Nuprl Lemma : consensus-ts5-archive-invariant
∀[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. ∀x:ts-reachable(consensus-ts5(V;A;W)). ∀v:V. ∀b:{a:Id| (a ∈ A)} . ∀i:ℤ.
(state fst(x) may consider v in inning i) supposing
((Estimate(fst(x);b)(i) = v ∈ V) and
(↑i ∈ dom(Estimate(fst(x);b))))))
Proof
Definitions occuring in Statement :
consensus-ts5: consensus-ts5(V;A;W),
cs-precondition: state s may consider v in inning i,
cs-estimate: Estimate(s;a),
fpf-ap: f(x),
fpf-dom: x ∈ dom(f),
Id: Id,
l_member: (x ∈ l),
list: T List,
int-deq: IntDeq,
assert: ↑b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
int: ℤ,
universe: Type,
equal: s = t ∈ T,
ts-reachable: ts-reachable(ts)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
ts-reachable: ts-reachable(ts),
consensus-ts5: consensus-ts5(V;A;W),
ts-type: ts-type(ts),
pi1: fst(t),
prop: ℙ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
so_apply: x[s],
guard: {T},
ts-init: ts-init(ts),
cs-estimate: Estimate(s;a),
pi2: snd(t),
fpf-empty: ⊗,
fpf-dom: x ∈ dom(f),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
false: False,
ts-rel: ts-rel(ts),
infix_ap: x f y,
consensus-rel-knowledge: consensus-rel-knowledge(V;A;W;x;y),
exists: ∃x:A. B[x],
consensus-rel-knowledge-step: consensus-rel-knowledge-step(V;A;W;x1;x2;y1;y2;a),
and: P ∧ Q,
or: P ∨ Q,
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),
decidable: Dec(P),
squash: ↓T,
true: True,
sq_type: SQType(T),
btrue: tt,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cs-precondition: state s may consider v in inning i,
cand: A c∧ B,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
cs-archive-blocked: in state s, ws' blocks ws from archiving v in inning i,
cs-knowledge-precondition: may consider v in inning i based on knowledge (s),
isl: isl(x),
outl: outl(x),
Id: Id
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. \mforall{}x:ts-reachable(consensus-ts5(V;A;W)). \mforall{}v:V.
\mforall{}b:\{a:Id| (a \mmember{} A)\} . \mforall{}i:\mBbbZ{}.
(state fst(x) may consider v in inning i) supposing
((Estimate(fst(x);b)(i) = v) and
(\muparrow{}i \mmember{} dom(Estimate(fst(x);b))))))
Date html generated:
2016_05_16-PM-00_25_13
Last ObjectModification:
2016_01_17-PM-04_04_10
Theory : event-ordering
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