Nuprl Lemma : small-arcsine
∀N:ℕ+. ∀a:ℝ. ((|a| ≤ (r1/r(N + 1))) ⇒ (|arcsine(a)| ≤ (r1/r(N))))
Proof
Definitions occuring in Statement :
arcsine: arcsine(x),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
cand: A c∧ B,
rge: x ≥ y,
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
less_than': less_than'(a;b),
le: A ≤ B,
subtract: n - m,
uiff: uiff(P;Q),
iproper: iproper(I),
less_than: a < b,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
subinterval: I ⊆ J ,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
i-member: r ∈ I,
rooint: (l, u),
rccint: [l, u],
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
sq_stable: SqStable(P),
arcsine_deriv: arcsine_deriv(x),
real: ℝ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
sq_type: SQType(T),
rsub: x - y
Lemmas referenced :
rleq_wf,
rabs_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
real_wf,
nat_plus_wf,
member_rooint_lemma,
rabs-rleq-iff,
rleq_weakening_equal,
rless_functionality_wrt_implies,
rminus_wf,
rmul-int,
rmul_over_rminus,
iff_weakening_equal,
rminus-int,
true_wf,
squash_wf,
rmul-rdiv-cancel2,
rminus_functionality,
req_weakening,
rless_functionality,
iff_transitivity,
rmul_wf,
rmul_preserves_rless,
int_term_value_mul_lemma,
itermMultiply_wf,
less_than_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
not-lt-2,
false_wf,
rless-int-fractions3,
mean-value-for-bounded-derivative,
rccint_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
i-finite_wf,
rinv_wf2,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul-rinv,
member_rccint_lemma,
arcsine_wf,
subtype_rel_sets,
i-member_wf,
rooint_wf,
arcsine_deriv_wf,
req_functionality,
arcsine_deriv_functionality,
req_wf,
set_wf,
derivative_functionality_wrt_subinterval,
derivative-arcsine,
mul_bounds_1b,
rsqrt_functionality_wrt_rless,
rsqrt_wf,
rleq_weakening_rless,
rsqrt0,
rnexp-rleq,
zero-rleq-rabs,
le_wf,
sq_stable__rleq,
rnexp_wf,
rnexp2-nonneg,
rleq_functionality,
req_inversion,
rabs-rnexp,
rabs-of-nonneg,
rnexp-positive,
rnexp2,
rnexp-rdiv,
rdiv_functionality,
rnexp-one,
rsub_wf,
rleq_functionality_wrt_implies,
rsub_functionality_wrt_rleq,
rmul_preserves_rleq,
radd_wf,
rleq-int,
sq_stable__less_than,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
uiff_transitivity,
rmul-rinv3,
real_term_value_add_lemma,
radd_functionality,
radd-int,
rsqrt_functionality_wrt_rleq,
rleq-int-fractions2,
mul_nat_plus,
zero-mul,
square-nonneg,
rsqrt_functionality,
rsqrt-of-square,
rless_transitivity1,
rsqrt-rdiv,
rneq_functionality,
equal_wf,
rsqrt-positive,
rmul_functionality,
rinv_functionality2,
rmul_preserves_rleq2,
less_than'_wf,
radd-zero-both,
radd_comm,
rdiv-zero,
rmul-int-rdiv,
rmul-distrib2,
rmul-identity1,
rminus-as-rmul,
radd-preserves-rleq,
subtype_base_sq,
int_subtype_base,
rmul-nonneg-case1,
intformeq_wf,
int_formula_prop_eq_lemma,
rabs-difference-symmetry,
rabs_functionality,
rinv-mul-as-rdiv,
rmul_comm,
rmul_functionality_wrt_rleq2,
rsqrt-unique,
mul_bounds_1a,
nat_plus_subtype_nat,
arcsine0,
rsub_functionality,
rabs-rminus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
addEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
sqequalRule,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
minusEquality,
universeEquality,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
applyEquality,
addLevel,
multiplyEquality,
dependent_set_memberEquality,
productEquality,
setEquality,
isect_memberFormation,
independent_pairEquality,
axiomEquality,
promote_hyp,
instantiate,
cumulativity,
inlFormation
Latex:
\mforall{}N:\mBbbN{}\msupplus{}. \mforall{}a:\mBbbR{}. ((|a| \mleq{} (r1/r(N + 1))) {}\mRightarrow{} (|arcsine(a)| \mleq{} (r1/r(N))))
Date html generated:
2017_10_04-PM-10_53_26
Last ObjectModification:
2017_07_28-AM-08_51_59
Theory : reals_2
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