| import Mathlib |
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| namespace OAI |
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| open intervalIntegral Filter |
| open scoped Topology |
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| noncomputable section |
| namespace CylinderCovering.RationalIntegrals |
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| theorem ceiling_integral (α c₀ c₂ : ℝ) : |
| (∫ y in (-1 / 4 : ℝ)..(1 / 4 : ℝ), α * (c₀ + c₂ * y ^ 2) * y ^ 2) = |
| α * (c₀ / 96 + c₂ / 2560) := by |
| have hf : (fun y : ℝ => α * (c₀ + c₂ * y ^ 2) * y ^ 2) = |
| (fun y : ℝ => (α * c₀) * y ^ 2 + (α * c₂) * y ^ 4) := by |
| funext y |
| ring |
| rw [hf, integral_add] |
| · simp only [integral_const_mul, integral_pow] |
| norm_num |
| ring |
| · exact (continuous_const.mul (continuous_id.pow 2)).intervalIntegrable _ _ |
| · exact (continuous_const.mul (continuous_id.pow 4)).intervalIntegrable _ _ |
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|
| theorem limiting_ceiling_integral : |
| (∫ y in (-1 / 4 : ℝ)..(1 / 4 : ℝ), ((1 / 16 : ℝ) + y ^ 2) * y ^ 2) = |
| (1 / 960 : ℝ) := by |
| have h := ceiling_integral 1 (1 / 16) 1 |
| norm_num at h ⊢ |
| exact h |
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|
| theorem limiting_energy_angular_integral : |
| (∫ z in (-1 / 2 : ℝ)..(1 / 2 : ℝ), |
| ((1 / 16 : ℝ) + z ^ 2 / 4) ^ 2 + (1 / 2 : ℝ) * (z / 2) ^ 2) = |
| (17 / 960 : ℝ) := by |
| have hf : (fun z : ℝ => ((1 / 16 : ℝ) + z ^ 2 / 4) ^ 2 + |
| (1 / 2 : ℝ) * (z / 2) ^ 2) = |
| (fun z : ℝ => (1 / 256 : ℝ) + (5 / 32 : ℝ) * z ^ 2 + |
| (1 / 16 : ℝ) * z ^ 4) := by |
| funext z |
| ring |
| rw [hf, integral_add, integral_add] |
| · simp only [integral_const, integral_const_mul, integral_pow] |
| norm_num |
| · exact continuous_const.intervalIntegrable _ _ |
| · exact (continuous_const.mul (continuous_id.pow 2)).intervalIntegrable _ _ |
| · exact (continuous_const.add (continuous_const.mul (continuous_id.pow 2))).intervalIntegrable _ _ |
| · exact (continuous_const.mul (continuous_id.pow 4)).intervalIntegrable _ _ |
|
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| theorem limiting_energy_integral : |
| (1 / 2 : ℝ) * (∫ r in (0 : ℝ)..(1 / 2 : ℝ), r) * |
| (∫ z in (-1 / 2 : ℝ)..(1 / 2 : ℝ), |
| ((1 / 16 : ℝ) + z ^ 2 / 4) ^ 2 + (1 / 2 : ℝ) * (z / 2) ^ 2) = |
| (17 / 15360 : ℝ) := by |
| rw [limiting_energy_angular_integral, integral_id] |
| norm_num |
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| theorem limiting_coefficient_gap : |
| (17 / 15360 : ℝ) - (1 / 960 : ℝ) = (1 / 15360 : ℝ) ∧ |
| 0 < (17 / 15360 : ℝ) - (1 / 960 : ℝ) := by |
| norm_num |
|
|
| theorem rational_ceiling_coefficient_formula (d : ℝ) : |
| (1 / (1 + 2 * d)) * |
| (((1 / 2 : ℝ) * (1 / 2 + 2 * d) / (16 * (1 / 2 + d) ^ 2)) / 96 + |
| (1 / (4 * (1 / 2 + d) ^ 2)) / 2560) = |
| (2 + 5 * d) / (1920 * (1 + 2 * d) ^ 3) := by |
| by_cases hd : 1 + 2 * d = 0 |
| · have he : d = -(1 / 2 : ℝ) := by linarith |
| subst d |
| norm_num |
| · have h : (1 / 2 : ℝ) + d ≠ 0 := by intro he; apply hd; linarith |
| field_simp |
| ring |
|
|
| theorem rational_ceiling_integral_tendsto : |
| Tendsto (fun d : ℝ => |
