openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Collision /SphereIsometryMeasure.lean
| import OAI.MathematicalPhysics.Boltzmann.Branching.NormalReflectionGeometry | |
| namespace OAI | |
| /-! # Orthogonal changes of coordinates preserve the actual sphere area measure -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open scoped Pointwise | |
| namespace BoltzmannNonuniqueness | |
| def sphereIsometry (R : Velocity ≃ₗᵢ[ℝ] Velocity) : Sphere ≃ᵐ Sphere where | |
| toFun n := ⟨R n.1,by simpa only [Metric.mem_sphere,dist_zero_right,R.norm_map] using n.property⟩ | |
| invFun n := ⟨R.symm n.1,by simpa only [Metric.mem_sphere,dist_zero_right,R.symm.norm_map] using n.property⟩ | |
| left_inv n := by apply Subtype.ext; exact R.symm_apply_apply _ | |
| right_inv n := by apply Subtype.ext; exact R.apply_symm_apply _ | |
| measurable_toFun := by | |
| exact (R.continuous.measurable.comp measurable_subtype_coe).subtype_mk | |
| measurable_invFun := by | |
| exact (R.symm.continuous.measurable.comp measurable_subtype_coe).subtype_mk | |
| theorem sphereIsometry_cone (R : Velocity ≃ₗᵢ[ℝ] Velocity) (S : Set Sphere) : | |
| (Ioo (0:ℝ) 1) • ((↑) '' ((sphereIsometry R) ⁻¹' S)) = | |
| R ⁻¹' ((Ioo (0:ℝ) 1) • ((↑) '' S)) := by | |
| ext x | |
| constructor | |
| · rintro ⟨r,hr,y,⟨n,hn,hny⟩,hxy⟩ | |
| subst y | |
| change R x ∈ (Ioo (0:ℝ) 1) • ((↑) '' S) | |
| refine ⟨r,hr,R n.1,⟨sphereIsometry R n,hn,rfl⟩,?_⟩ | |
| change r • R n.1 = R x | |
| change r • n.1 = x at hxy | |
| rw [← R.map_smul,hxy] | |
| · rintro ⟨r,hr,y,⟨n,hn,hny⟩,hxy⟩ | |
| subst y | |
| refine ⟨r,hr,R.symm n.1,⟨(sphereIsometry R).symm n,?_,rfl⟩,?_⟩ | |
| · change sphereIsometry R ((sphereIsometry R).symm n) ∈ S | |
| simpa only [MeasurableEquiv.apply_symm_apply] using hn | |
| · apply R.injective | |
| rw [R.map_smul,R.apply_symm_apply] | |
| exact hxy | |
| theorem sphereIsometry_measurePreserving (R : Velocity ≃ₗᵢ[ℝ] Velocity) : | |
| MeasurePreserving (sphereIsometry R) sphereArea sphereArea := by | |
| refine ⟨(sphereIsometry R).measurable,?_⟩ | |
| apply Measure.ext | |
| intro S hS | |
| rw [Measure.map_apply (sphereIsometry R).measurable hS,sphereArea, | |
| Measure.toSphere_apply' _ ((sphereIsometry R).measurable hS),Measure.toSphere_apply' _ hS, | |
| sphereIsometry_cone] | |
| congr 1 | |
| change volume (R.toHomeomorph ⁻¹' _) = volume _ | |
| rw [← R.toHomeomorph.measurableEmbedding.map_apply] | |
| change (volume.map R) _ = volume _ | |
| rw [R.measurePreserving.map_eq] | |
| end BoltzmannNonuniqueness | |
| end | |
| end OAI | |