openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Obstructions /PatchResidualTheorem.lean
| import OAI.Geometry.IsometricImmersion.Obstructions.BoundedClassNowhereDense | |
| import OAI.Geometry.IsometricImmersion.Obstructions.PatchObstructionSet | |
| import OAI.Geometry.IsometricImmersion.Metrics.MetricPatchOpen | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology | |
| namespace SmoothLocal.Perturbation | |
| open SmoothLocal.Geometry SmoothLocal.Flow SmoothLocal.Model | |
| theorem patchObstructionSet_dense | |
| {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) : | |
| Dense (patchObstructionSet g0 kappa) := by | |
| let : BaireSpace (metricPatchSet g0 kappa) := metricPatchSet_baire hg0 hV hSV kappa | |
| change Dense (⋂ index : ℕ × ℚ, (closure (boundedHeightClass g0 kappa index.1 index.2))ᶜ) | |
| apply dense_iInter_of_isOpen (fun _ => isClosed_closure.isOpen_compl) | |
| intro index | |
| exact (isClosed_isNowhereDense_iff_compl.mp ⟨isClosed_closure, | |
| (boundedHeightClass_isNowhereDense hg0 hV hSV hkappa hbackground index.1 index.2).closure⟩).2 | |
| theorem no_admissible_height_residual | |
| {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) : | |
| {eta : metricPatchSet g0 kappa | | |
| ∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric g0 eta.val) z} ∈ | |
| residual (metricPatchSet g0 kappa) := by | |
| let : BaireSpace (metricPatchSet g0 kappa) := metricPatchSet_baire hg0 hV hSV kappa | |
| apply mem_residual.mpr | |
| refine ⟨patchObstructionSet g0 kappa,?_,patchObstructionSet_isGdelta g0 kappa, | |
| patchObstructionSet_dense hg0 hV hSV hkappa hbackground⟩ | |
| intro eta heta | |
| exact patchObstructionSet_has_no_admissible_height heta | |
| theorem exists_patch_obstruction_in_open | |
| {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) | |
| {O : Set (metricPatchSet g0 kappa)} (hO : IsOpen O) (hne : O.Nonempty) : | |
| ∃ eta ∈ O, | |
| SmoothPositiveOn (perturbedMetric g0 eta.val) V ∧ | |
| (∀ p ∈ centralBox, gaussianCurvature (perturbedMetric g0 eta.val) p < -kappa/2) ∧ | |
| ∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric g0 eta.val) z := by | |
| obtain ⟨eta,hetaGood,hetaO⟩ := | |
| (patchObstructionSet_dense hg0 hV hSV hkappa hbackground).exists_mem_open hO hne | |
| exact ⟨eta,hetaO,perturbedMetric_smoothPositiveOn hg0 eta.val eta.property.1, | |
| eta.property.2,patchObstructionSet_has_no_admissible_height hetaGood⟩ | |
| theorem exists_patch_obstruction_in_neighborhood | |
| {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) | |
| {etaCenter : metricPatchSet g0 kappa} {O : Set (metricPatchSet g0 kappa)} | |
| (hO : O ∈ 𝓝 etaCenter) : | |
| ∃ eta ∈ O, ∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric g0 eta.val) z := by | |
| obtain ⟨W,hWO,hW,hcenter⟩ := mem_nhds_iff.mp hO | |
| obtain ⟨eta,hetaW,_,_,hno⟩ := | |
| exists_patch_obstruction_in_open hg0 hV hSV hkappa hbackground hW ⟨etaCenter,hcenter⟩ | |
| exact ⟨eta,hWO hetaW,hno⟩ | |
| end SmoothLocal.Perturbation | |
| end | |
| end OAI | |