| import Mathlib |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| open scoped Topology |
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| def coverageAngle (L : ℕ → ℝ) (M : ℝ) (n : ℕ) : ℝ := Real.sqrt (M*L n/(n:ℝ)) |
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| theorem coverageAngle_tendsto (L : ℕ → ℝ) (M : ℝ) |
| (hsub : Tendsto (fun n => L n/(n:ℝ)) atTop (𝓝 0)) : |
| Tendsto (coverageAngle L M) atTop (𝓝 0) := by |
| have H := (hsub.const_mul M).sqrt |
| change Tendsto (fun n => Real.sqrt (M*L n/(n:ℝ))) atTop (𝓝 0) |
| simpa only [← mul_div_assoc,mul_zero,Real.sqrt_zero] using H |
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| theorem coverageAngle_pos {L : ℕ → ℝ} {M : ℝ} (hM : 0 < M) {n : ℕ} |
| (hn : 0 < n) (hL : 0 < L n) : 0 < coverageAngle L M n := by |
| unfold coverageAngle |
| have hn' : (0:ℝ) < n := by exact_mod_cast hn |
| positivity |
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| theorem sin_eq_sinc_mul (x : ℝ) : Real.sin x = Real.sinc x*x := by |
| by_cases hx : x = 0 |
| · simp [hx] |
| · rw [Real.sinc_of_ne_zero hx,div_mul_cancel₀ _ hx] |
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| theorem coverageAngle_variance_tendsto (L : ℕ → ℝ) {M β : ℝ} (hM : 0 < M) |
| (hL : Tendsto L atTop atTop) |
| (hsub : Tendsto (fun n => L n/(n:ℝ)) atTop (𝓝 0)) : |
| Tendsto (fun n => (β*Real.sin (coverageAngle L M n))^2*(n:ℝ)/L n) |
| atTop (𝓝 (β^2*M)) := by |
| have hsinc : Tendsto (fun n => Real.sinc (coverageAngle L M n)) atTop (𝓝 (1:ℝ)) := by |
| simpa only [Function.comp_def,Real.sinc_zero] using |
| Real.continuous_sinc.continuousAt.tendsto.comp (coverageAngle_tendsto L M hsub) |
| have H := (hsinc.pow 2).const_mul (β^2*M) |
| simp only [one_pow,mul_one] at H |
| apply H.congr' |
| filter_upwards [hL.eventually (eventually_gt_atTop (0:ℝ)),eventually_gt_atTop (0:ℕ)] with n hLn hn |
| have hn' : (0:ℝ) < n := by exact_mod_cast hn |
| have hθ : coverageAngle L M n^2 = M*L n/(n:ℝ) := |
| Real.sq_sqrt (by positivity) |
| rw [sin_eq_sinc_mul,mul_pow,mul_pow,hθ] |
| field_simp |
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| theorem coverageAngle_loss_tendsto (L : ℕ → ℝ) {M β e : ℝ} (hM : 0 < M) |
| (hL : Tendsto L atTop atTop) |
| (hsub : Tendsto (fun n => L n/(n:ℝ)) atTop (𝓝 0)) : |
| Tendsto (fun n => β*(Real.cos (coverageAngle L M n)-1)*(n:ℝ)*e/L n) |
| atTop (𝓝 (-β*M*e/2)) := by |
| have hθ := coverageAngle_tendsto L M hsub |
| have hsinc : Tendsto (fun n => Real.sinc (coverageAngle L M n)) atTop (𝓝 (1:ℝ)) := by |
| simpa only [Function.comp_def,Real.sinc_zero] using Real.continuous_sinc.continuousAt.tendsto.comp hθ |
| have hcos : Tendsto (fun n => Real.cos (coverageAngle L M n)) atTop (𝓝 (1:ℝ)) := by |
| simpa only [Function.comp_def,Real.cos_zero] using Real.continuous_cos.continuousAt.tendsto.comp hθ |
| have H := ((hsinc.pow 2).const_mul (-β*M*e)).div (tendsto_const_nhds.add hcos) |
| (by norm_num : (1:ℝ)+1 ≠ 0) |
| norm_num only [one_pow,mul_one,one_add_one_eq_two] at H |
| apply H.congr' |
| filter_upwards [hL.eventually (eventually_gt_atTop (0:ℝ)),eventually_gt_atTop (0:ℕ), |
| hcos.eventually (eventually_gt_nhds (by norm_num : (0:ℝ)<1))] with n hLn hn hc |
| have hn' : (0:ℝ) < n := by exact_mod_cast hn |
| have hθsq : coverageAngle L M n^2 = M*L n/(n:ℝ) := Real.sq_sqrt (by positivity) |
| have htrig := Real.sin_sq_add_cos_sq (coverageAngle L M n) |
| rw [sin_eq_sinc_mul,mul_pow,hθsq] at htrig |
| have hden : 1+Real.cos (coverageAngle L M n) ≠ 0 := by positivity |
| have hcosEq : Real.cos (coverageAngle L M n)-1 = |
| -(Real.sinc (coverageAngle L M n)^2*(M*L n/(n:ℝ)))/(1+Real.cos (coverageAngle L M n)) := by |
| apply (eq_div_iff hden).mpr |
| nlinarith |
| change -β*M*e*Real.sinc (coverageAngle L M n)^2/(1+Real.cos (coverageAngle L M n)) = |
| β*(Real.cos (coverageAngle L M n)-1)*(n:ℝ)*e/L n |
| rw [hcosEq] |
| field_simp |
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| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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