Nuprl Lemma : real-unit-ball-totally-bounded1
∀n:ℕ. ∀k:ℕ+. ∀p:B(n). ∃q:unit-ball-approx(n;k * 8 * n). (d(p;approx-ball-to-ball(k * 8 * n;q)) ≤ (r1/r(k)))
Proof
Definitions occuring in Statement :
approx-ball-to-ball: approx-ball-to-ball(k;p),
unit-ball-approx: unit-ball-approx(n;k),
real-unit-ball: B(n),
real-vec-dist: d(x;y),
rdiv: (x/y),
rleq: x ≤ y,
int-to-real: r(n),
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
nat_plus: ℕ+,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
label: ...$L... t,
real-unit-ball: B(n),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
dot-product: x⋅y,
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
rneq: x ≠ y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
cand: A c∧ B,
nequal: a ≠ b ∈ T ,
unit-ball-approx: unit-ball-approx(n;k),
pi1: fst(t),
rge: x ≥ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
rless: x < y,
sq_exists: ∃x:A [B[x]],
approx-ball-to-ball: approx-ball-to-ball(k;p),
rleq: x ≤ y,
rnonneg: rnonneg(x),
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermConstant: "const",
rtermMultiply: left "*" right,
rtermVar: rtermVar(var),
pi2: snd(t),
sq_stable: SqStable(P)
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
real-unit-ball_wf,
nat_plus_wf,
istype-nat,
unit-ball-approx0,
multiply_nat_wf,
nat_plus_properties,
nat_properties,
decidable__le,
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,
intformeq_wf,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
subtype_rel_wf,
squash_wf,
true_wf,
istype-universe,
top_wf,
unit-ball-approx_wf,
iff_weakening_equal,
rsum-empty,
rleq_wf,
rsqrt_wf,
rleq_weakening_equal,
int-to-real_wf,
rdiv_wf,
rless-int,
decidable__lt,
rless_wf,
rleq-int-fractions2,
rleq_functionality,
rsqrt0,
req_weakening,
istype-less_than,
set_subtype_base,
less_than_wf,
rational-inner-approx-int,
mul_nat_plus,
int_seg_properties,
mul-swap,
mul-associates,
rabs_wf,
rneq-int,
int_entire_a,
rsub_wf,
int_seg_wf,
sum_wf,
real-vec-dist_wf,
approx-ball-to-ball_wf,
mul_bounds_1b,
rleq_functionality_wrt_implies,
real-vec-norm_wf,
square-rleq-implies,
real-vec-norm-nonneg,
nat_plus_subtype_nat,
rnexp_wf,
dot-product_wf,
real-vec-norm-squared,
real_wf,
rmul_wf,
rnexp2-nonneg,
req_functionality,
req_inversion,
rnexp2,
rabs-rnexp,
rabs-of-nonneg,
rsum_wf,
subtract_wf,
itermAdd_wf,
itermSubtract_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
rsum_functionality2,
item-rleq-rsum-of-nonneg,
subtract-add-cancel,
rabs-int,
subtype_rel_self,
absval_pos,
nat_wf,
le_wf,
absval-non-neg,
absval_wf,
iff_weakening_uiff,
req_transitivity,
rabs-rdiv,
rneq_wf,
rmul_preserves_rleq2,
rleq_weakening_rless,
rinv_wf2,
rleq-int,
rmul_functionality,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
absval_unfold,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
itermMinus_wf,
int_term_value_minus_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
assert_wf,
rsum_int,
rsum_functionality_wrt_rleq,
rnexp_functionality_wrt_rleq,
zero-rleq-rabs,
rleq_weakening,
exp_wf2,
exp-positive,
rless_functionality,
rnexp-int,
rnexp-rdiv,
rleq-implies-rleq,
rmul-int,
rsum_linearity2,
dot-product-comm,
mul_bounds_1a,
rnexp-one,
implies-real-vec-dist-rleq,
int-rdiv_wf,
nat_plus_inc_int_nzero,
le_witness_for_triv,
rabs_functionality,
rsub_functionality,
int-rdiv-req,
mul_preserves_le,
rsqrt_squared,
rneq_functionality,
assert-rat-term-eq2,
rtermDivide_wf,
rtermConstant_wf,
rtermMultiply_wf,
rtermVar_wf,
rdiv_functionality,
rmul_preserves_rleq,
sq_stable__rless,
rinv-of-rmul,
rinv-mul-as-rdiv,
rmul-neq-zero,
square-nonzero,
rmul-rinv3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
universeIsType,
dependent_set_memberEquality_alt,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
multiplyEquality,
productElimination,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
universeEquality,
imageMemberEquality,
baseClosed,
minusEquality,
inrFormation_alt,
closedConclusion,
promote_hyp,
productIsType,
equalityIstype,
sqequalBase,
baseApply,
functionExtensionality,
functionIsType,
addEquality,
equalityElimination,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isectIsTypeImplies,
applyLambdaEquality,
functionIsTypeImplies
Latex:
\mforall{}n:\mBbbN{}. \mforall{}k:\mBbbN{}\msupplus{}. \mforall{}p:B(n).
\mexists{}q:unit-ball-approx(n;k * 8 * n). (d(p;approx-ball-to-ball(k * 8 * n;q)) \mleq{} (r1/r(k)))
Date html generated:
2019_10_30-AM-11_28_55
Last ObjectModification:
2019_06_28-PM-01_56_25
Theory : real!vectors
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