Nuprl Lemma : Taylor-theorem-case1
∀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])
⇒ b - a ≠ r0
⇒ (∀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),
rneq: 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,
fact: (n)!,
int_seg: {i..j-},
nat: 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: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
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,
prop: ℙ,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
rneq: x ≠ y,
guard: {T},
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
so_lambda: λ2x.t[x],
so_apply: x[s],
subinterval: I ⊆ J ,
rev_uimplies: rev_uimplies(P;Q),
rat_term_to_real: rat_term_to_real(f;t),
rtermSubtract: left "-" right,
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rtermMultiply: left "*" right,
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
rtermConstant: "const",
rtermAdd: left "+" right,
pi2: snd(t),
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rdiv: (x/y),
Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]),
i-member: r ∈ I,
rccint: [l, u]
Lemmas referenced :
derivative-Taylor-approx,
Taylor-remainder_wf,
int_seg_properties,
nat_plus_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_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_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
istype-le,
istype-less_than,
subtype_rel_self,
real_wf,
i-member_wf,
int_seg_wf,
rless_wf,
int-to-real_wf,
rneq_wf,
rsub_wf,
finite-deriv-seq_wf,
req_wf,
rfun_wf,
istype-nat,
iproper_wf,
interval_wf,
rcc-subinterval,
rmin_wf,
rmax_wf,
rmin-i-member,
sq_stable__i-member,
rmax-i-member,
rleq_wf,
rmax_strict_ub,
rless-implies-rless,
itermSubtract_wf,
req-iff-rsub-is-0,
rmin_strict_lb,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
Rolles-theorem,
radd_wf,
Taylor-approx_wf,
subtype_rel_sets_simple,
rccint_wf,
rdiv_wf,
rmul_wf,
rnexp_wf,
fact_wf,
rless-int,
function-is-continuous,
req_functionality,
rsub_functionality,
req_weakening,
rmul_functionality,
rnexp_functionality,
rdiv_functionality,
derivative-sub,
derivative-const,
derivative-add,
derivative_functionality_wrt_subinterval,
istype-top,
member_rccint_lemma,
subtype_rel_dep_function,
top_wf,
derivative-rdiv-const,
derivative-const-mul,
derivative-id,
assert-rat-term-eq2,
rtermSubtract_wf,
rtermConstant_wf,
rtermAdd_wf,
rtermMultiply_wf,
rtermVar_wf,
rtermDivide_wf,
derivative_functionality,
rmin-max-cases,
radd-preserves-rless,
rless_functionality,
real_term_value_add_lemma,
rleq_weakening_equal,
rmin-rleq-rmax,
radd_functionality,
Taylor-approx_functionality,
trivial-Taylor-approx,
rmul_preserves_req,
rinv_wf2,
itermMultiply_wf,
req_transitivity,
rmul-rinv,
real_term_value_mul_lemma,
squash_wf,
true_wf,
iff_weakening_equal,
rabs-neq-zero,
rabs_wf,
rmul_preserves_rless,
rmul-rinv3,
rneq-int,
fact-non-zero,
rmul_preserves_rleq2,
zero-rleq-rabs,
rminus_wf,
itermMinus_wf,
rleq_functionality,
rabs_functionality,
real_term_value_minus_lemma,
req_inversion,
rabs-rmul
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
sqequalRule,
lambdaEquality_alt,
applyEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
productElimination,
imageElimination,
independent_pairFormation,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
addEquality,
productIsType,
functionEquality,
setEquality,
inhabitedIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
because_Cache,
functionIsType,
setIsType,
imageMemberEquality,
baseClosed,
inlFormation_alt,
inrFormation_alt,
applyLambdaEquality,
closedConclusion,
productEquality,
instantiate,
universeEquality
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \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{} b - a \mneq{} r0
{}\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:
2019_10_30-AM-10_10_34
Last ObjectModification:
2019_04_02-AM-09_42_21
Theory : reals
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