| import OAI.Combinatorics.CompleteCrossing.CirclePlacement |
| import Mathlib.Data.Fin.VecNotation |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| /-! Counting actual crossing points of the explicit two-page drawing. -/ |
|
|
| section |
|
|
| noncomputable section |
| open Set |
| open scoped Classical |
| namespace Paper170 |
|
|
| lemma twoPage_meeting_chords {n : ℕ} (C : CircleConfiguration n) |
| (D : AdmissibleDrawing (completeGraph n)) (hD : TwoPageDrawing C D) |
| (e f : CompleteEdge n) (x : Plane) |
| (he : x ∈ edgeInterior D.toContinuousDrawing e) |
| (hf : x ∈ edgeInterior D.toContinuousDrawing f) : |
| residueColor n (e.val.1.val + e.val.2.val) = residueColor n (f.val.1.val + f.val.2.val) ∧ |
| ∃ y, y ∈ chordInterior C.point e ∩ chordInterior C.point f ∧ |
| x = if residueColor n (e.val.1.val + e.val.2.val) then y else inversion y := by |
| rw [twoPage_edgeInterior C D hD e] at he |
| rw [twoPage_edgeInterior C D hD f] at hf |
| cases hec : residueColor n (e.val.1.val + e.val.2.val) <;> |
| cases hfc : residueColor n (f.val.1.val + f.val.2.val) |
| · simp only [hec, Bool.false_eq_true, ↓reduceIte] at he |
| simp only [hfc, Bool.false_eq_true, ↓reduceIte] at hf |
| obtain ⟨y, hy, rfl⟩ := he |
| obtain ⟨z, hz, hzy⟩ := hf |
| have hzy' : z = y := inversion_involutive.injective hzy |
| subst z |
| exact ⟨rfl, y, ⟨hy, hz⟩, rfl⟩ |
| · simp only [hec, Bool.false_eq_true, ↓reduceIte] at he |
| simp only [hfc, ↓reduceIte] at hf |
| have h1 := C.inverted_chord_outside e he |
| have h2 := C.chord_inside f hf |
| linarith |
| · simp only [hec, ↓reduceIte] at he |
| simp only [hfc, Bool.false_eq_true, ↓reduceIte] at hf |
| have h1 := C.chord_inside e he |
| have h2 := C.inverted_chord_outside f hf |
| linarith |
| · simp only [hec, ↓reduceIte] at he |
| simp only [hfc, ↓reduceIte] at hf |
| exact ⟨rfl, x, ⟨he, hf⟩, rfl⟩ |
|
|
| lemma chord_meeting_alternates {n : ℕ} (C : CircleConfiguration n) |
| (e f : CompleteEdge n) (hef : e ≠ f) (y : Plane) |
| (hy : y ∈ chordInterior C.point e ∩ chordInterior C.point f) : |
| ((e.val.1 < f.val.1 ∧ f.val.1 < e.val.2 ∧ e.val.2 < f.val.2) ∨ |
| (f.val.1 < e.val.1 ∧ e.val.1 < f.val.2 ∧ f.val.2 < e.val.2)) ∧ |
| y = lineIntersectionPoint (C.point e.val.1) (C.point e.val.2) |
| (C.point f.val.1) (C.point f.val.2) := by |
| rw [circle_chord_intersection C e f hef] at hy |
| split_ifs at hy with h |
| · exact ⟨h, hy⟩ |
| · exact hy.elim |
|
|
| lemma twoPage_meeting_alternates {n : ℕ} (C : CircleConfiguration n) |
| (D : AdmissibleDrawing (completeGraph n)) (hD : TwoPageDrawing C D) |
| (e f : CompleteEdge n) (hef : e ≠ f) (x : Plane) |
| (he : x ∈ edgeInterior D.toContinuousDrawing e) |
| (hf : x ∈ edgeInterior D.toContinuousDrawing f) : |
| residueColor n (e.val.1.val + e.val.2.val) = residueColor n (f.val.1.val + f.val.2.val) ∧ |
| ((e.val.1 < f.val.1 ∧ f.val.1 < e.val.2 ∧ e.val.2 < f.val.2) ∨ |
| (f.val.1 < e.val.1 ∧ e.val.1 < f.val.2 ∧ f.val.2 < e.val.2)) ∧ |
| x = if residueColor n (e.val.1.val + e.val.2.val) then |
| lineIntersectionPoint (C.point e.val.1) (C.point e.val.2) (C.point f.val.1) (C.point f.val.2) |
| else inversion (lineIntersectionPoint (C.point e.val.1) (C.point e.val.2) |
| (C.point f.val.1) (C.point f.val.2)) := by |
| obtain ⟨hc, y, hy, hx⟩ := twoPage_meeting_chords C D hD e f x he hf |
