lean/OAI/Geometry/Cannon/PlanarGraphs/TraceCard.lean

import Mathlib
import OAI.Geometry.Cannon.Boundary.BoundaryLocallyConnected
import OAI.Geometry.Cannon.PlaneTopology.PullChartUnionCompl
import OAI.Geometry.Cannon.Topology.ComponentRegionConnected
import OAI.Geometry.Cannon.Topology.Plane

namespace OAI

section
section
section
noncomputable section
open Set unitInterval Topology

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open Set Topology unitInterval

noncomputable section
open Set Metric Topology unitInterval

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open Set Metric Topology

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open Set unitInterval

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noncomputable section
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

namespace CannonIncidence
open Set Metric

variable {X ι : Type*} [MetricSpace X] [Fintype ι]

noncomputable def traceCard (E : ι → Set X) (S : Set X) : ℕ := by
  classical
  exact (Finset.univ.filter (fun i => (E i∩S).Nonempty)).card

omit [MetricSpace X] in
lemma traceCard_mono (E : ι → Set X) {S T : Set X} (h : S⊆T) :
    traceCard E S≤traceCard E T := by
  classical
  apply Finset.card_le_card
  intro i hi
  obtain ⟨_,x,hxE,hxS⟩ := Finset.mem_filter.mp hi
  exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,x,hxE,h hxS⟩

omit [MetricSpace X] in
lemma traceCard_finite_cover (E : ι → Set X) (T : Finset X) (U : X → Set X)
    (S : Set X) (hcover : S⊆⋃ x∈T,U x) (K : ℕ)
    (hK : ∀ x∈T, traceCard E (U x)≤K) : traceCard E S≤T.card*K := by
  classical
  let IX (S : Set X) := Finset.univ.filter (fun i => (E i∩S).Nonempty)
  have hsub : IX S⊆T.biUnion (fun x => IX (U x)) := by
    intro i hi
    obtain ⟨y,hyE,hyS⟩ := (Finset.mem_filter.mp hi).2
    obtain ⟨x,hx,hyU⟩ := mem_iUnion₂.mp (hcover hyS)
    exact Finset.mem_biUnion.mpr ⟨x,hx,Finset.mem_filter.mpr ⟨Finset.mem_univ _,y,hyE,hyU⟩⟩
  calc
    traceCard E S=(IX S).card := rfl
    _≤(T.biUnion (fun x => IX (U x))).card := Finset.card_le_card hsub
    _≤∑ x∈T,(IX (U x)).card := Finset.card_biUnion_le
    _≤∑ _x∈T,K := Finset.sum_le_sum (fun x hx => hK x hx)
    _=T.card*K := by simp

theorem uniform_larger_trace (Q : ℕ) (C R : ℝ)
    (hpack : CannonPacking.PolynomialPacking (X:=X) Q C) (hC : 0≤C) (hR : 0<R) :
    ∃ B : ℕ, ∀ (r A : ℝ), 0<r → 1≤A → (R+1)*r≤1 →
      ∀ T : Finset X, (∀ x∈T, ∀ y∈T, x≠y → r≤dist x y) →
      (∀ x : X, ∃ t∈T, dist x t≤r) →
      ∀ {κ : Type*} [Fintype κ], ∀ E : κ → Set X, ∀ K : ℕ, (∀ x, traceCard E (ball x (2*A*r))≤K) →
      ∀ x, traceCard E (ball x (R*r))≤B*K := by
  classical
  obtain ⟨B,hB⟩ := CannonPacking.exists_overlap_bound Q C (R+1) 1 hpack hC (by linarith) zero_lt_one
  refine ⟨B,?_⟩
  intro r A hr hA hscale T hsep hcover κ _ E K hK x
  let T' := T.filter (fun t => dist x t≤(R+1)*r)
  have hTc : T'.card≤B := by
    apply hB id T x r hr
    · simpa [max_eq_left (by linarith : (1:ℝ)≤R+1)] using hscale
    · simpa using hsep
  have hcov : ball x (R*r)⊆⋃ t∈T',ball t (2*A*r) := by
    intro y hy
    obtain ⟨t,ht,hyT⟩ := hcover y
    refine mem_iUnion₂.mpr ⟨t,Finset.mem_filter.mpr ⟨ht,?_⟩,?_⟩
    · have hh := dist_triangle x y t
      have hyx : dist x y<R*r := by simpa only [mem_ball,dist_comm] using hy
      linarith
    · change dist y t<2*A*r
      have hh : 2*r≤2*A*r := by nlinarith
      linarith
  exact (traceCard_finite_cover E T' (fun t => ball t (2*A*r)) _ hcov K
    (fun t _ => hK t)).trans (Nat.mul_le_mul_right K hTc)
end CannonIncidence

namespace CannonRelease
open Set Metric Topology unitInterval CannonGraph CannonRegularDisk CannonEmbeddedArc CannonEmbeddedArc.Arc

