Nuprl Lemma : arcsine-bounds2
∀x:{x:ℝ| (r0 < x) ∧ (x < (r(3)/r(4)))} . (|arcsine(x) - x| < (r1/r(10)))
Proof
Definitions occuring in Statement :
arcsine: arcsine(x),
rdiv: (x/y),
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
so_apply: x[s],
rfun: I ⟶ℝ,
top: Top,
subinterval: I ⊆ J ,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
not: ¬A,
rooint: (l, u),
i-member: r ∈ I,
subtype_rel: A ⊆r B,
real-fun: real-fun(f;a;b),
ifun: ifun(f;I),
sq_stable: SqStable(P),
rccint: [l, u],
arcsine_deriv: arcsine_deriv(x),
rsub: x - y,
rge: x ≥ y,
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
le: A ≤ B,
rgt: x > y,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
real: ℝ,
rmul: a * b,
int-to-real: r(n),
rinv: rinv(x),
mu-ge: mu-ge(f;n),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
absval: |i|,
btrue: tt,
eq_int: (i =z j),
accelerate: accelerate(k;f),
imax: imax(a;b),
canonical-bound: canonical-bound(r),
reg-seq-inv: reg-seq-inv(x),
le_int: i ≤z j,
bnot: ¬bb,
bfalse: ff,
reg-seq-mul: reg-seq-mul(x;y),
rminus: -(x),
halfpi: π/2,
cubic_converge: cubic_converge(b;m),
fastpi: fastpi(n),
primrec: primrec(n;b;c),
radd: a + b,
reg-seq-list-add: reg-seq-list-add(L),
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
cons: [a / b],
nil: [],
it: ⋅
Lemmas referenced :
rless-int,
rless-int-fractions3,
less_than_wf,
set_wf,
real_wf,
rless_wf,
int-to-real_wf,
rdiv_wf,
derivative-implies-strictly-increasing-closed,
rless-int-fractions2,
rsub_wf,
arcsine_wf,
member_rooint_lemma,
member_rccint_lemma,
rless_transitivity1,
rless_transitivity2,
i-member_wf,
rccint_wf,
arcsine_deriv_wf,
rooint_wf,
derivative-id,
derivative-sub,
derivative_functionality_wrt_subinterval,
rleq_wf,
derivative-arcsine,
req_wf,
req_weakening,
arcsine_deriv_functionality,
rsub_functionality,
equal_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
itermConstant_wf,
intformeq_wf,
satisfiable-full-omega-tt,
nat_plus_properties,
rneq-int,
subtype_rel_sets,
rleq_transitivity,
rleq_weakening,
req_inversion,
req_functionality,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
sq_stable__rleq,
arcsine-root-bounds,
rsqrt0,
rless_functionality,
rleq_weakening_rless,
rsqrt_wf,
rmul_wf,
rleq_weakening_equal,
rsqrt_functionality_wrt_rless,
radd-rminus-assoc,
radd-rminus-both,
radd_comm,
radd_functionality,
radd-ac,
req_transitivity,
radd-assoc,
rleq_functionality,
uiff_transitivity,
square-nonneg,
rminus_wf,
radd_wf,
radd-preserves-rleq,
rsqrt_functionality_wrt_rleq,
rleq_functionality_wrt_implies,
rsqrt1,
rmul-one-both,
rmul-rdiv-cancel2,
radd-int,
rmul_preserves_rleq,
sq_stable__rless,
rmul-is-positive,
radd-preserves-rless,
rmul_preserves_rless,
rleq-int-fractions2,
false_wf,
arcsine0,
rabs_wf,
rabs-of-nonneg,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
req-iff-rsub-is-0,
rless_functionality_wrt_implies,
rinv_wf2,
rinv-as-rdiv,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
rsin_wf,
halfpi_wf,
rsin-strict-bound,
arcsine_functionality_wrt_rless,
arcsine-rsin
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
minusEquality,
natural_numberEquality,
productElimination,
independent_functionElimination,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
hypothesis,
dependent_set_memberEquality,
isectElimination,
multiplyEquality,
because_Cache,
lambdaEquality,
productEquality,
independent_isectElimination,
inrFormation,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
setEquality,
computeAll,
intEquality,
dependent_pairFormation,
applyEquality,
imageElimination,
equalitySymmetry,
equalityTransitivity,
addEquality,
levelHypothesis,
addLevel,
inlFormation,
int_eqEquality,
dependent_set_memberFormation
Latex:
\mforall{}x:\{x:\mBbbR{}| (r0 < x) \mwedge{} (x < (r(3)/r(4)))\} . (|arcsine(x) - x| < (r1/r(10)))
Date html generated:
2017_10_04-PM-10_50_37
Last ObjectModification:
2017_07_28-AM-08_51_50
Theory : reals_2
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