Nuprl Lemma : compact-dist-nonneg
∀[X:Type]. ∀[d:metric(X)]. ∀[A:Type]. ∀[c:mcompact(A;d)]. ∀[x:X]. (r0 ≤ dist(x;A)) supposing A ⊆r X
Proof
Definitions occuring in Statement :
compact-dist: dist(x;A),
mcompact: mcompact(X;d),
metric: metric(X),
rleq: x ≤ y,
int-to-real: r(n),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
guard: {T},
dist-fun: dist-fun(d;x),
top: Top,
false: False,
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
prop: ℙ,
exists: ∃x:A. B[x],
and: P ∧ Q,
compact-dist: dist(x;A),
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
real_term_value_add_lemma,
req_weakening,
rless_functionality,
itermAdd_wf,
radd_wf,
rless_irreflexivity,
mdist_wf,
rless_transitivity1,
mdist-nonneg,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
istype-void,
real_term_value_sub_lemma,
istype-int,
real_polynomial_null,
req-iff-rsub-is-0,
itermMinus_wf,
itermVar_wf,
itermConstant_wf,
itermSubtract_wf,
rsub_wf,
istype-universe,
metric_wf,
subtype_rel_wf,
mcompact_wf,
le_witness_for_triv,
rless_wf,
rless-implies-rless,
rminus_wf,
metric-on-subtype,
compact-inf-property,
int-to-real_wf,
compact-dist_wf,
not-rless,
rmetric_wf,
real_wf,
mfun-subtype2,
dist-fun_wf
Rules used in proof :
equalityIstype,
voidElimination,
int_eqEquality,
approximateComputation,
universeEquality,
instantiate,
isectIsTypeImplies,
isect_memberEquality_alt,
inhabitedIsType,
functionIsTypeImplies,
lambdaEquality_alt,
universeIsType,
because_Cache,
independent_functionElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
lambdaFormation_alt,
natural_numberEquality,
sqequalRule,
independent_isectElimination,
applyEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
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]. (r0 \mleq{} dist(x;A)) supposing A \msubseteq{}r X
Date html generated:
2019_10_30-AM-07_12_42
Last ObjectModification:
2019_10_25-PM-04_56_23
Theory : reals
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