| import OAI.Combinatorics.CompleteCrossing.Drawing |
| import Mathlib.Order.Fin.Basic |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| /-! Vertex deletion, its crossing-count identity, and small complete graphs. -/ |
|
|
| section |
|
|
| /-! Source-faithful restriction of an actual complete-graph drawing by one vertex. -/ |
| noncomputable section |
| namespace Paper170 |
|
|
| structure VertexDeletion {n : ℕ} (D : AdmissibleDrawing (completeGraph (n + 1))) |
| (v : Fin (n + 1)) (A : AdmissibleDrawing (completeGraph n)) : Prop where |
| vertex_eq : ∀ i, A.vertex i = D.vertex (v.succAbove i) |
| edge_eq : ∀ (e : CompleteEdge n) (f : CompleteEdge (n + 1)), |
| f.val.1 = v.succAbove e.val.1 → f.val.2 = v.succAbove e.val.2 → |
| ∀ t, A.edge e t = D.edge f t |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open scoped Classical |
| namespace Paper170 |
|
|
| lemma crossingCount_zero_of_subsingleton {V E : Type} [Subsingleton E] {G : Graph V E} |
| (D : AdmissibleDrawing G) : crossingCount D = 0 := by |
| have h : crossingPoints D.toContinuousDrawing = ∅ := by |
| ext x |
| constructor |
| · rintro ⟨e, f, hef, _, _⟩ |
| exact (hef (Subsingleton.elim e f)).elim |
| · simp |
| simp [crossingCount, h] |
|
|
| lemma ordinaryCrossingNumber_zero_of_subsingleton {V E : Type} [Subsingleton E] |
| (G : Graph V E) : ordinaryCrossingNumber G = 0 := by |
| by_cases h : (Set.range (@crossingCount V E G)).Nonempty |
| · obtain ⟨_, D, rfl⟩ := h |
| apply Nat.eq_zero_of_le_zero |
| rw [← crossingCount_zero_of_subsingleton D] |
| exact Paper170Geometry.ordinaryCrossingNumber_le_count D |
| · simp [ordinaryCrossingNumber, Set.not_nonempty_iff_eq_empty.mp h] |
|
|
| theorem complete_graph_small_orders : |
| ordinaryCrossingNumber (completeGraph 1) = 0 ∧ |
| ordinaryCrossingNumber (completeGraph 2) = 0 := by |
| constructor |
| · have : Subsingleton (CompleteEdge 1) := ⟨fun e f => by |
| have := e.property |
| have h1 := e.val.1.isLt |
| have h2 := e.val.2.isLt |
| omega⟩ |
| exact ordinaryCrossingNumber_zero_of_subsingleton _ |
| · have : Subsingleton (CompleteEdge 2) := ⟨fun e f => by |
| apply Subtype.ext |
| apply Prod.ext <;> apply Fin.ext |
| all_goals |
| have he := e.property |
| have hf := f.property |
| have he1 := e.val.1.isLt |
| have he2 := e.val.2.isLt |
| have hf1 := f.val.1.isLt |
| have hf2 := f.val.2.isLt |
| omega⟩ |
| exact ordinaryCrossingNumber_zero_of_subsingleton _ |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set |
| open scoped BigOperators Classical |
| namespace Paper170 |
|
|
| def deletionEdge {n : ℕ} (v : Fin (n + 1)) (e : CompleteEdge n) : CompleteEdge (n + 1) := |
| ⟨(v.succAbove e.val.1, v.succAbove e.val.2), Fin.strictMono_succAbove v e.property⟩ |
|
|
| lemma deletionEdge_injective {n : ℕ} (v : Fin (n + 1)) : |
| Function.Injective (deletionEdge v) := by |
| intro e f h |
| apply Subtype.ext |
| apply Prod.ext |
| · exact Fin.succAbove_right_injective (congrArg (fun a => a.val.1) h) |
| · exact Fin.succAbove_right_injective (congrArg (fun a => a.val.2) h) |
|
|
| lemma deletionEdge_range {n : ℕ} (v : Fin (n + 1)) (e : CompleteEdge (n + 1)) : |
| (∃ f, deletionEdge v f = e) ↔ e.val.1 ≠ v ∧ e.val.2 ≠ v := by |
| constructor |
| · rintro ⟨f, rfl⟩ |
| exact ⟨Fin.succAbove_ne v _, Fin.succAbove_ne v _⟩ |
| · rintro ⟨h1, h2⟩ |
| obtain ⟨a, ha⟩ := Fin.exists_succAbove_eq h1 |
