Nuprl Lemma : Taylor-theorem
∀I:Interval
(iproper(I)
⇒ (∀n:ℕ+. ∀F:ℕn + 2 ⟶ I ⟶ℝ. ∀a,b:{a:ℝ| a ∈ I} .
((∀k:ℕn + 2. ∀x,y:{a:ℝ| a ∈ I} . ((x = y) ⇒ (F[k;x] = F[k;y])))
⇒ finite-deriv-seq(I;n + 1;i,x.F[i;x])
⇒ (∀e:ℝ
((r0 < e)
⇒ (∃c:ℝ
((rmin(a;b) ≤ c)
∧ (c ≤ rmax(a;b))
∧ (|Taylor-remainder(I;n;b;a;k,x.F[k;x]) - (b - c^n * (F[n + 1;c]/r((n)!))) * (b - a)| ≤ e))))))))
Proof
Definitions occuring in Statement :
Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]),
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rmin: rmin(x;y),
rmax: rmax(x;y),
rnexp: x^k1,
rsub: x - y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃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: $n,
fact: (n)!
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
subinterval: I ⊆ J ,
rsub: x - y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
Taylor-theorem-case2,
rless_wf,
int-to-real_wf,
real_wf,
finite-deriv-seq_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
le_wf,
int_seg_wf,
decidable__lt,
lelt_wf,
i-member_wf,
all_wf,
req_wf,
rfun_wf,
set_wf,
nat_plus_wf,
iproper_wf,
interval_wf,
rless-cases,
rabs_wf,
rsub_wf,
Taylor-theorem-case1,
nat_plus_subtype_nat,
rabs-positive-iff,
rless_functionality,
req_weakening,
rabs-difference-symmetry,
rcc-subinterval,
rmin_wf,
rmax_wf,
rmin-i-member,
sq_stable__i-member,
rmax-i-member,
rleq_wf,
rmin-rleq,
rleq-rmax,
Taylor-remainder_wf,
rmul_wf,
rnexp_wf,
rdiv_wf,
sq_stable__less_than,
member_rccint_lemma,
fact_wf,
rneq-int,
fact-non-zero,
radd_wf,
rminus_wf,
rnexp0,
req_functionality,
rnexp_functionality,
radd-rminus-both,
rleq_functionality,
rabs_functionality,
rsub_functionality,
rmul_functionality,
uiff_transitivity,
radd_functionality,
rminus_functionality,
req_transitivity,
rmul-distrib,
rmul_over_rminus,
rmul-zero-both,
rminus-radd,
rminus-zero,
radd-ac,
radd_comm,
rmul-int,
radd-assoc,
req_inversion,
rmul-identity1,
rmul-distrib2,
radd-int,
radd-zero-both
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
isectElimination,
natural_numberEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
applyEquality,
functionExtensionality,
because_Cache,
setEquality,
functionEquality,
imageMemberEquality,
baseClosed,
imageElimination,
productEquality,
minusEquality,
multiplyEquality
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}F:\mBbbN{}n + 2 {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}. \mforall{}a,b:\{a:\mBbbR{}| a \mmember{} I\} .
((\mforall{}k:\mBbbN{}n + 2. \mforall{}x,y:\{a:\mBbbR{}| a \mmember{} I\} . ((x = y) {}\mRightarrow{} (F[k;x] = F[k;y])))
{}\mRightarrow{} finite-deriv-seq(I;n + 1;i,x.F[i;x])
{}\mRightarrow{} (\mforall{}e:\mBbbR{}
((r0 < e)
{}\mRightarrow{} (\mexists{}c:\mBbbR{}
((rmin(a;b) \mleq{} c)
\mwedge{} (c \mleq{} rmax(a;b))
\mwedge{} (|Taylor-remainder(I;n;b;a;k,x.F[k;x]) - (b - c\^{}n * (F[n + 1;c]/r((n)!)))
* (b - a)| \mleq{} e))))))))
Date html generated:
2017_10_03-PM-00_41_03
Last ObjectModification:
2017_07_28-AM-08_46_16
Theory : reals
Home
Index