Nuprl Lemma : cosine-approx-for-small
∀a:{2...}. ∀N:ℕ+. ∀x:{x:ℝ| |x| ≤ (r1/r(a))} . (∃z:ℤ [(|cosine(x) - (r(z)/r(2 * N))| ≤ (r(2)/r(N)))])
Proof
Definitions occuring in Statement :
cosine: cosine(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+,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
set: {x:A| B[x]} ,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
uiff: uiff(P;Q),
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
sq_stable: SqStable(P),
sq_exists: ∃x:A [B[x]],
nat: ℕ,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rless: x < y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
rleq-int-fractions,
nat_plus_properties,
int_upper_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_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_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
decidable__le,
itermMultiply_wf,
int_term_value_mul_lemma,
sq_stable__rleq,
rabs_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
rleq_transitivity,
cosine-approx-lemma-ext,
cosine-approx_wf,
rleq_wf,
rsub_wf,
nat_properties,
real_wf,
nat_plus_wf,
istype-int_upper,
cosine_wf,
radd_wf,
rnexp_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-le,
fact_wf,
exp_wf2,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
rleq_functionality_wrt_implies,
cosine-poly-approx,
rleq_weakening_equal,
mul_bounds_1b,
mul_nat_plus,
radd_functionality_wrt_rleq,
rleq_functionality,
radd-int-fractions,
req_weakening,
rnexp-rleq,
zero-rleq-rabs,
rnexp-positive,
req_inversion,
rnexp-rdiv,
exp-positive,
rdiv_functionality,
rnexp-one,
rnexp-int,
exp_wf_nat_plus,
rmul_preserves_rleq,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
req_transitivity,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
because_Cache,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
multiplyEquality,
closedConclusion,
inrFormation_alt,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberFormation_alt,
setIsType,
addEquality,
applyEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
equalityIstype,
sqequalBase
Latex:
\mforall{}a:\{2...\}. \mforall{}N:\mBbbN{}\msupplus{}. \mforall{}x:\{x:\mBbbR{}| |x| \mleq{} (r1/r(a))\} . (\mexists{}z:\mBbbZ{} [(|cosine(x) - (r(z)/r(2 * N))| \mleq{} (r(2)/r(N)))]\000C)
Date html generated:
2019_10_29-AM-10_37_33
Last ObjectModification:
2019_02_02-PM-00_33_34
Theory : reals
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