| import Mathlib | |
| import OAI.Geometry.Borsuk.Projectors.Within | |
| import OAI.Geometry.Borsuk.FiniteCases.Cases145 | |
| namespace OAI | |
| open Set MeasureTheory | |
| namespace BorsukNine | |
| attribute [local instance] Classical.propDecidable | |
| theorem Admissible.support_ten_system {f : Vector4 → Fin 10 → ℝ} (hf : Admissible f) : | |
| FiniteObstruction.TenSystem (fun U => U.card = 3 ∧ SupportFace f U) | |
| (fun S => S.card = 4 ∧ SupportFace f S) := by | |
| classical | |
| constructor | |
| · intro S hS | |
| exact hS.1 | |
| · intro S U hS hUS hUc | |
| exact ⟨hUc,hS.2.mono hUS⟩ | |
| · intro U hU | |
| obtain ⟨S,hSc,hSf,hUS⟩ := hf.face_extends_four hU.2 | |
| exact ⟨S,⟨hSc,hSf⟩,hUS⟩ | |
| · intro S U hS hUS hUc | |
| obtain ⟨S',hS'c,hS'f,hUS',hne⟩ := hf.tetra_no_free_triangle hS.1 hS.2 hUS hUc | |
| exact ⟨S',⟨hS'c,hS'f⟩,hUS',hne⟩ | |
| · intro S U hS hUC hUc | |
| have hCc : (Sᶜ \ U).card = 3 := by | |
| rw [Finset.card_sdiff_of_subset hUC,Finset.card_compl,Fintype.card_fin,hS.1,hUc] | |
| simpa only [hUc,hCc,true_and] using (hf.facet_six_properties hS.1 hS.2).1 U hUC hUc | |
| · intro S P hS hPC hPc | |
| trans ((Sᶜ \ P).filter (fun w => SupportFace f (insert w P))).card | |
| · apply congrArg Finset.card | |
| ext w | |
| simp only [Finset.mem_filter] | |
| constructor | |
| · intro support_mem | |
| exact ⟨support_mem.1, support_mem.2.2⟩ | |
| · intro support_mem | |
| refine ⟨support_mem.1, ⟨?_, support_mem.2⟩⟩ | |
| rw [Finset.card_insert_of_notMem (Finset.mem_sdiff.mp support_mem.1).2, hPc] | |
| · exact (hf.facet_six_properties hS.1 hS.2).2 P hPC hPc | |
| theorem no_admissible_ten : ¬ ∃ f : Vector4 → Fin 10 → ℝ, Admissible f := by | |
| rintro ⟨f,hf⟩ | |
| have hempty : SupportFace f ∅ := ⟨EuclideanSpace.single 0 1,by simp,Finset.empty_subset _⟩ | |
| obtain ⟨S,hSc,hSf,_⟩ := hf.face_extends_four hempty | |
| exact FiniteObstruction.no_ten_system hf.support_ten_system ⟨S,hSc,hSf⟩ | |
| theorem main_theorem : | |
| IsCompact projectorSet ∧ | |
| projectorSet ⊆ traceOneSymmetric ∧ | |
| Metric.diam projectorSet = Real.sqrt 2 ∧ | |
| ¬ HasTenSmallCover := by | |
| refine ⟨isCompact_projectorSet,projectorSet_subset_traceOneSymmetric,diam_projectorSet,?_⟩ | |
| intro h | |
| exact no_admissible_ten (cover_to_admissible h) | |
| end BorsukNine | |
| end OAI | |