Nuprl Lemma : arcsine-root-bounds
∀t:ℝ. ((t ∈ (r(-1), r1)) ⇒ (r0 < (r1 - t * t)))
Proof
Definitions occuring in Statement :
rooint: (l, u),
i-member: r ∈ I,
rless: x < y,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
top: Top,
cand: A c∧ B,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
rsub: x - y
Lemmas referenced :
radd-preserves-rless,
int-to-real_wf,
rsub_wf,
rmul_wf,
i-member_wf,
rooint_wf,
real_wf,
rnexp_wf,
false_wf,
le_wf,
square-rless-1-iff,
member_rooint_lemma,
rabs-rless-iff,
rless_wf,
squash_wf,
true_wf,
rminus-int,
iff_weakening_equal,
radd_wf,
rminus_wf,
rless_functionality,
req_weakening,
radd-zero-both,
radd_functionality,
radd-rminus-both,
radd_comm,
radd-ac,
req_inversion,
rnexp2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
productElimination,
independent_functionElimination,
minusEquality,
because_Cache,
dependent_set_memberEquality,
sqequalRule,
independent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
addLevel,
levelHypothesis
Latex:
\mforall{}t:\mBbbR{}. ((t \mmember{} (r(-1), r1)) {}\mRightarrow{} (r0 < (r1 - t * t)))
Date html generated:
2016_10_26-PM-00_40_58
Last ObjectModification:
2016_09_12-PM-05_45_15
Theory : reals_2
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