| import OAI.Geometry.IsometricImmersion.Curvature.DiskCurvatureProfiles | |
| import OAI.Geometry.IsometricImmersion.Curvature.PatchCurvatureProfiles | |
| import Mathlib.Basic.Countable.Basic | |
| import Mathlib.Logic.Encodable.Basic | |
| namespace OAI | |
| noncomputable section | |
| open scoped ContDiff Topology BigOperators Matrix | |
| open Set Filter | |
| namespace SmoothLocal.Geometry | |
| abbrev PatchAddress := Σ n : ℕ, Σ k : ℕ, Σ R : OrientationLabel, | |
| {c : Coord // c ∈ orientedExteriorCenters n k R} | |
| instance : Countable PatchAddress := by | |
| classical | |
| let : ∀ (n k : ℕ) (R : OrientationLabel), | |
| Finite {c : Coord // c ∈ orientedExteriorCenters n k R} := | |
| fun n k R => (orientedExteriorCenters n k R).finite_toSet.to_subtype | |
| infer_instance | |
| abbrev AssemblyProfileIndex := ℕ ⊕ PatchAddress | |
| instance assemblyProfileEncodable : Encodable AssemblyProfileIndex := | |
| Encodable.ofCountable AssemblyProfileIndex | |
| def patchAddressCarrier (a : PatchAddress) : Set Coord := | |
| orientedClosedPatch a.1 a.2.1 a.2.2.1 a.2.2.2.val | |
| def patchAddressProfile (a : PatchAddress) : Coord → ℝ := | |
| orientedPatchProfile a.1 a.2.1 a.2.2.1 a.2.2.2.val | |
| theorem patchAddressCarriers_pairwise_disjoint : | |
| Pairwise (fun a b => Disjoint (patchAddressCarrier a) (patchAddressCarrier b)) := by | |
| intro a b hab | |
| rcases a with ⟨n, k, R, c, hc⟩ | |
| rcases b with ⟨m, l, S, d, hd⟩ | |
| by_cases hnm : n = m | |
| · subst m | |
| by_cases hkl : k = l | |
| · subst l | |
| by_cases hRS : R = S | |
| · subst S | |
| apply orientedClosedPatches_disjoint_same_layer n k R hc hd | |
| intro hcd | |
| subst d | |
| exact hab rfl | |
| · apply orientedClosedPatches_disjoint_different_layers n | |
| (a := (k, R)) (b := (k, S)) _ hc hd | |
| exact fun h => hRS (congrArg Prod.snd h) | |
| · apply orientedClosedPatches_disjoint_different_layers n | |
| (a := (k, R)) (b := (l, S)) _ hc hd | |
| exact fun h => hkl (congrArg Prod.fst h) | |
| · exact orientedClosedPatches_disjoint_different_disks hnm k l R S hc hd | |
| def assemblyCarrier : AssemblyProfileIndex → Set Coord | |
| | .inl n => accumulatingDisk n | |
| | .inr a => patchAddressCarrier a | |
| def assemblyProfile : AssemblyProfileIndex → Coord → ℝ | |
| | .inl n => accumulatingDiskProfile n | |
| | .inr a => patchAddressProfile a | |
| theorem assemblyProfile_contDiff (i : AssemblyProfileIndex) : ContDiff ℝ ∞ (assemblyProfile i) := by | |
| cases i with | |
| | inl n => exact accumulatingDiskProfile_contDiff n | |
| | inr a => exact orientedPatchProfile_contDiff _ _ _ _ | |
| theorem assemblyProfile_hasCompactSupport (i : AssemblyProfileIndex) : | |
| HasCompactSupport (assemblyProfile i) := by | |
| cases i with | |
| | inl n => exact accumulatingDiskProfile_hasCompactSupport n | |
| | inr a => exact orientedPatchProfile_hasCompactSupport _ _ _ _ | |
| theorem assemblyProfile_tsupport_subset (i : AssemblyProfileIndex) : | |
| tsupport (assemblyProfile i) ⊆ assemblyCarrier i := by | |
| cases i with | |
| | inl n => exact accumulatingDiskProfile_tsupport_subset n | |
| | inr a => exact orientedPatchProfile_tsupport_subset_patch _ _ _ _ | |
| theorem assemblyCarriers_pairwise_disjoint : | |
| Pairwise (fun i j => Disjoint (assemblyCarrier i) (assemblyCarrier j)) := by | |
| intro i j hij | |
| cases i with | |
