openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Assembly /SelectedPatchObstructions.lean
| import OAI.Geometry.IsometricImmersion.Coordinates.PatchAddressModel | |
| import OAI.Geometry.IsometricImmersion.Obstructions.SmallPatchObstruction | |
| import OAI.Geometry.IsometricImmersion.Metrics.PatchTensorPositivity | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise | |
| namespace SmoothLocal.Geometry | |
| open SmoothLocal.Perturbation | |
| theorem exists_obstructing_patch_tensor (a : PatchAddress) : | |
| ∃ eta : SymmetricPerturbation, | |
| eta ∈ patchTensorBudgetNeighborhood 1 a ∧ | |
| eta ∈ metricPatchSet (patchAddressInitialMetric a) (patchAddressAmplitude a) ∧ | |
| SmoothPositiveOn (perturbedMetric (patchAddressInitialMetric a) eta) patchAddressModelDomain ∧ | |
| ∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric (patchAddressInitialMetric a) eta) z := | |
| exists_supported_patch_obstruction_in_open (patchAddressInitialMetric_smoothPositive a) | |
| patchAddressModelDomain_isOpen modelSquare_subset_patchAddressModelDomain | |
| (patchAddressAmplitude_pos a) (fun p hp => patchAddressInitialMetric_curvature a hp) | |
| (patchTensorBudgetNeighborhood_isOpen 1 a) | |
| (zero_mem_patchTensorBudgetNeighborhood (by norm_num : (0 : ℝ) < 1) a) | |
| def selectedPatchTensor (a : PatchAddress) : SymmetricPerturbation := | |
| Classical.choose (exists_obstructing_patch_tensor a) | |
| theorem selectedPatchTensor_spec (a : PatchAddress) : | |
| selectedPatchTensor a ∈ patchTensorBudgetNeighborhood 1 a ∧ | |
| selectedPatchTensor a ∈ metricPatchSet (patchAddressInitialMetric a) (patchAddressAmplitude a) ∧ | |
| SmoothPositiveOn (perturbedMetric (patchAddressInitialMetric a) (selectedPatchTensor a)) | |
| patchAddressModelDomain ∧ | |
| ∀ z : Coord → ℝ, | |
| ¬ PatchAdmissibleHeight (perturbedMetric (patchAddressInitialMetric a) (selectedPatchTensor a)) z := | |
| Classical.choose_spec (exists_obstructing_patch_tensor a) | |
| theorem selectedPatchTensor_small (a : PatchAddress) : | |
| selectedPatchTensor a ∈ patchTensorBudgetNeighborhood 1 a := | |
| (selectedPatchTensor_spec a).1 | |
| theorem selected_local_metric_eq (a : PatchAddress) : | |
| localPatchMetric initialAssemblyMetric selectedPatchTensor a = | |
| perturbedMetric (patchAddressInitialMetric a) (selectedPatchTensor a) := rfl | |
| theorem selectedPatchTensor_local_positive (a : PatchAddress) (q : Coord) (hq : q ∈ modelSquare) : | |
| (localPatchMetric initialAssemblyMetric selectedPatchTensor a q).PosDef := by | |
| rw [selected_local_metric_eq] | |
| exact (selectedPatchTensor_spec a).2.1.1 q hq | |
| theorem selectedPatchTensor_local_no_height (a : PatchAddress) : | |
| ∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (localPatchMetric initialAssemblyMetric selectedPatchTensor a) z := by | |
| rw [selected_local_metric_eq] | |
| exact (selectedPatchTensor_spec a).2.2.2 | |
| end SmoothLocal.Geometry | |
| end | |
| end OAI | |