Nuprl Lemma : consensus-refinement3
∀[V:Type]
((∀v1,v2:V. Dec(v1 = v2 ∈ V))
⇒ {∃v,v':V. (¬(v = v' ∈ V))}
⇒ (∀L:V List. Dec(∃v:V. (¬(v ∈ L))))
⇒ (∀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) ⇒ ts-refinement(consensus-ts3(V);consensus-ts4(V;A;W);f)))
supposing ||W|| ≥ 1 ))
Proof
Definitions occuring in Statement :
cs-ref-map-constraints: cs-ref-map-constraints(V;A;W;f),
two-intersection: two-intersection(A;W),
consensus-ts4: consensus-ts4(V;A;W),
consensus-state4: ConsensusState,
consensus-ts3: consensus-ts3(T),
consensus-state3: consensus-state3(T),
Id: Id,
l_member: (x ∈ l),
length: ||as||,
list: T List,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
ge: i ≥ j ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
ts-refinement: ts-refinement(ts1;ts2;f)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
ge: i ≥ j ,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
prop: ℙ,
or: P ∨ Q,
less_than': less_than'(a;b),
true: True,
cons: [a / b],
top: Top,
two-intersection: two-intersection(A;W),
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
squash: ↓T,
ts-refinement: ts-refinement(ts1;ts2;f),
cand: A c∧ B,
infix_ap: x f y,
ts-reachable: ts-reachable(ts),
subtype_rel: A ⊆r B,
ts-type: ts-type(ts),
pi1: fst(t),
consensus-ts4: consensus-ts4(V;A;W),
consensus-state4: ConsensusState,
list: T List,
consensus-ts3: consensus-ts3(T),
nat: ℕ,
cs-ref-map-constraints: cs-ref-map-constraints(V;A;W;f),
decidable: Dec(P),
ts-init: ts-init(ts),
cs-inning: Inning(s;a),
pi2: snd(t),
satisfiable_int_formula: satisfiable_int_formula(fmla),
ts-rel: ts-rel(ts),
guard: {T},
less_than: a < b,
int_iseg: {i...j},
int_seg: {i..j-},
lelt: i ≤ j < k,
ts-stable: ts-stable(ts;x.P[x]),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
last: last(L),
select: L[n],
ts-final: ts-final(ts),
cs-estimate: Estimate(s;a),
cs-inning-committed: in state s, inning i has committed v,
cs-archived: by state s, a archived v in inning i,
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{}L:V List. Dec(\mexists{}v:V. (\mneg{}(v \mmember{} L))))
{}\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{} ts-refinement(consensus-ts3(V);consensus-ts4(V;A;W);f)))
supposing ||W|| \mgeq{} 1 ))
Date html generated:
2016_05_16-PM-00_15_10
Last ObjectModification:
2016_01_17-PM-04_05_15
Theory : event-ordering
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