Nuprl Lemma : cs-ref-map-changed
∀[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.
(two-intersection(A;W)
⇒ (∀f:ConsensusState ─→ (consensus-state3(V) List)
(cs-ref-map-constraints(V;A;W;f)
⇒ (∀x,y:ts-reachable(consensus-ts4(V;A;W)).
((x ts-rel(consensus-ts4(V;A;W)) y)
⇒ (∀i:ℕ
(∀v:V
((in state x, inning i could commit v ∧ (¬in state y, inning i could commit v ))
⇒ ((f y[i] = WITHDRAWN ∈ consensus-state3(V))
∨ ((f x[i] = INITIAL ∈ consensus-state3(V))
∧ ((f y[i] = INITIAL ∈ consensus-state3(V))
∨ (∃v':V
((∀j:ℕi. (¬(f x[j] = INITIAL ∈ consensus-state3(V))))
∧ ((f y[i] = CONSIDERING[v'] ∈ consensus-state3(V))
∨ (f y[i] = COMMITED[v'] ∈ consensus-state3(V)))
∧ (∀j:ℕi. ∀v'':V.
(((f x[j] = CONSIDERING[v''] ∈ consensus-state3(V))
∨ (f x[j] = COMMITED[v''] ∈ consensus-state3(V)))
⇒ (v'' = v' ∈ V)))))))))) supposing
(i < ||f y|| and
i < ||f x||)))))))))
Proof
Definitions occuring in Statement :
cs-ref-map-constraints: cs-ref-map-constraints(V;A;W;f),
two-intersection: two-intersection(A;W),
cs-inning-committable: in state s, inning i could commit v ,
consensus-ts4: consensus-ts4(V;A;W),
consensus-state4: ConsensusState,
cs-commited: COMMITED[v],
cs-considering: CONSIDERING[v],
cs-withdrawn: WITHDRAWN,
cs-initial: INITIAL,
consensus-state3: consensus-state3(T),
Id: Id,
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
decidable: Dec(P),
uimplies: b supposing a,
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,
or: 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-reachable: ts-reachable(ts),
ts-rel: ts-rel(ts)
Lemmas :
member-less_than,
length_wf,
consensus-state3_wf,
cs-inning-committable_wf,
not_wf,
less_than_wf,
nat_wf,
infix_ap_wf,
consensus-state4_wf,
ts-rel_wf,
subtype_rel_dep_function,
ts-type_wf,
subtype_rel_self,
l_member_wf,
fpf_wf,
ts-reachable_wf,
consensus-ts4_wf,
subtype_rel_wf,
cs-ref-map-constraints_wf,
list_wf,
two-intersection_wf,
Id_wf,
exists_wf,
equal_wf,
all_wf,
decidable_wf,
consensus-state3-cases,
select_wf,
sq_stable__le,
decidable__cs-inning-two-committable,
two-intersection-one-intersection,
int_seg_wf,
equal-wf-T-base,
less_than_transitivity2,
subtype_rel_set,
rel_star_wf,
ts-init_wf,
le_weakening2,
or_wf,
cs-considering_wf,
cs-commited_wf,
decidable__cs-inning-committable-some,
decidable__exists_int_seg,
cs-archived_wf,
list-subtype,
decidable__cs-archived,
l_all_iff,
sq_stable__l_member,
decidable__equal_Id,
cs-not-completed_wf,
set_wf,
l_exists_iff,
consensus-ts4-archived-invariant,
int_seg_subtype-nat,
false_wf,
consensus-ts4-archived-stable,
decidable__cs-not-completed,
subtype_base_sq,
atom2_subtype_base,
nat_properties,
decidable__le,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
add-commutes,
le_antisymmetry_iff,
cs-inning_wf,
add-associates,
less-iff-le,
add_functionality_wrt_le,
le-add-cancel2,
squash_wf,
true_wf,
fpf-domain_wf,
top_wf,
subtype-fpf2,
iff_weakening_equal,
le_transitivity,
le_weakening,
fpf-domain-join,
cs-estimate_wf,
consensus-state4-subtype,
fpf-single_wf,
int-deq_wf,
member_singleton,
less_than_transitivity1,
less_than_irreflexivity,
int_seg_properties,
decidable__cs-inning-committed,
cs-inning-committed-committable,
cs-inning-committable-step,
cs-inning-committed-single,
consensus-ts4-inning-committed-stable
\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.
(two-intersection(A;W)
{}\mRightarrow{} (\mforall{}f:ConsensusState {}\mrightarrow{} (consensus-state3(V) List)
(cs-ref-map-constraints(V;A;W;f)
{}\mRightarrow{} (\mforall{}x,y:ts-reachable(consensus-ts4(V;A;W)).
((x ts-rel(consensus-ts4(V;A;W)) y)
{}\mRightarrow{} (\mforall{}i:\mBbbN{}
(\mforall{}v:V
((in state x, inning i could commit v
\mwedge{} (\mneg{}in state y, inning i could commit v ))
{}\mRightarrow{} ((f y[i] = WITHDRAWN)
\mvee{} ((f x[i] = INITIAL)
\mwedge{} ((f y[i] = INITIAL)
\mvee{} (\mexists{}v':V
((\mforall{}j:\mBbbN{}i. (\mneg{}(f x[j] = INITIAL)))
\mwedge{} ((f y[i] = CONSIDERING[v']) \mvee{} (f y[i] = COMMITED[v']))
\mwedge{} (\mforall{}j:\mBbbN{}i. \mforall{}v'':V.
(((f x[j] = CONSIDERING[v''])
\mvee{} (f x[j] = COMMITED[v'']))
{}\mRightarrow{} (v'' = v')))))))))) supposing
(i < ||f y|| and
i < ||f x||)))))))))
Date html generated:
2015_07_17-AM-11_35_00
Last ObjectModification:
2015_07_16-AM-10_23_50
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