Nuprl Lemma : rlog-difference-bound
∀x,y:ℝ. ((r0 < x) ⇒ (x < y) ⇒ ((rlog(y) - rlog(x)) ≤ (y - x/x)))
Proof
Definitions occuring in Statement :
rlog: rlog(x),
rdiv: (x/y),
rleq: x ≤ y,
rless: 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,
uall: ∀[x:A]. B[x],
iproper: iproper(I),
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
and: P ∧ Q,
rfun: I ⟶ℝ,
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
sq_stable: SqStable(P),
subinterval: I ⊆ J ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
rge: x ≥ y,
ml-term-to-poly: ml-term-to-poly(t),
nil: [],
it: ⋅,
has-value: (a)↓,
req_int_terms: t1 ≡ t2
Lemmas referenced :
mean-value-for-bounded-derivative,
rccint_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
i-finite_wf,
rdiv_wf,
int-to-real_wf,
rless_wf,
real_wf,
i-member_wf,
rless_transitivity1,
member_rccint_lemma,
rlog_wf,
rleq_wf,
set_wf,
req_wf,
req_weakening,
rdiv_functionality,
req_functionality,
sq_stable__req,
derivative-rlog,
member_roiint_lemma,
sq_stable__rless,
roiint_wf,
derivative_functionality_wrt_subinterval,
rabs_wf,
sq_stable__rleq,
rmul-one-both,
rmul-rdiv-cancel,
rmul-ac,
rmul_comm,
rmul_functionality,
rmul-assoc,
req_inversion,
rmul-zero-both,
rmul-rdiv-cancel2,
uiff_transitivity,
rabs-of-nonneg,
rleq_functionality,
false_wf,
rleq-int,
rmul_wf,
rmul_preserves_rleq,
rleq_weakening_rless,
rleq_weakening_equal,
rsub_wf,
rless_transitivity2,
rleq_functionality_wrt_implies,
rsub_functionality_wrt_rleq,
rlog_functionality_wrt_rless,
rleq_weakening,
real_polynomial_null,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
evalall-sqequal,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
equal_wf,
req_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
independent_isectElimination,
inrFormation,
setEquality,
productElimination,
dependent_set_memberEquality,
rename,
setElimination,
lambdaEquality,
productEquality,
imageElimination,
baseClosed,
imageMemberEquality,
because_Cache,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
computeAll,
sqleReflexivity,
mlComputation,
int_eqEquality,
intEquality
Latex:
\mforall{}x,y:\mBbbR{}. ((r0 < x) {}\mRightarrow{} (x < y) {}\mRightarrow{} ((rlog(y) - rlog(x)) \mleq{} (y - x/x)))
Date html generated:
2017_10_04-PM-10_26_02
Last ObjectModification:
2017_07_28-AM-08_49_50
Theory : reals_2
Home
Index