[isabelle] Unnecessary assumption in HOL-Analysis.Convex.convex_on_alt



Hello,

I have stumbled upon this lemma in HOL-Analysis.Convex:

lemma convex_on_alt:
fixes C :: "'a::real_vector set"
assumes "convex C"
shows "convex_on C f ⟷
(∀x ∈ C. ∀ y ∈ C. ∀ μ :: real. μ ≥ 0 ∧ μ ≤ 1 ⟶
f (μ *⇩R x + (1 - μ) *⇩R y) ≤ μ * f x + (1 - μ) * f y)"

As this is a simple restatement of convex_on_def, I was somewhat puzzled
by the assumption of "convex C". As it turns out - it's not necessary.
I removed the assumes line and the proof succeeded without any
modification. I suppose it would be useful to reflect this in the
distribution?

Kind regards,
Jakub Kądziołka




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