theorem deriv_ne_zero_imp_pos_or_neg {f : ℝ → ℝ} {S : Set ℝ} (hS : Convex ℝ S) (hderiv : ∀ x ∈ S, DifferentiableAt ℝ f x) (hne : ∀ x ∈ S, deriv f x ≠ 0) : (∀ x ∈ S, 0 < deriv f x) ∨ ∀ x ∈ S, deriv f x < 0

theorem deriv_ne_zero_imp_strict_mono_or_anti {f f' : ℝ → ℝ} {b : ℝ} (hcont : ContinuousOn f (Set.Ici b)) (hderiv : ∀ x ∈ Set.Ioi b, HasDerivAt f (f' x) x) (hne : ∀ x ∈ Set.Ioi b, f' x ≠ 0) : StrictMonoOn f (Set.Ici b) ∨ StrictAntiOn f (Set.Ici b)

theorem claim (f f' : ℝ → ℝ) (hcont : ContinuousOn f (Set.Ici 0)) (hderiv : ∀ x ∈ Set.Ioi 0, HasDerivAt f (f' x) x) (a b : ℝ) (hab : 0 < a ∧ a < b ∧ f a = 0 ∧ f b = 0) (c : ℝ) (hlim : Filter.Tendsto f Filter.atTop (nhds 0)) (hc : (a < c ∧ c < b) ∧ f' c = 0) (h_deriv_ne_zero : ∀ (x : ℝ), 0 < x → ¬x = c → ¬f' x = 0) : False