Nuprl Lemma : consensus-reachable
[V:Type]
((v1,v2:V. Dec(v1 = v2))
{
v,v':V. (
(v = v'))}
(A:Id List. W:{a:Id| (a 
A)} List List.
(three-intersection(A;W)
(wsW.x:ts-reachable(consensus-ts4(V;A;W))
y:ConsensusState
((x ((
x,y.CR(in ws)[x, y] )^*) y) 
(v:V. i:
. in state y, inning i has committed v))))))
Proof not projected
Definitions occuring in Statement :
three-intersection: three-intersection(A;W),
cs-inning-committed: in state s, inning i has committed v,
consensus-ts4: consensus-ts4(V;A;W),
consensus-rel-mod: CR(in ws)[x, y] ,
consensus-state4: ConsensusState,
Id: Id,
decidable: Dec(P),
uall: [x:A]. B[x],
guard: {T},
infix_ap: x f y,
all: x:A. B[x],
exists: x:A. B[x],
not: A,
implies: P Q,
and: P Q,
set: {x:A| B[x]} ,
lambda: x.A[x],
list: type List,
int: ,
universe: Type,
equal: s = t,
l_all: (xL.P[x]),
l_member: (x l),
rel_star: R^*,
ts-reachable: ts-reachable(ts)
Definitions :
uall: [x:A]. B[x],
implies: P Q,
all: x:A. B[x],
exists: x:A. B[x],
l_all: (xL.P[x]),
member: t T,
length: ||as||,
ycomb: Y,
prop: 
,
so_lambda: 
x.t[x],
true: True,
top: Top,
cand: A c B,
infix_ap: x f y,
and: P Q,
false: False,
l_exists: (xL. P[x]),
le: A 
B,
not: A,
subtype: S 
T,
consensus-state4: ConsensusState,
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
pi1: fst(t),
cs-inning: Inning(s;a),
pi2: snd(t),
cs-estimate: Estimate(s;a),
ts-reachable: ts-reachable(ts),
consensus-ts4: consensus-ts4(V;A;W),
ts-type: ts-type(ts),
or: P 
Q,
consensus-rel: CR[x,y],
rel_implies: R1 => R2,
ts-rel: ts-rel(ts),
cs-precondition: state s may consider v in inning i,
ge: i 
j ,
reduce: reduce(f;k;as),
deq-member: deq-member(eq;x;L),
band: p 
q,
decidable: Dec(P),
squash: 
T,
nat_plus: 
,
eqof: eqof(d),
Id: Id,
bnot: b,
bor: p q,
iff: P 
Q,
rev_implies: P Q,
consensus-rel-mod: CR(in ws)[x, y] ,
int_seg: {i..j
},
lelt: i j < k,
guard: {T},
cs-archive-blocked: in state s, ws' blocks ws from archiving v in inning i,
assert: 
b,
fpf-ap: f(x),
fpf-single: x : v,
so_apply: x[s],
nat: ,
cs-inning-committed: in state s, inning i has committed v,
cs-archived: by state s, a archived v in inning i,
fpf-domain: fpf-domain(f),
one-intersection: one-intersection(A;W),
uiff: uiff(P;Q),
uimplies: b supposing a,
unit: Unit,
bool: 
,
sq_type: SQType(T),
three-intersection: three-intersection(A;W),
it: 
,
eq_id: a = b
Lemmas :
three-intersection-two-intersection,
two-intersection-one-intersection,
nil_member,
Id_wf,
l_member_wf,
length_wf,
non_neg_length,
length_wf_nat,
three-intersection_wf,
guard_wf,
exists_wf,
not_wf,
equal_wf,
all_wf,
decidable_wf,
ts-reachable_wf,
consensus-ts4_wf,
ts-type_wf,
less_than_wf,
nat_wf,
and_wf,
length-map,
subtype-top,
subtype-fpf2,
cs-estimate_wf,
fpf-domain_wf,
consensus-rel-mod_wf,
rel_star_wf,
consensus-state4_wf,
cs-inning_wf,
map_wf,
imax-list_wf,
le_wf,
member_map,
imax-list-ub,
id-deq_wf,
deq-member_wf,
ifthenelse_wf,
fresh-inning-reachable,
not_functionality_wrt_iff,
assert_of_bnot,
eqff_to_assert,
assert-deq-member,
eqtt_to_assert,
iff_weakening_uiff,
iff_transitivity,
bnot_wf,
assert_wf,
bool_wf,
strong-subtype-self,
strong-subtype-set3,
strong-subtype-deq-subtype,
bool_subtype_base,
subtype_base_sq,
bool_cases,
consensus-ts4-estimate-domain,
rel_star_transitivity,
ts-rel_wf,
ts-init_wf,
int-deq_wf,
fpf-single_wf,
fpf-join_wf,
cs-precondition_wf,
subtype_rel_simple_product,
subtype_rel_self,
subtype_rel_function,
top_wf,
fpf_wf,
rel_star_monotonic,
decidable__cs-archive-blocked,
cs-archive-blocked_wf,
decidable__l_exists,
decidable__lt,
int_subtype_base,
decidable__equal_Id,
decidable__l_member,
sq_stable_from_decidable,
property-from-l_member,
cs-inning-committed_wf,
nat_properties,
ge_wf,
rel_star_weakening,
first0,
band_wf,
firstn_wf,
select_wf,
decidable__cand,
eq_id_wf,
true_wf,
squash_wf,
bor_wf,
deq-member-firstn,
not_functionality_wrt_uiff,
assert-eq-id,
uiff_transitivity,
atom2_subtype_base,
bor_ff_simp,
rel_rel_star,
btrue_wf,
select_member,
lelt_wf,
l_member_set2,
or_wf,
pi1_wf_top,
Error :pi2_wf,
or_functionality_wrt_iff,
assert_of_bor,
bnot_thru_band,
assert_functionality_wrt_uiff,
and_functionality_wrt_iff,
assert_of_band,
list-subtype,
fpf-ap_wf,
fpf-dom_wf,
fpf-trivial-subtype-top,
fpf-join-dom-sq,
fpf-single-dom,
bfalse_wf,
iff_imp_equal_bool,
implies_functionality_wrt_iff,
fpf-join-ap-sq,
deq_wf,
member_firstn,
decidable__equal_int,
firstn_all,
cs-archived_wf,
fpf-domain-join,
cons_member,
member-fpf-domain
\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.
(three-intersection(A;W)
{}\mRightarrow{} (\mforall{}ws\mmember{}W.\mforall{}x:ts-reachable(consensus-ts4(V;A;W))
\mexists{}y:ConsensusState
((x rel\_star(ConsensusState; \mlambda{}x,y.CR(in ws)[x, y] ) y)
\mwedge{} (\mexists{}v:V. \mexists{}i:\mBbbZ{}. in state y, inning i has committed v))))))
Date html generated:
2012_01_23-PM-12_05_58
Last ObjectModification:
2011_12_31-AM-11_10_07
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