Nuprl Lemma : atan_approx-property
∀[a:{2...}]. ∀[x:ℝ]. ∀[N:ℕ+].
|arctangent(x) - (r(atan_approx(a;x;N))/r(2 * N))| ≤ (r(2)/r(N)) supposing |x| ≤ (r1/r(a))
Proof
Definitions occuring in Statement :
atan_approx: atan_approx(a;x;M),
arctangent: arctangent(x),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
sq_type: SQType(T),
real: ℝ,
sq_exists: ∃x:A [B[x]],
rless: x < y,
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
ireal-approx: j-approx(x;M;z),
true: True,
less_than': less_than'(a;b),
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rneq: x ≠ y,
squash: ↓T,
sq_stable: SqStable(P),
has-value: (a)↓,
so_apply: x[s],
prop: ℙ,
and: P ∧ Q,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
int_upper: {i...},
guard: {T},
nat: ℕ,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
atan_approx: atan_approx(a;x;M),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
less_than_transitivity1,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rinv-as-rdiv,
rinv_functionality2,
rinv-of-rmul,
rinv-mul-as-rdiv,
req_transitivity,
equal-wf-T-base,
int_formula_prop_eq_lemma,
intformeq_wf,
rneq-int,
rmul-int,
rneq_functionality,
req-iff-rsub-is-0,
itermSubtract_wf,
rinv_wf2,
rmul_wf,
rleq-int-fractions2,
rmul_preserves_rleq2,
exp-one,
set_subtype_base,
subtype_base_sq,
rnexp-rdiv,
req_inversion,
rnexp-int,
rless_functionality,
sq_stable__less_than,
exp-positive,
zero-rleq-rabs,
rnexp_functionality_wrt_rleq,
radd-int,
rdiv_functionality,
req_weakening,
radd-rdiv,
rleq_functionality,
radd_functionality_wrt_rleq,
rnexp_wf,
r-triangle-inequality2,
rleq_weakening_equal,
arctan-poly_wf,
radd_wf,
atan-approx_wf,
rleq_functionality_wrt_implies,
arctan-poly-approx,
rleq-int-fractions,
less_than_wf,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
false_wf,
add-is-int-iff,
int_subtype_base,
multiply-is-int-iff,
int_upper_wf,
real_wf,
rleq_wf,
nat_plus_wf,
atan_approx_wf,
arctangent_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
rsub_wf,
less_than'_wf,
equal_wf,
rless_wf,
int-to-real_wf,
rless-int,
rdiv_wf,
rabs_wf,
rleq_transitivity,
atan-approx-property,
int-value-type,
set-value-type,
sq_stable__le,
value-type-has-value,
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,
int_upper_properties,
nat_plus_properties,
nat_properties,
exp_wf2,
le_wf,
nat_wf,
set_wf,
atan-log_wf
Rules used in proof :
promote_hyp,
cumulativity,
instantiate,
closedConclusion,
baseApply,
axiomEquality,
minusEquality,
applyEquality,
independent_pairEquality,
equalitySymmetry,
equalityTransitivity,
productElimination,
inrFormation,
imageElimination,
baseClosed,
imageMemberEquality,
setEquality,
callbyvalueReduce,
lambdaFormation,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
addEquality,
multiplyEquality,
because_Cache,
lambdaEquality,
sqequalRule,
hypothesis,
rename,
setElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[a:\{2...\}]. \mforall{}[x:\mBbbR{}]. \mforall{}[N:\mBbbN{}\msupplus{}].
|arctangent(x) - (r(atan\_approx(a;x;N))/r(2 * N))| \mleq{} (r(2)/r(N)) supposing |x| \mleq{} (r1/r(a))
Date html generated:
2018_05_22-PM-03_05_45
Last ObjectModification:
2018_05_20-PM-11_20_12
Theory : reals_2
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