Nuprl Lemma : three-cs-archive-condition
∀V:Type. ∀A:Id List. ∀t:ℕ+. ∀f:(V List) ⟶ V. ∀s:ts-type(three-consensus-ts(V;A;t;f)).
((ts-init(three-consensus-ts(V;A;t;f)) (ts-rel(three-consensus-ts(V;A;t;f))^*) s)
⇒ (∀v:V. ∀a:{a:Id| (a ∈ A)} . ∀n:ℕ+. ∀L:consensus-rcv(V;A) List.
(L ≤ s a
⇒ archive-condition(V;A;t;f;n;v;L)
⇒ (∃as:{a:Id| (a ∈ A)} List
(no_repeats({a:Id| (a ∈ A)} ;as)
∧ (||as|| = ((2 * t) + 1) ∈ ℤ)
∧ (∃vs:V List
((||vs|| = ||as|| ∈ ℤ)
∧ (v = (f vs) ∈ V)
∧ (∀i:ℕ||as||
∃L:consensus-rcv(V;A) List. (L ≤ s as[i] ∧ archive-condition(V;A;t;f;n - 1;vs[i];L))))))))))
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),
consensus-rcv: consensus-rcv(V;A),
Id: Id,
iseg: l1 ≤ l2,
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
rel_star: R^*,
int_seg: {i..j-},
nat_plus: ℕ+,
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T,
ts-rel: ts-rel(ts),
ts-init: ts-init(ts),
ts-type: ts-type(ts)
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
ts-reachable: ts-reachable(ts),
so_apply: x[s],
uimplies: b supposing a,
ts-type: ts-type(ts),
pi1: fst(t),
three-consensus-ts: three-consensus-ts(V;A;t;f),
nat_plus: ℕ+,
and: P ∧ Q,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
ts-init: ts-init(ts),
pi2: snd(t),
iff: P ⇐⇒ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
bfalse: ff,
ts-rel: ts-rel(ts),
infix_ap: x f y,
nat: ℕ,
cand: A c∧ B,
Id: Id,
sq_type: SQType(T),
rev_implies: P ⇐ Q,
consensus-rcv: consensus-rcv(V;A),
cs-initial-rcv: Init[v],
cs-rcv-vote: Vote[a;i;v],
ge: i ≥ j ,
le: A ≤ B,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
true: True,
label: ...$L... t
Latex:
\mforall{}V:Type. \mforall{}A:Id List. \mforall{}t:\mBbbN{}\msupplus{}. \mforall{}f:(V List) {}\mrightarrow{} V. \mforall{}s:ts-type(three-consensus-ts(V;A;t;f)).
((ts-init(three-consensus-ts(V;A;t;f)) (ts-rel(three-consensus-ts(V;A;t;f))\^{}*) s)
{}\mRightarrow{} (\mforall{}v:V. \mforall{}a:\{a:Id| (a \mmember{} A)\} . \mforall{}n:\mBbbN{}\msupplus{}. \mforall{}L:consensus-rcv(V;A) List.
(L \mleq{} s a
{}\mRightarrow{} archive-condition(V;A;t;f;n;v;L)
{}\mRightarrow{} (\mexists{}as:\{a:Id| (a \mmember{} A)\} List
(no\_repeats(\{a:Id| (a \mmember{} A)\} ;as)
\mwedge{} (||as|| = ((2 * t) + 1))
\mwedge{} (\mexists{}vs:V List
((||vs|| = ||as||)
\mwedge{} (v = (f vs))
\mwedge{} (\mforall{}i:\mBbbN{}||as||
\mexists{}L:consensus-rcv(V;A) List
(L \mleq{} s as[i] \mwedge{} archive-condition(V;A;t;f;n - 1;vs[i];L))))))))))
Date html generated:
2016_05_16-PM-00_45_59
Last ObjectModification:
2016_01_17-PM-08_03_45
Theory : event-ordering
Home
Index