Nuprl Lemma : integral-from-Taylor-1
∀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))
⇒ (∀b:{b:ℝ| b ∈ (a - t, a + t)}
lim n→∞.b_∫-x Σ{(F[i;a]/r((i)!)) * t - a^i | 0≤i≤n} dt = λx.b_∫-x F[0;t] dt for x ∈ (a - t, a + t)))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
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: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
top: Top,
cand: A c∧ B,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
label: ...$L... t,
rless: x < y,
sq_exists: ∃x:A [B[x]],
real: ℝ,
req_int_terms: t1 ≡ t2
Lemmas referenced :
Taylor-series-converges,
fun-converges-to-integral,
rooint_wf,
rsub_wf,
radd_wf,
rsum_wf,
rmul_wf,
rdiv_wf,
int_seg_subtype_nat,
false_wf,
member_rooint_lemma,
trivial-rsub-rless,
sq_stable__rless,
int-to-real_wf,
trivial-rless-radd,
rless-implies-rless,
rless_wf,
fact_wf,
nat_plus_wf,
rless-int,
int_seg_properties,
nat_properties,
decidable__lt,
le_wf,
nat_plus_properties,
full-omega-unsat,
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,
rnexp_wf,
int_seg_wf,
nat_wf,
i-member_wf,
req_functionality,
rsum_functionality2,
rmul_functionality,
req_weakening,
rnexp_functionality,
rsub_functionality,
req_wf,
set_wf,
real_wf,
all_wf,
rleq_wf,
fun-converges-to_wf,
sq_stable__less_than,
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,
subtype_rel_self,
rfun_wf,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
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,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
productEquality,
inrFormation,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
setEquality,
functionEquality
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{} (\mforall{}b:\{b:\mBbbR{}| b \mmember{} (a - t, a + t)\}
lim n\mrightarrow{}\minfty{}.b\_\mint{}\msupminus{}x \mSigma{}\{(F[i;a]/r((i)!)) * t - a\^{}i | 0\mleq{}i\mleq{}n\} dt = \mlambda{}x.b\_\mint{}\msupminus{}x F[0;t] dt for x \mmember{} (a
- t, a + t)))
Date html generated:
2019_10_30-AM-11_39_52
Last ObjectModification:
2018_08_29-PM-06_07_15
Theory : reals_2
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