| obtain ⟨ho, rfl⟩ := chord_meeting_alternates C e f hef y hy |
| exact ⟨hc, ho, hx⟩ |
|
|
| lemma pagePoint_mem_edge {n : ℕ} (C : CircleConfiguration n) |
| (D : AdmissibleDrawing (completeGraph n)) (hD : TwoPageDrawing C D) |
| (e : CompleteEdge n) (y : Plane) (hy : y ∈ chordInterior C.point e) : |
| (if residueColor n (e.val.1.val + e.val.2.val) then y else inversion y) ∈ |
| edgeInterior D.toContinuousDrawing e := by |
| rw [twoPage_edgeInterior C D hD] |
| cases h : residueColor n (e.val.1.val + e.val.2.val) |
| · exact ⟨y, hy, rfl⟩ |
| · exact hy |
|
|
| lemma ordered_chord_meeting {n : ℕ} (C : CircleConfiguration n) |
| (e f : CompleteEdge n) |
| (ho : e.val.1 < f.val.1 ∧ f.val.1 < e.val.2 ∧ e.val.2 < f.val.2) : |
| lineIntersectionPoint (C.point e.val.1) (C.point e.val.2) |
| (C.point f.val.1) (C.point f.val.2) ∈ chordInterior C.point e ∩ chordInterior C.point f := by |
| have hef : e ≠ f := by intro h; subst f; exact (lt_irrefl _ ho.1) |
| rw [circle_chord_intersection C e f hef, ite_eq_left (Or.inl ho)] |
| exact Set.mem_singleton _ |
| |
| end Paper170 |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open Set |
| open scoped Classical |
| namespace Paper170 |
| |
| def tupleFirstEdge {n : ℕ} (v : Fin 4 → Fin n) (hv : StrictMono v) : CompleteEdge n := |
| ⟨(v 0, v 2), hv (by decide)⟩ |
|
|
| def tupleSecondEdge {n : ℕ} (v : Fin 4 → Fin n) (hv : StrictMono v) : CompleteEdge n := |
| ⟨(v 1, v 3), hv (by decide)⟩ |
|
|
| lemma tuple_edges_ne {n : ℕ} (v : Fin 4 → Fin n) (hv : StrictMono v) : |
| tupleFirstEdge v hv ≠ tupleSecondEdge v hv := by |
| intro h |
| exact (ne_of_lt (hv (by decide : (0 : Fin 4) < 1))) |
| (congrArg (fun e => e.val.1) h) |
|
|
| lemma tuple_point_membership {n : ℕ} (C : CircleConfiguration n) |
| (D : AdmissibleDrawing (completeGraph n)) (hD : TwoPageDrawing C D) |
| (v : Fin 4 → Fin n) (hv : v ∈ crossingTuples n) : |
| let hm := (Finset.mem_filter.mp hv).2.1 |
| crossingTuplePoint C v ∈ edgeInterior D.toContinuousDrawing (tupleFirstEdge v hm) ∧ |
| crossingTuplePoint C v ∈ edgeInterior D.toContinuousDrawing (tupleSecondEdge v hm) := by |
| let hm := (Finset.mem_filter.mp hv).2.1 |
| have hc := (Finset.mem_filter.mp hv).2.2 |
| have hy := ordered_chord_meeting C (tupleFirstEdge v hm) (tupleSecondEdge v hm) |
| ⟨hm (by decide : (0 : Fin 4) < 1), hm (by decide : (1 : Fin 4) < 2), |
| hm (by decide : (2 : Fin 4) < 3)⟩ |
| constructor |
| · exact pagePoint_mem_edge C D hD (tupleFirstEdge v hm) _ hy.1 |
| · have h := pagePoint_mem_edge C D hD (tupleSecondEdge v hm) _ hy.2 |
| change (if residueColor n ((v 1).val + (v 3).val) then _ else _) ∈ _ at h |
| rw [← hc] at h |
| exact h |
|
|
| lemma ordered_pair_tuple {n : ℕ} (C : CircleConfiguration n) (e f : CompleteEdge n) |
| (ho : e.val.1 < f.val.1 ∧ f.val.1 < e.val.2 ∧ e.val.2 < f.val.2) |
| (hc : residueColor n (e.val.1.val + e.val.2.val) = residueColor n (f.val.1.val + f.val.2.val)) : |
| ∃ v ∈ crossingTuples n, crossingTuplePoint C v = |
| if residueColor n (e.val.1.val + e.val.2.val) then |
| lineIntersectionPoint (C.point e.val.1) (C.point e.val.2) (C.point f.val.1) (C.point f.val.2) |
| else inversion (lineIntersectionPoint (C.point e.val.1) (C.point e.val.2) |
| (C.point f.val.1) (C.point f.val.2)) := by |
