Nuprl Lemma : rabs-rlog-difference-bound
∀x,y:ℝ. ((r0 < x) ⇒ (r0 < y) ⇒ (|rlog(y) - rlog(x)| ≤ (|y - x|/rmin(x;y))))
Proof
Definitions occuring in Statement :
rlog: rlog(x),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rmin: rmin(x;y),
rsub: x - y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
iproper: iproper(I),
i-finite: i-finite(I),
rciint: [l, ∞),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rfun: I ⟶ℝ,
top: Top,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
subinterval: I ⊆ J ,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
rge: x ≥ y
Lemmas referenced :
rmin_strict_ub,
mean-value-for-bounded-derivative,
rciint_wf,
rmin_wf,
true_wf,
false_wf,
rdiv_wf,
int-to-real_wf,
rless_wf,
real_wf,
rlog_wf,
member_rciint_lemma,
rless_transitivity1,
i-member_wf,
rleq_wf,
sq_stable__req,
req_functionality,
rdiv_functionality,
req_weakening,
req_wf,
set_wf,
derivative-rlog,
member_roiint_lemma,
derivative_functionality_wrt_subinterval,
roiint_wf,
sq_stable__rless,
sq_stable__rleq,
rabs_wf,
rmul_preserves_rleq,
rmul_wf,
rleq-int,
rleq_functionality,
rabs-of-nonneg,
uiff_transitivity,
rmul-rdiv-cancel2,
rmul-zero-both,
req_inversion,
rmul-assoc,
rmul_functionality,
rmul_comm,
rmul-ac,
rmul-rdiv-cancel,
rmul-one-both,
rsub_wf,
rmin-rleq,
equal_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
productElimination,
independent_functionElimination,
hypothesis,
independent_pairFormation,
isectElimination,
sqequalRule,
voidElimination,
productEquality,
natural_numberEquality,
independent_isectElimination,
inrFormation,
lambdaEquality,
setElimination,
rename,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
setEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}x,y:\mBbbR{}. ((r0 < x) {}\mRightarrow{} (r0 < y) {}\mRightarrow{} (|rlog(y) - rlog(x)| \mleq{} (|y - x|/rmin(x;y))))
Date html generated:
2017_10_04-PM-10_26_09
Last ObjectModification:
2017_07_28-AM-08_49_54
Theory : reals_2
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