lean/OAI/Combinatorics/Hamiltonian/MinorModel.lean

import OAI.Combinatorics.Hamiltonian.Corridors

namespace OAI

/-! Clean finite walks, disjoint grown branches, and lattice minors. -/

namespace Barnette

noncomputable section

structure MinorModel {V W : Type*} (G : SimpleGraph V) (H : SimpleGraph W) where
  branch : V → Set W
  branch_finite : ∀ v, (branch v).Finite
  branch_connected : ∀ v, (H.induce (branch v)).Connected
  branches_disjoint : ∀ v w, v ≠ w → Disjoint (branch v) (branch w)
  adjacency : ∀ v w, G.Adj v w → ∃ a ∈ branch v, ∃ b ∈ branch w, H.Adj a b

namespace WalkSelection

variable {W : Type*} {H : SimpleGraph W}

theorem shortest_join {A B : Set W}
    (hex : ∃ a ∈ A, ∃ b ∈ B, H.Reachable a b) :
    ∃ (a : W) (_ha : a ∈ A) (b : W) (_hb : b ∈ B) (p : H.Walk a b),
      p.IsPath ∧ (∀ x ∈ p.support, x ∈ A → x = a) ∧
        (∀ x ∈ p.support, x ∈ B → x = b) := by
  classical
  let P : ℕ → Prop := fun n => ∃ a ∈ A, ∃ b ∈ B, ∃ p : H.Walk a b, p.length = n
  have hp : ∃ n, P n := by
    obtain ⟨a, ha, b, hb, ⟨p⟩⟩ := hex
    exact ⟨p.length, a, ha, b, hb, p, rfl⟩
  obtain ⟨a, ha, b, hb, p, hlen⟩ := Nat.find_spec hp
  have minimal {x y : W} (hx : x ∈ A) (hy : y ∈ B) (q : H.Walk x y) :
      p.length ≤ q.length := by
    rw [hlen]
    exact Nat.find_min' hp ⟨x, hx, y, hy, q, rfl⟩
  have hpath : p.IsPath := by
    obtain ⟨q, hq⟩ := p.reachable.exists_walk_length_eq_dist
    apply p.isPath_of_length_eq_dist
    exact le_antisymm (hq ▸ minimal ha hb q) (SimpleGraph.dist_le p)
  refine ⟨a, ha, b, hb, p, hpath, ?_, ?_⟩
  · intro x hx hxA
    by_contra hxa
    have hshort := p.length_dropUntil_lt_length hx hxa
    exact (not_lt_of_ge (minimal hxA hb (p.dropUntil x hx))) hshort
  · intro x hx hxB
    by_contra hxb
    have hshort := p.length_takeUntil_lt_length hx hxb
    exact (not_lt_of_ge (minimal ha hxB (p.takeUntil x hx))) hshort

structure CleanConnection (A B C : Set W) where
  start : W
  finish : W
  start_mem : start ∈ A
  finish_mem : finish ∈ B
  walk : H.Walk start finish
  isPath : walk.IsPath
  support_subset : ∀ x ∈ walk.support, x ∈ C
  clean_start : ∀ x ∈ walk.support, x ∈ A → x = start
  clean_finish : ∀ x ∈ walk.support, x ∈ B → x = finish

theorem exists_clean_connection {A B C : Set W}
    (hC : (H.induce C).Connected) (hA : (A ∩ C).Nonempty) (hB : (B ∩ C).Nonempty) :
    Nonempty (CleanConnection (H := H) A B C) := by
  classical
  obtain ⟨a, ha, haC⟩ := hA
  obtain ⟨b, hb, hbC⟩ := hB
  obtain ⟨u, hu, v, hv, p, hpath, hcleanA, hcleanB⟩ :=
    shortest_join (H := H.induce C) (A := {x | x.val ∈ A}) (B := {x | x.val ∈ B})
      ⟨⟨a, haC⟩, ha, ⟨b, hbC⟩, hb, hC _ _⟩
  let f : H.induce C →g H := ⟨Subtype.val, fun h => h⟩
  let q : H.Walk u.val v.val := p.map f
  have hs : q.support = p.support.map Subtype.val := p.support_map _
  refine ⟨{
    start := u.val
    finish := v.val
    start_mem := hu
    finish_mem := hv
    walk := q
    isPath := hpath.map Subtype.val_injective
    support_subset := ?_
    clean_start := ?_
    clean_finish := ?_
  }⟩
  · intro x hx
    rw [hs] at hx
    obtain ⟨z, _, rfl⟩ := List.mem_map.mp hx
    exact z.property
  · intro x hx hxA
    rw [hs] at hx
    obtain ⟨z, hz, rfl⟩ := List.mem_map.mp hx
    exact congrArg Subtype.val (hcleanA z hz hxA)
  · intro x hx hxB
    rw [hs] at hx
    obtain ⟨z, hz, rfl⟩ := List.mem_map.mp hx
    exact congrArg Subtype.val (hcleanB z hz hxB)

