trans_iterate_Ico — Mathlib

English

If a ≤ c and b ∈ [a,c], the interval integrability on [a,c] follows from integrability on [a,b] and [b,c].

Русский

Если a ≤ c и b ∈ [a,c], интервальная интегрируемость на [a,c] следует из интегрируемости на [a,b] и [b,c].

LaTeX

$$$$\text{IntervalIntegrable}(f, μ, a, c) \iff \text{IntervalIntegrable}(f, μ, a, b) \wedge \text{IntervalIntegrable}(f, μ, b, c).$$$$

Lean4

theorem trans_iterate_Ico {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n)
    (hint : ∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a <| k + 1)) : IntervalIntegrable f μ (a m) (a n) :=
  by
  revert hint
  refine Nat.le_induction ?_ ?_ n hmn
  · simp
  · intro p hp IH h
    exact (IH fun k hk => h k (Ico_subset_Ico_right p.le_succ hk)).trans (h p (by simp [hp]))

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