openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Obstructions /SmallPatchObstruction.lean
| import OAI.Geometry.IsometricImmersion.Obstructions.PatchResidualTheorem | |
| import OAI.Geometry.IsometricImmersion.Metrics.AffineTensorNeighborhood | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise | |
| namespace SmoothLocal.Perturbation | |
| open SmoothLocal.Geometry SmoothLocal.Flow SmoothLocal.Model | |
| theorem model_zero_mem_metricPatchSet | |
| {g0 : MetricField} {V : Set Coord} {kappa : ℝ} | |
| (hg0 : SmoothPositiveOn g0 V) (hSV : modelSquare ⊆ V) (hkappa : 0 < kappa) | |
| (hbackground : ∀ p ∈ V, gaussianCurvature g0 p=modelCurvature kappa p) : | |
| (0 : SymmetricPerturbation) ∈ metricPatchSet g0 kappa := by | |
| apply zero_mem_metricPatchSet | |
| · intro p hp | |
| exact hg0.2 p (hSV hp) | |
| · intro p hp | |
| rw [hbackground p (hSV (centralBox_subset_modelSquare hp))] | |
| exact modelCurvature_central_square hkappa (fun i => ⟨hp.1 i,hp.2 i⟩) | |
| theorem exists_supported_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 SymmetricPerturbation} (hO : IsOpen O) (hzero : (0 : SymmetricPerturbation) ∈ O) : | |
| ∃ eta : SymmetricPerturbation, eta ∈ O ∧ eta ∈ metricPatchSet g0 kappa ∧ | |
| SmoothPositiveOn (perturbedMetric g0 eta) V ∧ | |
| ∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric g0 eta) z := by | |
| let zeroPatch : metricPatchSet g0 kappa := | |
| ⟨0,model_zero_mem_metricPatchSet hg0 hSV hkappa hbackground⟩ | |
| let W : Set (metricPatchSet g0 kappa) := Subtype.val ⁻¹' O | |
| have hW : IsOpen W := hO.preimage continuous_subtype_val | |
| have hWzero : zeroPatch ∈ W := hzero | |
| obtain ⟨eta,hetaW,hg,_,hno⟩ := | |
| exists_patch_obstruction_in_open hg0 hV hSV hkappa hbackground hW ⟨zeroPatch,hWzero⟩ | |
| exact ⟨eta.val,hetaW,eta.property,hg,hno⟩ | |
| theorem exists_small_affine_patch_obstruction | |
| {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) | |
| (b : Coord) (R : Matrix (Fin 2) (Fin 2) ℝ) (N : ℕ) | |
| {epsilon : ℝ} (hepsilon : 0 < epsilon) : | |
| ∃ eta : SymmetricPerturbation, | |
| eta ∈ metricPatchSet g0 kappa ∧ SmoothPositiveOn (perturbedMetric g0 eta) V ∧ | |
| (∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric g0 eta) z) ∧ | |
| (∀ k ≤ N, ∀ p : Coord, | |
| ‖iteratedFDeriv ℝ k (affinePushforwardMetric (perturbationTensor eta) b R) p‖ < epsilon) ∧ | |
| ∀ i j : Fin 2, ∀ k ≤ N, ∀ p : Coord, | |
| ‖iteratedFDeriv ℝ k | |
| (fun q => affinePushforwardMetric (perturbationTensor eta) b R q i j) p‖ < epsilon := by | |
| obtain ⟨eta,hetaO,hetaMetric,hg,hno⟩ := exists_supported_patch_obstruction_in_open | |
| hg0 hV hSV hkappa hbackground (affineTensorJetNeighborhood_isOpen R N epsilon) | |
| (zero_mem_affineTensorJetNeighborhood R N hepsilon) | |
| exact ⟨eta,hetaMetric,hg,hno, | |
| fun k hk p => affineTensorJetNeighborhood_physical_bounds b R N hetaO hk p, | |
| fun i j k hk p => affineTensorJetNeighborhood_coefficient_bounds b R N hetaO hk p i j⟩ | |
| end SmoothLocal.Perturbation | |
| end | |
| end OAI | |