Nuprl Lemma : Taylor-remainder-as-integral
∀I:Interval
(iproper(I)
⇒ (∀a,b:{a:ℝ| a ∈ I} . ∀n:ℕ. ∀F:ℕn + 2 ⟶ I ⟶ℝ.
((∀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])
⇒ (Taylor-remainder(I;n;b;a;k,x.F[k;x]) = a_∫-b (F[n + 1;t]/r((n)!)) * b - t^n dt))))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
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),
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],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
sq_stable: SqStable(P),
squash: ↓T,
top: Top,
uall: ∀[x:A]. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
false: False,
so_lambda: λ2x y.t[x; y],
label: ...$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,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
subinterval: I ⊆ J ,
Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]),
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
less_than': less_than'(a;b),
fact: (n)!,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
true: True,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
rccint: [l, u],
i-finite: i-finite(I),
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
bool: 𝔹,
unit: Unit,
it: ⋅,
cand: A c∧ B,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
nequal: a ≠ b ∈ T ,
rat_term_to_real: rat_term_to_real(f;t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermMultiply: left "*" right,
rtermDivide: num "/" denom,
rtermConstant: "const",
pi2: snd(t),
ge: i ≥ j ,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
subtract: n - m,
rsub: x - y
Lemmas referenced :
rmin-rmax-subinterval,
sq_stable__i-member,
fact0_redex_lemma,
rnexp_zero_lemma,
istype-void,
finite-deriv-seq_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
int_seg_properties,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
istype-less_than,
subtype_rel_self,
real_wf,
i-member_wf,
int_seg_wf,
req_wf,
rfun_wf,
req_witness,
Taylor-remainder_wf,
rmul_wf,
rdiv_wf,
subtype_rel_sets_simple,
rccint_wf,
rmin_wf,
rmax_wf,
int-to-real_wf,
fact_wf,
rless-int,
nat_plus_properties,
rless_wf,
rnexp_wf,
rsub_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
rmul_functionality,
req_weakening,
rnexp_functionality,
rsub_functionality,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
primrec-wf2,
all_wf,
istype-nat,
iproper_wf,
interval_wf,
rsum_wf,
int_seg_subtype_nat,
istype-false,
equal-wf-base,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
rsum-single,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
req_transitivity,
rinv1,
rmul-identity1,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
right-endpoint_wf,
left-endpoint_wf,
member_rccint_lemma,
rdiv_functionality,
ftc-integral,
integral_functionality,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermVar_wf,
rtermConstant_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rleq_wf,
itermMinus_wf,
rminus_wf,
radd_wf,
radd_functionality,
rminus_functionality,
real_term_value_add_lemma,
real_term_value_minus_lemma,
req_inversion,
fact-non-zero,
rneq-int,
nat_properties,
int_term_value_subtract_lemma,
subtract_wf,
integral-by-parts,
derivative-rdiv-const-alt,
derivative-rnexp-function,
istype-top,
derivative-add,
derivative-const,
derivative-const-mul,
derivative-id,
rmul-one,
derivative_functionality,
radd-zero-both,
req-int,
fact_unroll_1,
rmul-int,
rfun_subtype,
rinv-mul-as-rdiv,
rnexp0,
subtype_rel_function,
int_seg_subtype,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
subtract-add-cancel,
integral-rmul-const,
Taylor-approx_wf,
radd-preserves-req,
rsum-split-last,
squash_wf,
true_wf,
iff_weakening_equal,
rmul_preserves_req,
rmul-rinv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality_alt,
voidElimination,
universeIsType,
isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
applyEquality,
productElimination,
independent_pairFormation,
int_eqEquality,
productIsType,
functionEquality,
setEquality,
inhabitedIsType,
addEquality,
functionIsType,
because_Cache,
inrFormation_alt,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setIsType,
closedConclusion,
intEquality,
equalityIstype,
sqequalBase,
equalityElimination,
int_eqReduceTrueSq,
promote_hyp,
instantiate,
cumulativity,
int_eqReduceFalseSq,
productEquality,
minusEquality,
multiplyEquality,
equalityIsType1,
universeEquality
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}a,b:\{a:\mBbbR{}| a \mmember{} I\} . \mforall{}n:\mBbbN{}. \mforall{}F:\mBbbN{}n + 2 {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}.
((\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{} (Taylor-remainder(I;n;b;a;k,x.F[k;x]) = a\_\mint{}\msupminus{}b (F[n + 1;t]/r((n)!)) * b - t\^{}n dt))))
Date html generated:
2019_10_31-AM-06_24_13
Last ObjectModification:
2019_04_02-PM-10_38_44
Theory : reals_2
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