Nuprl Lemma : consensus-safety
∀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)
⇒ (∀s1,s2:ts-reachable(consensus-ts4(V;A;W)).
((s1 (ts-rel(consensus-ts4(V;A;W))^*) s2)
⇒ (∀v1,v2:V.
((∃i:ℕ. in state s1, inning i has committed v1)
⇒ (∃j:ℕ. in state s2, inning j has committed v2)
⇒ (v1 = v2 ∈ V))))))))
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),
Id: Id,
l_member: (x ∈ l),
length: ||as||,
list: T List,
rel_star: R^*,
nat: ℕ,
decidable: Dec(P),
guard: {T},
infix_ap: x f y,
ge: i ≥ j ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
ts-reachable: ts-reachable(ts),
ts-rel: ts-rel(ts),
ts-type: ts-type(ts)
Lemmas :
consensus-safety1,
exists_wf,
nat_wf,
cs-inning-committed_wf,
subtype_rel_set,
ts-type_wf,
consensus-ts4_wf,
consensus-state4_wf,
infix_ap_wf,
rel_star_wf,
ts-rel_wf,
ts-init_wf,
subtype_rel_self,
ts-reachable_wf,
subtype_rel_wf,
subtype_rel_dep_function,
two-intersection_wf,
ge_wf,
length_wf,
list_wf,
Id_wf,
l_member_wf,
all_wf,
decidable_wf,
not_wf,
equal_wf,
ts-refinement-reachable2,
consensus-ts1_wf,
consensus-state1_wf,
top_wf,
equal-wf-T-base,
cs-decided_wf,
eq_int_wf,
assert_wf,
bnot_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
and_wf,
outl_wf,
isl_wf,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse
\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{} (\mforall{}s1,s2:ts-reachable(consensus-ts4(V;A;W)).
((s1 rel\_star(ts-type(consensus-ts4(V;A;W)); ts-rel(consensus-ts4(V;A;W))) s2)
{}\mRightarrow{} (\mforall{}v1,v2:V.
((\mexists{}i:\mBbbN{}. in state s1, inning i has committed v1)
{}\mRightarrow{} (\mexists{}j:\mBbbN{}. in state s2, inning j has committed v2)
{}\mRightarrow{} (v1 = v2))))))))
Date html generated:
2015_07_17-AM-11_37_30
Last ObjectModification:
2015_01_28-AM-01_32_06
Home
Index