ts)
Lemmas :
ge_wf,
length_wf,
l_member_wf,
hd_wf,
exists_wf,
Id_wf,
consensus-rcv_wf,
cs-initial-rcv_wf,
l_exists_iff,
l_exists_wf,
decidable__equal_Id,
subtype_base_sq,
atom2_subtype_base,
list_wf,
list-cases,
length_of_nil_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
reduce_hd_cons_lemma,
equal-wf-base-T,
product_subtype_list,
list_ind_cons_lemma,
append_wf,
cons_wf,
nil_wf,
length_nil,
non_neg_length,
length_wf_nil,
length_cons,
length_wf_nat,
length_append,
subtype_rel_list,
top_wf,
iff_weakening_equal,
member_wf,
and_wf,
equal_wf,
iseg_wf,
iseg-append-one,
sq_stable_from_decidable,
archive-condition_wf,
squash_wf,
true_wf,
nat_wf,
nat_plus_subtype_nat,
archive-condition-append-init,
subtype_rel_self,
sq_stable__l_member,
cons_neq_nil,
decidable__le,
false_wf,
not-ge-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
archive-condition-append-vote,
cs-rcv-vote_wf,
values-for-distinct_wf,
id-deq_wf,
subtype_rel-deq,
set_wf,
votes-from-inning_wf,
subtract_wf,
listp_properties,
assert_of_lt_int,
assert_wf,
lt_int_wf,
member-values-for-distinct2,
member-votes-from-inning,
not-le-2,
less-iff-le,
zero-add,
minus-minus,
le-add-cancel,
le_wf,
member_append,
member_singleton,
or_wf,
pi2_wf,
three-cs-vote-invariant
\mforall{}[V:Type]
\mforall{}eq:EqDecider(V). \mforall{}A:Id List. \mforall{}t:\mBbbN{}\msupplus{}. \mforall{}f:(V List) {}\mrightarrow{} V.
((\mforall{}vs:V List. (f vs \mmember{} vs) supposing ||vs|| \mgeq{} 1 )
{}\mRightarrow{} (\mforall{}s:ts-reachable(three-consensus-ts(V;A;t;f)). \mforall{}v:V. \mforall{}a:\{a:Id| (a \mmember{} A)\} . \mforall{}n:\mBbbZ{}.
\mforall{}L:consensus-rcv(V;A) List.
(L \mleq{} s a
{}\mRightarrow{} archive-condition(V;A;t;f;n;v;L)
{}\mRightarrow{} (\mexists{}a\mmember{}A. (||s a|| \mgeq{} 1 ) \mwedge{} (hd(s a) = Init[v])))))
Date html generated:
2015_07_17-AM-11_53_15
Last ObjectModification:
2015_07_16-AM-10_12_24
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