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,
iseg: l1 ≤ l2,
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
hd: hd(l),
length: ||as||,
list: T List,
deq: EqDecider(T),
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)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
ge: i ≥ j ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s],
ts-reachable: ts-reachable(ts),
ts-type: ts-type(ts),
pi1: fst(t),
three-consensus-ts: three-consensus-ts(V;A;t;f),
guard: {T},
transition-system: transition-system{i:l},
nat: ℕ,
exists: ∃x:A. B[x],
top: Top,
infix_ap: x f y,
ts-init: ts-init(ts),
pi2: snd(t),
iff: P ⇐⇒ Q,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
false: False,
cons: [a / b],
bfalse: ff,
ts-rel: ts-rel(ts),
rev_implies: P ⇐ Q,
decidable: Dec(P),
Id: Id,
sq_type: SQType(T),
le: A ≤ B,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
less_than': less_than'(a;b),
true: True,
not: ¬A,
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
squash: ↓T,
sq_stable: SqStable(P),
consensus-rcv: consensus-rcv(V;A),
cs-initial-rcv: Init[v],
uiff: uiff(P;Q),
subtract: n - m,
cs-rcv-vote: Vote[a;i;v]
Latex:
\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:
2016_05_16-PM-00_46_34
Last ObjectModification:
2016_01_17-PM-08_04_22
Theory : event-ordering
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