deriv_nonpos — Mathlib

English

If g is monotone, then the derivative exists and is nonnegative: 0 ≤ deriv g x.

Русский

Если g монотонна, то её производная существует и неотрицательна: 0 ≤ deriv g x.

LaTeX

$$(Monotone g) \\rightarrow 0 \\le deriv g x$$

Lean4

/-- The derivative of an antitone function is nonpositive. -/
theorem deriv_nonpos (hg : Antitone g) : deriv g x ≤ 0 :=
  by
  rw [← derivWithin_univ]
  exact (hg.antitoneOn univ).derivWithin_nonpos

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