Nuprl Lemma : power-series-converges-to
∀a:ℕ ⟶ ℝ. ∀b:ℝ. ∀r:{r:ℝ| r0 < r} . ∀N:ℕ.
((∀n:{N...}. (|a[n + 1]| ≤ (|a[n]|/r)))
⇒ (∀g:(b - r, b + r) ⟶ℝ
((∀x:{x:ℝ| x ∈ (b - r, b + r)} . Σi.a[i] * x - b^i = g[x])
⇒ lim n→∞.Σ{a[i] * x - b^i | 0≤i≤n} = λx.g[x] for x ∈ (b - r, b + r))))
Proof
Definitions occuring in Statement :
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
rfun: I ⟶ℝ,
rooint: (l, u),
i-member: r ∈ I,
series-sum: Σn.x[n] = a,
rsum: Σ{x[k] | n≤k≤m},
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat: ℕ,
so_apply: x[s],
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 :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
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,
so_apply: x[s1;s2],
series-sum: Σn.x[n] = a,
top: Top,
int_upper: {i...},
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
rneq: x ≠ y,
sq_stable: SqStable(P),
squash: ↓T,
fun-series-converges: Σn.f[n; x]↓ for x ∈ I
Lemmas referenced :
power-series-converges,
fun-converges-converges-to,
rooint_wf,
rsub_wf,
radd_wf,
rsum_wf,
rmul_wf,
int_seg_subtype_nat,
false_wf,
rnexp_wf,
int_seg_wf,
real_wf,
i-member_wf,
nat_wf,
member_rooint_lemma,
set_wf,
all_wf,
series-sum_wf,
rfun_wf,
int_upper_wf,
rleq_wf,
rabs_wf,
int_upper_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
rdiv_wf,
int_upper_subtype_nat,
sq_stable__rless,
int-to-real_wf,
rless_wf,
fun-series-converges-absolutely-converges
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality,
natural_numberEquality,
applyEquality,
functionExtensionality,
addEquality,
independent_isectElimination,
independent_pairFormation,
setEquality,
dependent_set_memberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
inrFormation,
imageMemberEquality,
baseClosed,
imageElimination,
functionEquality
Latex:
\mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}b:\mBbbR{}. \mforall{}r:\{r:\mBbbR{}| r0 < r\} . \mforall{}N:\mBbbN{}.
((\mforall{}n:\{N...\}. (|a[n + 1]| \mleq{} (|a[n]|/r)))
{}\mRightarrow{} (\mforall{}g:(b - r, b + r) {}\mrightarrow{}\mBbbR{}
((\mforall{}x:\{x:\mBbbR{}| x \mmember{} (b - r, b + r)\} . \mSigma{}i.a[i] * x - b\^{}i = g[x])
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.\mSigma{}\{a[i] * x - b\^{}i | 0\mleq{}i\mleq{}n\} = \mlambda{}x.g[x] for x \mmember{} (b - r, b + r))))
Date html generated:
2016_10_26-AM-11_15_46
Last ObjectModification:
2016_08_27-PM-08_39_51
Theory : reals
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