lean/OAI/Geometry/Cannon/PlanarGraphs/Region.lean

import Mathlib

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

namespace CannonFaceIncidence
variable {X D : Type*} [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D]

def region (V : Set X) (π : V → D) (d : D) : Set X :=
  {x | ∃ h : x∈V, π ⟨x,h⟩=d}

def incident (V : Set X) (π : V → D) (x : X) : Set D :=
  {d | x∈closure (region V π d)}

omit [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D] in
lemma region_subset (V : Set X) (π : V → D) (d : D) : region V π d⊆V := by
  rintro x ⟨hx,_⟩
  exact hx

lemma region_isOpen {V : Set X} (hV : IsOpen V) {π : V → D} (hπ : Continuous π) (d : D) :
    IsOpen (region V π d) := by
  have he : region V π d=Subtype.val '' (π ⁻¹' {d}) := by
    ext x
    simp only [region,mem_ofPred_eq,mem_image,mem_preimage,mem_singleton_iff]
    constructor
    · rintro ⟨hx,he⟩
      exact ⟨⟨x,hx⟩,he,rfl⟩
    · rintro ⟨x,hx,rfl⟩
      exact ⟨x.property,hx⟩
  rw [he]
  exact hV.isOpenMap_subtype_val _ (hπ.isOpen_preimage _ (isOpen_discrete _))

lemma label_on_connected {V S : Set X} {π : V → D} (hπ : Continuous π)
    (hS : IsConnected S) (hSV : S⊆V) : ∃ d : D, S⊆region V π d := by
  let _ : PreconnectedSpace S := isPreconnected_iff_preconnectedSpace.mp hS.isPreconnected
  obtain ⟨x,hx⟩ := hS.nonempty
  let f : S → D := fun y => π ⟨y.val,hSV y.property⟩
  have hf : Continuous f := hπ.comp (by fun_prop)
  refine ⟨f ⟨x,hx⟩,fun y hy => ⟨hSV hy,?_⟩⟩
  exact (IsLocallyConstant.iff_continuous f).mpr hf |>.apply_eq_of_preconnectedSpace ⟨y,hy⟩ ⟨x,hx⟩

theorem sector_incidence {V W B : Set X} (hW : IsOpen W) (hBW : B⊆W)
    {π : V → D} (hπ : Continuous π) {ι : Type*} [Finite ι]
    (S : ι → Set X) (hS : ∀ i, IsConnected (S i))
    (hcover : W∩V=⋃ i, S i) (hclose : ∀ i, B⊆closure (S i)) :
    ∃ labels : ι → D, (∀ i, S i⊆region V π (labels i)) ∧
      ∀ x∈B, incident V π x=range labels := by
  classical
  have hSV (i : ι) : S i⊆V := by
    intro x hx
    exact (hcover.symm ▸ mem_iUnion_of_mem i hx).2
  choose labels hlabels using fun i => label_on_connected hπ (hS i) (hSV i)
  refine ⟨labels,hlabels,?_⟩
  intro x hx
  ext d
  constructor
  · intro hd
    by_contra hn
    have hnone : W∩region V π d=∅ := by
      apply eq_empty_iff_forall_notMem.mpr
      rintro y ⟨hyW,hyR⟩
      have hyWV : y∈W∩V := ⟨hyW,region_subset V π d hyR⟩
      rw [hcover] at hyWV
      obtain ⟨i,hi⟩ := mem_iUnion.mp hyWV
      obtain ⟨hy,he⟩ := hlabels i hi
      obtain ⟨hy',he'⟩ := hyR
      apply hn
      exact ⟨i,he.symm.trans he'⟩
    have hm : (W∩region V π d).Nonempty :=
      (mem_closure_iff_nhds.mp hd) W (hW.mem_nhds (hBW hx))
    rw [hnone] at hm
    exact not_nonempty_empty hm
  · rintro ⟨i,rfl⟩
    exact closure_mono (hlabels i) (hclose i hx)

omit [TopologicalSpace D] [DiscreteTopology D] in

theorem incident_constant_on_preconnected {V A : Set X} {π : V → D}
    (hA : IsPreconnected A)
    (hloc : ∀ x∈A, ∃ W : Set X, IsOpen W ∧ x∈W ∧
      ∀ y∈A∩W, incident V π y=incident V π x) :
    ∀ x∈A, ∀ y∈A, incident V π x=incident V π y := by
  let _ : PreconnectedSpace A := isPreconnected_iff_preconnectedSpace.mp hA
  let f : A → Set D := fun x => incident V π x.val
  have hf : IsLocallyConstant f := by
    apply (IsLocallyConstant.iff_eventually_eq f).mpr
    intro x
    obtain ⟨W,hW,hxW,heq⟩ := hloc x.val x.property
    filter_upwards [(continuous_subtype_val.isOpen_preimage W hW).mem_nhds hxW] with y hy
    exact heq y.val ⟨y.property,hy⟩
  intro x hx y hy
  exact hf.apply_eq_of_preconnectedSpace ⟨x,hx⟩ ⟨y,hy⟩

end CannonFaceIncidence

namespace CannonFaceIncidence
open Set Metric Topology

def openBox (r : ℝ) : Set (ℝ × ℝ) := Ioo (-r) r ×ˢ Ioo (-r) r
def halfBox (r : ℝ) (b : Bool) : Set (ℝ × ℝ) :=
  Ioo (-r) r ×ˢ (if b then Ioo 0 r else Ioo (-r) 0)

