Nuprl Lemma : iter-arcsine-contraction-property2
∀a:{a:ℝ| (r(-1) < a) ∧ (a < r1)} . ∀n:ℕ. (|arcsine-contraction^n(a) - arcsine(a)| ≤ |a - arcsine(a)|^3^n)
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],
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,
squash: ↓T,
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
less_than': less_than'(a;b),
cand: A c∧ B,
guard: {T},
req_int_terms: t1 ≡ t2,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
iter-arcsine-contraction: arcsine-contraction^n(a),
decidable: Dec(P),
or: P ∨ Q,
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,
true: True,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
subtract: n - m
Lemmas referenced :
sq_stable__rleq,
int-to-real_wf,
rsub_wf,
rmul_wf,
radd-preserves-rleq,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
le_witness_for_triv,
subtract-1-ge-0,
istype-nat,
real_wf,
rless_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
rleq_wf,
rnexp_wf,
istype-le,
rminus_wf,
rleq_weakening,
rless_transitivity2,
rleq_weakening_rless,
itermMinus_wf,
req-iff-rsub-is-0,
rleq_functionality,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
iff_transitivity,
iff_weakening_uiff,
req_inversion,
rnexp2,
req_weakening,
square-rleq-1-iff,
rabs-rleq-iff,
real_term_value_minus_lemma,
fun_exp0_lemma,
exp0_lemma,
rabs_wf,
arcsine_wf,
member_rooint_lemma,
rleq_weakening_equal,
rnexp1,
fun_exp_unroll,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
arcsine-contraction_wf,
iter-arcsine-contraction_wf,
subtract_wf,
int_term_value_subtract_lemma,
exp_wf4,
rleq_functionality_wrt_implies,
arcsine-contraction-Taylor2,
rleq-implies-rleq,
rabs-rless-iff,
squash_wf,
true_wf,
rminus-int,
subtype_rel_self,
iff_weakening_equal,
square-nonneg,
rsqrt_wf,
rleq_transitivity,
rsqrt_functionality_wrt_rleq,
rsqrt1,
zero-rleq-rabs,
rnexp_functionality_wrt_rleq,
multiply_nat_wf,
rnexp-mul,
exp_step,
decidable__lt,
mul-commutes,
exp_wf2,
int_term_value_add_lemma,
add-commutes
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
setElimination,
thin,
rename,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_functionElimination,
because_Cache,
productElimination,
independent_isectElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
intWeakElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
setIsType,
productIsType,
minusEquality,
dependent_set_memberEquality_alt,
productEquality,
unionElimination,
equalityElimination,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
applyEquality,
universeEquality,
addEquality
Latex:
\mforall{}a:\{a:\mBbbR{}| (r(-1) < a) \mwedge{} (a < r1)\} . \mforall{}n:\mBbbN{}.
(|arcsine-contraction\^{}n(a) - arcsine(a)| \mleq{} |a - arcsine(a)|\^{}3\^{}n)
Date html generated:
2019_10_31-AM-06_12_53
Last ObjectModification:
2019_05_21-PM-01_18_44
Theory : reals_2
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