Nuprl Lemma : Taylor-series-around-zero-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])
⇒ (∀r:{r:ℝ| r0 ≤ r} . lim k→∞.r^k * (F[k + 1;x]/r((k)!)) = λx.r0 for x ∈ (-∞, ∞))
⇒ 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,
rnexp: x^k1,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
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: $n,
fact: (n)!
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_apply: x[s],
label: ...$L... t,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
rsub: x - y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
Taylor-series-converges-everywhere,
int-to-real_wf,
fun-converges-to_functionality,
riiint_wf,
rsum_wf,
rmul_wf,
rdiv_wf,
int_seg_subtype_nat,
false_wf,
fact_wf,
nat_plus_wf,
rless-int,
int_seg_properties,
nat_properties,
decidable__lt,
le_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
rless_wf,
rnexp_wf,
rsub_wf,
int_seg_wf,
real_wf,
i-member_wf,
nat_wf,
set_wf,
all_wf,
rleq_wf,
fun-converges-to_wf,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
rneq-int,
fact-non-zero,
infinite-deriv-seq_wf,
req_wf,
rsum_functionality,
radd_wf,
rminus_wf,
req_weakening,
uiff_transitivity,
req_functionality,
radd_functionality,
rminus-zero,
radd_comm,
radd-zero-both,
rmul_functionality,
rnexp_functionality
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesis,
lambdaFormation,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
lambdaEquality,
setElimination,
rename,
because_Cache,
applyEquality,
functionExtensionality,
addEquality,
independent_isectElimination,
independent_pairFormation,
inrFormation,
productElimination,
dependent_set_memberEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
Error :applyLambdaEquality,
voidElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
setEquality,
functionEquality
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{}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;r0]/r((i)!)) * x\^{}i | 0\mleq{}i\mleq{}k\} = \mlambda{}x.F[0;x] for x \mmember{} (-\minfty{}, \minfty{}))
Date html generated:
2016_10_26-AM-11_51_03
Last ObjectModification:
2016_08_30-AM-11_21_35
Theory : reals
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