Nuprl Lemma : rabs-rexp-difference-bound
∀x,y:ℝ. (|e^x - e^y| ≤ (e^rmax(x;y) * |x - y|))
Proof
Definitions occuring in Statement :
rexp: e^x,
rleq: x ≤ y,
rabs: |x|,
rmax: rmax(x;y),
rsub: x - y,
rmul: a * b,
real: ℝ,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
rgt: x > y
Lemmas referenced :
rleq-iff-all-rless,
rabs_wf,
rsub_wf,
rexp_wf,
rmul_wf,
rmax_wf,
rless-cases,
radd_wf,
trivial-rless-radd,
sq_stable__rless,
int-to-real_wf,
rexp-rless,
rexp-difference-bound,
rabs-difference-bound-iff,
rless-implies-rless,
set_wf,
real_wf,
rless_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rsub_functionality_wrt_rleq,
rleq_weakening_rless,
rexp_functionality_wrt_rless,
rleq_weakening,
itermConstant_wf,
rleq_functionality,
rabs-of-nonneg,
radd_functionality,
req_weakening,
rmul_functionality,
rabs-difference-symmetry,
radd_functionality_wrt_rless1,
rleq-rmax,
rexp-rleq,
trivial-rleq-radd,
radd_functionality_wrt_rleq,
rmul_functionality_wrt_rleq2,
rleq_wf,
rexp-positive,
zero-rleq-rabs,
rmul-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
dependent_functionElimination,
setElimination,
rename,
independent_functionElimination,
natural_numberEquality,
because_Cache,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
independent_pairFormation,
lambdaEquality,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
inlFormation,
productEquality
Latex:
\mforall{}x,y:\mBbbR{}. (|e\^{}x - e\^{}y| \mleq{} (e\^{}rmax(x;y) * |x - y|))
Date html generated:
2017_10_04-PM-10_35_55
Last ObjectModification:
2017_07_28-AM-08_50_44
Theory : reals_2
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