Nuprl Lemma : real-unit-ball-totally-bounded
∀n:ℕ. ∀k:ℕ+. (∃L:{p:B(n)| rational-vec(n;p)} List [(∀p:B(n). ∃i:ℕ||L||. (d(p;L[i]) ≤ (r1/r(k))))])
Proof
Definitions occuring in Statement :
real-unit-ball: B(n),
rational-vec: rational-vec(n;x),
real-vec-dist: d(x;y),
rdiv: (x/y),
rleq: x ≤ y,
int-to-real: r(n),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
unit-ball-approx: unit-ball-approx(n;k),
uall: ∀[x:A]. B[x],
nat: ℕ,
nat_plus: ℕ+,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
implies: P ⇒ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
iff: P ⇐⇒ Q,
sq_exists: ∃x:A [B[x]],
real-unit-ball: B(n),
le: A ≤ B,
rneq: x ≠ y,
guard: {T},
rev_implies: P ⇐ Q,
sq_type: SQType(T),
ext-eq: A ≡ B,
rational-vec: rational-vec(n;x),
less_than': less_than'(a;b),
approx-ball-to-ball: approx-ball-to-ball(k;p),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
l_member: (x ∈ l),
cand: A c∧ B,
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
dot-product: x⋅y,
subtract: n - m,
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
real-unit-ball-totally-bounded1,
finite-decidable-subset,
int_seg_wf,
le_wf,
sum_wf,
finite-function,
nsub_finite,
int_seg_finite,
decidable__squash,
decidable__le,
finite-iff-listable,
unit-ball-approx_wf,
multiply_nat_wf,
nat_plus_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
itermMultiply_wf,
int_term_value_mul_lemma,
nat_plus_wf,
istype-nat,
map_wf,
nat_plus_subtype_nat,
real-unit-ball_wf,
rational-vec_wf,
length_wf,
rleq_wf,
real-vec-dist_wf,
select_wf,
int_seg_properties,
decidable__lt,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
real-unit-ball-0,
approx-ball-to-ball_wf,
mul_nat_plus,
istype-less_than,
intformeq_wf,
int_formula_prop_eq_lemma,
int-rdiv-req,
int_entire_a,
nequal_wf,
req_wf,
int-rdiv_wf,
rneq-int,
length-map,
select-map,
subtype_rel_list,
top_wf,
rsum-empty,
squash_wf,
true_wf,
real_wf,
real-vec_wf,
subtype_rel_self,
iff_weakening_equal,
rsqrt_wf,
rleq_weakening_equal,
rleq-int-fractions2,
rleq_functionality,
rsqrt0,
req_weakening
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
functionEquality,
isectElimination,
natural_numberEquality,
setElimination,
rename,
minusEquality,
multiplyEquality,
productElimination,
imageElimination,
addEquality,
because_Cache,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
universeIsType,
functionIsType,
independent_functionElimination,
dependent_set_memberEquality_alt,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
dependent_set_memberFormation_alt,
setEquality,
productIsType,
closedConclusion,
inrFormation_alt,
instantiate,
cumulativity,
intEquality,
equalityIstype,
baseClosed,
sqequalBase,
baseApply,
imageMemberEquality,
setIsType,
universeEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}k:\mBbbN{}\msupplus{}. (\mexists{}L:\{p:B(n)| rational-vec(n;p)\} List [(\mforall{}p:B(n). \mexists{}i:\mBbbN{}||L||. (d(p;L[i]) \mleq{} (r1/r(k))))])
Date html generated:
2019_10_30-AM-11_29_00
Last ObjectModification:
2019_06_28-PM-01_56_28
Theory : real!vectors
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