Nuprl Lemma : derivative-arcsine
d(arcsine(x))/dx = λx.arcsine_deriv(x) on (r(-1), r1)
Proof
Definitions occuring in Statement :
arcsine: arcsine(x),
arcsine_deriv: arcsine_deriv(x),
derivative: d(f[x])/dx = λz.g[z] on I,
rooint: (l, u),
int-to-real: r(n),
minus: -n,
natural_number: $n
Definitions unfolded in proof :
arcsine: arcsine(x),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
top: Top,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
derivative-of-integral,
rooint_wf,
int-to-real_wf,
member_rooint_lemma,
rless-int,
rless_wf,
arcsine_deriv_wf,
i-member_wf,
real_wf,
req_wf,
set_wf,
all_wf,
req_weakening,
req_functionality,
arcsine_deriv_functionality
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
minusEquality,
natural_numberEquality,
hypothesis,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
dependent_set_memberEquality,
productEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
lambdaFormation,
because_Cache,
functionEquality,
applyEquality,
independent_isectElimination
Latex:
d(arcsine(x))/dx = \mlambda{}x.arcsine\_deriv(x) on (r(-1), r1)
Date html generated:
2016_10_26-PM-00_41_25
Last ObjectModification:
2016_09_12-PM-05_45_38
Theory : reals_2
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