Nuprl Lemma : integral-from-Taylor
∀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→∞.Σ{(F[i;a]/r((i)!)) * (x - a^i + 1 - b - a^i + 1/r(i + 1)) | 0≤i≤n} = 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: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
top: Top,
cand: A c∧ B,
uiff: uiff(P;Q),
prop: ℙ,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
rless: x < y,
sq_exists: ∃x:A [B[x]],
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
so_apply: x[s],
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
subinterval: I ⊆ J ,
i-member: r ∈ I,
rccint: [l, u],
rooint: (l, u),
label: ...$L... t,
req_int_terms: t1 ≡ t2,
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
i-finite: i-finite(I),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
subtract: n - m,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermSubtract: left "-" right,
rtermVar: rtermVar(var),
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
integral-from-Taylor-1,
fun-converges-to_functionality,
rooint_wf,
rsub_wf,
radd_wf,
rsum_wf,
rmul_wf,
rdiv_wf,
int_seg_subtype_nat,
istype-false,
member_rooint_lemma,
istype-void,
trivial-rsub-rless,
trivial-rless-radd,
rless_wf,
int-to-real_wf,
fact_wf,
rless-int,
int_seg_properties,
nat_properties,
nat_plus_properties,
decidable__lt,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
intformless_wf,
int_formula_prop_less_lemma,
rnexp_wf,
int_seg_wf,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
rsum_functionality2,
rmul_functionality,
req_weakening,
rnexp_functionality,
rsub_functionality,
req_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
istype-nat,
sq_stable__rless,
rless-implies-rless,
itermAdd_wf,
int_term_value_add_lemma,
rcc-subinterval,
rmin-i-member,
rmax-i-member,
rleq_wf,
subtype_rel_sets_simple,
real_wf,
member_rccint_lemma,
rmin-rmax-subinterval,
fun-converges-to_wf,
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,
integral-rsum,
all_wf,
le_wf,
nat_plus_wf,
false_wf,
nat_wf,
right-endpoint_wf,
left-endpoint_wf,
set_wf,
rsum_functionality,
integral-rmul-const,
ftc-total-integral,
riiint_wf,
derivative-rdiv-const-alt,
derivative-function-rsub-const,
add-subtract-cancel,
less_than_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
derivative-rnexp,
assert-rat-term-eq2,
rtermSubtract_wf,
rtermDivide_wf,
rtermVar_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
closedConclusion,
natural_numberEquality,
applyEquality,
addEquality,
independent_isectElimination,
independent_pairFormation,
isect_memberEquality_alt,
voidElimination,
productElimination,
productIsType,
universeIsType,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
inrFormation_alt,
imageElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
applyLambdaEquality,
setIsType,
imageMemberEquality,
baseClosed,
productEquality,
functionIsType,
functionEquality,
setEquality,
intEquality,
dependent_pairFormation,
inrFormation,
voidEquality,
isect_memberEquality,
lambdaFormation,
functionExtensionality,
lambdaEquality,
dependent_set_memberEquality,
equalityIstype,
minusEquality
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{}.\mSigma{}\{(F[i;a]/r((i)!))
* (x - a\^{}i + 1 - b - a\^{}i + 1/r(i + 1)) | 0\mleq{}i\mleq{}n\} = \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_40_01
Last ObjectModification:
2019_04_03-AM-00_21_45
Theory : reals_2
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