Nuprl Lemma : arcsine-approx_wf
∀a:{a:ℝ| ((r(-3)/r(4)) < a) ∧ (a < (r(3)/r(4)))} . ∀n:ℕ+. (arcsine-approx(a;n) ∈ {x:ℝ| |x - arcsine(a)| ≤ (r1/r(n))} )
Proof
Definitions occuring in Statement :
arcsine-approx: arcsine-approx(a;n),
arcsine: arcsine(x),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
arcsine-approx: arcsine-approx(a;n),
has-value: (a)↓,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
nat_plus: ℕ+,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rless: x < y,
sq_exists: ∃x:A [B[x]],
ge: i ≥ j ,
cand: A c∧ B,
guard: {T},
sq_type: SQType(T),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
rneq: x ≠ y,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
le: A ≤ B,
uiff: uiff(P;Q),
stable: Stable{P},
real: ℝ,
sq_stable: SqStable(P),
i-member: r ∈ I,
rooint: (l, u),
req_int_terms: t1 ≡ t2,
rdiv: (x/y)
Lemmas referenced :
value-type-has-value,
int-value-type,
cubic_converge_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
intformand_wf,
itermMultiply_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
nat_wf,
set-value-type,
le_wf,
rless-int,
rless-int-fractions3,
nat_properties,
decidable__lt,
istype-less_than,
rless-int-fractions2,
rless_transitivity2,
int-to-real_wf,
rleq_weakening_rless,
rless_wf,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
approx-iter-arcsine_wf,
rleq_wf,
rabs_wf,
rsub_wf,
iter-arcsine-contraction_wf,
int-rdiv_wf,
set_subtype_base,
less_than_wf,
nequal_wf,
rdiv_wf,
arcsine_wf,
member_rooint_lemma,
nat_plus_wf,
real_wf,
rleq_functionality_wrt_implies,
radd_wf,
rleq_weakening_equal,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
rnexp_wf,
exp_wf4,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
iter-arcsine-contraction-property2,
stable__rleq,
false_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
zero-rleq-rabs,
rneq-int,
rnexp_functionality,
rnexp_functionality_wrt_rleq,
arcsine-bounds2,
exp_wf2,
rnexp-positive,
req_inversion,
rnexp-rdiv,
rdiv_functionality,
rnexp-int,
rleq-int-fractions,
sq_stable__less_than,
squash_wf,
true_wf,
rneq_wf,
exp-one,
subtype_rel_self,
iff_weakening_equal,
req-int-fractions,
not-rless,
rminus_wf,
rless-implies-rless,
itermSubtract_wf,
itermMinus_wf,
req-iff-rsub-is-0,
i-member_wf,
rooint_wf,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req_functionality,
rminus_functionality,
rsub_functionality,
arcsine-rminus,
rabs_functionality,
istype-nat,
rabs-rminus,
radd-preserves-rless,
rless_functionality,
rmul_wf,
rinv_wf2,
itermAdd_wf,
real_term_value_add_lemma,
real_term_value_mul_lemma,
rleq_antisymmetry,
rleq-implies-rleq,
uiff_transitivity,
arcsine_functionality,
arcsine0,
rleq_weakening,
rabs-of-nonneg,
rnexp0,
exp_wf_nat_plus,
rleq-int-fractions2,
mul_bounds_1b,
mul_nat_plus,
int_term_value_add_lemma,
radd-int-fractions
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
setElimination,
thin,
rename,
sqequalRule,
callbyvalueReduce,
sqequalHypSubstitution,
productElimination,
introduction,
extract_by_obid,
isectElimination,
intEquality,
independent_isectElimination,
hypothesis,
multiplyEquality,
natural_numberEquality,
hypothesisEquality,
dependent_functionElimination,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
universeIsType,
int_eqEquality,
independent_pairFormation,
inhabitedIsType,
because_Cache,
minusEquality,
imageMemberEquality,
baseClosed,
productIsType,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
setEquality,
closedConclusion,
applyLambdaEquality,
imageElimination,
equalityIstype,
baseApply,
applyEquality,
sqequalBase,
inrFormation_alt,
setIsType,
unionEquality,
functionEquality,
functionIsType,
unionIsType,
addEquality,
universeEquality
Latex:
\mforall{}a:\{a:\mBbbR{}| ((r(-3)/r(4)) < a) \mwedge{} (a < (r(3)/r(4)))\} . \mforall{}n:\mBbbN{}\msupplus{}.
(arcsine-approx(a;n) \mmember{} \{x:\mBbbR{}| |x - arcsine(a)| \mleq{} (r1/r(n))\} )
Date html generated:
2019_10_31-AM-06_13_08
Last ObjectModification:
2019_05_21-PM-01_46_08
Theory : reals_2
Home
Index