Nuprl Lemma : near-log_wf
∀[a:{a:ℝ| r0 < a} ]. ∀[N:ℕ+]. (near-log(a;N) ∈ {r:ℕ+ × ℤ| let j,c = r in |(r(c))/j - rlog(a)| ≤ (r1/r(N))} )
Proof
Definitions occuring in Statement :
near-log: near-log(a;N),
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+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
spread: spread def,
product: x:A × B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
sq_exists: ∃x:A [B[x]],
uall: ∀[x:A]. B[x],
member: t ∈ T,
near-log: near-log(a;N),
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ
Lemmas referenced :
near-log-exists-ext,
rleq_wf,
rabs_wf,
rsub_wf,
int-rdiv_wf,
int-to-real_wf,
rlog_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_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_var_lemma,
int_formula_prop_wf,
rless_wf,
nat_plus_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
because_Cache,
sqequalHypSubstitution,
hypothesisEquality,
lambdaEquality_alt,
productElimination,
setElimination,
rename,
dependent_set_memberEquality_alt,
independent_pairEquality,
universeIsType,
isectElimination,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
inrFormation_alt,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
inhabitedIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
setIsType
Latex:
\mforall{}[a:\{a:\mBbbR{}| r0 < a\} ]. \mforall{}[N:\mBbbN{}\msupplus{}].
(near-log(a;N) \mmember{} \{r:\mBbbN{}\msupplus{} \mtimes{} \mBbbZ{}| let j,c = r in |(r(c))/j - rlog(a)| \mleq{} (r1/r(N))\} )
Date html generated:
2019_10_31-AM-06_09_50
Last ObjectModification:
2019_01_28-AM-10_09_09
Theory : reals_2
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