  CannonFaceIncidence CannonFaceLabels CannonPlaneLinear CannonIncidence
variable {G : Type*} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)

theorem boundary_uniform_controlled_faces (φ : Boundary D ≃ₜ SphereTwo) :
    let _ := visualMetricSpace D hδ
    ∃ A L r₀ : ℝ, 1≤A ∧ 1≤L ∧ 0<r₀ ∧ ∃ K J : ℕ,
      ∀ r : ℝ, 0<r → r≤r₀ → ∀ T : Finset (Boundary D),
      (∀ x∈T, ∀ y∈T, x≠y → r≤dist x y) →
      (∀ x : Boundary D, ∃ y∈T, dist x y≤r) →
      ∀ p : Boundary D, p∉T →
      ∃ e : ℂ → Boundary D, IsOpenEmbedding e ∧ range e={p}ᶜ ∧
      ∃ Q : Finset (ℂ × ℂ), ∃ n : ℕ, 0<n ∧ ∃ F : Fin n → Arc (Boundary D),
        Pairwise (fun i j => Compatible (F i) (F j)) ∧
        (∀ j, diam (range (F j))≤A*r) ∧
        (∀ x, intersectionCount F (ball x (2*A*r))≤K) ∧
        (∀ x, intersectionCount F (ball x ((L+1)*r))≤J) ∧
        IsPathConnected (⋃ j,range (F j)) ∧ (T:Set (Boundary D))⊆(⋃ j,range (F j)) ∧
        (⋃ j,range (F j))⊆e '' (⋃ q∈Q,segment ℝ q.1 q.2) ∧
        Finite (ConnectedComponents ((⋃ j,range (F j))ᶜ : Set (Boundary D))) ∧
        (∀ R : Set (Boundary D), IsPreconnected R → R⊆(⋃ j,range (F j))ᶜ →
          ∀ x∈R, ∀ y∈R, dist x y<L*r) ∧
        (let Faces := ConnectedComponents ((⋃ j,range (F j))ᶜ : Set (Boundary D))
         let π : ((⋃ j,range (F j))ᶜ : Set (Boundary D)) → Faces := ConnectedComponents.mk
         ∃ (S : Faces → Set (Fin n)) (labels : Fin n → Bool → Faces),
           (∀ d, closure (region _ π d)=region _ π d∪⋃ i∈S d,range (F i)) ∧
           (∀ d i, i∈S d ↔ d∈range (labels i)) ∧
           (∀ d, Nat.card (S d)≤J)) := by
  classical
  let _ := visualMetricSpace D hδ
  let _ : LocallyPathConnectedSpace SphereTwo := CannonPaths.sphere_locallyPathConnected
  let _ : LocallyConnectedSpace (Boundary D) := φ.locallyConnectedSpace
  let _ : PathConnectedSpace SphereTwo := CannonPaths.sphere_pathConnected
  let _ : PathConnectedSpace (Boundary D) := φ.symm.surjective.pathConnectedSpace φ.symm.continuous
  let s : SphereTwo := Classical.choice inferInstance
  let _ : Nontrivial (Boundary D) := ⟨⟨φ.symm s,φ.symm (-s),
    fun h => (ne_neg_of_mem_unit_sphere ℝ s) (φ.symm.injective h)⟩⟩
  obtain ⟨a,b,hab⟩ := exists_pair_ne (Boundary D)
  have hdist : 0<dist a b := dist_pos.mpr hab
  dsimp only
  obtain ⟨A,L,rG,hA,hL,hrG,K,hgraph⟩ := boundary_uniform_finite_face_graph D hδ φ
  obtain ⟨Q₀,C,hQ₀,hC,hpoly⟩ := relative_polynomial_packing D hδ
  obtain ⟨B,hB⟩ := uniform_larger_trace Q₀ C (L+1) hpoly hC.le (by linarith)
  refine ⟨A,L,min rG (min (dist a b/4) (1/(L+2))),hA,hL,
    lt_min hrG (lt_min (by positivity) (by positivity)),K,B*K,?_⟩
  intro r hr hrs T hsep hcover p hpT
  have hrG' : r≤rG := hrs.trans (min_le_left ..)
  have hrD : r≤dist a b/4 := hrs.trans ((min_le_right ..).trans (min_le_left ..))
  have hrL : r≤1/(L+2) := hrs.trans ((min_le_right ..).trans (min_le_right ..))
  have hscale : (L+2)*r≤1 := by have hh := (le_div_iff₀ (by linarith : 0<L+2)).mp hrL; nlinarith
  have hTnt : (T:Set (Boundary D)).Nontrivial := by
    obtain ⟨u,hu,hau⟩ := hcover a
    obtain ⟨v,hv,hbv⟩ := hcover b
    refine ⟨u,hu,v,hv,?_⟩
    intro heq
    subst v
    have hh := dist_triangle a u b
    rw [dist_comm u b] at hh
    linarith
  obtain ⟨e₀,he₀,he₀R,P,n,F,hFc,hFd,hFt,hconn,hFS,hfinite,hsmall⟩ :=
    hgraph r hr hrG' T hsep hcover p hpT
  let V : Set (Boundary D) := ⋃ j,range (F j)
  have hV : IsClosed V := isClosed_iUnion_of_finite (fun j => (F j).compact_range.isClosed)
  have hTV : (T:Set (Boundary D))⊆V := CannonConnectedSupport.finite_extras_absorbed
