Nuprl Lemma : derivative-log-contraction-bound
∀a:{a:ℝ| r0 < a} . ∀[x:ℝ]. (|(a - e^x/a + e^x)^2| ≤ r1)
Proof
Definitions occuring in Statement :
rexp: e^x,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rnexp: x^k1,
rsub: x - y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
less_than': less_than'(a;b),
prop: ℙ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
rev_implies: P ⇐ Q,
rge: x ≥ y,
rgt: x > y,
iff: P ⇐⇒ Q,
exp: i^n,
primrec: primrec(n;b;c),
subtract: n - m,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
rsub: x - y,
less_than: a < b,
true: True
Lemmas referenced :
real_wf,
less_than'_wf,
rsub_wf,
int-to-real_wf,
rabs_wf,
rnexp_wf,
false_wf,
le_wf,
rdiv_wf,
rexp_wf,
radd_wf,
rless_wf,
nat_plus_wf,
set_wf,
sq_stable__rless,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
radd_functionality_wrt_rless1,
rexp-positive,
rless_functionality,
req_weakening,
radd-zero-both,
radd_comm,
rnexp-rleq,
zero-rleq-rabs,
exp_wf2,
rleq-int,
rleq_functionality,
rabs-rnexp,
rleq_functionality_wrt_implies,
rnexp-int,
rabs-of-nonneg,
req_inversion,
rless_transitivity1,
rleq_weakening,
rabs-rdiv,
rabs-difference-bound-rleq,
rleq_wf,
rmul_wf,
rminus_wf,
radd-preserves-rleq,
uiff_transitivity,
radd_functionality,
rminus-radd,
req_transitivity,
radd-assoc,
rmul-identity1,
rmul-distrib2,
rminus-as-rmul,
rmul_functionality,
radd-int,
rmul-zero-both,
rmul_preserves_rleq,
rless-int,
rmul-nonneg-case1,
rmul-int,
rmul-assoc,
rmul_comm,
radd-ac,
radd-rminus-both,
rdiv_functionality,
rmul-rdiv-cancel2,
rmul-distrib,
rmul-one-both,
rabs_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
isect_memberFormation,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
voidElimination,
isectElimination,
applyEquality,
natural_numberEquality,
dependent_set_memberEquality,
independent_pairFormation,
setElimination,
rename,
because_Cache,
independent_isectElimination,
inrFormation,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
addLevel,
addEquality,
multiplyEquality
Latex:
\mforall{}a:\{a:\mBbbR{}| r0 < a\} . \mforall{}[x:\mBbbR{}]. (|(a - e\^{}x/a + e\^{}x)\^{}2| \mleq{} r1)
Date html generated:
2016_10_26-PM-00_30_24
Last ObjectModification:
2016_09_19-AM-10_01_16
Theory : reals_2
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