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,
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,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
ts-reachable: ts-reachable(ts)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
three-cs-decided: three-cs-decided(V;A;t;f;s;v),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
ts-reachable: ts-reachable(ts),
so_lambda: λ2x.t[x],
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),
ge: i ≥ j ,
or: P ∨ Q,
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
cons: [a / b],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
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{}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:
2016_05_16-PM-00_48_04
Last ObjectModification:
2016_01_17-PM-07_57_58
Theory : event-ordering
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