| import OAI.Geometry.IsometricImmersion.Assembly.SelectedPatchObstructions | |
| import OAI.Geometry.IsometricImmersion.Assembly.AssembledImmersionHeight | |
| namespace OAI | |
| noncomputable section | |
| open Set | |
| open scoped ContDiff Topology Matrix | |
| namespace SmoothLocal.Geometry | |
| open SmoothLocal.Perturbation | |
| def counterexampleMetric : MetricField := | |
| assembledPatchMetric initialAssemblyMetric selectedPatchTensor | |
| theorem counterexampleMetric_smoothPositive : | |
| SmoothPositiveOn counterexampleMetric square := | |
| assembledPatchMetric_smoothPositive_of_local_metrics (by norm_num : (0 : ℝ) < 1) | |
| initialAssemblyMetric_smoothPositiveOn selectedPatchTensor selectedPatchTensor_small | |
| selectedPatchTensor_local_positive | |
| theorem counterexampleMetric_no_local_isometric_immersion | |
| {U : Set Coord} (hU : IsOpen U) (hUS : U ⊆ square) (h0 : (0 : Coord) ∈ U) | |
| (F : Coord → Ambient) : ¬ IsometricOn counterexampleMetric F U := by | |
| intro hF | |
| obtain ⟨a,_,_,_,_,_,_,_,_,_,hadm⟩ := | |
| every_assembled_local_immersion_has_actual_local_height (by norm_num : (0 : ℝ) < 1) | |
| selectedPatchTensor selectedPatchTensor_small selectedPatchTensor_local_positive hU hUS h0 F hF 0 0 | |
| exact selectedPatchTensor_local_no_height a _ hadm | |
| end SmoothLocal.Geometry | |
| end | |
| end OAI | |