Nuprl Lemma : arcsine-shift
∀[x:{x:ℝ| x ∈ (r(-1), r1)} ]. arcsine(x) = (π/2 - arcsine(rsqrt(r1 - x * x))) supposing r0 < x
Proof
Definitions occuring in Statement :
arcsine: arcsine(x),
halfpi: π/2,
rsqrt: rsqrt(x),
rooint: (l, u),
i-member: r ∈ I,
rless: x < y,
rsub: x - y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
cand: A c∧ B,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
less_than: a < b,
rsub: x - y,
rge: x ≥ y,
i-member: r ∈ I,
rooint: (l, u),
pi: π
Lemmas referenced :
radd-preserves-rleq,
rsub_wf,
int-to-real_wf,
rmul_wf,
sq_stable__req,
arcsine_wf,
i-member_wf,
rooint_wf,
halfpi_wf,
member_rooint_lemma,
radd-preserves-rless,
rsqrt_functionality_wrt_rless,
rleq_wf,
rless_wf,
set_wf,
real_wf,
radd_wf,
rminus_wf,
sq_stable__rleq,
rnexp_wf,
false_wf,
le_wf,
squash_wf,
true_wf,
rminus-int,
iff_weakening_equal,
rleq_weakening_rless,
rsqrt_wf,
req_wf,
rless_transitivity2,
rsqrt_nonneg,
rless-int,
arcsine-unique,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd-ac,
req_weakening,
radd_functionality,
radd-rminus-both,
radd-zero-both,
iff_transitivity,
iff_weakening_uiff,
req_inversion,
rnexp2,
square-rleq-1-iff,
rabs-rleq-iff,
rless_functionality,
req_transitivity,
radd-rminus-assoc,
radd-assoc,
rmul-is-positive,
rsqrt1,
rless_functionality_wrt_implies,
rleq_weakening_equal,
arcsine-bounds,
rsub_functionality_wrt_rleq,
halfpi-positive,
rmul-zero-both,
rmul_functionality,
radd-int,
rminus-as-rmul,
rmul-identity1,
rmul-distrib2,
arcsine-root-bounds,
sq_stable__rless,
rsqrt0,
arcsine_functionality_wrt_rless,
arcsine0,
trivial-rsub-rless,
equal_wf,
rsin-arcsine,
rsin_wf,
rsin-rcos-pythag,
rcos_wf,
radd-preserves-req,
req_functionality,
rnexp_functionality,
rsqrt-rnexp-2,
square-req-iff,
rcos-nonneg,
rccint_wf,
member_rccint_lemma,
rless_transitivity1,
rsin-shift-half-pi,
rsin-shift-pi,
pi_wf,
int-rmul_wf,
rmul_comm,
int-rmul-req,
rsin_functionality,
rsin-rminus,
rminus-radd,
rminus-rminus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
natural_numberEquality,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
dependent_set_memberEquality,
minusEquality,
hypothesisEquality,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
independent_pairFormation,
lambdaFormation,
productEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
inlFormation,
setEquality,
addLevel,
levelHypothesis,
addEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (r(-1), r1)\} ]. arcsine(x) = (\mpi{}/2 - arcsine(rsqrt(r1 - x * x))) supposing r0 < x
Date html generated:
2017_10_04-PM-10_48_50
Last ObjectModification:
2017_07_28-AM-08_51_36
Theory : reals_2
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