Nuprl Lemma : concave-on_wf
∀[I:Interval]. ∀[f:I ⟶ℝ]. (concave-on(I;x.f[x]) ∈ ℙ)
Proof
Definitions occuring in Statement :
concave-on: concave-on(I;x.f[x]),
rfun: I ⟶ℝ,
interval: Interval,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
concave-on: concave-on(I;x.f[x]),
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
rfun: I ⟶ℝ
Lemmas referenced :
member_rccint_lemma,
all_wf,
real_wf,
i-member_wf,
rleq_wf,
int-to-real_wf,
radd_wf,
rmul_wf,
rsub_wf,
i-member-convex,
rfun_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
lambdaEquality,
because_Cache,
functionEquality,
hypothesisEquality,
productEquality,
natural_numberEquality,
applyEquality,
dependent_set_memberEquality,
productElimination,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[I:Interval]. \mforall{}[f:I {}\mrightarrow{}\mBbbR{}]. (concave-on(I;x.f[x]) \mmember{} \mBbbP{})
Date html generated:
2018_05_22-PM-02_20_05
Last ObjectModification:
2017_10_20-PM-01_33_08
Theory : reals
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