Nuprl Lemma : consensus-reachable
∀[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.
(three-intersection(A;W)
⇒ (∀ws∈W.∀x:ts-reachable(consensus-ts4(V;A;W))
∃y:ConsensusState
((x ((λx,y. CR(in ws)[x, y] )^*) y) ∧ (∃v:V. ∃i:ℤ. in state y, inning i has committed v))))))
Proof
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,
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
rel_star: R^*,
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: λx.A[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T,
ts-reachable: ts-reachable(ts)
Lemmas :
three-intersection-two-intersection,
two-intersection-one-intersection,
l_all_iff,
l_member_wf,
all_wf,
ts-reachable_wf,
consensus-ts4_wf,
subtype_rel_wf,
ts-type_wf,
exists_wf,
consensus-state4_wf,
infix_ap_wf,
subtype_rel_set,
rel_star_wf,
consensus-rel-mod_wf,
subtype_rel_dep_function,
ts-rel_wf,
ts-init_wf,
subtype_rel_self,
cs-inning-committed_wf,
set_wf,
Id_wf,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
product_subtype_list,
length_of_cons_lemma,
length_wf,
non_neg_length,
length_wf_nat,
three-intersection_wf,
list_wf,
not_wf,
equal_wf,
decidable_wf,
imax-list_wf,
map_wf,
cs-inning_wf,
map_length,
equal-wf-T-base,
int_subtype_base,
fpf-domain_wf,
cs-estimate_wf,
top_wf,
consensus-state4-subtype,
imax-list-ub,
l_exists_iff,
le_wf,
member_map,
le_weakening,
decidable__lt,
false_wf,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
less_than_wf,
deq-member_wf,
id-deq_wf,
subtype_rel-deq,
sq_stable__l_member,
decidable__equal_Id,
bool_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
assert-deq-member,
fresh-inning-reachable,
eqtt_to_assert,
assert_wf,
bnot_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
consensus-ts4-estimate-domain,
fpf_wf,
rel_star_transitivity,
rel_star_monotonic,
or_wf,
cs-precondition_wf,
fpf-join_wf,
fpf-single_wf,
int-deq_wf,
decidable__l_exists,
cs-archive-blocked_wf,
decidable__cs-archive-blocked,
l_all_wf2,
decidable__equal_int,
not-equal-2,
less-iff-le,
lelt_wf,
list_subtype_base,
set_subtype_base,
atom2_subtype_base,
select_wf,
sq_stable__le,
subtype_rel_list,
primrec-wf2,
nat_wf,
first0,
reduce_nil_lemma,
rel_star_weakening,
eta_conv,
iff_weakening_equal,
firstn_wf,
subtract_wf,
assert_of_band,
decidable__cand,
decidable__le,
not-le-2,
zero-add,
minus-minus,
add-zero,
decidable__l_member,
decidable__not,
eq_id_wf,
le-add-cancel-alt,
assert-eq-id,
assert_elim,
and_wf,
eq_id_self,
band_wf,
not_assert_elim,
deq-member-firstn,
bor_wf,
deq_wf,
bfalse_wf,
rel_rel_star,
list-subtype,
l_member-settype,
pi1_wf_top,
squash_wf,
true_wf,
subtype_rel_product,
subtype_top,
pi2_wf,
member_wf,
equal-wf-base-T,
bnot_thru_band,
assert_of_bor,
fpf-dom_wf,
fpf-ap_wf,
subtype-fpf2,
fpf-join-dom-sq,
deq_member_cons_lemma,
intdeq_reduce_lemma,
deq_member_nil_lemma,
eq_int_wf,
assert_of_eq_int,
less_than_transitivity1,
less_than_irreflexivity,
neg_assert_of_eq_int,
assert_functionality_wrt_uiff,
bor_ff_simp,
fpf-join-ap-sq,
fpf-join-dom,
equal-wf-base,
fpf-single-dom,
ifthenelse_wf,
member_firstn,
le_weakening2,
le-add-cancel2,
less_than_transitivity2,
firstn_all,
iff_imp_equal_bool,
iff_wf,
cs-archived_wf,
fpf-domain-join,
cons_member,
fpf_ap_single_lemma,
uiff_transitivity,
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:
2015_07_17-AM-11_39_52
Last ObjectModification:
2015_07_16-AM-10_24_50
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