Nuprl Lemma : consensus-safety1
∀V:Type
((∀v1,v2:V. Dec(v1 = v2 ∈ V))
⇒ {∃v,v':V. (¬(v = v' ∈ V))}
⇒ (∀L:V List. Dec(∃v:V. (¬(v ∈ L))))
⇒ (∀A:Id List. ∀W:{a:Id| (a ∈ A)} List List.
((||W|| ≥ 1 )
⇒ two-intersection(A;W)
⇒ (∃f:ConsensusState ─→ consensus-state1(V)
((∀v:V. ∀s:ts-reachable(consensus-ts4(V;A;W)).
((f s) = Decided[v] ∈ consensus-state1(V) ⇐⇒ ∃i:ℕ. in state s, inning i has committed v))
∧ ts-refinement(consensus-ts1(V);consensus-ts4(V;A;W);f))))))
Proof
Definitions occuring in Statement :
two-intersection: two-intersection(A;W),
cs-inning-committed: in state s, inning i has committed v,
consensus-ts4: consensus-ts4(V;A;W),
consensus-state4: ConsensusState,
consensus-ts1: consensus-ts1(T),
cs-decided: Decided[v],
consensus-state1: consensus-state1(V),
Id: Id,
l_member: (x ∈ l),
length: ||as||,
list: T List,
nat: ℕ,
decidable: Dec(P),
guard: {T},
ge: i ≥ j ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ─→ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
ts-refinement: ts-refinement(ts1;ts2;f),
ts-reachable: ts-reachable(ts)
Lemmas :
consensus-refinement3,
consensus-ts4-ref-map,
consensus-refinement2,
consensus-refinement1,
two-intersection_wf,
ge_wf,
length_wf,
list_wf,
Id_wf,
l_member_wf,
all_wf,
decidable_wf,
exists_wf,
not_wf,
equal_wf,
ts-refinement-composition,
consensus-ts1_wf,
consensus-ts2_wf,
consensus-ts3_wf,
cs-ref-map3_wf,
consensus-state3_wf,
cs-is-decided_wf,
bool_wf,
eqtt_to_assert,
subtype_rel_sum,
top_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cs-undecided_wf,
consensus-state2_wf,
consensus-ts4_wf,
compose_wf,
consensus-state1_wf,
consensus-state4_wf,
ts-reachable_wf,
subtype_rel_wf,
ts-type_wf,
iff_wf,
subtype_rel_set,
infix_ap_wf,
rel_star_wf,
ts-rel_wf,
ts-init_wf,
subtype_rel_self,
cs-decided_wf,
nat_wf,
cs-inning-committed_wf,
ts-refinement_wf,
subtype_rel_dep_function,
ts-refinement-reachable,
colist_wf,
has-value_wf-partial,
set-value-type,
le_wf,
int-value-type,
colength_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
bnot_wf,
assert_of_bnot,
consensus-state2-cases,
cs-decided_wf2,
assert_functionality_wrt_uiff,
squash_wf,
true_wf,
bfalse_wf,
and_wf,
isl_wf,
btrue_wf,
btrue_neq_bfalse,
cs-ref-map3-decided,
two-intersection-one-intersection,
l_all_iff,
consensus-ts4-estimate-domain,
cs-inning_wf,
less_than_wf,
cs-commited_wf,
select_wf,
sq_stable__le,
equal-wf-base-T
\mforall{}V:Type
((\mforall{}v1,v2:V. Dec(v1 = v2))
{}\mRightarrow{} \{\mexists{}v,v':V. (\mneg{}(v = v'))\}
{}\mRightarrow{} (\mforall{}L:V List. Dec(\mexists{}v:V. (\mneg{}(v \mmember{} L))))
{}\mRightarrow{} (\mforall{}A:Id List. \mforall{}W:\{a:Id| (a \mmember{} A)\} List List.
((||W|| \mgeq{} 1 )
{}\mRightarrow{} two-intersection(A;W)
{}\mRightarrow{} (\mexists{}f:ConsensusState {}\mrightarrow{} consensus-state1(V)
((\mforall{}v:V. \mforall{}s:ts-reachable(consensus-ts4(V;A;W)).
((f s) = Decided[v] \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}. in state s, inning i has committed v))
\mwedge{} ts-refinement(consensus-ts1(V);consensus-ts4(V;A;W);f))))))
Date html generated:
2015_07_17-AM-11_37_24
Last ObjectModification:
2015_01_28-AM-01_34_20
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