Nuprl Lemma : consensus-refinement5
∀[V:Type]
∀A:Id List. ∀W:{a:Id| (a ∈ A)} List List.
((1 < ||W|| ∧ two-intersection(A;W))
⇒ ts-refinement(consensus-ts5(V;A;W);consensus-ts6(V;A;W);λs.cs-events-to-state(A; s)))
Proof
Definitions occuring in Statement :
consensus-ts6: consensus-ts6(V;A;W),
cs-events-to-state: cs-events-to-state(A; s),
consensus-ts5: consensus-ts5(V;A;W),
two-intersection: two-intersection(A;W),
Id: Id,
l_member: (x ∈ l),
length: ||as||,
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
natural_number: $n,
universe: Type,
ts-refinement: ts-refinement(ts1;ts2;f)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
ts-refinement: ts-refinement(ts1;ts2;f),
member: t ∈ T,
prop: ℙ,
consensus-ts6: consensus-ts6(V;A;W),
ts-init: ts-init(ts),
consensus-ts5: consensus-ts5(V;A;W),
ts-rel: ts-rel(ts),
ts-type: ts-type(ts),
pi1: fst(t),
pi2: snd(t),
consensus-state4: ConsensusState,
cs-events-to-state: cs-events-to-state(A; s),
bfalse: ff,
consensus-state5: Knowledge(ConsensusState),
consensus-accum-state: consensus-accum-state(A;L),
list_accum: list_accum,
nil: [],
it: ⋅,
spreadn: spread3,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
top: Top,
uimplies: b supposing a,
cand: A c∧ B,
ts-reachable: ts-reachable(ts),
consensus-state6: consensus-state6(V;A),
infix_ap: x f y,
one-consensus-event: y = x after e@a,
consensus-event-constraint: e@a allowed in state x,
exists: ∃x:A. B[x],
or: P ∨ Q,
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),
cs-knowledge: Knowledge(x;a),
cs-estimate: Estimate(s;a),
cs-inning: Inning(s;a),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
consensus-rel-knowledge-inning-step: consensus-rel-knowledge-inning-step(V;A;W;x1;x2;y1;y2;a),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
inning-event: NextInning,
consensus-accum: consensus-accum(s;e),
consensus-rel-knowledge-archive-step: consensus-rel-knowledge-archive-step(V;A;W;x1;x2;y1;y2;a),
archive-event: Archive(v),
consensus-rel-add-knowledge-step: consensus-rel-add-knowledge-step(V;A;W;x1;x2;y1;y2;a),
consensus-message: consensus-message(b;i;z),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
unit: Unit,
ts-final: ts-final(ts),
bool: 𝔹,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
deq-member: x ∈b L,
reduce: reduce(f;k;as),
list_ind: list_ind,
decidable: Dec(P),
eq_id: a = b,
bor: p ∨bq,
Id: Id,
deq: EqDecider(T),
consensus-event: consensus-event(V;A),
cons: [a / b],
fpf-empty: ⊗,
fpf-domain: fpf-domain(f),
mk_fpf: mk_fpf(L;f),
cs-knowledge-precondition: may consider v in inning i based on knowledge (s),
fpf-ap: f(x),
isl: isl(x),
outl: outl(x),
fpf-dom: x ∈ dom(f),
less_than: a < b,
less_than': less_than'(a;b),
sq_stable: SqStable(P),
two-intersection: two-intersection(A;W),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Latex:
\mforall{}[V:Type]
\mforall{}A:Id List. \mforall{}W:\{a:Id| (a \mmember{} A)\} List List.
((1 < ||W|| \mwedge{} two-intersection(A;W))
{}\mRightarrow{} ts-refinement(consensus-ts5(V;A;W);consensus-ts6(V;A;W);\mlambda{}s.cs-events-to-state(A; s)))
Date html generated:
2016_05_16-PM-00_31_21
Last ObjectModification:
2016_01_17-PM-04_03_22
Theory : event-ordering
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