Nuprl Lemma : near-arcsine_wf
∀a:{a:ℝ| a ∈ (r(-1), r1)} . ∀N:ℕ+. (near-arcsine(a;N) ∈ {y:ℝ| |y - arcsine(a)| ≤ (r1/r(N))} )
Proof
Definitions occuring in Statement :
near-arcsine: near-arcsine(a;N),
arcsine: arcsine(x),
rooint: (l, u),
i-member: r ∈ I,
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
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,
sq_exists: ∃x:{A| B[x]},
subtype_rel: A ⊆r B,
top: Top,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
so_apply: x[s],
near-arcsine: near-arcsine(a;N),
near-arcsine-exists-ext
Lemmas referenced :
near-arcsine-exists-ext,
member_rooint_lemma,
all_wf,
real_wf,
i-member_wf,
rooint_wf,
int-to-real_wf,
nat_plus_wf,
sq_exists_wf,
rleq_wf,
rabs_wf,
rsub_wf,
arcsine_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
introduction,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesisEquality,
isectElimination,
setEquality,
minusEquality,
natural_numberEquality,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality,
independent_isectElimination,
inrFormation,
productElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}a:\{a:\mBbbR{}| a \mmember{} (r(-1), r1)\} . \mforall{}N:\mBbbN{}\msupplus{}. (near-arcsine(a;N) \mmember{} \{y:\mBbbR{}| |y - arcsine(a)| \mleq{} (r1/r(N))\} )
Date html generated:
2016_10_26-PM-00_49_56
Last ObjectModification:
2016_10_13-PM-03_20_56
Theory : reals_2
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