Nuprl Lemma : iter-arcsine-contraction-property
∀a:{a:ℝ| (r(-1) < a) ∧ (a < r1)} . ((r0 ≤ a) ⇒ (∀n:ℕ. (|arcsine-contraction^n(a) - arcsine(a)| ≤ |a - arcsine(a)|^3^n))\000C)
Proof
Definitions occuring in Statement :
iter-arcsine-contraction: arcsine-contraction^n(a),
arcsine: arcsine(x),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
exp: i^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
squash: ↓T,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
iter-arcsine-contraction: arcsine-contraction^n(a),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
compose: f o g,
nequal: a ≠ b ∈ T ,
rge: x ≥ y,
rsub: x - y
Lemmas referenced :
sq_stable__rleq,
int-to-real_wf,
rsub_wf,
rmul_wf,
radd-preserves-rleq,
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,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
less_than'_wf,
rnexp_wf,
exp_wf4,
nat_plus_properties,
rabs_wf,
arcsine_wf,
member_rooint_lemma,
iter-arcsine-contraction_wf,
rless_wf,
nat_plus_wf,
le_wf,
false_wf,
decidable__le,
subtract_wf,
intformnot_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
rleq_wf,
set_wf,
real_wf,
rminus_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,
rnexp1,
rleq_weakening_equal,
exp0_lemma,
fun_exp0_lemma,
fun_exp_unroll,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
arcsine-contraction_wf,
rleq_functionality_wrt_implies,
arcsine-contraction-Taylor,
rleq-implies-rleq,
sq_stable__rless,
radd-rminus-assoc,
radd-rminus-both,
radd_comm,
radd_functionality,
radd-ac,
req_transitivity,
radd-assoc,
uiff_transitivity,
square-nonneg,
rsqrt_functionality_wrt_rleq,
rleq_transitivity,
req_wf,
rsqrt_wf,
rsqrt1,
rnexp_functionality_wrt_rleq,
zero-rleq-rabs,
rnexp-mul,
mul_bounds_1a,
exp_wf2,
mul-commutes,
exp_step
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
setElimination,
thin,
rename,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_functionElimination,
because_Cache,
productElimination,
independent_isectElimination,
dependent_functionElimination,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
intWeakElimination,
dependent_pairFormation,
independent_pairFormation,
independent_pairEquality,
applyEquality,
dependent_set_memberEquality,
productEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
universeEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
setEquality,
multiplyEquality
Latex:
\mforall{}a:\{a:\mBbbR{}| (r(-1) < a) \mwedge{} (a < r1)\}
((r0 \mleq{} a) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. (|arcsine-contraction\^{}n(a) - arcsine(a)| \mleq{} |a - arcsine(a)|\^{}3\^{}n)))
Date html generated:
2017_10_04-PM-10_50_15
Last ObjectModification:
2017_07_28-AM-08_51_46
Theory : reals_2
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