| import Mathlib |
| import OAI.Geometry.Cannon.PlanarGraphs.Support |
| import OAI.Geometry.Cannon.PlaneTopology.NoncutEdgeTwoFaces |
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| namespace OAI |
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| section |
| section |
| section |
| noncomputable section |
| open Set unitInterval Topology |
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| noncomputable section |
| noncomputable section |
| noncomputable section |
| open Set Topology unitInterval |
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| noncomputable section |
| open Set Metric Topology unitInterval |
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| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| noncomputable section |
| open Set unitInterval |
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| noncomputable section |
| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| open Set Topology |
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| noncomputable section |
| open Set Topology |
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| noncomputable section |
| open Set Topology |
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| noncomputable section |
| open Set Topology unitInterval |
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| noncomputable section |
| noncomputable section |
| noncomputable section |
| noncomputable section |
| namespace CannonPlanarEuler |
| open Set Topology unitInterval CannonEmbeddedArc CannonEmbeddedArc.Arc CannonFaceIncidence CannonPolygonal CannonSegmentCones |
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| variable {ι : Type*} [Fintype ι] [DecidableEq ι] [Nonempty ι] |
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| theorem triangular_counts (F : ι → Arc ℂ) |
| (hF : Pairwise (fun i j => Compatible (F i) (F j))) |
| (Q : Finset (ℂ × ℂ)) (hQ : ∀ i,range (F i)⊆⋃q∈Q,segment ℝ q.1 q.2) |
| (hK : ∀ q : ℂ,IsPreconnected ((⋃ i,range (F i))\{q})) |
| (htri : ∀ d : Faces F,∃ i j k : ι,i≠j ∧ j≠k ∧ k≠i ∧ |
| closure (face F d)=face F d∪(range (F i)∪range (F j)∪range (F k))) : |
| (∀ d : Faces F,(faceVertices F d).card=3) ∧ |
| Nat.card (Faces F)+4=2*(vertices F Finset.univ).card ∧ |
| Nat.card ι+6=3*(vertices F Finset.univ).card := by |
| classical |
| let _ : Finite (Faces F) := CannonEmbeddedArc.Arc.faces_finite F hF Q hQ |
| let _ := Fintype.ofFinite (Faces F) |
| have hthree (d : Faces F) : Nat.card (regionEdges F ConnectedComponents.mk d)=3 := by |
| obtain ⟨i,j,k,hij,hjk,hki,hcl⟩ := htri d |
| rw [triangle_region_edges F hF Q hQ d i j k hcl,Nat.card_coe_set_eq] |
| have hnotMem : i∉({j,k} : Set ι) := by |
| rintro (heq|heq) |
| · exact hij heq |
| · exact hki heq.symm |
| rw [Set.ncard_insert_of_notMem hnotMem,Set.ncard_pair hjk] |
| have hI := noncut_total_face_incidence F hF Q hQ hK |
| simp only [hthree,Finset.sum_const,Finset.card_univ,smul_eq_mul,Nat.card_eq_fintype_card] at hI |
| obtain ⟨z,hz⟩ := (complex_finite_segments_compl_dense Q).nonempty |
| have hzK : z∉⋃ i,range (F i) := fun h=>hz (iUnion_subset hQ h) |
| have hconn : IsPreconnected (support F Finset.univ) := by |
| have hh := hK z |
| rw [sdiff_singleton_eq_self hzK] at hh |
| simpa only [support,Finset.mem_univ,iUnion_true] using hh |
| have hEuler := planar_euler F hF Q hQ hconn |
| have hfaces : faces F Finset.univ=Nat.card (Faces F) := by |
| unfold faces support Faces |
| have he : (⋃ i∈(Finset.univ : Finset ι),range (F i))=⋃ i,range (F i) := by simp |
| rw [he] |
| rw [hfaces,Nat.card_eq_fintype_card] at hEuler |
| refine ⟨fun d=>(noncut_facial_vertices_edges F hF Q hQ hK d).trans (hthree d),?_,?_⟩ <;> |
| rw [Nat.card_eq_fintype_card] <;> omega |
| end CannonPlanarEuler |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end OAI |
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