| import Mathlib |
| import OAI.Geometry.Cannon.PlanarGraphs.Region |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| open Set unitInterval Topology |
|
|
| noncomputable section |
| noncomputable section |
| noncomputable section |
| open Set Topology unitInterval |
|
|
| noncomputable section |
| open Set Metric Topology unitInterval |
|
|
| noncomputable section |
| open Set Metric Topology |
|
|
| noncomputable section |
| noncomputable section |
| open Set unitInterval |
|
|
| noncomputable section |
| noncomputable section |
| open Set Metric Topology |
|
|
| noncomputable section |
| open Set Metric Topology |
|
|
| namespace CannonFaceIncidence |
| variable {X D : Type*} [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D] |
|
|
| lemma closure_region_inside {V : Set X} (hV : IsOpen V) {π : V → D} |
| (hπ : Continuous π) {d : D} {x : X} (hx : x∈closure (region V π d)) |
| (hxV : x∈V) : x∈region V π d := by |
| let d₀ := π ⟨x,hxV⟩ |
| have hR := region_isOpen hV hπ d₀ |
| have hxR : x∈region V π d₀ := ⟨hxV,rfl⟩ |
| obtain ⟨y,⟨hyV,hyd₀⟩,⟨hyV',hyd⟩⟩ := |
| (mem_closure_iff_nhds.mp hx) _ (hR.mem_nhds hxR) |
| exact ⟨hxV,hyd₀.symm.trans hyd⟩ |
|
|
| theorem region_boundary_hits {K T : Set X} (hK : IsClosed K) (hT : IsClosed T) |
| (hTK : T⊆K) (hconn : IsPreconnected Tᶜ) (hne : (K\T).Nonempty) |
| {π : (Kᶜ : Set X) → D} (hπ : Continuous π) (d : D) |
| (hd : (region Kᶜ π d).Nonempty) : |
| ∃ x∈K, x∉T ∧ d∈incident Kᶜ π x := by |
| by_contra hn |
| have hbound : closure (region Kᶜ π d)⊆region Kᶜ π d∪T := by |
| intro x hx |
| by_cases hxK : x∈K |
| · right |
| by_contra hxT |
| exact hn ⟨x,hxK,hxT,hx⟩ |
| · exact Or.inl (closure_region_inside hK.isOpen_compl hπ hx hxK) |
| have hclosed : IsClosed (region Kᶜ π d∪T) := by |
| apply isClosed_of_closure_subset |
| rw [closure_union,hT.closure_eq] |
| exact union_subset hbound subset_union_right |
| let _ : PreconnectedSpace (Tᶜ : Set X) := isPreconnected_iff_preconnectedSpace.mp hconn |
| let C : Set (Tᶜ : Set X) := Subtype.val ⁻¹' region Kᶜ π d |
| have hC : IsClopen C := by |
| constructor |
| · have he : C=Subtype.val ⁻¹' (region Kᶜ π d∪T) := by |
| ext x |
| simp only [C,mem_preimage,mem_union] |
| exact or_iff_left x.property |>.symm |
| rw [he] |
| exact hclosed.preimage continuous_subtype_val |
| · exact (region_isOpen hK.isOpen_compl hπ d).preimage continuous_subtype_val |
| obtain ⟨x,hx⟩ := hd |
| have hxT : x∉T := fun hxT => (region_subset Kᶜ π d hx) (hTK hxT) |
| have hCe : C=univ := hC.eq_univ ⟨⟨x,hxT⟩,hx⟩ |
| obtain ⟨y,hyK,hyT⟩ := hne |
| have hyC : (⟨y,hyT⟩ : (Tᶜ : Set X))∈C := hCe.symm ▸ mem_univ _ |
| exact (region_subset Kᶜ π d hyC) hyK |
| end CannonFaceIncidence |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
