English
If s is subsingleton, HasConstantSpeedOnWith f s l holds for any l.
Русский
Если s — подселективное множество, то HasConstantSpeedOnWith f s l верно для любого l.
LaTeX
$$hasConstantSpeedOnWith_of_subsingleton f hs l$$
Lean4
/-- If both `f` and `f ∘ φ` have unit speed (on `t` and `s` respectively) and `φ`
monotonically maps `s` onto `t`, then `φ` is just a translation (on `s`).
-/
theorem unique_unit_speed {φ : ℝ → ℝ} (φm : MonotoneOn φ s) (hfφ : HasUnitSpeedOn (f ∘ φ) s)
(hf : HasUnitSpeedOn f (φ '' s)) ⦃x : ℝ⦄ (xs : x ∈ s) : EqOn φ (fun y => y - x + φ x) s :=
by
dsimp only [HasUnitSpeedOn] at hf hfφ
convert HasConstantSpeedOnWith.ratio one_ne_zero φm hfφ hf xs using 3
simp