| obtain ⟨b, hb⟩ := Fin.exists_succAbove_eq h2 |
| have hab : a < b := (Fin.strictMono_succAbove v).lt_iff_lt.mp (by |
| rw [ha, hb] |
| exact e.property) |
| exact ⟨⟨(a,b), hab⟩, Subtype.ext (Prod.ext ha hb)⟩ |
|
|
| def deleteContinuousDrawing {n : ℕ} (D : ContinuousDrawing (completeGraph (n + 1))) |
| (v : Fin (n + 1)) : ContinuousDrawing (completeGraph n) where |
| vertex i := D.vertex (v.succAbove i) |
| vertex_injective := D.vertex_injective.comp Fin.succAbove_right_injective |
| edge e := D.edge (deletionEdge v e) |
| edge_injective e := D.edge_injective (deletionEdge v e) |
| interior_avoids_vertices e t ht i := |
| D.interior_avoids_vertices (deletionEdge v e) t ht (v.succAbove i) |
|
|
| lemma deletion_crossings_subset {n : ℕ} (D : ContinuousDrawing (completeGraph (n + 1))) |
| (v : Fin (n + 1)) : crossingPoints (deleteContinuousDrawing D v) ⊆ crossingPoints D := by |
| rintro x ⟨e, f, hef, he, hf⟩ |
| exact ⟨deletionEdge v e, deletionEdge v f, |
| fun h => hef (deletionEdge_injective v h), he, hf⟩ |
|
|
| def deleteDrawing {n : ℕ} (D : AdmissibleDrawing (completeGraph (n + 1))) |
| (v : Fin (n + 1)) : AdmissibleDrawing (completeGraph n) where |
| toContinuousDrawing := deleteContinuousDrawing D.toContinuousDrawing v |
| crossings_finite := D.crossings_finite.subset (deletion_crossings_subset _ v) |
| crossings_proper e f hef x he hf := D.crossings_proper (deletionEdge v e) (deletionEdge v f) |
| (fun h => hef (deletionEdge_injective v h)) x he hf |
| no_triples e f g hef heg hfg := D.no_triples (deletionEdge v e) (deletionEdge v f) |
| (deletionEdge v g) (fun h => hef (deletionEdge_injective v h)) |
| (fun h => heg (deletionEdge_injective v h)) (fun h => hfg (deletionEdge_injective v h)) |
|
|
| lemma deleteDrawing_spec {n : ℕ} (D : AdmissibleDrawing (completeGraph (n + 1))) |
| (v : Fin (n + 1)) : VertexDeletion D v (deleteDrawing D v) := by |
| refine ⟨fun _ => rfl, ?_⟩ |
| intro e f h1 h2 t |
| have h : f = deletionEdge v e := Subtype.ext (Prod.ext h1 h2) |
| subst f |
| rfl |
|
|
| lemma deletion_crossing_iff {n : ℕ} (D : AdmissibleDrawing (completeGraph (n + 1))) |
| (v : Fin (n + 1)) (x : Plane) (e f : CompleteEdge (n + 1)) (hef : e ≠ f) |
| (he : x ∈ edgeInterior D.toContinuousDrawing e) |
| (hf : x ∈ edgeInterior D.toContinuousDrawing f) : |
| x ∈ crossingPoints (deleteDrawing D v).toContinuousDrawing ↔ |
| e.val.1 ≠ v ∧ e.val.2 ≠ v ∧ f.val.1 ≠ v ∧ f.val.2 ≠ v := by |
| constructor |
| · rintro ⟨a, b, hab, ha, hb⟩ |
| have ha' := (Paper170Geometry.crossing_incident_iff D x e f hef he hf |
| (deletionEdge v a)).mp ha |
| have hb' := (Paper170Geometry.crossing_incident_iff D x e f hef he hf |
| (deletionEdge v b)).mp hb |
| have hab' : deletionEdge v a ≠ deletionEdge v b := |
| fun h => hab (deletionEdge_injective v h) |
| have hae := (deletionEdge_range v (deletionEdge v a)).mp ⟨a, rfl⟩ |
| have hbe := (deletionEdge_range v (deletionEdge v b)).mp ⟨b, rfl⟩ |
| rcases ha' with ha' | ha' <;> rcases hb' with hb' | hb' |
| · exact (hab' (ha'.trans hb'.symm)).elim |
| · simpa [ha', hb', and_assoc] using And.intro hae hbe |
| · simpa [ha', hb', and_assoc] using And.intro hbe hae |
| · exact (hab' (ha'.trans hb'.symm)).elim |
| · rintro ⟨he1, he2, hf1, hf2⟩ |
| obtain ⟨a, ha⟩ := (deletionEdge_range v e).mpr ⟨he1, he2⟩ |
| obtain ⟨b, hb⟩ := (deletionEdge_range v f).mpr ⟨hf1, hf2⟩ |
| refine ⟨a, b, ?_, ?_, ?_⟩ |
