Nuprl Lemma : realvec-max-ibs-property
∀k:ℕ. ∀p:ℝ^k.
((r0 < mdist(max-metric(k);p;λi.r0) ⇐⇒ ∃n:ℕ. ((realvec-max-ibs(k;p) n) = 1 ∈ ℤ))
∧ (∀n:ℕ. (((realvec-max-ibs(k;p) n) = 0 ∈ ℤ) ⇒ (mdist(max-metric(k);p;λi.r0) ≤ (r(4)/r(n + 1))))))
Proof
Definitions occuring in Statement :
realvec-max-ibs: realvec-max-ibs(n;p),
max-metric: max-metric(n),
real-vec: ℝ^n,
mdist: mdist(d;x;y),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
real-vec: ℝ^n,
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
realvec-max-ibs: realvec-max-ibs(n;p),
implies: P ⇒ Q,
or: P ∨ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
false: False,
rneq: x ≠ y,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
incr-binary-seq: IBS,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2
Lemmas referenced :
int-to-real_wf,
int_seg_wf,
mdist-nonneg,
real-vec_wf,
max-metric_wf,
rless_ibs_property,
rless_transitivity1,
mdist_wf,
rless_irreflexivity,
req_inversion,
rleq_transitivity,
rdiv_wf,
rless-int,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
rless_wf,
rleq_weakening,
rless_ibs_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
istype-nat,
rabs_wf,
rsub_wf,
rminus_wf,
itermSubtract_wf,
itermMinus_wf,
req_wf,
squash_wf,
true_wf,
real_wf,
rabs-rminus,
rabs-of-nonneg,
subtype_rel_self,
iff_weakening_equal,
req_functionality,
rabs_functionality,
req_weakening,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
lambdaEquality_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
productElimination,
hypothesis,
universeIsType,
natural_numberEquality,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
because_Cache,
independent_pairFormation,
promote_hyp,
independent_functionElimination,
unionElimination,
independent_isectElimination,
voidElimination,
closedConclusion,
addEquality,
inrFormation_alt,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
equalityIstype,
applyEquality,
inhabitedIsType,
intEquality,
baseClosed,
sqequalBase,
imageElimination,
imageMemberEquality,
instantiate,
universeEquality
Latex:
\mforall{}k:\mBbbN{}. \mforall{}p:\mBbbR{}\^{}k.
((r0 < mdist(max-metric(k);p;\mlambda{}i.r0) \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}. ((realvec-max-ibs(k;p) n) = 1))
\mwedge{} (\mforall{}n:\mBbbN{}. (((realvec-max-ibs(k;p) n) = 0) {}\mRightarrow{} (mdist(max-metric(k);p;\mlambda{}i.r0) \mleq{} (r(4)/r(n + 1))))))
Date html generated:
2019_10_30-AM-10_16_13
Last ObjectModification:
2019_07_03-PM-04_22_00
Theory : real!vectors
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