| ∫ y in (-1 / 4 : ℝ)..(1 / 4 : ℝ), |
| (1 / (1 + 2 * d)) * |
| (((1 / 2 : ℝ) * (1 / 2 + 2 * d) / (16 * (1 / 2 + d) ^ 2)) + |
| (1 / (4 * (1 / 2 + d) ^ 2)) * y ^ 2) * y ^ 2) |
| (𝓝 0) (𝓝 (1 / 960 : ℝ)) := by |
| simp_rw [ceiling_integral, rational_ceiling_coefficient_formula] |
| have h : ContinuousAt (fun d : ℝ => |
| (2 + 5 * d) / (1920 * (1 + 2 * d) ^ 3)) 0 := by |
| fun_prop (disch := norm_num) |
| convert h.tendsto using 1 |
| norm_num |
|
|
| theorem rational_ceiling_integral_tendsto_right : |
| Tendsto (fun d : ℝ => |
| ∫ y in (-1 / 4 : ℝ)..(1 / 4 : ℝ), |
| (1 / (1 + 2 * d)) * |
| (((1 / 2 : ℝ) * (1 / 2 + 2 * d) / (16 * (1 / 2 + d) ^ 2)) + |
| (1 / (4 * (1 / 2 + d) ^ 2)) * y ^ 2) * y ^ 2) |
| (𝓝[Set.Ioi 0] 0) (𝓝 (1 / 960 : ℝ)) := |
| rational_ceiling_integral_tendsto.mono_left nhdsWithin_le_nhds |
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| theorem uncentered_ceiling_integral (c : ℝ) : |
| (∫ u in (-c)..c, (c ^ 2 * u ^ 2 / 2 + u ^ 4 / 2)) = (8 / 15 : ℝ) * c ^ 5 := by |
| have he : (fun u : ℝ => c ^ 2 * u ^ 2 / 2 + u ^ 4 / 2) = |
| (fun u : ℝ => (c ^ 2 / 2) * u ^ 2 + (1 / 2 : ℝ) * u ^ 4) := by |
| funext u |
| ring |
| rw [he, intervalIntegral.integral_add] |
| · rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul, |
| integral_pow, integral_pow] |
| norm_num |
| ring |
| · exact (continuous_const.mul (continuous_id.pow 2)).intervalIntegrable _ _ |
| · exact (continuous_const.mul (continuous_id.pow 4)).intervalIntegrable _ _ |
|
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| theorem uncentered_energy_integral (c : ℝ) : |
| (1 / 4 : ℝ) * (∫ b in (-c)..c, ((c ^ 2 + b ^ 2) ^ 2 / 4 + b ^ 2)) = |
| c ^ 3 / 6 + 7 * c ^ 5 / 30 := by |
| have he : (fun b : ℝ => (c ^ 2 + b ^ 2) ^ 2 / 4 + b ^ 2) = |
| (fun b : ℝ => c ^ 4 / 4 + (c ^ 2 / 2 + 1) * b ^ 2 + (1 / 4 : ℝ) * b ^ 4) := by |
| funext b |
| ring |
| rw [he, intervalIntegral.integral_add, intervalIntegral.integral_add] |
| · rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul, |
| integral_pow, integral_pow, intervalIntegral.integral_const] |
| norm_num |
| ring |
| · exact continuous_const.intervalIntegrable _ _ |
| · exact (continuous_const.mul (continuous_id.pow 2)).intervalIntegrable _ _ |
| · exact (continuous_const.add (continuous_const.mul (continuous_id.pow 2))).intervalIntegrable _ _ |
| · exact (continuous_const.mul (continuous_id.pow 4)).intervalIntegrable _ _ |
|
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| theorem uncentered_limiting_gap {c : ℝ} (hc : c ^ 2 = (1 / 2 : ℝ)) : |
| (1 / 4 : ℝ) * (∫ b in (-c)..c, ((c ^ 2 + b ^ 2) ^ 2 / 4 + b ^ 2)) - |
| (∫ u in (-c)..c, (c ^ 2 * u ^ 2 / 2 + u ^ 4 / 2)) = c ^ 3 / 60 := by |
| rw [uncentered_energy_integral, uncentered_ceiling_integral] |
| have h5 : c ^ 5 = c ^ 3 / 2 := by |
| calc c ^ 5 = c ^ 3 * c ^ 2 := by ring |
| _ = c ^ 3 / 2 := by rw [hc]; ring |
| rw [h5] |
| ring |
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| end CylinderCovering.RationalIntegrals |
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| end |
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| end OAI |
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