| · intro h |
| exact hef (ha.symm.trans ((congrArg (deletionEdge v) h).trans hb)) |
| · change x ∈ edgeInterior D.toContinuousDrawing (deletionEdge v a) |
| simpa [ha] using he |
| · change x ∈ edgeInterior D.toContinuousDrawing (deletionEdge v b) |
| simpa [hb] using hf |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open Set |
| open scoped BigOperators Classical |
| namespace Paper170 |
|
|
| lemma crossing_endpoints_distinct {n : ℕ} (N : NormalizedDrawing (completeGraph n)) |
| (x : Plane) (e f : CompleteEdge n) (hef : e ≠ f) |
| (he : x ∈ edgeInterior N.toContinuousDrawing e) |
| (hf : x ∈ edgeInterior N.toContinuousDrawing f) : |
| e.val.1 ≠ f.val.1 ∧ e.val.1 ≠ f.val.2 ∧ e.val.2 ≠ f.val.1 ∧ e.val.2 ≠ f.val.2 := by |
| have hn : ¬ (e.val.1 = f.val.1 ∨ e.val.1 = f.val.2 ∨ |
| e.val.2 = f.val.1 ∨ e.val.2 = f.val.2) := by |
| intro h |
| exact Set.disjoint_left.mp (N.no_adjacent e f hef h) he hf |
| tauto |
|
|
| lemma deletion_survivor_card {n : ℕ} (N : NormalizedDrawing (completeGraph (n + 1))) |
| (x : Plane) (hx : x ∈ crossingPoints N.toContinuousDrawing) : |
| (Finset.univ.filter (fun v : Fin (n + 1) => |
| x ∈ crossingPoints (deleteDrawing N.toAdmissibleDrawing v).toContinuousDrawing)).card = |
| n + 1 - 4 := by |
| obtain ⟨e, f, hef, he, hf⟩ := hx |
| rcases crossing_endpoints_distinct N x e f hef he hf with ⟨h13, h14, h23, h24⟩ |
| have h12 : e.val.1 ≠ e.val.2 := ne_of_lt e.property |
| have h34 : f.val.1 ≠ f.val.2 := ne_of_lt f.property |
| simp_rw [deletion_crossing_iff N.toAdmissibleDrawing _ x e f hef he hf] |
| have hset : Finset.univ.filter (fun v : Fin (n + 1) => |
| e.val.1 ≠ v ∧ e.val.2 ≠ v ∧ f.val.1 ≠ v ∧ f.val.2 ≠ v) = |
| Finset.univ \ {e.val.1, e.val.2, f.val.1, f.val.2} := by |
| ext v |
| simp [eq_comm] |
| rw [hset, Finset.card_sdiff_of_subset (Finset.subset_univ _)] |
| simp [h12, h13, h14, h23, h24, h34] |
|
|
| lemma deleteDrawing_count_filter {n : ℕ} (D : AdmissibleDrawing (completeGraph (n + 1))) |
| (v : Fin (n + 1)) : |
| crossingCount (deleteDrawing D v) = (D.crossings_finite.toFinset.filter fun x => |
| x ∈ crossingPoints (deleteDrawing D v).toContinuousDrawing).card := by |
| unfold crossingCount |
| congr 1 |
| ext x |
| simp only [Set.Finite.mem_toFinset, Finset.mem_filter] |
| exact ⟨fun h => ⟨deletion_crossings_subset D.toContinuousDrawing v h, h⟩, And.right⟩ |
|
|
| theorem vertex_deletion_count (n : ℕ) (_hn : 4 ≤ n) |
| (N : NormalizedDrawing (completeGraph (n + 1))) : |
| ∃ A : Fin (n + 1) → AdmissibleDrawing (completeGraph n), |
| (∀ v, VertexDeletion N.toAdmissibleDrawing v (A v)) ∧ |
| (∑ v, crossingCount (A v)) = (n + 1 - 4) * crossingCount N.toAdmissibleDrawing := by |
| refine ⟨deleteDrawing N.toAdmissibleDrawing, deleteDrawing_spec N.toAdmissibleDrawing, ?_⟩ |
| simp_rw [deleteDrawing_count_filter, Finset.card_filter] |
| rw [Finset.sum_comm] |
| calc |
| (∑ x ∈ N.crossings_finite.toFinset, |
| ∑ v : Fin (n + 1), if x ∈ crossingPoints |
| (deleteDrawing N.toAdmissibleDrawing v).toContinuousDrawing then 1 else 0) = |
| ∑ _x ∈ N.crossings_finite.toFinset, (n + 1 - 4) := by |
| apply Finset.sum_congr rfl |
| intro x hx |
| simpa only [Finset.card_filter] using |
| deletion_survivor_card N x (by simpa using hx) |
| _ = (n + 1 - 4) * crossingCount N.toAdmissibleDrawing := by |
| simp [crossingCount, Nat.mul_comm] |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
