Nuprl Lemma : compact-metric-to-metric-continuity
∀X:Type. ∀d:metric(X). (mcompact(X;d) ⇒ (∀Y:Type. ∀dY:metric(Y). ∀f:FUN(X ⟶ Y). UC(f:X ⟶ Y)))
Proof
Definitions occuring in Statement :
mcompact: mcompact(X;d),
m-unif-cont: UC(f:X ⟶ Y),
mfun: FUN(X ⟶ Y),
metric: metric(X),
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
req_int_terms: t1 ≡ t2,
bfalse: ff,
subtype_rel: A ⊆r B,
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
subtract: n - m,
prod-metric: prod-metric(k;d),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rneq: x ≠ y,
rmetric: rmetric(),
mdist: mdist(d;x;y),
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
nat_plus: ℕ+,
int_seg: {i..j-},
pi2: snd(t),
pi1: fst(t),
prod-metric-space: prod-metric-space(k;X),
mk-metric-space: X with d,
false: False,
not: ¬A,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
m-unif-cont: UC(f:X ⟶ Y),
squash: ↓T,
is-mfun: f:FUN(X;Y),
metric: metric(X),
meq: x ≡ y,
sq_stable: SqStable(P),
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
mfun: FUN(X ⟶ Y),
mcompact: mcompact(X;d),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
mdist-nonneg,
rabs-of-nonneg,
rsub_functionality,
rabs_functionality,
rabs_wf,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
mdist-symm,
mdist-same,
req-iff-rsub-is-0,
itermAdd_wf,
itermSubtract_wf,
rsub_wf,
rleq-implies-rleq,
rsum-single,
radd_functionality,
req_weakening,
rsum-split-first,
rleq_functionality,
int_subtype_base,
istype-false,
radd_wf,
rsum_wf,
rless_wf,
int_term_value_var_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformand_wf,
rless-int,
rdiv_wf,
rleq_wf,
eq_int_wf,
ifthenelse_wf,
mdist_functionality,
rmetric-meq,
prod-metric-meq,
rmetric_wf,
real_wf,
is-mfun_wf,
istype-less_than,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
nat_plus_properties,
mdist_wf,
prod-metric_wf,
compact-metric-to-real-continuity,
nat_plus_wf,
mk-metric-space_wf,
prod-metric-space-complete,
int_seg_wf,
istype-le,
istype-void,
m-TB-product,
istype-universe,
mcompact_wf,
metric_wf,
mfun_wf,
int-to-real_wf,
req_witness,
sq_stable__meq,
meq_wf,
sq_stable__all
Rules used in proof :
equalityTransitivity,
equalitySymmetry,
sqequalBase,
baseApply,
equalityIstype,
setIsType,
addEquality,
int_eqEquality,
inrFormation_alt,
closedConclusion,
functionIsType,
productIsType,
isect_memberEquality_alt,
dependent_pairFormation_alt,
approximateComputation,
independent_isectElimination,
unionElimination,
independent_pairEquality,
voidElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
universeEquality,
instantiate,
imageElimination,
baseClosed,
imageMemberEquality,
functionIsTypeImplies,
natural_numberEquality,
dependent_functionElimination,
inhabitedIsType,
because_Cache,
independent_functionElimination,
universeIsType,
applyEquality,
hypothesis,
functionEquality,
lambdaEquality_alt,
sqequalRule,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
setElimination,
cut,
rename,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}X:Type. \mforall{}d:metric(X). (mcompact(X;d) {}\mRightarrow{} (\mforall{}Y:Type. \mforall{}dY:metric(Y). \mforall{}f:FUN(X {}\mrightarrow{} Y). UC(f:X {}\mrightarrow{} Y)))
Date html generated:
2019_10_31-AM-05_59_13
Last ObjectModification:
2019_10_30-AM-11_43_29
Theory : reals
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