namespace CleanConnection
variable {A B C : Set W} (p : CleanConnection (H := H) A B C)

def leftSet : Set W := {x | x ∈ p.walk.dropLast.support}

theorem leftSet_finite : p.leftSet.Finite := p.walk.dropLast.support.finite_toSet

theorem leftSet_connected : (H.induce p.leftSet).Connected :=
  p.walk.dropLast.connected_induce_support

theorem start_mem_leftSet : p.start ∈ p.leftSet := p.walk.dropLast.start_mem_support

theorem leftSet_subset : p.leftSet ⊆ C := by
  intro x hx
  apply p.support_subset x
  exact (p.walk.isSubwalk_take _).support_subset hx

theorem leftSet_disjoint_finish (hAB : Disjoint A B) : Disjoint p.leftSet B := by
  have hne : p.start ≠ p.finish := by
    intro heq
    exact Set.disjoint_left.mp hAB p.start_mem (heq ▸ p.finish_mem)
  have hn : ¬p.walk.Nil := SimpleGraph.Walk.not_nil_of_ne hne
  have hd := p.isPath.support_nodup
  rw [← p.walk.support_dropLast_concat hn] at hd
  have hfinish : p.finish ∉ p.leftSet := by
    intro h
    exact hd.disjoint h (by simp)
  rw [Set.disjoint_left]
  intro x hx hxB
  have hs : x ∈ p.walk.support := (p.walk.isSubwalk_take _).support_subset hx
  have heq := p.clean_finish x hs hxB
  exact hfinish (heq ▸ hx)

theorem leftSet_adj_finish (hAB : Disjoint A B) :
    ∃ a ∈ p.leftSet, H.Adj a p.finish := by
  have hne : p.start ≠ p.finish := by
    intro heq
    exact Set.disjoint_left.mp hAB p.start_mem (heq ▸ p.finish_mem)
  exact ⟨p.walk.penultimate, p.walk.dropLast.end_mem_support,
    p.walk.adj_penultimate (SimpleGraph.Walk.not_nil_of_ne hne)⟩

end CleanConnection

end WalkSelection
end
end Barnette

namespace Barnette

namespace CorridorModel
noncomputable section
variable {V W : Type*} {G : SimpleGraph V} {H : SimpleGraph W}

private theorem exists_edgeDart (e : G.edgeSet) : ∃ d : G.Dart, d.edge = e.val := by
  rcases e with ⟨e, he⟩
  induction e using Sym2.inductionOn with
  | _ a b => exact ⟨⟨(a, b), he⟩, rfl⟩

def edgeDart (e : G.edgeSet) : G.Dart := Classical.choose (exists_edgeDart e)

@[simp] theorem edgeDart_edge (e : G.edgeSet) : (edgeDart e).edge = e.val :=
  Classical.choose_spec (exists_edgeDart e)

variable (M : CorridorModel G H)

def connection (e : G.edgeSet) : WalkSelection.CleanConnection (H := H)
    (M.branch (edgeDart e).fst) (M.branch (edgeDart e).snd) (M.corridor (edgeDart e)) :=
  Classical.choice (WalkSelection.exists_clean_connection
    (M.corridor_connected _) (M.incident_meets _).1 (M.incident_meets _).2)

def extension (e : G.edgeSet) : Set W := (M.connection e).leftSet

theorem extension_finite (e : G.edgeSet) : (M.extension e).Finite :=
  (M.connection e).leftSet_finite

theorem extension_connected (e : G.edgeSet) : (H.induce (M.extension e)).Connected :=
  (M.connection e).leftSet_connected

theorem extension_subset (e : G.edgeSet) : M.extension e ⊆ M.corridor (edgeDart e) :=
  (M.connection e).leftSet_subset

theorem extension_disjoint_branch (e : G.edgeSet) (v : V) (hv : v ≠ (edgeDart e).fst) :
    Disjoint (M.extension e) (M.branch v) := by
  by_cases hs : v = (edgeDart e).snd
  · subst v
    exact (M.connection e).leftSet_disjoint_finish
      (M.branches_disjoint _ _ (edgeDart e).fst_ne_snd)
  · exact ((M.nonincident_disjoint _ _ hv hs).mono_right (M.extension_subset e)).symm

theorem extensions_disjoint (e f : G.edgeSet) (hef : e ≠ f) :
    Disjoint (M.extension e) (M.extension f) := by
  have hne : (edgeDart e).edge ≠ (edgeDart f).edge := by
    simpa only [edgeDart_edge] using (fun h => hef (Subtype.ext h) : e.val ≠ f.val)
  exact (M.corridors_disjoint _ _ hne).mono (M.extension_subset e) (M.extension_subset f)

def grownBranch (v : V) : Set W := M.branch v ∪
  ⋃ e : {e : G.edgeSet // (edgeDart e).fst = v}, M.extension e.val

theorem branch_subset_grown (v : V) : M.branch v ⊆ M.grownBranch v :=
  Set.subset_union_left

