Nuprl Lemma : third-derivative-log-contraction-nonneg
∀a:{a:ℝ| r0 < a} . ∀x:ℝ.
((|x - rlog(a)| ≤ r1) ⇒ (r0 ≤ (((r(16) * a^2) * e^x^2) + ((r(-4) * a^3) * e^x) + ((r(-4) * a) * e^x^3)/a + e^x^4)))
Proof
Definitions occuring in Statement :
rlog: rlog(x),
rexp: e^x,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
nat: ℕ,
le: A ≤ B,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
rge: x ≥ y,
rgt: x > y,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
iff: P ⇐⇒ Q,
rdiv: (x/y),
cand: A c∧ B,
rsub: x - y,
rleq: x ≤ y,
rnonneg: rnonneg(x),
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
int-to-real: r(n),
subtract: n - m
Lemmas referenced :
rnexp-positive,
radd_wf,
rexp_wf,
nat_plus_subtype_nat,
nat_plus_wf,
rmul_preserves_rleq,
rdiv_wf,
rmul_wf,
rnexp_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rlog_wf,
rless_wf,
int-to-real_wf,
real_wf,
set_wf,
sq_stable__rless,
less_than_wf,
false_wf,
le_wf,
rinv_wf2,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
radd_functionality_wrt_rless1,
rexp-positive,
rless_functionality,
req_weakening,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rleq_functionality,
itermMultiply_wf,
real_term_value_mul_lemma,
req_transitivity,
radd_functionality,
rmul_functionality,
rmul-rinv3,
rabs-difference-bound-rleq,
rexp_functionality_wrt_rleq,
radd-preserves-rleq,
rexp-rlog,
rexp-radd,
rminus_wf,
squash_wf,
true_wf,
radd_comm_eq,
iff_weakening_equal,
rleq-implies-rleq,
equal_wf,
rmul_preserves_rleq2,
less_than'_wf,
rexp-rminus,
rless-int-fractions2,
rmul_preserves_rless,
rless-int,
rless-cases,
rinv-mul-as-rdiv,
rmul-rinv,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
uiff_transitivity,
rmul-rdiv-cancel2,
rmul-rdiv-cancel,
rmul-assoc,
req_inversion,
req_functionality,
rmul-ac,
rmul-distrib,
rmul_comm,
req_wf,
rmul-nonneg,
rmul-int,
rleq-int,
radd_comm,
rnexp2,
rmul-zero-both,
radd-int,
rmul-distrib2,
radd-zero-both,
radd-ac,
radd-assoc,
rnexp_step
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
because_Cache,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
setElimination,
rename,
hypothesis,
hypothesisEquality,
independent_functionElimination,
applyEquality,
sqequalRule,
independent_isectElimination,
inrFormation,
productElimination,
dependent_set_memberEquality,
natural_numberEquality,
lambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
minusEquality,
equalityTransitivity,
equalitySymmetry,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
promote_hyp,
universeEquality,
productEquality,
isect_memberFormation,
independent_pairEquality,
axiomEquality,
multiplyEquality,
unionElimination,
dependent_set_memberFormation,
addEquality,
inlFormation
Latex:
\mforall{}a:\{a:\mBbbR{}| r0 < a\} . \mforall{}x:\mBbbR{}.
((|x - rlog(a)| \mleq{} r1)
{}\mRightarrow{} (r0 \mleq{} (((r(16) * a\^{}2) * e\^{}x\^{}2) + ((r(-4) * a\^{}3) * e\^{}x) + ((r(-4) * a) * e\^{}x\^{}3)/a + e\^{}x\^{}4)))
Date html generated:
2017_10_04-PM-10_30_02
Last ObjectModification:
2017_07_28-AM-08_50_25
Theory : reals_2
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