Nuprl Lemma : arctan-poly-approx
∀[x:ℝ]. ∀[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],
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
absval: |i|,
rdiv: (x/y),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
so_apply: x[s],
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
sq_type: SQType(T),
true: True,
lelt: i ≤ j < k,
nequal: a ≠ b ∈ T ,
int_seg: {i..j-},
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
arctan-poly: arctan-poly(x;k),
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
sq_stable: SqStable(P),
real: ℝ,
sq_exists: ∃x:A [B[x]],
rless: x < y,
stable: Stable{P},
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
rneq: x ≠ y,
prop: ℙ,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat_plus: ℕ+,
nat: ℕ,
false: False,
implies: P ⇒ Q,
not: ¬A,
and: P ∧ Q,
le: A ≤ B,
all: ∀x:A. B[x],
rnonneg: rnonneg(x),
rleq: x ≤ y,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
minus-zero,
rneq_wf,
rinv-mul-as-rdiv,
rabs-int,
rmul_comm,
rnexp0,
int-rdiv_functionality,
arctangent0,
rmul-zero-both,
less_than_wf,
rsum-zero-req,
arctangent_functionality,
not-rless,
rleq_antisymmetry,
rabs-of-nonpos,
rabs_functionality,
arctangent-rminus,
rsub_functionality,
iff_weakening_equal,
subtype_rel_self,
rabs-rminus,
squash_wf,
rminus-rminus,
rnexp-rminus,
assert-isOdd,
isOdd_wf,
real_term_value_var_lemma,
real_term_value_mul_lemma,
rmul-rinv3,
req_transitivity,
rminus_functionality,
int-rdiv-req,
rsum-rminus,
req_inversion,
req_functionality,
rinv_wf2,
rmul_wf,
rmul_preserves_req,
rsum_functionality,
int_seg_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
true_wf,
subtype_base_sq,
eq_int_wf,
nequal_wf,
int_subtype_base,
equal-wf-base,
int_formula_prop_eq_lemma,
intformeq_wf,
int_seg_properties,
int-rdiv_wf,
rsum_wf,
real_term_value_minus_lemma,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
itermMinus_wf,
itermSubtract_wf,
rleq_weakening,
rminus_functionality_wrt_rleq,
rminus_wf,
rabs-of-nonneg,
rnexp_functionality,
rdiv_functionality,
req_weakening,
rleq_functionality,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
sq_stable__less_than,
rleq_weakening_rless,
arctan-poly-approx-1,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
rleq_wf,
not_wf,
or_wf,
false_wf,
stable__rleq,
real_wf,
nat_wf,
nat_plus_wf,
arctan-poly_wf,
arctangent_wf,
rless_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
rless-int,
int-to-real_wf,
rabs_wf,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_mul_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,
itermVar_wf,
itermMultiply_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
nat_plus_properties,
rnexp_wf,
rdiv_wf,
rsub_wf,
less_than'_wf
Rules used in proof :
universeEquality,
promote_hyp,
equalityElimination,
cumulativity,
instantiate,
addLevel,
remainderEquality,
closedConclusion,
baseApply,
imageElimination,
baseClosed,
imageMemberEquality,
lambdaFormation,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
inrFormation,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
hypothesis,
rename,
setElimination,
natural_numberEquality,
multiplyEquality,
addEquality,
dependent_set_memberEquality,
applyEquality,
isectElimination,
extract_by_obid,
because_Cache,
independent_pairEquality,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[k:\mBbbN{}]. (|arctangent(x) - arctan-poly(x;k)| \mleq{} (|x|\^{}(2 * k) + 3/r((2 * k) + 3)))
Date html generated:
2018_05_22-PM-03_04_48
Last ObjectModification:
2018_05_20-PM-11_15_16
Theory : reals_2
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