Nuprl Lemma : arctan-poly-approx-1
∀[x:{x:ℝ| r0 ≤ x} ]. ∀[k:ℕ]. (|arctangent(x) - arctan-poly(x;k)| ≤ (x^(2 * k) + 3/r((2 * k) + 3)))
Proof
Definitions occuring in Statement :
arctan-poly: arctan-poly(x;k),
arctangent: arctangent(x),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
rneq: x ≠ y,
or: P ∨ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
decidable: Dec(P),
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
ge: i ≥ j ,
guard: {T},
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
arctangent: arctangent(x),
rleq: x ≤ y,
rnonneg: rnonneg(x),
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
rge: x ≥ y,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
arctan-poly: arctan-poly(x;k),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
isOdd: isOdd(n),
nat_plus: ℕ+,
req_int_terms: t1 ≡ t2,
rat_term_to_real: rat_term_to_real(f;t),
rtermMinus: rtermMinus(num),
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rtermVar: rtermVar(var),
pi1: fst(t),
pi2: snd(t),
rsub: x - y,
rtermAdd: left "+" right,
rtermConstant: "const",
rtermSubtract: left "-" right,
rdiv: (x/y),
sq_stable: SqStable(P),
rccint: [l, u],
i-member: r ∈ I
Lemmas referenced :
rnexp2-nonneg,
real_wf,
rless_wf,
int-to-real_wf,
rminus_wf,
rnexp_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
integral-rsub,
rdiv_wf,
radd_wf,
nat_properties,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
rdiv_functionality,
req_weakening,
radd_functionality,
rnexp_functionality,
req_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
rsum_wf,
int_seg_properties,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_seg_wf,
rsum_functionality2,
rminus_functionality,
le_witness_for_triv,
istype-nat,
rleq_wf,
trivial-rless-radd,
rless-int,
squash_wf,
true_wf,
rminus-int,
subtype_rel_self,
iff_weakening_equal,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rminus_functionality_wrt_rleq,
arctangent_wf,
arctan-poly_wf,
integral_wf,
rsub_functionality,
int_seg_subtype_nat,
istype-false,
integral-rsum,
all_wf,
le_wf,
false_wf,
set_wf,
member_rccint_lemma,
rsum_functionality,
remainder_wfa,
subtype_base_sq,
int_subtype_base,
nequal_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
int-rdiv_wf,
intformeq_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
integral_functionality,
rmul_wf,
rmul_functionality,
ifthenelse_wf,
isOdd_wf,
rnexp-rminus,
req_inversion,
rnexp-mul,
btrue_wf,
bfalse_wf,
rnexp-minus-one,
modulus-is-rem,
rem_bounds_1,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
rminus-as-rmul,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
integral-rmul-const,
int-rdiv-req,
integral-rnexp-from-0,
rmul_preserves_req,
req_witness,
req-implies-req,
rsub_wf,
itermMinus_wf,
real_term_value_minus_lemma,
assert-rat-term-eq2,
rtermDivide_wf,
rtermVar_wf,
rtermMinus_wf,
rabs_wf,
rleq_functionality,
rabs_functionality,
rneq_wf,
rminus-rminus-eq,
rtermSubtract_wf,
rtermConstant_wf,
rtermAdd_wf,
rless_functionality,
real_term_value_add_lemma,
partial-geometric-series,
rinv_wf2,
req_transitivity,
rmul-rinv3,
rnexp-rmul,
rabs-rminus,
rmul-identity1,
equal-wf-base,
equal_wf,
sq_stable__rleq,
Riemann-integral_wf,
integral-is-Riemann,
rabs-of-nonneg,
Riemann-integral-nonneg,
rnexp-nonneg,
rmul-zero-both,
rmul_preserves_rleq,
Riemann-integral-rleq,
trivial-rleq-radd,
rmul-rinv,
rmul_preserves_rleq2,
rinv-mul-as-rdiv,
rleq_functionality_wrt_implies,
decidable__equal_int
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
universeIsType,
inlFormation_alt,
isectElimination,
natural_numberEquality,
dependent_set_memberEquality_alt,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
isect_memberFormation_alt,
setElimination,
rename,
closedConclusion,
because_Cache,
inrFormation_alt,
setIsType,
productElimination,
imageElimination,
int_eqEquality,
independent_pairFormation,
addEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
imageMemberEquality,
baseClosed,
applyEquality,
instantiate,
universeEquality,
minusEquality,
functionEquality,
functionExtensionality,
setEquality,
lambdaFormation,
lambdaEquality,
dependent_set_memberEquality,
productEquality,
voidEquality,
isect_memberEquality,
cumulativity,
intEquality,
equalityIstype,
sqequalBase,
equalityElimination,
multiplyEquality,
baseApply,
promote_hyp,
productIsType,
applyLambdaEquality,
dependent_pairFormation,
inrFormation,
addLevel,
remainderEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| r0 \mleq{} x\} ]. \mforall{}[k:\mBbbN{}]. (|arctangent(x) - arctan-poly(x;k)| \mleq{} (x\^{}(2 * k) + 3/r((2 * k) + 3)))
Date html generated:
2019_10_31-AM-06_05_35
Last ObjectModification:
2019_04_03-AM-00_28_09
Theory : reals_2
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