Nuprl Lemma : Taylor-theorem-case2
∀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) ⇒ (∃d:ℝ. ((r0 < d) ∧ ((|a - b| < d) ⇒ (|Taylor-remainder(I;n;b;a;k,x.F[k;x])| ≤ 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,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
req: x = y,
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: $n
Definitions unfolded in proof :
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
ge: i ≥ j ,
subtract: n - m,
int_upper: {i...},
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
int_nzero: ℤ-o,
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
primrec: primrec(n;b;c),
fact: (n)!,
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]),
subinterval: I ⊆ J ,
i-member: r ∈ I,
rnonneg: rnonneg(x),
rleq: x ≤ y,
rneq: x ≠ y,
true: True,
less_than: a < b,
rccint: [l, u],
i-approx: i-approx(I;n),
continuous: f[x] continuous for x ∈ I,
cand: A c∧ B,
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
less_than': less_than'(a;b),
le: A ≤ B,
squash: ↓T,
sq_stable: SqStable(P),
real: ℝ,
sq_exists: ∃x:A [B[x]],
rless: x < y,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_apply: x[s1;s2],
rfun: I ⟶ℝ,
label: ...$L... t,
so_lambda: λ2x y.t[x; y],
and: P ∧ Q,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
rinv-as-rdiv,
rmul-rinv3,
rmul-rinv,
rinv-of-rmul,
rinv_functionality2,
rmul-int,
rneq_functionality,
subtract-add-cancel,
rsum-one,
bfalse_wf,
ifthenelse_wf,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
rsum-constant,
mul_preserves_le,
nat_properties,
upper_subtype_nat,
exp_preserves_le,
nat_wf,
int_term_value_subtract_lemma,
subtract_wf,
exp_step,
exp-one,
rleq-int-fractions,
rdiv_functionality,
rnexp-rdiv,
req_inversion,
rnexp-int,
exp_wf3,
int_nzero-rational,
equal_functionality_wrt_subtype_rel2,
int-subtype-rationals,
exp_wf_nat_plus,
rationals_wf,
not_functionality_wrt_implies,
exp-positive,
exp_wf2,
rless_transitivity1,
le-add-cancel,
add_functionality_wrt_le,
zero-add,
minus-zero,
minus-add,
add-commutes,
condition-implies-le,
not-lt-2,
zero-rleq-rabs,
rnexp-rless,
rabs-rnexp,
rless_functionality,
int_upper_properties,
int_upper_wf,
rleq_weakening,
rabs-rmul-rleq,
rsum_functionality_wrt_rleq,
rabs-difference-symmetry,
rleq_weakening_rless,
rleq-rmax,
rmin-rleq,
rabs-rsum,
radd_functionality_wrt_rleq,
real_term_value_minus_lemma,
real_term_value_add_lemma,
r-triangle-inequality,
rleq_functionality_wrt_implies,
iff_weakening_equal,
rabs-rminus,
rleq_weakening_equal,
itermMinus_wf,
rminus_wf,
equal_wf,
radd_functionality,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-identity1,
rinv1,
rmul_functionality,
req_transitivity,
req_functionality,
rsum-split-first,
rsub_functionality,
rabs_functionality,
rleq_functionality,
req_weakening,
req-iff-rsub-is-0,
itermSubtract_wf,
rinv_wf2,
rnexp_zero_lemma,
fact0_redex_lemma,
equal-wf-T-base,
int_formula_prop_eq_lemma,
intformeq_wf,
nequal_wf,
subtype_rel_sets,
mul_nzero,
true_wf,
equal-wf-base,
int_subtype_base,
subtype_base_sq,
int_entire_a,
radd_wf,
rnexp_wf,
rmul_wf,
rsum_wf,
rless-int-fractions2,
rmin_strict_ub,
Taylor-remainder_wf,
mul_bounds_1b,
member_rccint_lemma,
squash_wf,
less_than'_wf,
sq_stable__rleq,
sq_stable__all,
sq_stable__rless,
int_term_value_mul_lemma,
itermMultiply_wf,
rless-int,
rsub_wf,
i-approx_wf,
sq_stable__and,
mul_nat_plus,
icompact_wf,
rmin-rleq-rmax,
rccint-icompact,
less_than_wf,
rccint_wf,
continuous_functionality_wrt_subinterval,
function-is-continuous,
rmax-i-member,
sq_stable__i-member,
rmin-i-member,
rmax_wf,
rmin_wf,
rcc-subinterval,
small-reciprocal-real,
rleq_transitivity,
r-bound-property,
int_seg_properties,
rmaximum_ub,
rleq_wf,
fact-non-zero,
rneq-int,
false_wf,
int_seg_subtype_nat,
fact_wf,
rdiv_wf,
rabs_wf,
sq_stable__less_than,
rmaximum_wf,
r-bound_wf,
interval_wf,
iproper_wf,
nat_plus_wf,
rfun_wf,
set_wf,
subtype_rel_self,
req_wf,
all_wf,
i-member_wf,
lelt_wf,
decidable__lt,
int_seg_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_plus_properties,
finite-deriv-seq_wf,
real_wf,
int-to-real_wf,
rless_wf
Rules used in proof :
promote_hyp,
equalityElimination,
universeEquality,
cumulativity,
instantiate,
addLevel,
applyLambdaEquality,
productEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
independent_pairEquality,
inrFormation,
multiplyEquality,
imageElimination,
baseClosed,
imageMemberEquality,
functionEquality,
setEquality,
productElimination,
because_Cache,
functionExtensionality,
applyEquality,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
rename,
setElimination,
addEquality,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
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{}d:\mBbbR{}
((r0 < d)
\mwedge{} ((|a - b| < d) {}\mRightarrow{} (|Taylor-remainder(I;n;b;a;k,x.F[k;x])| \mleq{} e)))))))))
Date html generated:
2018_05_22-PM-02_48_48
Last ObjectModification:
2018_05_21-AM-01_22_31
Theory : reals
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