Nuprl Lemma : derivative-Taylor-approx
∀I:Interval
(iproper(I)
⇒ (∀n:ℕ. ∀F:ℕn + 2 ⟶ I ⟶ℝ. ∀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])
⇒ d(Taylor-approx(n;a;b;i,x.F[i;x]))/da = λx.b - x^n * (F[n + 1;x]/r((n)!)) on I)))
Proof
Definitions occuring in Statement :
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
derivative: d(f[x])/dx = λz.g[z] on I,
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: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
int_upper: {i...},
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
subtract: n - m,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
req_int_terms: t1 ≡ t2,
true: True,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
less_than: a < b,
squash: ↓T,
rdiv: (x/y)
Lemmas referenced :
finite-deriv-seq_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
int_seg_wf,
req_wf,
subtype_rel_self,
real_wf,
i-member_wf,
rfun_wf,
istype-nat,
iproper_wf,
interval_wf,
derivative-rsum,
istype-false,
int_upper_wf,
rmul_wf,
rdiv_wf,
int-to-real_wf,
fact_wf,
int_seg_subtype_nat,
rless-int,
int_seg_properties,
nat_plus_properties,
rless_wf,
rnexp_wf,
rsub_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
radd_wf,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
intformeq_wf,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
eqff_to_assert,
int_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
derivative-mul,
req_weakening,
rdiv_functionality,
req_functionality,
rmul_functionality,
rnexp_functionality,
rsub_functionality,
derivative-rdiv-const,
set_subtype_base,
lelt_wf,
rnexp_zero_lemma,
derivative-const,
itermMultiply_wf,
req-iff-rsub-is-0,
derivative_functionality,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
simple-chain-rule,
riiint_wf,
derivative-rnexp,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
istype-top,
subtype_rel_dep_function,
derivative-sub,
derivative-id,
rsum_wf,
rsum_functionality,
rneq-int,
fact-non-zero,
real_term_value_add_lemma,
real_term_value_var_lemma,
fact0_redex_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
eq_int_eq_true,
btrue_wf,
iff_weakening_equal,
bfalse_wf,
bnot_wf,
assert_elim,
btrue_neq_bfalse,
rsum-telescopes,
add-subtract-cancel,
rmul_comm,
rmul-assoc,
nat_plus_wf,
less_than_wf,
fact_unroll,
lt_int_wf,
assert_of_lt_int,
iff_weakening_uiff,
assert_wf,
decidable__equal_int,
int_term_value_mul_lemma,
mul_bounds_1b,
add-swap,
minus-one-mul,
minus-one-mul-top,
le-add-cancel2,
rneq_functionality,
rmul-int,
rinv_wf2,
req_inversion,
rdiv-rdiv,
req_transitivity,
rmul-rinv3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalRule,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
addEquality,
setElimination,
rename,
hypothesis,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
applyEquality,
productElimination,
productIsType,
because_Cache,
inhabitedIsType,
functionIsType,
functionEquality,
setEquality,
setIsType,
equalityTransitivity,
equalitySymmetry,
inrFormation_alt,
applyLambdaEquality,
closedConclusion,
equalityElimination,
equalityIsType4,
baseApply,
baseClosed,
promote_hyp,
instantiate,
cumulativity,
equalityIsType1,
intEquality,
minusEquality,
imageMemberEquality,
imageElimination,
universeEquality,
multiplyEquality
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}F:\mBbbN{}n + 2 {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}. \mforall{}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{} d(Taylor-approx(n;a;b;i,x.F[i;x]))/da = \mlambda{}x.b - x\^{}n * (F[n + 1;x]/r((n)!)) on I)))
Date html generated:
2019_10_30-AM-10_09_49
Last ObjectModification:
2018_11_12-PM-01_59_29
Theory : reals
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