Nuprl Lemma : cs-ref-map-step
∀[V:Type]
({∃v1,v2:V. (¬(v1 = v2 ∈ V))}
⇒ (∀A:Id List. ∀W:{a:Id| (a ∈ A)} List List.
∀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)
⇒ {(||f x|| ≤ ||f y||)
∧ (∃i:ℤ
((∀j:ℕ||f x||. f y[j] = f x[j] ∈ consensus-state3(V) supposing ¬(j = i ∈ ℤ))
∧ (||f y|| = (||f x|| + 1) ∈ ℤ) ∧ (f y[||f x||] = INITIAL ∈ consensus-state3(V))
supposing ||f x|| < ||f y||))})))
supposing ||W|| ≥ 1 ))
Proof
Definitions occuring in Statement :
cs-ref-map-constraints: cs-ref-map-constraints(V;A;W;f),
consensus-ts4: consensus-ts4(V;A;W),
consensus-state4: ConsensusState,
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-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
infix_ap: x f y,
ge: i ≥ j ,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ─→ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T,
ts-reachable: ts-reachable(ts),
ts-rel: ts-rel(ts)
Lemmas :
decidable__le,
length_wf,
consensus-state3_wf,
length_wf_nat,
decidable__lt,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
add-commutes,
add_functionality_wrt_le,
le-add-cancel,
consensus-ts4-inning-rel,
rel_rel_star,
ts-type_wf,
consensus-ts4_wf,
ts-rel_wf,
subtype_rel_self,
Id_wf,
l_member_wf,
fpf_wf,
le_transitivity,
cs-inning_wf,
le_wf,
less_than_irreflexivity,
all_wf,
int_seg_wf,
isect_wf,
not_wf,
equal-wf-T-base,
int_subtype_base,
equal_wf,
select_wf,
sq_stable__le,
less_than_transitivity1,
less_than_wf,
zero-le-nat,
decidable__equal_Id,
set_wf,
cs-estimate_wf,
member_wf,
fpf-domain_wf,
top_wf,
consensus-state4-subtype,
iff_wf,
subtype-fpf2,
subtype_base_sq,
atom2_subtype_base,
member_singleton,
or_wf,
fpf-single_wf,
fpf-domain-join,
int-deq_wf,
int_seg_properties,
fpf-ap_wf,
squash_wf,
assert_wf,
fpf-dom_wf,
subtype_top,
deq_wf,
member-fpf-dom,
fpf-join-ap-sq,
bool_wf,
bnot_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
member-fpf-domain,
fpf-join-dom,
equal-wf-base-T,
fpf-single-dom,
le_antisymmetry_iff,
add-associates,
not-equal-2,
le-add-cancel2,
le_weakening,
cs-ref-map-equal,
lelt_wf,
iff_weakening_equal,
cs-not-completed_wf,
cs-archived_wf,
list_wf,
cs-inning-committable_wf,
cs-inning-committed_wf,
nat_wf,
non_neg_length,
decidable__equal_int,
less-iff-le,
zero-add,
add-mul-special,
zero-mul,
add-zero,
hd_wf,
hd_member,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
consensus-ts4-estimate-domain,
infix_ap_wf,
rel_star_wf,
ts-init_wf,
exists_wf
\mforall{}[V:Type]
(\{\mexists{}v1,v2:V. (\mneg{}(v1 = v2))\}
{}\mRightarrow{} (\mforall{}A:Id List. \mforall{}W:\{a:Id| (a \mmember{} A)\} List List.
\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{} \{(||f x|| \mleq{} ||f y||)
\mwedge{} (\mexists{}i:\mBbbZ{}
((\mforall{}j:\mBbbN{}||f x||. f y[j] = f x[j] supposing \mneg{}(j = i))
\mwedge{} (||f y|| = (||f x|| + 1)) \mwedge{} (f y[||f x||] = INITIAL)
supposing ||f x|| < ||f y||))\})))
supposing ||W|| \mgeq{} 1 ))
Date html generated:
2015_07_17-AM-11_35_25
Last ObjectModification:
2015_07_16-AM-10_16_51
Home
Index