translationNumber_lt_of_forall_lt_add

English

If f is continuous, then f^n is continuous for every natural n.

Русский

Если f непрерывно, то f^n непрерывно для каждого натурального n.

LaTeX

$$$$ \\forall f:\\ CircleDeg1Lift, \\ hf : Continuous f, \\ \\forall n \\in \\mathbb{N}, \\ Continuous \\big(f^n\\big). $$$$

Lean4

theorem translationNumber_lt_of_forall_lt_add (hf : Continuous f) {z : ℝ} (hz : ∀ x, f x < x + z) : τ f < z :=
  by
  obtain ⟨x, -, hx⟩ : ∃ x ∈ Icc (0 : ℝ) 1, ∀ y ∈ Icc (0 : ℝ) 1, f y - y ≤ f x - x :=
    isCompact_Icc.exists_isMaxOn (nonempty_Icc.2 zero_le_one) (hf.sub continuous_id).continuousOn
  refine lt_of_le_of_lt ?_ (sub_lt_iff_lt_add'.2 <| hz x)
  apply translationNumber_le_of_le_add
  simp only [← sub_le_iff_le_add']
  exact f.forall_map_sub_of_Icc (fun a => a ≤ f x - x) hx

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