Nuprl Lemma : arcsine-nonneg
∀[x:{x:ℝ| x ∈ [r0, r1)} ]. (r0 ≤ arcsine(x))
Proof
Definitions occuring in Statement :
arcsine: arcsine(x),
rcoint: [l, u),
i-member: r ∈ I,
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
guard: {T},
rge: x ≥ y,
uimplies: b supposing a,
true: True,
less_than': less_than'(a;b),
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
squash: ↓T,
implies: P ⇒ Q,
sq_stable: SqStable(P),
top: Top,
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q)
Lemmas referenced :
rleq_weakening_equal,
rless_functionality_wrt_implies,
rless-int,
rcoint_wf,
i-member_wf,
real_wf,
set_wf,
rless_wf,
int-to-real_wf,
member_rcoint_lemma,
sq_stable__rless,
arcsine-bounds,
member_rooint_lemma,
arcsine_wf,
sq_stable__rleq,
halfpi_wf,
rminus_wf,
rooint_wf,
rsin-strictly-increasing,
not-rless,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
itermMultiply_wf,
itermSubtract_wf,
halfpi-positive,
rmul-zero-both,
rmul_wf,
rless_functionality,
rmul_reverses_rless_iff,
rsin0,
rsin-arcsine,
rsin_wf,
rless_irreflexivity,
rless_transitivity1
Rules used in proof :
independent_isectElimination,
because_Cache,
lambdaEquality,
productEquality,
independent_pairFormation,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
productElimination,
independent_functionElimination,
natural_numberEquality,
minusEquality,
isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
hypothesis,
sqequalHypSubstitution,
hypothesisEquality,
rename,
thin,
setElimination,
dependent_set_memberEquality,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
setEquality,
applyEquality,
equalitySymmetry,
equalityTransitivity,
lambdaFormation,
intEquality,
int_eqEquality,
approximateComputation
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} [r0, r1)\} ]. (r0 \mleq{} arcsine(x))
Date html generated:
2017_10_04-PM-10_48_24
Last ObjectModification:
2017_08_01-PM-10_03_47
Theory : reals_2
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