Nuprl Lemma : near-arcsine-exists
∀a:{a:ℝ| a ∈ (r(-1), r1)} . ∀N:ℕ+. (∃y:{ℝ| (|y - arcsine(a)| ≤ (r1/r(N)))})
Proof
Definitions occuring in Statement :
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],
sq_exists: ∃x:{A| B[x]},
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
squash: ↓T,
implies: P ⇒ Q,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
top: Top,
member: t ∈ T,
and: P ∧ Q,
rooint: (l, u),
i-member: r ∈ I,
all: ∀x:A. B[x],
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
true: True,
less_than': less_than'(a;b),
less_than: a < b,
so_apply: x[s1;s2],
sq_exists: ∃x:{A| B[x]},
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
nat_plus: ℕ+,
so_lambda: λ2x y.t[x; y],
cand: A c∧ B,
rsub: x - y,
le: A ≤ B,
nat: ℕ,
rgt: x > y,
rge: x ≥ y,
real: ℝ,
subtype_rel: A ⊆r B,
it: ⋅,
nil: [],
int-rdiv: (a)/k1,
reg-seq-mul: reg-seq-mul(x;y),
bfalse: ff,
bnot: ¬bb,
le_int: i ≤z j,
reg-seq-inv: reg-seq-inv(x),
canonical-bound: canonical-bound(r),
imax: imax(a;b),
eq_int: (i =z j),
btrue: tt,
absval: |i|,
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
mu-ge: mu-ge(f;n),
rinv: rinv(x),
rmul: a * b,
rdiv: (x/y),
cons: [a / b],
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
reg-seq-list-add: reg-seq-list-add(L),
accelerate: accelerate(k;f),
radd: a + b,
int-to-real: r(n),
rless: x < y,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
subtract: n - m
Lemmas referenced :
rooint_wf,
i-member_wf,
real_wf,
set_wf,
nat_plus_wf,
int-to-real_wf,
sq_stable__rless,
member_rooint_lemma,
nequal_wf,
true_wf,
equal-wf-base,
int_subtype_base,
subtype_base_sq,
int-rdiv_wf,
less_than_wf,
rless_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,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
arcsine_wf,
rsub_wf,
rabs_wf,
rleq_wf,
sq_exists_wf,
by-nearby-cases-ext,
rleq_weakening_rless,
rless_transitivity2,
arcsine-shift,
radd-ac,
radd_comm,
radd-rminus-both,
radd_functionality,
radd-zero-both,
req_weakening,
rless_functionality,
rminus_wf,
radd_wf,
rmul_wf,
radd-preserves-rless,
rnexp2,
req_inversion,
square-rless-1-iff,
le_wf,
false_wf,
rnexp_wf,
rabs-of-nonneg,
rmul_functionality_wrt_rless2,
radd_functionality_wrt_rless1,
rleq_weakening_equal,
rless_functionality_wrt_implies,
int-rdiv-req,
rless-int-fractions2,
rsqrt0,
rsqrt_wf,
rsqrt_functionality_wrt_rless,
rmul-int-fractions,
req_functionality,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
itermMultiply_wf,
intformeq_wf,
decidable__equal_int,
req-int-fractions,
rleq-int-fractions2,
rsqrt-unique,
halfpi_wf,
rational-approx-property,
mul_nat_plus,
arcsine-approx_wf,
rmul-is-positive,
radd-assoc,
radd-rminus-assoc,
req_transitivity,
rabs-rleq-iff,
square-rleq-1-iff,
iff_weakening_uiff,
iff_transitivity,
rleq_functionality,
uiff_transitivity,
iff_weakening_equal,
rminus-int,
squash_wf,
radd-preserves-rleq,
rsqrt1,
rsqrt_nonneg,
req_wf,
rational-approx_wf,
rsub_functionality,
equal_wf,
rminus-rminus,
rmul_functionality,
rminus-as-rmul,
rminus-radd,
rabs_functionality,
r-triangle-inequality-rsub,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
rabs-difference-symmetry,
sq_stable__rleq,
radd-int,
rdiv_functionality,
radd-rdiv,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
sq_stable__less_than,
rleq-int-fractions,
rless-int-fractions,
rabs-difference-bound-rleq,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rminus_functionality_wrt_rleq,
rabs-rminus,
arcsine-rminus,
int_term_value_add_lemma,
itermAdd_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
not-lt-2,
rminus-zero,
small-arcsine
Rules used in proof :
cut,
lambdaEquality,
because_Cache,
imageElimination,
baseClosed,
imageMemberEquality,
productElimination,
independent_functionElimination,
hypothesisEquality,
natural_numberEquality,
isectElimination,
rename,
setElimination,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
extract_by_obid,
introduction,
sqequalHypSubstitution,
independent_pairFormation,
sqequalRule,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
equalitySymmetry,
equalityTransitivity,
cumulativity,
instantiate,
addLevel,
dependent_set_memberEquality,
computeAll,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
inrFormation,
independent_isectElimination,
functionEquality,
productEquality,
minusEquality,
levelHypothesis,
multiplyEquality,
applyEquality,
addEquality,
dependent_set_memberFormation,
promote_hyp,
inlFormation,
universeEquality,
setEquality,
applyLambdaEquality
Latex:
\mforall{}a:\{a:\mBbbR{}| a \mmember{} (r(-1), r1)\} . \mforall{}N:\mBbbN{}\msupplus{}. (\mexists{}y:\{\mBbbR{}| (|y - arcsine(a)| \mleq{} (r1/r(N)))\})
Date html generated:
2017_10_04-PM-10_53_54
Last ObjectModification:
2017_07_28-AM-08_52_03
Theory : reals_2
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