    hV T.finite_toSet hconn.isConnected.isPreconnected (hTnt.mono subset_union_right)
  have heq : V∪(T:Set (Boundary D))=V := union_eq_left.mpr hTV
  change IsPathConnected (V∪(T:Set (Boundary D))) at hconn
  change V∪(T:Set (Boundary D))⊆e₀ '' _ at hFS
  change Finite (ConnectedComponents ((V∪(T:Set (Boundary D)))ᶜ : Set (Boundary D))) at hfinite
  change ∀ R : Set (Boundary D), IsPreconnected R → R⊆(V∪(T:Set (Boundary D)))ᶜ →
    ∀ x∈R, ∀ y∈R, dist x y<L*r at hsmall
  rw [heq] at hconn hFS hfinite hsmall
  have hn : 0<n := by
    by_contra hn
    have hn0 : n=0 := Nat.eq_zero_of_not_pos hn
    obtain ⟨x,hx⟩ := hTnt.nonempty
    obtain ⟨i,hi⟩ := mem_iUnion.mp (hTV hx)
    exact (Nat.not_lt_zero i.val) (hn0 ▸ i.isLt)
  let e : ℂ → Boundary D := e₀ ∘ complexPlaneL
  have he : IsOpenEmbedding e := he₀.comp complexPlaneL.toHomeomorph.isOpenEmbedding
  have heR : range e={p}ᶜ := by rw [show range e=range e₀ from (complexPlaneL.surjective.range_comp e₀),he₀R]
  obtain ⟨Q,hQ⟩ := finite_segments_pull P
  have hVS : V⊆e '' (⋃ q∈Q,segment ℝ q.1 q.2) := by
    intro x hx
    obtain ⟨y,hy,rfl⟩ := hFS hx
    refine ⟨complexPlaneL.symm y,?_,by simp [e,Function.comp_def]⟩
    rw [←hQ]
    simpa using hy
  have hrange (i : Fin n) : range (F i)⊆range e :=
    (subset_iUnion (fun j => range (F j)) i).trans (hVS.trans (image_subset_range ..))
  have hQs (i : Fin n) : e ⁻¹' range (F i)⊆⋃ q∈Q,segment ℝ q.1 q.2 := by
    intro z hz
    obtain ⟨w,hw,heq⟩ := hVS (mem_iUnion.mpr ⟨i,hz⟩)
    exact he.injective heq ▸ hw
  have hwide (x : Boundary D) : intersectionCount F (ball x ((L+1)*r))≤B*K := by
    have hh := hB r A hr hA (by convert hscale using 1; ring) T hsep hcover
      (fun i => range (F i)) K (fun x => by simpa only [traceCard,intersectionCount_eq_card] using hFt x) x
    simpa only [traceCard,intersectionCount_eq_card] using hh
  let Faces := ConnectedComponents (Vᶜ : Set (Boundary D))
  let _ : DiscreteTopology Faces := complement_components_discrete hV
  let π : (Vᶜ : Set (Boundary D)) → Faces := ConnectedComponents.mk
  have hπ : Continuous π := ConnectedComponents.continuous_coe
  obtain ⟨S,labels,hcl,hlabels⟩ := chart_tame_face_structure e he F hrange hFc Q hQs hπ
  refine ⟨e,he,heR,Q,n,hn,F,hFc,hFd,hFt,hwide,hconn,hTV,hVS,hfinite,hsmall,
    S,labels,hcl,hlabels,?_⟩
  intro d
  have hR := component_region_connected Vᶜ d
  obtain ⟨x,hx⟩ := hR.nonempty
  have hRd : region Vᶜ π d⊆closedBall x (L*r) := by
    intro y hy
    simpa only [mem_closedBall,dist_comm] using (hsmall _ hR.isPreconnected (region_subset Vᶜ π d) x hx y hy).le
  have hclose : closure (region Vᶜ π d)⊆closedBall x (L*r) :=
    closure_minimal hRd isClosed_closedBall
  have hsub : S d⊆{i | (range (F i)∩ball x ((L+1)*r)).Nonempty} := by
    intro i hi
    refine ⟨F i 0,mem_range_self 0,?_⟩
    have hh : F i 0∈closure (region Vᶜ π d) := by
      rw [hcl d]
      exact Or.inr (mem_iUnion₂.mpr ⟨i,hi,mem_range_self 0⟩)
    have hd := hclose hh
    change dist (F i 0) x≤L*r at hd
    change dist (F i 0) x<(L+1)*r
    nlinarith
  calc
    Nat.card (S d)≤Nat.card {i | (range (F i)∩ball x ((L+1)*r)).Nonempty} :=
      Nat.card_le_card_of_injective (Set.inclusion hsub) (Set.inclusion_injective hsub)
    _=intersectionCount F (ball x ((L+1)*r)) := by
      simp [intersectionCount_eq_card,Nat.card_eq_fintype_card,Fintype.card_subtype]
    _≤B*K := hwide x
end CannonRelease

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

end OAI