Nuprl Lemma : three-cs-safety2
∀[V:Type]
∀eq:EqDecider(V). ∀A:Id List. ∀t:ℕ+. ∀f:(V List) ─→ V.
((∀vs:V List. (f vs ∈ vs) supposing ||vs|| ≥ 1 )
⇒ (∀v:V. ∀s:ts-reachable(three-consensus-ts(V;A;t;f)).
(three-cs-decided(V;A;t;f;s;v) ⇒ (∃a∈A. (||s a|| ≥ 1 ) ∧ (hd(s a) = Init[v] ∈ consensus-rcv(V;A))))))
Proof
Definitions occuring in Statement :
three-cs-decided: three-cs-decided(V;A;t;f;s;v),
three-consensus-ts: three-consensus-ts(V;A;t;f),
cs-initial-rcv: Init[v],
consensus-rcv: consensus-rcv(V;A),
Id: Id,
deq: EqDecider(T),
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,
apply: f a,
function: x:A ─→ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
ts-reachable: ts-reachable(ts)
Lemmas :
three-cs-archive-invariant,
three-cs-decided_wf,
subtype_rel_set,
ts-type_wf,
three-consensus-ts_wf,
nat_plus_subtype_nat,
Id_wf,
l_member_wf,
list_wf,
consensus-rcv_wf,
infix_ap_wf,
rel_star_wf,
ts-rel_wf,
ts-init_wf,
subtype_rel_self,
ts-reachable_wf,
subtype_rel_wf,
all_wf,
isect_wf,
ge_wf,
length_wf,
nat_plus_wf,
deq_wf,
set_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
l_all_iff,
cons_wf,
exists_wf,
iseg_wf,
archive-condition_wf,
cons_member
\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{}v:V. \mforall{}s:ts-reachable(three-consensus-ts(V;A;t;f)).
(three-cs-decided(V;A;t;f;s;v) {}\mRightarrow{} (\mexists{}a\mmember{}A. (||s a|| \mgeq{} 1 ) \mwedge{} (hd(s a) = Init[v])))))
Date html generated:
2015_07_17-AM-11_54_07
Last ObjectModification:
2015_01_28-AM-00_43_37
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