Nuprl Lemma : compact-dist-zero
∀[X:Type]
∀d:metric(X). ∀A:Type.
∀c:mcompact(A;d). ∀x:X. (dist(x;A) = r0 ⇐⇒ ∀n:ℕ+. ∃a:A. (mdist(d;x;a) < (r1/r(n)))) supposing A ⊆r X
Proof
Definitions occuring in Statement :
compact-dist: dist(x;A),
mcompact: mcompact(X;d),
mdist: mdist(d;x;y),
metric: metric(X),
rdiv: (x/y),
rless: x < y,
req: x = y,
int-to-real: r(n),
nat_plus: ℕ+,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
sq_exists: ∃x:A [B[x]],
rless: x < y,
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
dist-fun: dist-fun(d;x),
so_apply: x[s],
so_lambda: λ2x.t[x],
mfun: FUN(X ⟶ Y),
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
istype: istype(T),
rev_implies: P ⇐ Q,
prop: ℙ,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
compact-dist: dist(x;A),
subtype_rel: A ⊆r B,
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
rless_irreflexivity,
rleq_weakening_rless,
rless_transitivity2,
rless_transitivity1,
small-reciprocal-real,
rleq_antisymmetry,
compact-dist-nonneg,
not-rless,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
radd_functionality,
req_weakening,
rless_functionality,
int_term_value_mul_lemma,
itermMultiply_wf,
rless-int-fractions2,
itermAdd_wf,
itermSubtract_wf,
radd_wf,
subtype_rel_dep_function,
is-mfun_wf,
subtype_rel_set,
istype-universe,
metric_wf,
subtype_rel_wf,
mcompact_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
mdist_wf,
rless_wf,
nat_plus_wf,
int-to-real_wf,
compact-dist_wf,
req_wf,
metric-on-subtype,
compact-inf-property,
rmetric_wf,
real_wf,
mfun-subtype2,
dist-fun_wf
Rules used in proof :
dependent_set_memberEquality_alt,
multiplyEquality,
functionEquality,
universeEquality,
instantiate,
inhabitedIsType,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
independent_functionElimination,
inrFormation_alt,
because_Cache,
setElimination,
closedConclusion,
productIsType,
lambdaEquality_alt,
functionIsType,
natural_numberEquality,
universeIsType,
independent_pairFormation,
productElimination,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
independent_isectElimination,
applyEquality,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
rename,
thin,
hypothesis,
axiomEquality,
sqequalRule,
introduction,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X). \mforall{}A:Type.
\mforall{}c:mcompact(A;d). \mforall{}x:X. (dist(x;A) = r0 \mLeftarrow{}{}\mRightarrow{} \mforall{}n:\mBbbN{}\msupplus{}. \mexists{}a:A. (mdist(d;x;a) < (r1/r(n))))
supposing A \msubseteq{}r X
Date html generated:
2019_10_30-AM-07_13_25
Last ObjectModification:
2019_10_25-PM-05_46_39
Theory : reals
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