Nuprl Lemma : Taylor-series-bounded-converges-everywhere
∀F:ℕ ⟶ ℝ ⟶ ℝ
((∀k:ℕ. ∀x,y:ℝ. ((x = y) ⇒ (F[k;x] = F[k;y])))
⇒ infinite-deriv-seq((-∞, ∞);i,x.F[i;x])
⇒ (∀m:ℕ. ∃c:ℝ. ∃N:ℕ. ∀k:{N...}. ∀x:{x:ℝ| |x| ≤ r(m)} . (|F[k;x]| ≤ c))
⇒ lim k→∞.Σ{(F[i;r0]/r((i)!)) * x^i | 0≤i≤k} = λx.F[0;x] for x ∈ (-∞, ∞))
Proof
Definitions occuring in Statement :
infinite-deriv-seq: infinite-deriv-seq(I;i,x.F[i; x]),
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
riiint: (-∞, ∞),
rsum: Σ{x[k] | n≤k≤m},
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rnexp: x^k1,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat: ℕ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
fact: (n)!
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
rfun: I ⟶ℝ,
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
i-approx: i-approx(I;n),
riiint: (-∞, ∞),
top: Top,
nat_plus: ℕ+,
uimplies: b supposing a,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
int_upper: {i...},
sq_stable: SqStable(P),
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
le: A ≤ B,
rge: x ≥ y,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
rneq: x ≠ y,
real: ℝ,
rnonneg: rnonneg(x),
rleq: x ≤ y,
subtract: n - m,
converges-to: lim n→∞.x[n] = y,
rsub: x - y,
rdiv: (x/y)
Lemmas referenced :
Taylor-series-around-zero-converges-everywhere,
set_wf,
real_wf,
rleq_wf,
int-to-real_wf,
all_wf,
nat_wf,
exists_wf,
int_upper_wf,
rabs_wf,
int_upper_subtype_nat,
infinite-deriv-seq_wf,
riiint_wf,
i-member_wf,
req_wf,
member_rccint_lemma,
subtype_rel_set,
nat_plus_wf,
icompact_wf,
rccint_wf,
nat_plus_subtype_nat,
radd_wf,
nat_properties,
nat_plus_properties,
sq_stable__icompact,
decidable__le,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
zero-rleq-rabs,
rleq_transitivity,
rless_wf,
rleq_weakening_equal,
rleq-int,
intformand_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
rless-int,
radd-preserves-rleq,
rminus_wf,
rleq_functionality,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
false_wf,
rabs-of-nonneg,
req_weakening,
rless_functionality_wrt_implies,
radd_functionality_wrt_rleq,
rless_functionality,
rleq_functionality_wrt_implies,
rleq_weakening_rless,
equal_wf,
rmul_wf,
rmul-is-positive,
decidable__lt,
rmul-rdiv-cancel2,
rmul-zero-both,
rmul_preserves_rless,
rdiv_wf,
small-reciprocal-real,
rdiv-factorial-limit-zero,
fact-non-zero,
rneq-int,
fact_wf,
int_term_value_add_lemma,
sq_stable__less_than,
int_upper_properties,
rnexp_wf,
rsub_wf,
less_than'_wf,
less_than_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
radd-zero-both,
radd_comm,
rminus-zero,
radd_functionality,
req_functionality,
uiff_transitivity,
rabs_functionality,
rabs-rdiv,
rleq_weakening,
rless_transitivity1,
req_inversion,
rmul_preserves_rleq,
rabs-rabs,
rnexp_functionality,
req_transitivity,
rabs-rnexp,
rmul_preserves_rleq2,
rinv_wf2,
itermMultiply_wf,
real_term_value_mul_lemma,
rmul_functionality,
rinv-as-rdiv,
rinv-of-rmul,
rmul-rinv,
rmul-int-rdiv2,
rmul_preserves_req,
rabs-rmul,
rmul-rinv3,
rneq_wf,
iff_weakening_equal,
squash_wf,
true_wf,
imax_wf,
imax_nat_plus,
intformeq_wf,
int_formula_prop_eq_lemma,
imax_ub,
rabs-rleq-iff,
rminus-int,
int_upper_subtype_int_upper,
rmul-identity1,
rmul_functionality_wrt_rleq2
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
sqequalRule,
lambdaEquality,
natural_numberEquality,
because_Cache,
setElimination,
rename,
setEquality,
applyEquality,
functionExtensionality,
functionEquality,
isect_memberEquality,
voidElimination,
voidEquality,
minusEquality,
independent_isectElimination,
productElimination,
dependent_pairFormation,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
int_eqEquality,
intEquality,
computeAll,
independent_pairFormation,
productEquality,
equalityTransitivity,
equalitySymmetry,
inlFormation,
addLevel,
inrFormation,
axiomEquality,
applyLambdaEquality,
independent_pairEquality,
addEquality,
isect_memberFormation,
multiplyEquality,
universeEquality
Latex:
\mforall{}F:\mBbbN{} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}
((\mforall{}k:\mBbbN{}. \mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (F[k;x] = F[k;y])))
{}\mRightarrow{} infinite-deriv-seq((-\minfty{}, \minfty{});i,x.F[i;x])
{}\mRightarrow{} (\mforall{}m:\mBbbN{}. \mexists{}c:\mBbbR{}. \mexists{}N:\mBbbN{}. \mforall{}k:\{N...\}. \mforall{}x:\{x:\mBbbR{}| |x| \mleq{} r(m)\} . (|F[k;x]| \mleq{} c))
{}\mRightarrow{} lim k\mrightarrow{}\minfty{}.\mSigma{}\{(F[i;r0]/r((i)!)) * x\^{}i | 0\mleq{}i\mleq{}k\} = \mlambda{}x.F[0;x] for x \mmember{} (-\minfty{}, \minfty{}))
Date html generated:
2017_10_03-PM-00_43_08
Last ObjectModification:
2017_07_28-AM-08_46_53
Theory : reals
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