Nuprl Lemma : implies-convex-on
∀[I:Interval]. ∀[f:I ⟶ℝ].
((∀x,y:ℝ. ((x ∈ I) ⇒ (y ∈ I) ⇒ (x = y) ⇒ (f[x] = f[y])))
⇒ (∀x,y:ℝ.
((x < y)
⇒ (∀t:ℝ
((x ∈ I)
⇒ (y ∈ I)
⇒ (t ∈ [r0, r1])
⇒ (f[(t * x) + ((r1 - t) * y)] ≤ ((t * f[x]) + ((r1 - t) * f[y])))))))
⇒ convex-on(I;x.f[x]))
Proof
Definitions occuring in Statement :
convex-on: convex-on(I;x.f[x]),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
convex-on: convex-on(I;x.f[x]),
all: ∀x:A. B[x],
i-member: r ∈ I,
rccint: [l, u],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
stable: Stable{P},
uimplies: b supposing a,
or: P ∨ Q,
top: Top,
cand: A c∧ B,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
guard: {T}
Lemmas referenced :
i-member-convex,
i-member_wf,
rccint_wf,
int-to-real_wf,
real_wf,
all_wf,
rless_wf,
rleq_wf,
radd_wf,
rmul_wf,
rsub_wf,
req_wf,
less_than'_wf,
nat_plus_wf,
rfun_wf,
interval_wf,
stable__rleq,
false_wf,
or_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
member_rccint_lemma,
rleq-implies-rleq,
trivial-rsub-rleq,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
radd-preserves-rleq,
rleq_functionality,
radd-zero,
itermAdd_wf,
itermMultiply_wf,
real_term_value_add_lemma,
real_term_value_mul_lemma,
rleq_transitivity,
rleq_weakening,
rleq_antisymmetry,
not-rless,
req_functionality,
radd_functionality,
req_weakening,
rmul_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
productElimination,
isectElimination,
natural_numberEquality,
lambdaEquality,
because_Cache,
functionEquality,
applyEquality,
dependent_set_memberEquality,
independent_pairEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
independent_isectElimination,
unionElimination,
voidEquality,
independent_pairFormation,
approximateComputation,
int_eqEquality,
intEquality
Latex:
\mforall{}[I:Interval]. \mforall{}[f:I {}\mrightarrow{}\mBbbR{}].
((\mforall{}x,y:\mBbbR{}. ((x \mmember{} I) {}\mRightarrow{} (y \mmember{} I) {}\mRightarrow{} (x = y) {}\mRightarrow{} (f[x] = f[y])))
{}\mRightarrow{} (\mforall{}x,y:\mBbbR{}.
((x < y)
{}\mRightarrow{} (\mforall{}t:\mBbbR{}
((x \mmember{} I)
{}\mRightarrow{} (y \mmember{} I)
{}\mRightarrow{} (t \mmember{} [r0, r1])
{}\mRightarrow{} (f[(t * x) + ((r1 - t) * y)] \mleq{} ((t * f[x]) + ((r1 - t) * f[y])))))))
{}\mRightarrow{} convex-on(I;x.f[x]))
Date html generated:
2018_05_22-PM-02_19_36
Last ObjectModification:
2017_10_21-PM-08_45_01
Theory : reals
Home
Index