Nuprl Lemma : consensus-refinement4
∀[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.
((1 < ||W|| ∧ two-intersection(A;W)) ⇒ ts-refinement(consensus-ts4(V;A;W);consensus-ts5(V;A;W);λs.(fst(s))))))
Proof
Definitions occuring in Statement :
consensus-ts5: consensus-ts5(V;A;W),
two-intersection: two-intersection(A;W),
consensus-ts4: consensus-ts4(V;A;W),
Id: Id,
l_member: (x ∈ l),
length: ||as||,
list: T List,
less_than: a < b,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
lambda: λx.A[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],
and: P ∧ Q,
ts-refinement: ts-refinement(ts1;ts2;f),
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
consensus-ts5: consensus-ts5(V;A;W),
ts-init: ts-init(ts),
consensus-ts4: consensus-ts4(V;A;W),
ts-rel: ts-rel(ts),
pi2: snd(t),
pi1: fst(t),
subtype_rel: A ⊆r B,
consensus-state4: ConsensusState,
uimplies: b supposing a,
l_all: (∀x∈L.P[x]),
l_contains: A ⊆ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
top: Top,
eq_id: a = b,
decidable: Dec(P),
bor: p ∨bq,
consensus-rel: CR[x,y],
infix_ap: x f y,
cs-inning: Inning(s;a),
cs-estimate: Estimate(s;a),
cand: A c∧ B,
fpf-empty: ⊗,
fpf-domain: fpf-domain(f),
true: True,
sq_stable: SqStable(P),
squash: ↓T,
ts-type: ts-type(ts),
ts-reachable: ts-reachable(ts),
consensus-rel-knowledge: consensus-rel-knowledge(V;A;W;x;y),
consensus-rel-knowledge-step: consensus-rel-knowledge-step(V;A;W;x1;x2;y1;y2;a),
consensus-rel-knowledge-inning-step: consensus-rel-knowledge-inning-step(V;A;W;x1;x2;y1;y2;a),
consensus-rel-knowledge-archive-step: consensus-rel-knowledge-archive-step(V;A;W;x1;x2;y1;y2;a),
consensus-rel-add-knowledge-step: consensus-rel-add-knowledge-step(V;A;W;x1;x2;y1;y2;a),
cs-knowledge-precondition: may consider v in inning i based on knowledge (s),
cs-precondition: state s may consider v in inning i,
le: A ≤ B,
cs-archive-blocked: in state s, ws' blocks ws from archiving v in inning i,
isl: isl(x),
outl: outl(x),
satisfiable_int_formula: satisfiable_int_formula(fmla),
rev_uimplies: rev_uimplies(P;Q),
ts-final: ts-final(ts),
cs-knowledge: Knowledge(x;a)
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.
((1 < ||W|| \mwedge{} two-intersection(A;W))
{}\mRightarrow{} ts-refinement(consensus-ts4(V;A;W);consensus-ts5(V;A;W);\mlambda{}s.(fst(s))))))
Date html generated:
2016_05_16-PM-00_26_32
Last ObjectModification:
2016_01_17-PM-04_00_05
Theory : event-ordering
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