Nuprl Lemma : cosine-poly-approx
∀[x:{x:ℝ| |x| ≤ (r1/r(2))} ]. ∀[k:ℕ]. ∀[N:ℕ+].
(|cosine(x) - (r(cosine-approx(x;k;N))/r(2 * N))| ≤ ((|x|^(2 * k) + 2/r(((2 * k) + 2)!)) + (r1/r(N))))
Proof
Definitions occuring in Statement :
cosine-approx: cosine-approx(x;k;N),
cosine: cosine(x),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
fact: (n)!,
nat_plus: ℕ+,
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
sq_stable: SqStable(P),
ireal-approx: j-approx(x;M;z),
rev_uimplies: rev_uimplies(P;Q),
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
le: A ≤ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
so_apply: x[s],
rge: x ≥ y,
cand: A c∧ B,
uiff: uiff(P;Q),
stable: Stable{P}
Lemmas referenced :
cosine-approx-property,
sq_stable__rleq,
rabs_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
rleq_functionality_wrt_implies,
rsub_wf,
cosine_wf,
cosine-approx_wf,
nat_plus_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
radd_wf,
rsum_wf,
int-rmul_wf,
fastexp_wf,
int_seg_subtype_nat,
istype-false,
int-rdiv_wf,
fact_wf,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
rnexp_wf,
int_seg_wf,
rleq_weakening_equal,
r-triangle-inequality2,
itermAdd_wf,
int_term_value_add_lemma,
radd_functionality_wrt_rleq,
nat_plus_wf,
istype-nat,
real_wf,
rleq_wf,
cosine-poly-approx-1,
zero-rleq-rabs,
rleq-int-fractions3,
istype-less_than,
rleq_transitivity,
rleq_functionality,
radd_functionality,
req_weakening,
rabs_functionality,
rsub_functionality,
nat_plus_inc_int_nzero,
stable_req,
false_wf,
not_wf,
req_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rabs-of-nonpos,
rleq_weakening_rless,
rminus_wf,
req_functionality,
cosine_functionality,
cosine-rminus,
not-rless,
rabs-of-nonneg,
rnexp-nonneg,
rnexp2-nonneg,
rsum_functionality2,
int-rmul_functionality,
int-rdiv_functionality,
req_inversion,
rabs-rnexp,
rnexp-mul
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
independent_functionElimination,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
inrFormation_alt,
dependent_functionElimination,
because_Cache,
productElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
universeIsType,
imageElimination,
multiplyEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
minusEquality,
applyEquality,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
addEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setIsType,
productIsType,
unionEquality,
functionEquality,
functionIsType,
unionIsType
Latex:
\mforall{}[x:\{x:\mBbbR{}| |x| \mleq{} (r1/r(2))\} ]. \mforall{}[k:\mBbbN{}]. \mforall{}[N:\mBbbN{}\msupplus{}].
(|cosine(x) - (r(cosine-approx(x;k;N))/r(2 * N))| \mleq{} ((|x|\^{}(2 * k) + 2/r(((2 * k) + 2)!))
+ (r1/r(N))))
Date html generated:
2019_10_29-AM-10_37_16
Last ObjectModification:
2019_02_02-PM-00_23_11
Theory : reals
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