Nuprl Lemma : arcsine-contraction-Taylor
∀[a:{a:ℝ| (r(-1) < a) ∧ (a < r1)} ]. ∀[x:ℝ].
∀c:ℝ
|arcsine-contraction(a;x) - arcsine(a)| ≤ (c * |x - arcsine(a)|^3)
supposing (r0 ≤ a) ∧ (a ≤ c) ∧ (rsqrt(r1 - a * a) ≤ c)
Proof
Definitions occuring in Statement :
arcsine-contraction: arcsine-contraction(a;x),
arcsine: arcsine(x),
rsqrt: rsqrt(x),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
top: Top,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
subinterval: I ⊆ J ,
true: True,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
squash: ↓T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
subtype_rel: A ⊆r B,
cand: A c∧ B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
less_than: a < b,
so_lambda: λ2x y.t[x; y],
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
so_apply: x[s1;s2],
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
arcsine-contraction: arcsine-contraction(a;x),
eq_int: (i =z j),
decidable: Dec(P),
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
Taylor-remainder: Taylor-remainder(I;n;b;a;i,x.F[i; x]),
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
rsub: x - y,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
pointwise-req: x[k] = y[k] for k ∈ [n,m],
rneq: x ≠ y,
real: ℝ,
fact: (n)!,
primrec: primrec(n;b;c),
subtract: n - m,
rdiv: (x/y),
rge: x ≥ y
Lemmas referenced :
sq_stable__rleq,
rabs_wf,
rsub_wf,
arcsine-contraction_wf,
arcsine_wf,
member_rooint_lemma,
rmul_wf,
rnexp_wf,
false_wf,
le_wf,
radd-preserves-rleq,
int-to-real_wf,
rleq_functionality,
radd_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rleq-iff-all-rless,
member_riiint_lemma,
rless_wf,
rminus_wf,
halfpi_wf,
deriviative-rcos,
deriviative-rsin,
derivative-minus,
riiint_wf,
rsin_wf,
real_wf,
i-member_wf,
rcos_wf,
derivative-minus-minus,
set_wf,
less_than'_wf,
nat_plus_wf,
rleq_wf,
rsqrt_wf,
squash_wf,
true_wf,
rminus-int,
iff_weakening_equal,
rleq_weakening_rless,
iff_transitivity,
iff_weakening_uiff,
req_inversion,
rnexp2,
req_weakening,
square-rleq-1-iff,
rabs-rleq-iff,
Taylor-theorem,
iproper-riiint,
less_than_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_seg_wf,
req_wf,
sq_stable__rless,
rminus_functionality,
rsin_functionality,
rcos_functionality,
rmul_functionality,
rsub_functionality,
radd_functionality,
req_functionality,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_seg_cases,
int_seg_subtype,
int_seg_properties,
int_subtype_base,
decidable__equal_int,
derivative-const,
derivative-const-mul,
derivative-sub,
derivative-id,
derivative-add,
member_rccint_lemma,
rcos-nonneg,
rsqrt-unique,
arcsine-bounds,
rsin-rcos-pythag,
rsin-arcsine,
rnexp_functionality,
radd-zero-both,
radd_comm,
rmul-zero-both,
radd-int,
rmul-distrib2,
rmul-identity1,
radd-assoc,
rminus-as-rmul,
req_transitivity,
uiff_transitivity,
radd-preserves-req,
rsum_wf,
ifthenelse_wf,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
rsum-split-first,
rsum-zero-req,
nequal-le-implies,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
rless-int,
int_seg_subtype_nat,
fact_wf,
rdiv_wf,
rsum_functionality,
rmul-rdiv-cancel2,
rless_transitivity2,
rnexp_zero_lemma,
fact0_redex_lemma,
rdiv-zero,
rdiv_functionality,
rminus-zero,
rmul-int,
radd-ac,
rminus-radd,
rmul_over_rminus,
rsqrt_squared,
fact-non-zero,
decidable__le,
sq_stable__less_than,
rneq-int,
radd-rminus-both,
rminus-rminus,
rmul_comm,
rabs_functionality,
set_subtype_base,
rleq-int,
rmul_preserves_rleq,
rinv_wf2,
rabs-rdiv,
rneq_functionality,
rabs-of-nonneg,
rmul-rinv3,
rsqrt_functionality,
itermMinus_wf,
real_term_value_minus_lemma,
rleq_functionality_wrt_implies,
r-triangle-inequality,
rleq_weakening_equal,
rabs-rmul,
zero-rleq-rabs,
rsqrt_nonneg,
radd_functionality_wrt_rleq,
rmul_functionality_wrt_rleq2,
rabs-rcos-rleq,
rabs-rsin-rleq,
rabs-difference-bound-rleq,
rmin_wf,
rmin_ub,
trivial-rsub-rleq,
rmax_wf,
rmax_lb,
trivial-rleq-radd,
rabs-bounds,
rabs-difference-symmetry,
r-triangle-inequality2,
rabs-rnexp,
rnexp-rleq,
rmul-nonneg-case1,
rnexp-nonneg,
rnexp2-nonneg,
rnexp_step,
rleq_transitivity,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
setElimination,
rename,
extract_by_obid,
isectElimination,
dependent_set_memberEquality,
because_Cache,
independent_pairFormation,
hypothesis,
hypothesisEquality,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
independent_functionElimination,
independent_isectElimination,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
productEquality,
setEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairEquality,
applyEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
addEquality,
hypothesis_subsumption,
applyLambdaEquality,
inrFormation,
multiplyEquality,
inlFormation
Latex:
\mforall{}[a:\{a:\mBbbR{}| (r(-1) < a) \mwedge{} (a < r1)\} ]. \mforall{}[x:\mBbbR{}].
\mforall{}c:\mBbbR{}
|arcsine-contraction(a;x) - arcsine(a)| \mleq{} (c * |x - arcsine(a)|\^{}3)
supposing (r0 \mleq{} a) \mwedge{} (a \mleq{} c) \mwedge{} (rsqrt(r1 - a * a) \mleq{} c)
Date html generated:
2017_10_04-PM-10_49_55
Last ObjectModification:
2017_07_28-AM-08_51_39
Theory : reals_2
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