Nuprl Lemma : consensus-reachable
∀[V:Type]
((∀v1,v2:V. Dec(v1 = v2 ∈ V))
⇒ {∃v,v':V. (¬(v = v' ∈ V))}
⇒ (∀A:Id List. ∀W:{a:Id| (a ∈ A)} List List.
(three-intersection(A;W)
⇒ (∀ws∈W.∀x:ts-reachable(consensus-ts4(V;A;W))
∃y:ConsensusState
((x ((λx,y. CR(in ws)[x, y] )^*) y) ∧ (∃v:V. ∃i:ℤ. in state y, inning i has committed v))))))
Proof
Definitions occuring in Statement :
three-intersection: three-intersection(A;W),
cs-inning-committed: in state s, inning i has committed v,
consensus-ts4: consensus-ts4(V;A;W),
consensus-rel-mod: CR(in ws)[x, y] ,
consensus-state4: ConsensusState,
Id: Id,
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
rel_star: R^*,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
guard: {T},
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T,
ts-reachable: ts-reachable(ts)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
ts-reachable: ts-reachable(ts),
infix_ap: x f y,
ts-type: ts-type(ts),
pi1: fst(t),
consensus-ts4: consensus-ts4(V;A;W),
consensus-state4: ConsensusState,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
one-intersection: one-intersection(A;W),
or: P ∨ Q,
not: ¬A,
false: False,
cons: [a / b],
top: Top,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
guard: {T},
decidable: Dec(P),
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
cand: A c∧ B,
le: A ≤ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
cs-inning: Inning(s;a),
cs-estimate: Estimate(s;a),
pi2: snd(t),
ts-rel: ts-rel(ts),
rel_implies: R1 => R2,
consensus-rel-mod: CR(in ws)[x, y] ,
consensus-rel: CR[x,y],
cs-archive-blocked: in state s, ws' blocks ws from archiving v in inning i,
three-intersection: three-intersection(A;W),
cs-precondition: state s may consider v in inning i,
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
Id: Id,
ge: i ≥ j ,
deq-member: x ∈b L,
band: p ∧b q,
eqof: eqof(d),
eq_id: a = b,
bor: p ∨bq,
fpf-single: x : v,
fpf-dom: x ∈ dom(f),
cs-inning-committed: in state s, inning i has committed v,
cs-archived: by state s, a archived v in inning i,
fpf-domain: fpf-domain(f)
Latex:
\mforall{}[V:Type]
((\mforall{}v1,v2:V. Dec(v1 = v2))
{}\mRightarrow{} \{\mexists{}v,v':V. (\mneg{}(v = v'))\}
{}\mRightarrow{} (\mforall{}A:Id List. \mforall{}W:\{a:Id| (a \mmember{} A)\} List List.
(three-intersection(A;W)
{}\mRightarrow{} (\mforall{}ws\mmember{}W.\mforall{}x:ts-reachable(consensus-ts4(V;A;W))
\mexists{}y:ConsensusState
((x rel\_star(ConsensusState; \mlambda{}x,y. CR(in ws)[x, y] ) y)
\mwedge{} (\mexists{}v:V. \mexists{}i:\mBbbZ{}. in state y, inning i has committed v))))))
Date html generated:
2016_05_16-PM-00_19_58
Last ObjectModification:
2016_01_17-PM-04_04_41
Theory : event-ordering
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