Nuprl Lemma : consensus-ts4-ref-map1
∀[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.
(two-intersection(A;W)
⇒ {∀s:ConsensusState. ∀i:ℤ.
∃x:consensus-state3(V)
((x = INITIAL ∈ consensus-state3(V)
⇐⇒ ∃v,v':V
((¬(v = v' ∈ V)) ∧ in state s, inning i could commit v ∧ in state s, inning i could commit v' ))
∧ (x = WITHDRAWN ∈ consensus-state3(V) ⇐⇒ ¬(∃v:V. in state s, inning i could commit v ))
∧ (∀v:V
((x = COMMITED[v] ∈ consensus-state3(V) ⇐⇒ in state s, inning i has committed v)
∧ (x = CONSIDERING[v] ∈ consensus-state3(V)
⇐⇒ in state s, inning i could commit v
∧ (¬in state s, inning i has committed v)
∧ (∀v':V. (in state s, inning i could commit v' ⇒ (v' = v ∈ V)))))))})))
Proof
Definitions occuring in Statement :
two-intersection: two-intersection(A;W),
cs-inning-committable: in state s, inning i could commit v ,
cs-inning-committed: in state s, inning i has committed v,
consensus-state4: ConsensusState,
cs-commited: COMMITED[v],
cs-considering: CONSIDERING[v],
cs-withdrawn: WITHDRAWN,
cs-initial: INITIAL,
consensus-state3: consensus-state3(T),
Id: Id,
l_member: (x ∈ l),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: 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],
two-intersection: two-intersection(A;W),
one-intersection: one-intersection(A;W),
l_all: (∀x∈L.P[x]),
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cs-commited: COMMITED[v],
cs-considering: CONSIDERING[v],
isl: isl(x),
consensus-state3: consensus-state3(T),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
cs-inning-committed: in state s, inning i has committed v,
cs-inning-committable: in state s, inning i could commit v
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.
(two-intersection(A;W)
{}\mRightarrow{} \{\mforall{}s:ConsensusState. \mforall{}i:\mBbbZ{}.
\mexists{}x:consensus-state3(V)
((x = INITIAL
\mLeftarrow{}{}\mRightarrow{} \mexists{}v,v':V
((\mneg{}(v = v'))
\mwedge{} in state s, inning i could commit v
\mwedge{} in state s, inning i could commit v' ))
\mwedge{} (x = WITHDRAWN \mLeftarrow{}{}\mRightarrow{} \mneg{}(\mexists{}v:V. in state s, inning i could commit v ))
\mwedge{} (\mforall{}v:V
((x = COMMITED[v] \mLeftarrow{}{}\mRightarrow{} in state s, inning i has committed v)
\mwedge{} (x = CONSIDERING[v]
\mLeftarrow{}{}\mRightarrow{} in state s, inning i could commit v
\mwedge{} (\mneg{}in state s, inning i has committed v)
\mwedge{} (\mforall{}v':V. (in state s, inning i could commit v' {}\mRightarrow{} (v' = v)))))))\})))
Date html generated:
2016_05_16-PM-00_06_22
Last ObjectModification:
2016_01_17-PM-04_03_07
Theory : event-ordering
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