Nuprl Lemma : atan-approx-property
∀[k:ℕ]. ∀[x:ℝ]. ∀[N:ℕ+]. ((|x| ≤ (r1/r(2))) ⇒ 1-approx(arctan-poly(x;k);N;atan-approx(k;x;N)))
Proof
Definitions occuring in Statement :
atan-approx: atan-approx(k;x;N),
arctan-poly: arctan-poly(x;k),
ireal-approx: j-approx(x;M;z),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
real: ℝ,
has-value: (a)↓,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
arctan-poly: arctan-poly(x;k),
lelt: i ≤ j < k,
int_seg: {i..j-},
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
subtract: n - m,
primrec: primrec(n;b;c),
exp: i^n,
subtype_rel: A ⊆r B,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
nat_plus: ℕ+,
false: False,
not: ¬A,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
ireal-approx: j-approx(x;M;z),
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
prop: ℙ,
atan-approx: atan-approx(k;x;N),
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
ireal-approx_functionality,
ireal-approx-rmul2,
mul-commutes,
ireal-approx_wf,
ireal-approx-1,
set-value-type,
add-is-int-iff,
add_nat_plus,
absval_wf,
int-value-type,
value-type-has-value,
poly-approx_wf,
real_term_value_minus_lemma,
rminus_functionality,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-rinv3,
int-rdiv-req,
rnexp-mul,
rnexp-add1,
int-rdiv_functionality,
rnexp_functionality,
rmul_functionality,
uiff_transitivity,
rsum_linearity2,
req_functionality,
itermMinus_wf,
req-iff-rsub-is-0,
itermSubtract_wf,
rinv_wf2,
rmul_preserves_req,
req_wf,
rminus_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
rsum_functionality,
int_seg_wf,
int_seg_subtype_nat,
int_seg_properties,
rsum_wf,
mul_nat_plus,
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,
eq_int_wf,
nequal_wf,
equal-wf-base,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
int-rdiv_wf,
poly-approx-property,
rnexp2,
rabs_functionality,
rmul_wf,
exp-one,
int_subtype_base,
less_than_wf,
set_subtype_base,
subtype_base_sq,
req_weakening,
rneq_functionality,
rnexp-int,
rdiv_functionality,
rnexp-rdiv,
req_transitivity,
rabs-rnexp,
req_inversion,
rleq_functionality,
exp_wf2,
rnexp_wf,
le_wf,
false_wf,
zero-rleq-rabs,
rnexp_functionality_wrt_rleq,
nat_wf,
real_wf,
nat_plus_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
atan-approx_wf,
arctan-poly_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
nat_plus_properties,
rsub_wf,
less_than'_wf,
rless_wf,
rless-int,
int-to-real_wf,
rdiv_wf,
rabs_wf,
rleq_wf
Rules used in proof :
divideEquality,
pointwiseFunctionality,
callbyvalueReduce,
applyLambdaEquality,
promote_hyp,
equalityElimination,
addLevel,
remainderEquality,
closedConclusion,
baseApply,
addEquality,
cumulativity,
instantiate,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
multiplyEquality,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
rename,
setElimination,
applyEquality,
independent_pairEquality,
lambdaEquality,
baseClosed,
imageMemberEquality,
independent_pairFormation,
independent_functionElimination,
productElimination,
because_Cache,
dependent_functionElimination,
inrFormation,
sqequalRule,
independent_isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[x:\mBbbR{}]. \mforall{}[N:\mBbbN{}\msupplus{}]. ((|x| \mleq{} (r1/r(2))) {}\mRightarrow{} 1-approx(arctan-poly(x;k);N;atan-approx(k;x;N)))
Date html generated:
2018_05_22-PM-03_05_04
Last ObjectModification:
2018_05_20-PM-11_17_27
Theory : reals_2
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