theorem extension_subset_grown (e : G.edgeSet) : M.extension e ⊆ M.grownBranch (edgeDart e).fst :=
  fun _ hx => Or.inr (Set.mem_iUnion.mpr ⟨⟨e, rfl⟩, hx⟩)

theorem grownBranch_connected (v : V) : (H.induce (M.grownBranch v)).Connected := by
  obtain ⟨⟨a, ha⟩⟩ := (M.branch_connected v).nonempty
  apply H.induce_connected_of_patches a (M.branch_subset_grown v ha)
  intro x hx
  rcases hx with hx | hx
  · exact ⟨M.branch v, M.branch_subset_grown v, ha, hx, M.branch_connected v _ _⟩
  · obtain ⟨⟨e, he⟩, hx⟩ := Set.mem_iUnion.mp hx
    have hext : M.extension e ⊆ M.grownBranch v := by
      simpa only [he] using M.extension_subset_grown e
    have hi : (M.branch v ∩ M.extension e).Nonempty := by
      refine ⟨(M.connection e).start, ?_, (M.connection e).start_mem_leftSet⟩
      simpa only [he] using (M.connection e).start_mem
    exact ⟨M.branch v ∪ M.extension e,
      Set.union_subset (M.branch_subset_grown v) hext, Or.inl ha, Or.inr hx,
      SimpleGraph.induce_union_connected (M.branch_connected v).preconnected
        (M.extension_connected e).preconnected hi _ _⟩

theorem grownBranches_disjoint (v w : V) (hvw : v ≠ w) :
    Disjoint (M.grownBranch v) (M.grownBranch w) := by
  rw [Set.disjoint_left]
  intro x hx hy
  rcases hx with hx | hx <;> rcases hy with hy | hy
  · exact Set.disjoint_left.mp (M.branches_disjoint v w hvw) hx hy
  · obtain ⟨⟨e, he⟩, hy⟩ := Set.mem_iUnion.mp hy
    exact Set.disjoint_left.mp (M.extension_disjoint_branch e v (he ▸ hvw)) hy hx
  · obtain ⟨⟨e, he⟩, hx⟩ := Set.mem_iUnion.mp hx
    exact Set.disjoint_left.mp (M.extension_disjoint_branch e w (he ▸ hvw.symm)) hx hy
  · obtain ⟨⟨e, he⟩, hx⟩ := Set.mem_iUnion.mp hx
    obtain ⟨⟨f, hf⟩, hy⟩ := Set.mem_iUnion.mp hy
    have hef : e ≠ f := by
      intro h
      subst f
      exact hvw (he.symm.trans hf)
    exact Set.disjoint_left.mp (M.extensions_disjoint e f hef) hx hy

theorem grownBranch_adjacency (v w : V) (hvw : G.Adj v w) :
    ∃ a ∈ M.grownBranch v, ∃ b ∈ M.grownBranch w, H.Adj a b := by
  let e : G.edgeSet := ⟨s(v, w), hvw⟩
  have hAB := M.branches_disjoint _ _ (edgeDart e).fst_ne_snd
  obtain ⟨a, ha, hadj⟩ := (M.connection e).leftSet_adj_finish hAB
  have ha' : a ∈ M.grownBranch (edgeDart e).fst := M.extension_subset_grown e ha
  have hb' : (M.connection e).finish ∈ M.grownBranch (edgeDart e).snd :=
    M.branch_subset_grown _ (M.connection e).finish_mem
  have hd : (edgeDart e).edge = s(v, w) := edgeDart_edge e
  rcases SimpleGraph.dart_edge_eq_mk'_iff'.mp hd with ⟨h₁, h₂⟩ | ⟨h₂, h₁⟩
  · exact ⟨a, h₁ ▸ ha', (M.connection e).finish, h₂ ▸ hb', hadj⟩
  · exact ⟨(M.connection e).finish, h₁ ▸ hb', a, h₂ ▸ ha', hadj.symm⟩

variable [Fintype V]

theorem grownBranch_finite (v : V) : (M.grownBranch v).Finite := by
  classical
  exact (M.branch_finite v).union (Set.finite_iUnion (fun e => M.extension_finite e.val))

def toMinorModel : MinorModel G H where
  branch := M.grownBranch
  branch_finite := M.grownBranch_finite
  branch_connected := M.grownBranch_connected
  branches_disjoint := M.grownBranches_disjoint
  adjacency := M.grownBranch_adjacency

end
end CorridorModel
end Barnette

namespace Barnette.PlaneEmbedding
noncomputable section
variable {V : Type*} [Fintype V] {G : SimpleGraph V}

theorem exists_lattice_minor_model (E : PlaneEmbedding G) :
    Nonempty (MinorModel G SquareGrid.graph) := by
  classical
  obtain ⟨M⟩ := E.exists_lattice_corridor_model
  exact ⟨M.toMinorModel⟩
end
end Barnette.PlaneEmbedding

end OAI