openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Obstructions /BoundedClassNowhereDense.lean
| import OAI.Geometry.IsometricImmersion.Pulses.OriginalPulseExclusion | |
| import OAI.Geometry.IsometricImmersion.Obstructions.BoundedClassTopology | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise | |
| namespace SmoothLocal.Perturbation | |
| open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.Flow SmoothLocal.Model | |
| theorem boundedHeightClass_isNowhereDense | |
| {g0 : MetricField} {V : Set Coord} {kappa : ℝ} | |
| (hg0 : SmoothPositiveOn g0 V) (hV : IsOpen V) (hSV : modelSquare ⊆ V) | |
| (hkappa : 0 < kappa) | |
| (hbackground : ∀ p ∈ V, gaussianCurvature g0 p=modelCurvature kappa p) | |
| (M : ℕ) (q0 : ℚ) : | |
| IsNowhereDense (boundedHeightClass g0 kappa M q0) := by | |
| change interior (closure (boundedHeightClass g0 kappa M q0)) = ∅ | |
| apply Set.eq_empty_iff_forall_notMem.mpr | |
| intro etaStar hStar | |
| let O := interior (closure (boundedHeightClass g0 kappa M q0)) | |
| have hO : IsOpen O := isOpen_interior | |
| have hOclosure : O ⊆ closure (boundedHeightClass g0 kappa M q0) := interior_subset | |
| have hparameters := boundedHeightClass_closure_parameters (hOclosure hStar) | |
| have hgStar : SmoothPositiveOn (perturbedMetric g0 etaStar.val) V := | |
| perturbedMetric_smoothPositiveOn hg0 etaStar.val etaStar.property.1 | |
| obtain ⟨r,hr,_,hLr,hexclusion⟩ := exists_original_pulse_exclusion | |
| hg0 hgStar hV hSV M hkappa hparameters.1 hparameters.2.le hbackground | |
| obtain ⟨N,hN,happroximants⟩ := bounded_class_open_closure_pulse_family_at_radius | |
| hg0 hV hSV M q0 hO hStar hOclosure hr hLr | |
| obtain ⟨delta,hdelt,_,hnoheight⟩ := hexclusion N hN | |
| obtain ⟨tau,happroxTau,hnoheightTau⟩ := ((happroximants delta hdelt).and hnoheight).exists | |
| obtain ⟨eta,z,hclass,hcenter,haccuracy⟩ := happroxTau.2 | |
| exact hnoheightTau eta z hclass hcenter haccuracy | |
| end SmoothLocal.Perturbation | |
| end | |
| end OAI | |