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