Nuprl Lemma : near-log-exists-ext
∀a:{a:ℝ| r0 < a} . ∀N:ℕ+. ∃m:ℕ+. (∃z:ℤ [(|(r(z))/m - rlog(a)| ≤ (r1/r(N)))])
Proof
Definitions occuring in Statement :
rlog: rlog(x),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
int-rdiv: (a)/k1,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: 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,
int: ℤ
Definitions unfolded in proof :
member: t ∈ T,
subtract: n - m,
so_lambda: λ2x.t[x],
genrec-ap: genrec-ap,
near-log-exists,
nearby-cases,
r-archimedean,
iff_weakening_uiff,
rleq_functionality,
near-inverse-of-increasing-function-ext,
sq_stable__and,
sq_stable__rleq,
decidable__lt,
rleq_functionality_wrt_implies,
canonical-bound-property,
decidable__squash,
decidable__and,
decidable__less_than',
decidable_functionality,
squash_elim,
sq_stable_from_decidable,
any: any x,
iff_preserves_decidability,
sq_stable__from_stable,
stable__from_decidable,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
so_apply: x[s],
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
near-log-exists,
lifting-strict-callbyvalue,
istype-void,
strict4-decide,
lifting-strict-decide,
lifting-strict-less,
strict4-spread,
lifting-strict-spread,
nearby-cases,
r-archimedean,
iff_weakening_uiff,
rleq_functionality,
near-inverse-of-increasing-function-ext,
sq_stable__and,
sq_stable__rleq,
decidable__lt,
rleq_functionality_wrt_implies,
canonical-bound-property,
decidable__squash,
decidable__and,
decidable__less_than',
decidable_functionality,
squash_elim,
sq_stable_from_decidable,
iff_preserves_decidability,
sq_stable__from_stable,
stable__from_decidable
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination
Latex:
\mforall{}a:\{a:\mBbbR{}| r0 < a\} . \mforall{}N:\mBbbN{}\msupplus{}. \mexists{}m:\mBbbN{}\msupplus{}. (\mexists{}z:\mBbbZ{} [(|(r(z))/m - rlog(a)| \mleq{} (r1/r(N)))])
Date html generated:
2019_10_31-AM-06_09_42
Last ObjectModification:
2019_02_05-PM-04_50_45
Theory : reals_2
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