Nuprl Lemma : near-log-exists
∀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 :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
guard: {T},
nat: ℕ,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
int_upper: {i...},
rless: x < y,
sq_exists: ∃x:A [B[x]],
real: ℝ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rneq: x ≠ y,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
rgt: x > y,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
rsub: x - y
Lemmas referenced :
r-archimedean,
sq_stable__rleq,
int-to-real_wf,
rleq_transitivity,
rleq-int,
nat_properties,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_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_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
set-value-type,
equal_wf,
less_than_wf,
int-value-type,
subtype_base_sq,
nat_plus_wf,
set_subtype_base,
int_subtype_base,
rless-int,
rexp-of-nonneg-stronger,
rless_transitivity1,
rleq_weakening_rless,
real_exp_wf,
real_wf,
rleq_wf,
rexp_wf,
int_upper_wf,
radd_wf,
le_wf,
int_upper_properties,
sq_stable__less_than,
decidable__le,
istype-le,
itermAdd_wf,
int_term_value_add_lemma,
nat_plus_subtype_nat,
itermMultiply_wf,
int_term_value_mul_lemma,
iff_weakening_uiff,
rleq_functionality,
req_weakening,
real_exp-req,
radd-int,
le_functionality,
le_weakening,
multiply_functionality_wrt_le,
near-inverse-of-increasing-function-ext,
rleq_functionality_wrt_implies,
rexp_functionality_wrt_rless,
rleq_weakening_equal,
rleq_weakening,
rsub_wf,
rdiv_wf,
rless_wf,
int-rdiv_wf,
nequal_wf,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rsub_functionality,
rmul_wf,
rexp-difference-bound,
rmul_preserves_req,
rinv_wf2,
req_functionality,
int-rdiv-req,
req_transitivity,
rmul_functionality,
rinv1,
rmul-identity1,
real_term_value_mul_lemma,
rexp_functionality_wrt_rleq,
radd-preserves-rleq,
real_term_value_add_lemma,
rmul_functionality_wrt_rleq2,
rmul_comm,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
rmul-rinv3,
rleq-int-fractions,
int-rdiv-one,
rexp_functionality,
rexp0,
rleq-implies-rleq,
radd_functionality_wrt_rless2,
sq_stable__and,
rabs_wf,
nat_plus_inc_int_nzero,
le_witness_for_triv,
rlog_wf,
rexp-positive,
rlog-rexp,
subtype_rel_sets_simple,
rabs_functionality,
rmin_ub,
req_inversion,
rleq-int-fractions2,
istype-false,
rmin_wf,
rabs-rlog-difference-bound,
rmul_preserves_rleq,
rinv-mul-as-rdiv,
rmul-rinv,
sq_stable__rless,
rmin_strict_ub,
rlog1,
nearby-cases,
rless_transitivity2,
rminus_wf,
squash_wf,
true_wf,
rabs-rminus,
subtype_rel_self,
iff_weakening_equal,
rlog-inv,
itermMinus_wf,
rminus_functionality,
radd_functionality,
rminus-int,
real_term_value_minus_lemma,
rabs-difference-bound-rleq,
rinv-as-rdiv,
rmul_preserves_rleq2,
rinv-of-rmul,
rmul-is-positive,
rleq-int-fractions3,
rmul-neq-zero,
square-nonzero,
rneq_functionality,
rmul-int,
int_entire_a
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
because_Cache,
isectElimination,
natural_numberEquality,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
independent_isectElimination,
promote_hyp,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
cutEval,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
inhabitedIsType,
intEquality,
instantiate,
cumulativity,
applyEquality,
setIsType,
minusEquality,
productIsType,
addEquality,
multiplyEquality,
applyLambdaEquality,
closedConclusion,
inrFormation_alt,
sqequalBase,
inlFormation_alt,
productEquality,
independent_pairEquality,
functionIsTypeImplies,
dependent_set_memberFormation_alt,
universeEquality,
baseApply
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_36
Last ObjectModification:
2019_02_05-AM-11_00_11
Theory : reals_2
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