import Mathlib
import OAI.Geometry.Cannon.PlanarGraphs.Support
import OAI.Geometry.Cannon.PlaneTopology.NoncutEdgeTwoFaces

namespace OAI

section
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open Set unitInterval Topology

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open Set Topology unitInterval

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open Set Metric Topology unitInterval

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open Set Metric Topology

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open Set unitInterval

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open Set Metric Topology

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open Set Metric Topology

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open Set Metric Topology

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open Set Topology

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open Set Topology

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open Set Topology

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open Set Topology unitInterval

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namespace CannonPlanarEuler
open Set Topology unitInterval CannonEmbeddedArc CannonEmbeddedArc.Arc CannonFaceIncidence CannonPolygonal CannonSegmentCones

variable {ι : Type*} [Fintype ι] [DecidableEq ι] [Nonempty ι]

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

end
end
end
end
end
end
end
end
end
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end
end

end OAI