Nuprl Lemma : Taylor-series-converges
∀a:ℝ. ∀t:{t:ℝ| r0 < t} . ∀F:ℕ ⟶ (a - t, a + t) ⟶ℝ.
((∀k:ℕ. ∀x,y:{x:ℝ| x ∈ (a - t, a + t)} . ((x = y) ⇒ (F[k;x] = F[k;y])))
⇒ infinite-deriv-seq((a - t, a + t);i,x.F[i;x])
⇒ (∀r:{r:ℝ| (r0 ≤ r) ∧ (r < t)} . lim k→∞.r^k * (F[k + 1;x]/r((k)!)) = λx.r0 for x ∈ (a - t, a + t))
⇒ lim k→∞.Σ{(F[i;a]/r((i)!)) * x - a^i | 0≤i≤k} = λx.F[0;x] for x ∈ (a - t, a + t))
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,
rfun: I ⟶ℝ,
rooint: (l, u),
i-member: r ∈ I,
rsum: Σ{x[k] | n≤k≤m},
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rnexp: x^k1,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: 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,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
iproper: iproper(I),
right-endpoint: right-endpoint(I),
left-endpoint: left-endpoint(I),
i-finite: i-finite(I),
rooint: (l, u),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
endpoints: endpoints(I),
outl: outl(x),
pi1: fst(t),
pi2: snd(t),
less_than: a < b,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
subtract: n - m,
le: A ≤ B,
icompact: icompact(I),
i-nonvoid: i-nonvoid(I),
i-member: r ∈ I,
i-approx: i-approx(I;n),
rccint: [l, u],
rneq: x ≠ y,
guard: {T},
cand: A c∧ B,
rless: x < y,
sq_exists: ∃x:A [B[x]],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rdiv: (x/y),
int_upper: {i...},
subinterval: I ⊆ J ,
int_seg: {i..j-},
lelt: i ≤ j < k,
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
infinite-deriv-seq: infinite-deriv-seq(I;i,x.F[i; x]),
Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]),
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x])
Lemmas referenced :
nat_plus_wf,
icompact_wf,
i-approx_wf,
rooint_wf,
rsub_wf,
radd_wf,
rleq_wf,
int-to-real_wf,
rless_wf,
fun-converges-to_wf,
rmul_wf,
rnexp_wf,
rdiv_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-le,
fact_wf,
rneq-int,
fact-non-zero,
i-member_wf,
istype-nat,
infinite-deriv-seq_wf,
subtype_rel_self,
real_wf,
req_wf,
rfun_wf,
iproper-approx,
sq_stable__icompact,
radd-preserves-rless,
true_wf,
itermSubtract_wf,
itermMultiply_wf,
sq_stable__rless,
rmul_preserves_rless,
rless-int,
rless_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
decidable__lt,
istype-false,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
istype-less_than,
iproper_wf,
nat_plus_properties,
intformless_wf,
int_formula_prop_less_lemma,
trivial-rsub-rless,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
less_than_wf,
int_subtype_base,
rless-int-fractions2,
int_term_value_mul_lemma,
member_rccint_lemma,
rleq_transitivity,
rminus_wf,
radd-preserves-rleq,
itermMinus_wf,
rmul_preserves_rleq,
iff_weakening_uiff,
rleq_functionality,
req_weakening,
real_term_value_minus_lemma,
r-bound_wf,
rccint_wf,
rabs_wf,
sq_stable__rleq,
rabs-difference-bound-rleq,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
r-bound-property,
trivial-rleq-radd,
rleq-int-fractions2,
rsub_functionality_wrt_rleq,
radd_functionality_wrt_rleq,
rinv_wf2,
req_transitivity,
rinv-as-rdiv,
mul_nat_plus,
i-approx-is-subinterval,
istype-int_upper,
rsum_wf,
int_seg_subtype_nat,
member_rooint_lemma,
trivial-rless-radd,
rless-implies-rless,
int_seg_wf,
subtype_rel_sets_simple,
int_upper_properties,
rleq-implies-rleq,
Taylor-theorem,
less_than_transitivity1,
subtype_rel_dep_function,
nat_wf,
int_seg_properties,
rfun_subtype,
derivative_functionality_wrt_subinterval,
rcc-subinterval,
rmin_wf,
rmax_wf,
i-member-iff,
rmin-i-member,
rmax-i-member,
Taylor-remainder_wf,
upper_subtype_nat,
sq_stable__le,
le_weakening2,
rmin_ub,
rmax_lb,
mul_bounds_1b,
req_functionality,
rabs_functionality,
req-int-fractions,
decidable__equal_int,
rabs-rmul-rleq,
rmul_comm,
rmul-int-rdiv,
rnexp-rleq,
zero-rleq-rabs,
rabs-rnexp,
rsub_functionality,
rabs-of-nonneg,
radd_functionality,
trivial-rsub-rleq,
rabs-rmul,
rleq-int-fractions,
uimplies_transitivity,
rdiv_functionality,
radd-int,
radd-rdiv,
r-triangle-inequality,
rabs-difference-symmetry,
i-approx-monotonic,
i-member-approx
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
setElimination,
thin,
rename,
setIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
because_Cache,
sqequalRule,
functionIsType,
inhabitedIsType,
productIsType,
natural_numberEquality,
lambdaEquality_alt,
applyEquality,
dependent_set_memberEquality_alt,
addEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
productElimination,
functionEquality,
setEquality,
imageMemberEquality,
baseClosed,
imageElimination,
closedConclusion,
minusEquality,
equalityIsType1,
inrFormation_alt,
equalityIsType4,
intEquality,
multiplyEquality,
promote_hyp,
productEquality,
applyLambdaEquality,
baseApply
Latex:
\mforall{}a:\mBbbR{}. \mforall{}t:\{t:\mBbbR{}| r0 < t\} . \mforall{}F:\mBbbN{} {}\mrightarrow{} (a - t, a + t) {}\mrightarrow{}\mBbbR{}.
((\mforall{}k:\mBbbN{}. \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} (a - t, a + t)\} . ((x = y) {}\mRightarrow{} (F[k;x] = F[k;y])))
{}\mRightarrow{} infinite-deriv-seq((a - t, a + t);i,x.F[i;x])
{}\mRightarrow{} (\mforall{}r:\{r:\mBbbR{}| (r0 \mleq{} r) \mwedge{} (r < t)\}
lim k\mrightarrow{}\minfty{}.r\^{}k * (F[k + 1;x]/r((k)!)) = \mlambda{}x.r0 for x \mmember{} (a - t, a + t))
{}\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{} (a - t, a + t))
Date html generated:
2019_10_30-AM-10_11_33
Last ObjectModification:
2018_11_14-AM-11_40_16
Theory : reals
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