openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Approximation /PositiveAccessAngles.lean
| import OAI.MathematicalPhysics.Boltzmann.Compactness.LocalMomentIntegrability | |
| namespace OAI | |
| /-! # A fixed positive set of angular pairs reaching the target velocity ball -/ | |
| noncomputable section | |
| open MeasureTheory Set Metric | |
| open scoped ENNReal | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| def accessAngles (κ ρ : ℝ) : Set (Sphere × Sphere) := | |
| twoCollisionMap beamAxis ⁻¹' ball (κ • firstAxis) ρ ∩ firstAngleSet beamAxis | |
| theorem accessAngles_measurable (κ ρ : ℝ) : MeasurableSet (accessAngles κ ρ) := | |
| ((twoCollisionMap_measurable beamAxis) measurableSet_ball).inter | |
| (firstAngleSet_measurable beamAxis) | |
| theorem accessAngles_positive : ∃ δ : ℝ, 0 < δ ∧ δ ≤ 1/4 ∧ | |
| ∀ κ ρ : ℝ, 0 < κ → 0 < ρ → κ+ρ ≤ δ → | |
| 0 < (sphereProbability.prod sphereProbability) (accessAngles κ ρ) := by | |
| obtain ⟨δ,B,hδ,hδsmall,_hB0,_hBt,hvolume⟩ := localTwoCollisionLaw_volume_domination | |
| refine ⟨δ,hδ,hδsmall,?_⟩ | |
| intro κ ρ hκ hρ hκρ | |
| have hsub : ball (κ • firstAxis) ρ ⊆ ball (0 : V) δ := by | |
| intro v hv | |
| have ht := dist_triangle v (κ • firstAxis) 0 | |
| simp only [dist_zero_right,norm_smul,Real.norm_eq_abs,abs_of_pos hκ,axis_norms.1,mul_one] at ht | |
| change dist v 0 < δ | |
| rw [dist_zero_right] | |
| have hv' : dist v (κ • firstAxis) < ρ := hv | |
| linarith | |
| have hv := hvolume beamAxis (mem_ball_self hδ) (ball (κ • firstAxis) ρ) | |
| rw [Measure.restrict_apply measurableSet_ball,inter_eq_left.mpr hsub, | |
| Measure.smul_apply,smul_eq_mul] at hv | |
| have hp : 0 < volume (ball (κ • firstAxis) ρ) := measure_ball_pos volume _ hρ | |
| have hl : 0 < localTwoCollisionLaw beamAxis (ball (κ • firstAxis) ρ) := by | |
| by_contra hz | |
| have hz' : localTwoCollisionLaw beamAxis (ball (κ • firstAxis) ρ) = 0 := | |
| bot_unique (not_lt.mp hz) | |
| rw [hz',mul_zero] at hv | |
| exact (not_le_of_gt hp) hv | |
| simpa only [localTwoCollisionLaw,Measure.map_apply (twoCollisionMap_measurable beamAxis) | |
| measurableSet_ball,Measure.restrict_apply | |
| ((twoCollisionMap_measurable beamAxis) measurableSet_ball),accessAngles] using hl | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end OAI | |