Nuprl Lemma : logseq-converges
∀a:{a:ℝ| r0 < a} . ∀b:{b:ℝ| |b - rlog(a)| ≤ (r1/r(10))} . lim n→∞.logseq(a;b;n) = rlog(a)
Proof
Definitions occuring in Statement :
logseq: logseq(a;b;n),
rlog: rlog(x),
converges-to: lim n→∞.x[n] = y,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
so_apply: x[s],
exists: ∃x:A. B[x],
int_upper: {i...},
le: A ≤ B,
false: False,
not: ¬A,
subtype_rel: A ⊆r B,
nat: ℕ,
nat_plus: ℕ+,
sq_stable: SqStable(P),
sq_exists: ∃x:{A| B[x]},
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q)
Lemmas referenced :
nat_plus_wf,
set_wf,
real_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rlog_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
cubic_converge_wf,
false_wf,
le_wf,
nat_plus_subtype_nat,
nat_wf,
exp_wf2,
exp_wf4,
equal_wf,
sq_stable__le,
all_wf,
logseq_wf,
nat_properties,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
exp-positive,
exp_wf_nat_plus,
less_than_wf,
rleq_functionality_wrt_implies,
logseq-property,
rleq_weakening_equal,
rleq-int-fractions,
less_than_transitivity1,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
trivial-int-eq1,
squash_wf,
true_wf,
exp_add,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
iff_weakening_equal,
mul_preserves_le,
multiply-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
natural_numberEquality,
independent_isectElimination,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
applyEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
dependent_set_memberFormation,
functionEquality,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
multiplyEquality,
universeEquality,
Error :applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}a:\{a:\mBbbR{}| r0 < a\} . \mforall{}b:\{b:\mBbbR{}| |b - rlog(a)| \mleq{} (r1/r(10))\} . lim n\mrightarrow{}\minfty{}.logseq(a;b;n) = rlog(a)
Date html generated:
2016_10_26-PM-00_37_24
Last ObjectModification:
2016_09_18-PM-09_53_53
Theory : reals_2
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