openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /VelocityDetection /HeatKernelsIntegrableOnInvSqrt.lean
| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| namespace VelocityDetection.HeatKernels | |
| open scoped BigOperators Topology ContDiff | |
| open Set Function Filter | |
| open Set Function Filter MeasureTheory | |
| open scoped Topology BigOperators ContDiff | |
| open scoped Topology ContDiff BigOperators | |
| open scoped Topology ContDiff ZeroAtInfty | |
| open scoped Topology ContDiff ZeroAtInfty BigOperators | |
| open scoped Topology | |
| theorem integrableOn_inv_sqrt {T : ℝ} (hT : 0 ≤ T) : | |
| IntegrableOn (fun s : ℝ => (Real.sqrt s)⁻¹) (Ioc (0 : ℝ) T) := by | |
| have hi : IntegrableOn (fun s : ℝ => s ^ (-(1 / 2) : ℝ)) (Ioc (0 : ℝ) T) := | |
| (intervalIntegrable_iff_integrableOn_Ioc_of_le hT).mp | |
| (intervalIntegral.intervalIntegrable_rpow' (by norm_num : (-1 : ℝ) < -(1 / 2))) | |
| apply hi.congr | |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs | |
| rw [Real.rpow_neg hs.1.le, Real.sqrt_eq_rpow] | |
| theorem integral_inv_sqrt {T : ℝ} (hT : 0 ≤ T) : | |
| (∫ s in Ioc (0 : ℝ) T, (Real.sqrt s)⁻¹) = 2 * Real.sqrt T := by | |
| calc | |
| (∫ s in Ioc (0 : ℝ) T, (Real.sqrt s)⁻¹) = ∫ s in Ioc (0 : ℝ) T, s ^ (-(1 / 2) : ℝ) := by | |
| apply setIntegral_congr_fun measurableSet_Ioc | |
| intro s hs | |
| change (Real.sqrt s)⁻¹ = s ^ (-(1 / 2) : ℝ) | |
| rw [Real.rpow_neg hs.1.le, Real.sqrt_eq_rpow] | |
| _ = ∫ s in (0 : ℝ)..T, s ^ (-(1 / 2) : ℝ) := | |
| (intervalIntegral.integral_of_le hT).symm | |
| _ = 2 * Real.sqrt T := by | |
| rw [integral_rpow (Or.inl (by norm_num : (-1 : ℝ) < -(1 / 2)))] | |
| norm_num [show -(1 / 2 : ℝ) + 1 = 1 / 2 by norm_num, ← Real.sqrt_eq_rpow] | |
| ring | |
| end VelocityDetection.HeatKernels | |
| end | |
| end OAI | |