Nuprl Lemma : cs-possible-state-reachable
∀[V:Type]. ∀[A:Id List]. ∀[W:{a:Id| (a ∈ A)} List List]. ∀[v:V].
(∀[L:{a:Id| (a ∈ A)} List]
(<λa.<0, if a ∈b L then 0 : v else ⊗ fi >, λa.mk_fpf(A;λb.<0, ff>)>
∈ ts-reachable(consensus-ts5(V;A;W)))) supposing
(two-intersection(A;W) and
1 < ||W||)
Proof
Definitions occuring in Statement :
consensus-ts5: consensus-ts5(V;A;W),
two-intersection: two-intersection(A;W),
mk_fpf: mk_fpf(L;f),
fpf-single: x : v,
fpf-empty: ⊗,
id-deq: IdDeq,
Id: Id,
l_member: (x ∈ l),
length: ||as||,
deq-member: x ∈b L,
list: T List,
ifthenelse: if b then t else f fi ,
bfalse: ff,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
lambda: λx.A[x],
pair: <a, b>,
natural_number: $n,
universe: Type,
ts-reachable: ts-reachable(ts)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
ts-reachable: ts-reachable(ts),
prop: ℙ,
consensus-ts5: consensus-ts5(V;A;W),
ts-type: ts-type(ts),
pi1: fst(t),
bfalse: ff,
consensus-state5: Knowledge(ConsensusState),
consensus-state4: ConsensusState,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
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,
infix_ap: x f y,
ts-init: ts-init(ts),
ts-rel: ts-rel(ts),
pi2: snd(t),
bor: p ∨bq,
decidable: Dec(P),
eq_id: a = b,
squash: ↓T,
true: True,
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),
deq: EqDecider(T),
cand: A c∧ B,
cs-inning: Inning(s;a),
cs-estimate: Estimate(s;a),
consensus-rel-knowledge-archive-step: consensus-rel-knowledge-archive-step(V;A;W;x1;x2;y1;y2;a),
cs-knowledge: Knowledge(x;a),
exposed-bfalse: exposed-bfalse,
fpf-single: x : v,
fpf-domain: fpf-domain(f),
fpf-empty: ⊗,
cs-knowledge-precondition: may consider v in inning i based on knowledge (s),
isl: isl(x),
outl: outl(x),
less_than: a < b,
less_than': less_than'(a;b),
cons: [a / b],
mk_fpf: mk_fpf(L;f),
fpf-dom: x ∈ dom(f),
sq_stable: SqStable(P),
two-intersection: two-intersection(A;W),
fpf-ap: f(x)
Latex:
\mforall{}[V:Type]. \mforall{}[A:Id List]. \mforall{}[W:\{a:Id| (a \mmember{} A)\} List List]. \mforall{}[v:V].
(\mforall{}[L:\{a:Id| (a \mmember{} A)\} List]
(<\mlambda{}a.ɘ, if a \mmember{}\msubb{} L then 0 : v else \motimes{} fi >, \mlambda{}a.mk\_fpf(A;\mlambda{}b.ɘ, ff>)>
\mmember{} ts-reachable(consensus-ts5(V;A;W)))) supposing
(two-intersection(A;W) and
1 < ||W||)
Date html generated:
2016_05_16-PM-00_25_36
Last ObjectModification:
2016_01_17-PM-03_57_43
Theory : event-ordering
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