Nuprl Lemma : cs-ref-map-changed
∀[V:Type]
((∀v1,v2:V. Dec(v1 = v2 ∈ V))
⇒ {∃v,v':V. (¬(v = v' ∈ V))}
⇒ (∀A:Id List. ∀W:{a:Id| (a ∈ A)} List List.
(two-intersection(A;W)
⇒ (∀f:ConsensusState ⟶ (consensus-state3(V) List)
(cs-ref-map-constraints(V;A;W;f)
⇒ (∀x,y:ts-reachable(consensus-ts4(V;A;W)).
((x ts-rel(consensus-ts4(V;A;W)) y)
⇒ (∀i:ℕ
(∀v:V
((in state x, inning i could commit v ∧ (¬in state y, inning i could commit v ))
⇒ ((f y[i] = WITHDRAWN ∈ consensus-state3(V))
∨ ((f x[i] = INITIAL ∈ consensus-state3(V))
∧ ((f y[i] = INITIAL ∈ consensus-state3(V))
∨ (∃v':V
((∀j:ℕi. (¬(f x[j] = INITIAL ∈ consensus-state3(V))))
∧ ((f y[i] = CONSIDERING[v'] ∈ consensus-state3(V))
∨ (f y[i] = COMMITED[v'] ∈ consensus-state3(V)))
∧ (∀j:ℕi. ∀v'':V.
(((f x[j] = CONSIDERING[v''] ∈ consensus-state3(V))
∨ (f x[j] = COMMITED[v''] ∈ consensus-state3(V)))
⇒ (v'' = v' ∈ V)))))))))) supposing
(i < ||f y|| and
i < ||f x||)))))))))
Proof
Definitions occuring in Statement :
cs-ref-map-constraints: cs-ref-map-constraints(V;A;W;f),
two-intersection: two-intersection(A;W),
cs-inning-committable: in state s, inning i could commit v ,
consensus-ts4: consensus-ts4(V;A;W),
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cs-withdrawn: WITHDRAWN,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),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
ts-reachable: ts-reachable(ts),
ts-rel: ts-rel(ts)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
ts-reachable: ts-reachable(ts),
consensus-ts4: consensus-ts4(V;A;W),
ts-type: ts-type(ts),
pi1: fst(t),
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
prop: ℙ,
infix_ap: x f y,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
cs-ref-map-constraints: cs-ref-map-constraints(V;A;W;f),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
cs-inning-committable: in state s, inning i could commit v ,
sq_stable: SqStable(P),
ts-rel: ts-rel(ts),
pi2: snd(t),
le: A ≤ B,
less_than': less_than'(a;b),
ts-stable: ts-stable(ts;x.P[x]),
consensus-rel: CR[x,y],
Id: Id,
sq_type: SQType(T),
cs-not-completed: in state s, a has not completed inning i,
true: True,
fpf-single: x : v,
fpf-domain: fpf-domain(f)
Latex:
\mforall{}[V:Type]
((\mforall{}v1,v2:V. Dec(v1 = v2))
{}\mRightarrow{} \{\mexists{}v,v':V. (\mneg{}(v = v'))\}
{}\mRightarrow{} (\mforall{}A:Id List. \mforall{}W:\{a:Id| (a \mmember{} A)\} List List.
(two-intersection(A;W)
{}\mRightarrow{} (\mforall{}f:ConsensusState {}\mrightarrow{} (consensus-state3(V) List)
(cs-ref-map-constraints(V;A;W;f)
{}\mRightarrow{} (\mforall{}x,y:ts-reachable(consensus-ts4(V;A;W)).
((x ts-rel(consensus-ts4(V;A;W)) y)
{}\mRightarrow{} (\mforall{}i:\mBbbN{}
(\mforall{}v:V
((in state x, inning i could commit v
\mwedge{} (\mneg{}in state y, inning i could commit v ))
{}\mRightarrow{} ((f y[i] = WITHDRAWN)
\mvee{} ((f x[i] = INITIAL)
\mwedge{} ((f y[i] = INITIAL)
\mvee{} (\mexists{}v':V
((\mforall{}j:\mBbbN{}i. (\mneg{}(f x[j] = INITIAL)))
\mwedge{} ((f y[i] = CONSIDERING[v']) \mvee{} (f y[i] = COMMITED[v']))
\mwedge{} (\mforall{}j:\mBbbN{}i. \mforall{}v'':V.
(((f x[j] = CONSIDERING[v''])
\mvee{} (f x[j] = COMMITED[v'']))
{}\mRightarrow{} (v'' = v')))))))))) supposing
(i < ||f y|| and
i < ||f x||)))))))))
Date html generated:
2016_05_16-PM-00_10_59
Last ObjectModification:
2016_01_17-PM-04_03_41
Theory : event-ordering
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