Nuprl Lemma : approx-iter-arcsine_wf
∀[a:{a:ℝ| (r(-1) < a) ∧ (a < r1)} ]. ∀[n:ℕ]. ∀[k:ℕ+].
(approx-iter-arcsine(a;k;n) ∈ {y:ℤ| |arcsine-contraction^n(a) - (r(y))/2 * k| ≤ (r(2)/r(k))} )
Proof
Definitions occuring in Statement :
approx-iter-arcsine: approx-iter-arcsine(a;k;n),
iter-arcsine-contraction: arcsine-contraction^n(a),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
int-rdiv: (a)/k1,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
multiply: n * m,
minus: -n,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
approx-iter-arcsine: approx-iter-arcsine(a;k;n),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
iter-arcsine-contraction: arcsine-contraction^n(a),
real: ℝ,
int_nzero: ℤ-o,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rational-approx: (x within 1/n),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
squash: ↓T,
true: True,
sq_type: SQType(T),
bfalse: ff,
has-value: (a)↓,
less_than: a < b,
less_than': less_than'(a;b),
sq_stable: SqStable(P),
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b,
compose: f o g,
cand: A c∧ B,
le: A ≤ B
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_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,
nat_plus_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
set_wf,
real_wf,
rless_wf,
int-to-real_wf,
fun_exp0_lemma,
rleq_wf,
rabs_wf,
rsub_wf,
int-rdiv_wf,
nat_plus_properties,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
equal-wf-base,
int_subtype_base,
nequal_wf,
rdiv_wf,
rless-int,
decidable__lt,
rational-approx_wf,
rleq-int-fractions,
rleq_functionality_wrt_implies,
rational-approx-property,
rleq_weakening_equal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
equal_wf,
squash_wf,
true_wf,
eq_int_eq_false,
bfalse_wf,
iff_weakening_equal,
value-type-has-value,
int-value-type,
mul_nat_plus,
iter-arcsine-contraction_wf,
sq_stable__rleq,
le_wf,
equal-wf-T-base,
fun_exp_unroll,
rneq-int,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
arcsine-contraction_wf,
arcsine-contraction-difference,
rmul_wf,
zero-rleq-rabs,
rleq-int,
false_wf,
rleq_functionality,
rmul_comm,
req_weakening,
rmul_functionality_wrt_rleq2,
uiff_transitivity,
rmul-int-rdiv,
radd_wf,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
itermAdd_wf,
int_term_value_add_lemma,
req_transitivity,
radd-rdiv,
rdiv_functionality,
radd-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
callbyvalueReduce,
sqleReflexivity,
unionElimination,
because_Cache,
productEquality,
minusEquality,
dependent_set_memberEquality,
applyEquality,
multiplyEquality,
baseApply,
closedConclusion,
baseClosed,
inrFormation,
productElimination,
instantiate,
cumulativity,
imageElimination,
universeEquality,
imageMemberEquality,
applyLambdaEquality,
setEquality,
equalityElimination,
promote_hyp,
inlFormation,
addEquality
Latex:
\mforall{}[a:\{a:\mBbbR{}| (r(-1) < a) \mwedge{} (a < r1)\} ]. \mforall{}[n:\mBbbN{}]. \mforall{}[k:\mBbbN{}\msupplus{}].
(approx-iter-arcsine(a;k;n) \mmember{} \{y:\mBbbZ{}| |arcsine-contraction\^{}n(a) - (r(y))/2 * k| \mleq{} (r(2)/r(k))\} )
Date html generated:
2017_10_04-PM-10_50_07
Last ObjectModification:
2017_07_28-AM-08_51_42
Theory : reals_2
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