Nuprl Lemma : cs-inning-committable-some1
∀[V:Type]
((∃v,v':V. (¬(v = v' ∈ V)))
⇒ (∀v,v':V. Dec(v = v' ∈ V))
⇒ (∀A:Id List. ∀W:{a:Id| (a ∈ A)} List List.
(one-intersection(A;W)
⇒ (∀s:ConsensusState. ∀i:ℤ.
∃L:V List
(∃v:V. in state s, inning i could commit v
⇐⇒ (∃v∈L. in state s, inning i could commit v )
∨ (∃ws∈W. ∀a:{a:Id| (a ∈ A)} . ((a ∈ ws) ⇒ in state s, a has not completed inning i)))))))
Proof
Definitions occuring in Statement :
one-intersection: one-intersection(A;W),
cs-inning-committable: in state s, inning i could commit v ,
cs-not-completed: in state s, a has not completed inning i,
consensus-state4: ConsensusState,
Id: Id,
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
set: {x:A| B[x]} ,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T}
Latex:
\mforall{}[V:Type]
((\mexists{}v,v':V. (\mneg{}(v = v')))
{}\mRightarrow{} (\mforall{}v,v':V. Dec(v = v'))
{}\mRightarrow{} (\mforall{}A:Id List. \mforall{}W:\{a:Id| (a \mmember{} A)\} List List.
(one-intersection(A;W)
{}\mRightarrow{} (\mforall{}s:ConsensusState. \mforall{}i:\mBbbZ{}.
\mexists{}L:V List
(\mexists{}v:V. in state s, inning i could commit v
\mLeftarrow{}{}\mRightarrow{} (\mexists{}v\mmember{}L. in state s, inning i could commit v )
\mvee{} (\mexists{}ws\mmember{}W. \mforall{}a:\{a:Id| (a \mmember{} A)\}
((a \mmember{} ws) {}\mRightarrow{} in state s, a has not completed inning i)))))))
Date html generated:
2016_05_16-PM-00_05_11
Last ObjectModification:
2015_12_29-PM-01_23_48
Theory : event-ordering
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