Nuprl Lemma : cs-inning-committable-some2
∀[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
⇐⇒ (in state s, inning i could commit v ∧ (v ∈ L))
∨ (∃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,
and: P ∧ Q,
set: {x:A| B[x]} ,
int: ℤ,
universe: Type,
equal: s = t ∈ T
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],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
fpf-dom: x ∈ dom(f),
deq-member: x ∈b L,
reduce: reduce(f;k;as),
list_ind: list_ind,
pi1: fst(t),
cs-estimate: Estimate(s;a),
pi2: snd(t),
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
cand: A c∧ B,
guard: {T},
decidable: Dec(P),
cs-inning-committable: in state s, inning i could commit v ,
cs-not-completed: in state s, a has not completed inning i,
le: A ≤ B,
not: ¬A,
false: False,
cs-archived: by state s, a archived v in inning i,
sq_stable: SqStable(P),
squash: ↓T,
fpf-domain: fpf-domain(f)
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
\mforall{}v:V
(in state s, inning i could commit v
\mLeftarrow{}{}\mRightarrow{} (in state s, inning i could commit v \mwedge{} (v \mmember{} L))
\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_00
Last ObjectModification:
2016_01_17-PM-03_54_37
Theory : event-ordering
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