Nuprl Lemma : Taylor-series-converges-everywhere
∀a:ℝ. ∀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])
⇒ (∀r:{r:ℝ| r0 ≤ r} . lim k→∞.r^k * (F[k + 1;x]/r((k)!)) = λx.r0 for x ∈ (-∞, ∞))
⇒ lim k→∞.Σ{(F[i;a]/r((i)!)) * x - a^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,
rnexp: x^k1,
rsub: x - y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
fact: (n)!,
nat: ℕ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
nat_plus: ℕ+,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
label: ...$L... t,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
rfun: I ⟶ℝ,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
top: Top,
member: t ∈ T,
riiint: (-∞, ∞),
i-approx: i-approx(I;n),
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
implies: P ⇒ Q,
all: ∀x:A. B[x],
rge: x ≥ y,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
cand: A c∧ B,
true: True,
squash: ↓T,
sq_exists: ∃x:A [B[x]],
rless: x < y,
infinite-deriv-seq: infinite-deriv-seq(I;i,x.F[i; x]),
subinterval: I ⊆ J ,
sq_stable: SqStable(P),
icompact: icompact(I),
i-nonvoid: i-nonvoid(I),
rneq: x ≠ y,
int_upper: {i...},
rooint: (l, u),
i-member: r ∈ I,
lelt: i ≤ j < k,
int_seg: {i..j-},
less_than': less_than'(a;b),
le: A ≤ B,
rccint: [l, u]
Lemmas referenced :
infinite-deriv-seq_wf,
fact-non-zero,
rneq-int,
fact_wf,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
rdiv_wf,
rnexp_wf,
rmul_wf,
fun-converges-to_wf,
int-to-real_wf,
rleq_wf,
all_wf,
riiint_wf,
i-approx_wf,
icompact_wf,
nat_plus_wf,
rooint_wf,
i-member_wf,
req_wf,
nat_wf,
req_witness,
set_wf,
subtype_rel_self,
radd_wf,
rsub_wf,
rless_wf,
real_wf,
subtype_rel_dep_function,
member_rooint_lemma,
Taylor-series-converges,
member_rccint_lemma,
rccint_wf,
subinterval_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
nat_plus_properties,
rless-int,
zero-rleq-rabs,
rabs_wf,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rless_functionality,
req_weakening,
radd-int,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_inversion,
rabs-as-rmax,
rminus_wf,
rleq-rmax,
rcc-subinterval,
rsub_functionality_wrt_rleq,
int_term_value_minus_lemma,
itermMinus_wf,
req_transitivity,
squash_wf,
true_wf,
rminus-int,
real_term_value_minus_lemma,
rminus_functionality,
radd-ac,
radd-rminus-both,
radd_functionality,
radd-zero-both,
member_riiint_lemma,
derivative_functionality_wrt_subinterval,
sq_stable__icompact,
less_than_wf,
icompact-is-subinterval,
i-approx-finite,
i-approx-closed,
rleq-int,
int_upper_properties,
nat_plus_subtype_nat,
int_upper_subtype_nat,
int_upper_wf,
subtype_rel_sets,
i-approx-containing2,
int_seg_wf,
int_seg_properties,
false_wf,
int_seg_subtype_nat,
rsum_wf
Rules used in proof :
functionEquality,
computeAll,
independent_pairFormation,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
addEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_functionElimination,
independent_isectElimination,
because_Cache,
productEquality,
setEquality,
lambdaEquality,
isectElimination,
applyEquality,
functionExtensionality,
hypothesisEquality,
productElimination,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
rename,
thin,
setElimination,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
minusEquality,
equalityTransitivity,
equalitySymmetry,
addLevel,
levelHypothesis,
imageElimination,
imageMemberEquality,
baseClosed,
inrFormation,
applyLambdaEquality
Latex:
\mforall{}a:\mBbbR{}. \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{}r:\{r:\mBbbR{}| r0 \mleq{} r\} . lim k\mrightarrow{}\minfty{}.r\^{}k * (F[k + 1;x]/r((k)!)) = \mlambda{}x.r0 for x \mmember{} (-\minfty{}, \minfty{}))
{}\mRightarrow{} lim k\mrightarrow{}\minfty{}.\mSigma{}\{(F[i;a]/r((i)!)) * x - a\^{}i | 0\mleq{}i\mleq{}k\} = \mlambda{}x.F[0;x] for x \mmember{} (-\minfty{}, \minfty{}))
Date html generated:
2018_05_22-PM-02_49_16
Last ObjectModification:
2017_10_20-PM-05_31_27
Theory : reals
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