openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Obstructions /PatchAdmissibilityRestriction.lean
| import OAI.Geometry.IsometricImmersion.Metrics.PatchTensorRestriction | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Function | |
| open scoped ContDiff Topology BigOperators Matrix Matrix.Norms.Elementwise | |
| namespace SmoothLocal.Geometry | |
| open SmoothLocal.Perturbation | |
| theorem PatchAdmissibleHeight.congr_metric_on_open | |
| {g h : MetricField} {z : Coord → ℝ} (hadm : PatchAdmissibleHeight g z) | |
| {W : Set Coord} (hW : IsOpen W) (hSW : modelSquare ⊆ W) (heq : EqOn g h W) : | |
| PatchAdmissibleHeight h z := by | |
| obtain ⟨U,hU,hSU,hg,hz,hD,hE,hyy,hq⟩ := hadm | |
| have hevent (p : Coord) (hp : p ∈ modelSquare) : g =ᶠ[𝓝 p] h := by | |
| filter_upwards [hW.mem_nhds (hSW hp)] with q hq | |
| exact heq hq | |
| have hmetric : SmoothPositiveOn h (U ∩ W) := by | |
| constructor | |
| · intro i j | |
| apply ((hg.1 i j).mono inter_subset_left).congr | |
| intro p hp | |
| exact congrFun (congrFun (heq hp.2).symm i) j | |
| · intro p hp | |
| rw [← heq hp.2] | |
| exact hg.2 p hp.1 | |
| refine ⟨U ∩ W,hU.inter hW,(fun p hp => ⟨hSU hp,hSW hp⟩),hmetric, | |
| hz.mono inter_subset_left,?_,?_,?_,?_⟩ | |
| · intro p hp | |
| rw [← covHessian_eq_of_metric_eventuallyEq (hevent p hp) z, | |
| ← gaussianCurvature_eq_of_eventuallyEq (hevent p hp), | |
| ← heightEnergy_eq_of_metric_eventuallyEq (hevent p hp) z] | |
| exact hD p hp | |
| · intro p hp | |
| rw [← heightEnergy_eq_of_metric_eventuallyEq (hevent p hp) z] | |
| exact hE p hp | |
| · intro p hp | |
| rw [← covHessian_eq_of_metric_eventuallyEq (hevent p hp) z] | |
| exact hyy p hp | |
| · intro p hp | |
| unfold hessianQuotient | |
| rw [← covHessian_eq_of_metric_eventuallyEq (hevent p hp) z] | |
| exact hq p hp | |
| theorem assembledPatchMetric_admissible_local | |
| (g : MetricField) (eta : PatchAddress → SymmetricPerturbation) (a : PatchAddress) | |
| {z : Coord → ℝ} | |
| (hadm : PatchAdmissibleHeight | |
| (affinePullbackMetric (assembledPatchMetric g eta) (patchAddressCenter a) (patchAddressMatrix a)) z) : | |
| PatchAdmissibleHeight (localPatchMetric g eta a) z := by | |
| apply hadm.congr_metric_on_open Metric.isOpen_ball (W := Metric.ball (0 : Coord) 4) | |
| · intro p hp | |
| have hnorm : ‖p‖ ≤ 3 := by | |
| apply (pi_norm_le_iff_of_nonneg (by norm_num : (0 : ℝ) ≤ 3)).mpr | |
| intro i | |
| simpa only [Real.norm_eq_abs] using (abs_le.mpr ⟨hp.1 i,hp.2 i⟩) | |
| simpa only [Metric.mem_ball,dist_zero_right] using (lt_of_le_of_lt hnorm (by norm_num : (3 : ℝ) < 4)) | |
| · intro p hp | |
| apply assembledPatchMetric_pullback_eq_on_closedPatchBox g eta a | |
| apply (mem_closedPatchBox_iff_norm_le p).mpr | |
| exact (by simpa only [Metric.mem_ball,dist_zero_right] using hp : ‖p‖ < 4).le | |
| end SmoothLocal.Geometry | |
| end | |
| end OAI | |