nevanlinna/Nevanlinna/periodic_integrability.lean

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import Mathlib.MeasureTheory.Integral.Periodic
theorem periodic_integrability₁
{E : Type u_1} [NormedAddCommGroup E] [NormedSpace E]
{f : → E}
{t T : }
{n : }
(h₁f : Function.Periodic f T)
(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) :
IntervalIntegrable f MeasureTheory.volume t (t + n * T) := by
induction' n with n hn
simp
apply IntervalIntegrable.trans (b := t + n * T)
exact hn
let A := IntervalIntegrable.comp_sub_right h₂f (n * T)
have : f = fun x ↦ f (x - n * T) := by simp [Function.Periodic.sub_nat_mul_eq h₁f n]
simp_rw [← this] at A
have : (t + T + ↑n * T) = (t + ↑(n + 1) * T) := by simp; ring
simp_rw [this] at A
exact A
theorem periodic_integrability₂
{E : Type u_1} [NormedAddCommGroup E] [NormedSpace E]
{f : → E}
{t T : }
{n : }
(h₁f : Function.Periodic f T)
(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) :
IntervalIntegrable f MeasureTheory.volume (t - n * T) t := by
induction' n with n hn
simp
--
apply IntervalIntegrable.trans (b := t - n * T)
--
have A := IntervalIntegrable.comp_add_right h₂f (((n + 1): ) * T)
have : f = fun x ↦ f (x + ((n + 1) : ) * T) := by
funext x
have : x = x + ↑(n + 1) * T - ↑(n + 1) * T := by ring
nth_rw 1 [this]
rw [Function.Periodic.sub_nat_mul_eq h₁f]
simp_rw [← this] at A
have : (t + T - ↑(n + 1) * T) = (t - ↑n * T) := by simp; ring
simp_rw [this] at A
exact A
--
exact hn
theorem periodic_integrability₃
{E : Type u_1} [NormedAddCommGroup E] [NormedSpace E]
{f : → E}
{t T : }
{n₁ n₂ : }
(h₁f : Function.Periodic f T)
(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) :
IntervalIntegrable f MeasureTheory.volume (t - n₁ * T) (t + n₂ * T) := by
apply IntervalIntegrable.trans (b := t)
exact periodic_integrability₂ h₁f h₂f
exact periodic_integrability₁ h₁f h₂f
theorem periodic_integrability4
{E : Type u_1} [NormedAddCommGroup E] [NormedSpace E]
{f : → E}
{t T : }
{a₁ a₂ : }
(h₁f : Function.Periodic f T)
(hT : 0 < T)
(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) :
IntervalIntegrable f MeasureTheory.volume a₁ a₂ := by
obtain ⟨n₁, hn₁⟩ : ∃ n₁ : , t - n₁ * T ≤ min a₁ a₂ := by
obtain ⟨n₁, hn₁⟩ := exists_nat_ge ((t -min a₁ a₂) / T)
use n₁
rw [sub_le_iff_le_add]
rw [div_le_iff₀ hT] at hn₁
rw [sub_le_iff_le_add] at hn₁
rw [add_comm]
exact hn₁
obtain ⟨n₂, hn₂⟩ : ∃ n₂ : , max a₁ a₂ ≤ t + n₂ * T := by
obtain ⟨n₂, hn₂⟩ := exists_nat_ge ((max a₁ a₂ - t) / T)
use n₂
rw [← sub_le_iff_le_add]
rw [div_le_iff₀ hT] at hn₂
linarith
have : Set.uIcc a₁ a₂ ⊆ Set.uIcc (t - n₁ * T) (t + n₂ * T) := by
apply Set.uIcc_subset_uIcc_iff_le.mpr
constructor
· calc min (t - ↑n₁ * T) (t + ↑n₂ * T)
_ ≤ (t - ↑n₁ * T) := by exact min_le_left (t - ↑n₁ * T) (t + ↑n₂ * T)
_ ≤ min a₁ a₂ := by exact hn₁
· calc max a₁ a₂
_ ≤ t + n₂ * T := by exact hn₂
_ ≤ max (t - ↑n₁ * T) (t + ↑n₂ * T) := by exact le_max_right (t - ↑n₁ * T) (t + ↑n₂ * T)
apply IntervalIntegrable.mono_set _ this
exact periodic_integrability₃ h₁f h₂f