lemma openBox_eq_ball (r : ℝ) : openBox r=ball (0,0) r := by
  rw [←ball_prod_same,Real.ball_eq_Ioo]
  simp [openBox]

lemma halfBox_connected {r : ℝ} (hr : 0<r) (b : Bool) : IsConnected (halfBox r b) := by
  cases b
  · exact (isConnected_Ioo (by linarith : -r<r)).prod (isConnected_Ioo (neg_neg_of_pos hr))
  · exact (isConnected_Ioo (by linarith : -r<r)).prod (isConnected_Ioo hr)

lemma halfBox_closure {r : ℝ} (hr : 0<r) {p : ℝ × ℝ}
    (hp : p∈openBox r) (hp₂ : p.2=0) (b : Bool) : p∈closure (halfBox r b) := by
  have hrr : -r<r := by linarith
  cases b <;> simp only [halfBox,Bool.false_eq_true,↓reduceIte,closure_prod_eq]
  · rw [closure_Ioo hrr.ne,closure_Ioo (neg_neg_of_pos hr).ne]
    exact ⟨⟨hp.1.1.le,hp.1.2.le⟩,by rw [hp₂]; exact ⟨(neg_neg_of_pos hr).le,le_rfl⟩⟩
  · rw [closure_Ioo hrr.ne,closure_Ioo hr.ne]
    exact ⟨⟨hp.1.1.le,hp.1.2.le⟩,by rw [hp₂]; exact ⟨le_rfl,hr.le⟩⟩

variable {X D : Type*} [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D]

theorem flat_chart_incidence {V W : Set X} {π : V → D} (hπ : Continuous π)
    (H : (ℝ × ℝ) ≃ₜ X) (hW : IsOpen W) (h₀ : H (0,0)∈W)
    (hflat : ∀ z∈W, z∈V ↔ (H.symm z).2≠0) :
    ∃ U : Set X, IsOpen U ∧ H (0,0)∈U ∧ U⊆W ∧
      ∃ labels : Bool → D, ∀ x∈U∩Vᶜ, incident V π x=range labels := by
  obtain ⟨r,hr,hball⟩ := Metric.mem_nhds_iff.mp
    ((hW.preimage H.continuous).mem_nhds h₀)
  let C := openBox r
  let U := H '' C
  have hCU : C⊆H ⁻¹' W := by rw [show C=openBox r from rfl,openBox_eq_ball]; exact hball
  have hUW : U⊆W := by rintro _ ⟨p,hp,rfl⟩; exact hCU hp
  have hU : IsOpen U := H.isOpenMap C (by rw [show C=openBox r from rfl,openBox_eq_ball]; exact isOpen_ball)
  have h₀U : H (0,0)∈U := ⟨(0,0),by rw [show C=openBox r from rfl,openBox_eq_ball]; exact mem_ball_self hr,rfl⟩
  let S : Bool → Set X := fun b => H '' halfBox r b
  have hSc (b : Bool) : IsConnected (S b) :=
    (halfBox_connected hr b).image H H.continuous.continuousOn
  have hcover : U∩V=⋃ b, S b := by
    ext x
    constructor
    · rintro ⟨⟨p,hp,rfl⟩,hpV⟩
      have hp₂ : p.2≠0 := by simpa using (hflat (H p) (hCU hp)).mp hpV
      rcases lt_or_gt_of_ne hp₂ with hl|hr'
      · exact mem_iUnion.mpr ⟨false,⟨p,⟨hp.1,hp.2.1,hl⟩,rfl⟩⟩
      · exact mem_iUnion.mpr ⟨true,⟨p,⟨hp.1,hr',hp.2.2⟩,rfl⟩⟩
    · intro hx
      obtain ⟨b,p,hp,rfl⟩ := mem_iUnion.mp hx
      have hpC : p∈C := by
        cases b
        · exact ⟨hp.1,hp.2.1,hp.2.2.trans hr⟩
        · exact ⟨hp.1,(neg_neg_of_pos hr).trans hp.2.1,hp.2.2⟩
      refine ⟨⟨p,hpC,rfl⟩,(hflat (H p) (hCU hpC)).mpr ?_⟩
      simp only [H.symm_apply_apply]
      cases b
      · exact hp.2.2.ne
      · exact hp.2.1.ne'
  have hclose (b : Bool) : U∩Vᶜ⊆closure (S b) := by
    rintro x ⟨⟨p,hp,hpx⟩,hxV⟩
    subst x
    have hp₂ : p.2=0 := by
      by_contra h
      apply hxV
      apply (hflat (H p) (hCU hp)).mpr
      simpa using h
    rw [show S b=H '' halfBox r b from rfl,←H.image_closure]
    exact ⟨p,halfBox_closure hr hp hp₂ b,rfl⟩
  obtain ⟨labels,_,hlabels⟩ := sector_incidence hU inter_subset_left hπ S hSc hcover hclose
  exact ⟨U,hU,h₀U,hUW,labels,hlabels⟩
end CannonFaceIncidence

end
end
end
end
end
end
end
end
end
end
end
end
end
end

end OAI