Nuprl Lemma : arcsine-rsin
∀[x:{x:ℝ| x ∈ (-(π/2), π/2)} ]. (arcsine(rsin(x)) = x)
Proof
Definitions occuring in Statement :
arcsine: arcsine(x),
halfpi: π/2,
rsin: rsin(x),
rooint: (l, u),
i-member: r ∈ I,
req: x = y,
rminus: -(x),
real: ℝ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
rdiv: (x/y),
rneq: x ≠ y,
le: A ≤ B,
nat: ℕ,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
arcsine_deriv: arcsine_deriv(x),
r-ap: f(x),
rfun-eq: rfun-eq(I;f;g),
increasing-on-interval: f[x] increasing for x ∈ I,
real-fun: real-fun(f;a;b),
ifun: ifun(f;I),
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
i-finite: i-finite(I),
or: P ∨ Q,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
pi2: snd(t),
pi1: fst(t),
outl: outl(x),
rooint: (l, u),
endpoints: endpoints(I),
left-endpoint: left-endpoint(I),
right-endpoint: right-endpoint(I),
iproper: iproper(I),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
uimplies: b supposing a,
guard: {T},
cand: A c∧ B,
and: P ∧ Q,
so_apply: x[s],
prop: ℙ,
rfun: I ⟶ℝ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
top: Top,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
rsin0,
arcsine_functionality,
arcsine0,
halfpi-positive,
radd-zero,
radd-rminus,
rless_functionality,
radd-preserves-rless,
rcos-positive,
rsqrt-of-square,
rsqrt_functionality,
square-nonneg,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
rmul-rinv,
rmul_functionality,
req_transitivity,
rmul-identity1,
rmul-one,
rinv_wf2,
rdiv_wf,
rmul_preserves_req,
equal_wf,
rsqrt_wf,
rsqrt-positive,
arcsine-root-bounds,
rnexp2,
req_inversion,
radd_functionality,
le_wf,
false_wf,
rnexp_wf,
rsin-rcos-pythag,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
itermConstant_wf,
itermVar_wf,
itermMultiply_wf,
itermAdd_wf,
itermSubtract_wf,
radd_comm,
radd_wf,
rsub_wf,
radd-preserves-req,
derivative_functionality,
rmul_wf,
rleq_weakening_rless,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
member_rccint_lemma,
derivative-rsin,
subinterval-riiint,
riiint_wf,
derivative_functionality_wrt_subinterval,
rcos-nonneg,
rccint_wf,
derivative-implies-increasing-simple,
all_wf,
rleq_wf,
monotone-maps-compact,
i-finite_wf,
rless-int,
derivative-arcsine,
arcsine_deriv_functionality,
req_wf,
req_weakening,
rcos_functionality,
req_functionality,
arcsine_deriv_wf,
rcos_wf,
chain-rule,
set_wf,
req_witness,
derivative-id,
rless_wf,
rsin_wf,
arcsine_wf,
i-member_wf,
real_wf,
int-to-real_wf,
halfpi-interval-proper,
halfpi_wf,
rminus_wf,
rooint_wf,
antiderivatives-equal,
member_rooint_lemma,
rsin-strict-bound
Rules used in proof :
dependent_pairFormation,
universeEquality,
instantiate,
imageElimination,
applyEquality,
inrFormation,
equalitySymmetry,
equalityTransitivity,
intEquality,
int_eqEquality,
approximateComputation,
functionEquality,
inlFormation,
baseClosed,
imageMemberEquality,
independent_isectElimination,
lambdaFormation,
productElimination,
isect_memberFormation,
because_Cache,
minusEquality,
productEquality,
rename,
setElimination,
dependent_set_memberEquality,
independent_pairFormation,
hypothesisEquality,
setEquality,
natural_numberEquality,
lambdaEquality,
independent_functionElimination,
isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalReflexivity,
sqequalRule,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (-(\mpi{}/2), \mpi{}/2)\} ]. (arcsine(rsin(x)) = x)
Date html generated:
2018_05_22-PM-03_07_52
Last ObjectModification:
2018_05_20-PM-11_35_49
Theory : reals_2
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