Nuprl Lemma : arcsin-shift
∀[x:{x:ℝ| x ∈ [r(-1), r1]} ]. arcsin(x) = (π/2 - arcsin(rsqrt(r1 - x * x))) supposing r0 ≤ x
Proof
Definitions occuring in Statement :
arcsin: arcsin(a),
halfpi: π/2,
rsqrt: rsqrt(x),
rccint: [l, u],
i-member: r ∈ I,
rleq: 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,
sq_stable: SqStable(P),
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
squash: ↓T,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
cand: A c∧ B,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
req_int_terms: t1 ≡ t2,
or: P ∨ Q,
stable: Stable{P},
less_than: a < b
Lemmas referenced :
radd-preserves-rleq,
int-to-real_wf,
rsub_wf,
rmul_wf,
rsqrt_functionality_wrt_rleq,
rleq_wf,
real_wf,
i-member_wf,
rccint_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
sq_stable__rleq,
member_rccint_lemma,
istype-void,
rnexp_wf,
istype-le,
rminus_wf,
squash_wf,
true_wf,
rminus-int,
subtype_rel_self,
iff_weakening_equal,
square-nonneg,
rsqrt_wf,
rleq_transitivity,
rsqrt_nonneg,
rleq-int,
istype-false,
arcsin_wf,
rleq_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
iff_transitivity,
iff_weakening_uiff,
req_inversion,
rnexp2,
req_weakening,
square-rleq-1-iff,
rabs-rleq-iff,
rsqrt1,
sq_stable__req,
halfpi_wf,
stable_req,
false_wf,
rless_wf,
not_wf,
req_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
member_rooint_lemma,
rless_transitivity2,
rless-int,
rless_transitivity1,
rmul_preserves_rless,
rsqrt-rless-iff,
trivial-rsub-rless,
rless_functionality,
arcsine_wf,
arcsine-shift,
req_functionality,
arcsin-is-arcsine,
rsub_functionality,
not-rless,
rleq_antisymmetry,
rleq-implies-rleq,
rleq_weakening_equal,
arcsin1,
rsqrt_functionality,
rmul_functionality,
arcsin_functionality,
uiff_transitivity,
arcsin0,
rsqrt-is-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
setElimination,
rename,
because_Cache,
productElimination,
independent_isectElimination,
dependent_set_memberEquality_alt,
hypothesisEquality,
universeIsType,
setIsType,
minusEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
lambdaFormation_alt,
productEquality,
productIsType,
applyEquality,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
instantiate,
universeEquality,
approximateComputation,
int_eqEquality,
unionEquality,
functionEquality,
functionIsType,
unionIsType,
unionElimination,
applyLambdaEquality,
closedConclusion
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} [r(-1), r1]\} ]. arcsin(x) = (\mpi{}/2 - arcsin(rsqrt(r1 - x * x))) supposing r0 \mleq{} x
Date html generated:
2019_10_31-AM-06_15_48
Last ObjectModification:
2019_05_24-PM-05_05_01
Theory : reals_2
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