Nuprl Lemma : consensus-safety
∀V:Type
((∀v1,v2:V. Dec(v1 = v2 ∈ V))
⇒ {∃v,v':V. (¬(v = v' ∈ V))}
⇒ (∀L:V List. Dec(∃v:V. (¬(v ∈ L))))
⇒ (∀A:Id List. ∀W:{a:Id| (a ∈ A)} List List.
((||W|| ≥ 1 )
⇒ two-intersection(A;W)
⇒ (∀s1,s2:ts-reachable(consensus-ts4(V;A;W)).
((s1 (ts-rel(consensus-ts4(V;A;W))^*) s2)
⇒ (∀v1,v2:V.
((∃i:ℕ. in state s1, inning i has committed v1)
⇒ (∃j:ℕ. in state s2, inning j has committed v2)
⇒ (v1 = v2 ∈ V))))))))
Proof
Definitions occuring in Statement :
two-intersection: two-intersection(A;W),
cs-inning-committed: in state s, inning i has committed v,
consensus-ts4: consensus-ts4(V;A;W),
Id: Id,
l_member: (x ∈ l),
length: ||as||,
list: T List,
rel_star: R^*,
nat: ℕ,
decidable: Dec(P),
guard: {T},
infix_ap: x f y,
ge: i ≥ j ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
ts-reachable: ts-reachable(ts),
ts-rel: ts-rel(ts),
ts-type: ts-type(ts)
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
ts-reachable: ts-reachable(ts),
infix_ap: x f y,
so_apply: x[s],
uimplies: b supposing a,
ts-type: ts-type(ts),
pi1: fst(t),
consensus-ts4: consensus-ts4(V;A;W),
consensus-state4: ConsensusState,
nat: ℕ,
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
consensus-state1: consensus-state1(V),
consensus-ts1: consensus-ts1(T),
ts-rel: ts-rel(ts),
pi2: snd(t),
rel_star: R^*,
rel_exp: R^n,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
not: ¬A,
bfalse: ff,
cs-decided: Decided[v],
outl: outl(x),
isl: isl(x),
assert: ↑b,
true: True,
cs-undecided: UNDECIDED,
false: False
Latex:
\mforall{}V:Type
((\mforall{}v1,v2:V. Dec(v1 = v2))
{}\mRightarrow{} \{\mexists{}v,v':V. (\mneg{}(v = v'))\}
{}\mRightarrow{} (\mforall{}L:V List. Dec(\mexists{}v:V. (\mneg{}(v \mmember{} L))))
{}\mRightarrow{} (\mforall{}A:Id List. \mforall{}W:\{a:Id| (a \mmember{} A)\} List List.
((||W|| \mgeq{} 1 )
{}\mRightarrow{} two-intersection(A;W)
{}\mRightarrow{} (\mforall{}s1,s2:ts-reachable(consensus-ts4(V;A;W)).
((s1 rel\_star(ts-type(consensus-ts4(V;A;W)); ts-rel(consensus-ts4(V;A;W))) s2)
{}\mRightarrow{} (\mforall{}v1,v2:V.
((\mexists{}i:\mBbbN{}. in state s1, inning i has committed v1)
{}\mRightarrow{} (\mexists{}j:\mBbbN{}. in state s2, inning j has committed v2)
{}\mRightarrow{} (v1 = v2))))))))
Date html generated:
2016_05_16-PM-00_15_52
Last ObjectModification:
2015_12_29-PM-01_26_44
Theory : event-ordering
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