Nuprl Lemma : arcsine-contraction-difference
∀[a:{a:ℝ| (r(-1) < a) ∧ (a < r1)} ]. ∀[x,y:ℝ].
(|arcsine-contraction(a;x) - arcsine-contraction(a;y)| ≤ (r(3) * |x - y|))
Proof
Definitions occuring in Statement :
arcsine-contraction: arcsine-contraction(a;x),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
prop: ℙ,
sq_stable: SqStable(P),
implies: P ⇒ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
subtype_rel: A ⊆r B,
arcsine-contraction: arcsine-contraction(a;x),
squash: ↓T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
real: ℝ,
nat: ℕ,
less_than': less_than'(a;b),
cand: A c∧ B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rsub: x - y,
top: Top,
rge: x ≥ y,
or: P ∨ Q
Lemmas referenced :
sq_stable__rleq,
rabs_wf,
rsub_wf,
arcsine-contraction_wf,
rless_wf,
int-to-real_wf,
rmul_wf,
radd-preserves-rleq,
mean-value-for-bounded-derivative,
riiint_wf,
iproper-riiint,
real_wf,
i-member_wf,
radd_wf,
rminus_wf,
rsin_wf,
rsqrt_wf,
rleq_wf,
rcos_wf,
req_functionality,
radd_functionality,
rsub_functionality,
rmul_functionality,
req_weakening,
rminus_functionality,
rsin_functionality,
rcos_functionality,
req_wf,
set_wf,
less_than'_wf,
nat_plus_wf,
rnexp_wf,
false_wf,
le_wf,
squash_wf,
true_wf,
rminus-int,
iff_weakening_equal,
rleq_weakening_rless,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd-ac,
radd-rminus-both,
radd-zero-both,
iff_transitivity,
iff_weakening_uiff,
req_inversion,
rnexp2,
square-rleq-1-iff,
rabs-rleq-iff,
derivative-add,
derivative-id,
derivative-sub,
derivative-const-mul,
deriviative-rcos,
deriviative-rsin,
member_riiint_lemma,
rleq_functionality_wrt_implies,
rleq_transitivity,
r-triangle-inequality,
radd_functionality_wrt_rleq,
rleq_weakening_equal,
r-triangle-inequality-rsub,
rleq_weakening,
rabs-rmul,
req_transitivity,
rabs-rminus,
zero-rleq-rabs,
rmul_comm,
rmul_functionality_wrt_rleq2,
rabs-rsin-rleq,
rmul-one-both,
rsqrt_nonneg,
square-rleq-implies,
rleq-int,
square-nonneg,
trivial-rleq-radd,
rabs-rcos-rleq,
rabs-of-nonneg,
rsqrt-rnexp-2,
rmul-int,
radd-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
productEquality,
minusEquality,
natural_numberEquality,
because_Cache,
independent_functionElimination,
productElimination,
independent_isectElimination,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
independent_pairFormation,
setEquality,
applyEquality,
lambdaFormation,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
universeEquality,
voidEquality,
inlFormation,
multiplyEquality,
addEquality
Latex:
\mforall{}[a:\{a:\mBbbR{}| (r(-1) < a) \mwedge{} (a < r1)\} ]. \mforall{}[x,y:\mBbbR{}].
(|arcsine-contraction(a;x) - arcsine-contraction(a;y)| \mleq{} (r(3) * |x - y|))
Date html generated:
2016_10_26-PM-00_44_01
Last ObjectModification:
2016_10_10-AM-08_58_45
Theory : reals_2
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