| | inl n => | |
| cases j with | |
| | inl m => exact accumulatingDisks_pairwise_disjoint (fun h => hij (congrArg Sum.inl h)) | |
| | inr a => | |
| exact (orientedClosedPatch_disjoint_every_disk a.1 a.2.1 a.2.2.1 a.2.2.2.property n).symm | |
| | inr a => | |
| cases j with | |
| | inl n => exact orientedClosedPatch_disjoint_every_disk a.1 a.2.1 a.2.2.1 a.2.2.2.property n | |
| | inr b => exact patchAddressCarriers_pairwise_disjoint (fun h => hij (congrArg Sum.inr h)) | |
| theorem zero_not_mem_assemblyCarrier (i : AssemblyProfileIndex) : | |
| (0 : Coord) ∉ assemblyCarrier i := by | |
| cases i with | |
| | inl n => exact zero_not_mem_accumulatingDisk n | |
| | inr a => | |
| intro h | |
| exact zero_not_mem_enlargedAccumulatingDisk a.1 | |
| (orientedClosedPatch_subset_enlarged a.1 a.2.1 a.2.2.1 a.2.2.2.property h) | |
| def enumeratedAssemblyProfile (j : ℕ) : Coord → ℝ := | |
| match Encodable.decode₂ AssemblyProfileIndex j with | |
| | none => 0 | |
| | some i => assemblyProfile i | |
| theorem enumeratedAssemblyProfile_encode (i : AssemblyProfileIndex) : | |
| enumeratedAssemblyProfile (Encodable.encode i) = assemblyProfile i := by | |
| simp only [enumeratedAssemblyProfile, Encodable.decode₂_encode] | |
| theorem enumeratedAssemblyProfile_contDiff (j : ℕ) : | |
| ContDiff ℝ ∞ (enumeratedAssemblyProfile j) := by | |
| cases h : Encodable.decode₂ AssemblyProfileIndex j with | |
| | none => | |
| simp only [enumeratedAssemblyProfile, h] | |
| change ContDiff ℝ ∞ (fun _ : Coord => (0 : ℝ)) | |
| exact contDiff_const | |
| | some i => simpa only [enumeratedAssemblyProfile, h] using assemblyProfile_contDiff i | |
| theorem enumeratedAssemblyProfile_hasCompactSupport (j : ℕ) : | |
| HasCompactSupport (enumeratedAssemblyProfile j) := by | |
| cases h : Encodable.decode₂ AssemblyProfileIndex j with | |
| | none => simpa only [enumeratedAssemblyProfile, h] using | |
| (HasCompactSupport.zero : HasCompactSupport (0 : Coord → ℝ)) | |
| | some i => simpa only [enumeratedAssemblyProfile, h] using assemblyProfile_hasCompactSupport i | |
| theorem zero_not_mem_enumeratedAssemblyProfile_tsupport (j : ℕ) : | |
| (0 : Coord) ∉ tsupport (enumeratedAssemblyProfile j) := by | |
| cases h : Encodable.decode₂ AssemblyProfileIndex j with | |
| | none => simp [enumeratedAssemblyProfile, h] | |
| | some i => | |
| simpa only [enumeratedAssemblyProfile, h] using | |
| (fun hp => zero_not_mem_assemblyCarrier i (assemblyProfile_tsupport_subset i hp)) | |
| theorem enumeratedAssemblyProfile_zero_other (i : AssemblyProfileIndex) {p : Coord} | |
| (hp : p ∈ assemblyCarrier i) {j : ℕ} (hj : j ≠ Encodable.encode i) : | |
| enumeratedAssemblyProfile j p = 0 := by | |
| cases hd : Encodable.decode₂ AssemblyProfileIndex j with | |
| | none => simp [enumeratedAssemblyProfile, hd] | |
| | some a => | |
| have ha : a ≠ i := by | |
| intro hai | |
| subst a | |
| exact hj (Encodable.decode₂_eq_some.mp hd).symm | |
| have hnot : p ∉ tsupport (assemblyProfile a) := fun hpa => | |
| Set.disjoint_left.mp (assemblyCarriers_pairwise_disjoint ha) | |
| (assemblyProfile_tsupport_subset a hpa) hp | |
| have hz : assemblyProfile a p = 0 := by | |
| by_contra hne | |
| exact hnot (subset_tsupport (assemblyProfile a) hne) | |
| simpa only [enumeratedAssemblyProfile, hd] using hz | |
| end SmoothLocal.Geometry | |
| end | |
| end OAI | |