Nuprl Lemma : second-deriv-nonneg-convex
∀I:Interval
(iproper(I)
⇒ (∀f,g,h:I ⟶ℝ.
((∀x,y:{a:ℝ| a ∈ I} . ((x = y) ⇒ (h[x] = h[y])))
⇒ d(f[x])/dx = λx.g[x] on I
⇒ d(g[x])/dx = λx.h[x] on I
⇒ (∀x:{a:ℝ| a ∈ I} . (r0 ≤ h[x]))
⇒ convex-on(I;x.f[x]))))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
convex-on: convex-on(I;x.f[x]),
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
rleq: x ≤ y,
req: x = y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
label: ...$L... t,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
subinterval: I ⊆ J ,
top: Top,
cand: A c∧ B,
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
iff: P ⇐⇒ Q,
convex-on: convex-on(I;x.f[x]),
i-member: r ∈ I,
rccint: [l, u],
true: True,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q
Lemmas referenced :
Taylor-theorem-for-2,
all_wf,
real_wf,
i-member_wf,
rleq_wf,
int-to-real_wf,
derivative_wf,
req_wf,
rfun_wf,
iproper_wf,
interval_wf,
rleq-iff-all-rless,
radd_wf,
rmul_wf,
rsub_wf,
set_wf,
rless_wf,
sq_stable__rless,
rmin-rmax-subinterval,
sq_stable__i-member,
stable__rleq,
member_rccint_lemma,
false_wf,
or_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rmax-req,
rleq_weakening_rless,
radd-preserves-rleq,
rleq_functionality,
radd-zero,
rleq_transitivity,
rmax_wf,
rleq_weakening,
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,
rmul-nonneg,
rmul-nonneg-case1,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rsub_functionality_wrt_rleq,
itermConstant_wf,
rmin-req,
req_inversion,
rmin_wf,
equal_wf,
rmul_preserves_rleq2,
rmul-zero-both,
rmul_comm,
rleq-implies-rleq,
itermMultiply_wf,
real_term_value_mul_lemma,
rleq_antisymmetry,
not-rless,
req_weakening,
rmul_functionality,
rsub_functionality,
rabs_wf,
rabs-difference-bound-rleq,
radd_comm,
i-member-convex,
rccint_wf,
radd_functionality,
squash_wf,
true_wf,
iff_weakening_equal,
rminus_wf,
itermMinus_wf,
real_term_value_minus_lemma,
radd_functionality_wrt_rleq
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
setEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
functionEquality,
productElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
unionElimination,
approximateComputation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
inlFormation,
productEquality,
inrFormation,
universeEquality,
promote_hyp
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}f,g,h:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{a:\mBbbR{}| a \mmember{} I\} . ((x = y) {}\mRightarrow{} (h[x] = h[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.g[x] on I
{}\mRightarrow{} d(g[x])/dx = \mlambda{}x.h[x] on I
{}\mRightarrow{} (\mforall{}x:\{a:\mBbbR{}| a \mmember{} I\} . (r0 \mleq{} h[x]))
{}\mRightarrow{} convex-on(I;x.f[x]))))
Date html generated:
2018_05_22-PM-02_50_16
Last ObjectModification:
2017_10_21-PM-09_45_19
Theory : reals
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