| let v : Fin 4 → Fin n := ![e.val.1, f.val.1, e.val.2, f.val.2] |
| refine ⟨v, Finset.mem_filter.mpr ⟨Finset.mem_univ _, ?_, ?_⟩, rfl⟩ |
| · apply Fin.strictMono_iff_lt_succ.mpr |
| intro i |
| fin_cases i |
| · exact ho.1 |
| · exact ho.2.1 |
| · exact ho.2.2 |
| · exact hc |
|
|
| lemma crossingTuplePoint_surjOn {n : ℕ} (C : CircleConfiguration n) |
| (D : AdmissibleDrawing (completeGraph n)) (hD : TwoPageDrawing C D) : |
| SurjOn (crossingTuplePoint C) (crossingTuples n : Set (Fin 4 → Fin n)) |
| (crossingPoints D.toContinuousDrawing) := by |
| rintro x ⟨e, f, hef, he, hf⟩ |
| obtain ⟨hc, ho, hx⟩ := twoPage_meeting_alternates C D hD e f hef x he hf |
| rcases ho with ho | ho |
| · obtain ⟨v, hv, hp⟩ := ordered_pair_tuple C e f ho hc |
| exact ⟨v, hv, hp.trans hx.symm⟩ |
| · obtain ⟨_, _, hx'⟩ := twoPage_meeting_alternates C D hD f e hef.symm x hf he |
| obtain ⟨v, hv, hp⟩ := ordered_pair_tuple C f e ho hc.symm |
| exact ⟨v, hv, hp.trans hx'.symm⟩ |
|
|
| lemma crossingTuplePoint_injOn {n : ℕ} (C : CircleConfiguration n) |
| (D : AdmissibleDrawing (completeGraph n)) (hD : TwoPageDrawing C D) : |
| InjOn (crossingTuplePoint C) (crossingTuples n : Set (Fin 4 → Fin n)) := by |
| intro v hv w hw hpoint |
| let hv' := (Finset.mem_filter.mp hv).2.1 |
| let hw' := (Finset.mem_filter.mp hw).2.1 |
| have hpv := tuple_point_membership C D hD v hv |
| have hpw := tuple_point_membership C D hD w hw |
| rw [← hpoint] at hpw |
| have hinc := Paper170Geometry.crossing_incident_iff D (crossingTuplePoint C v) |
| (tupleFirstEdge v hv') (tupleSecondEdge v hv') (tuple_edges_ne v hv') hpv.1 hpv.2 |
| have he := (hinc (tupleFirstEdge w hw')).mp hpw.1 |
| have hf := (hinc (tupleSecondEdge w hw')).mp hpw.2 |
| rcases he with he | he <;> rcases hf with hf | hf |
| · exact (tuple_edges_ne w hw' (he.trans hf.symm)).elim |
| · have h0 := congrArg (fun e => e.val.1) he |
| have h2 := congrArg (fun e => e.val.2) he |
| have h1 := congrArg (fun e => e.val.1) hf |
| have h3 := congrArg (fun e => e.val.2) hf |
| funext i |
| fin_cases i |
| · exact h0.symm |
| · exact h1.symm |
| · exact h2.symm |
| · exact h3.symm |
| · have h0 : w 0 = v 1 := congrArg (fun e => e.val.1) he |
| have h1 : w 1 = v 0 := congrArg (fun e => e.val.1) hf |
| have hlt := hw' (by decide : (0 : Fin 4) < 1) |
| rw [h0, h1] at hlt |
| exact (lt_asymm (hv' (by decide : (0 : Fin 4) < 1)) hlt).elim |
| · exact (tuple_edges_ne w hw' (he.trans hf.symm)).elim |
|
|
| theorem two_page_crossing_bijection {n : ℕ} (C : CircleConfiguration n) |
| (D : AdmissibleDrawing (completeGraph n)) (hD : TwoPageDrawing C D) : |
| BijOn (crossingTuplePoint C) (crossingTuples n : Set (Fin 4 → Fin n)) |
| (crossingPoints D.toContinuousDrawing) := by |
| refine ⟨?_, crossingTuplePoint_injOn C D hD, crossingTuplePoint_surjOn C D hD⟩ |
| intro v hv |
| have hp := tuple_point_membership C D hD v hv |
| exact ⟨_, _, tuple_edges_ne _ _, hp.1, hp.2⟩ |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace Paper170 |
|
|
| theorem two_page_crossing_count {n : ℕ} (C : CircleConfiguration n) |
| (D : AdmissibleDrawing (completeGraph n)) (hD : TwoPageDrawing C D) : |
| crossingCount D = (crossingTuples n).card := by |
| have h := (two_page_crossing_bijection C D hD).ncard_eq |
| simpa [crossingCount, Set.ncard_eq_toFinset_card _ D.crossings_finite] using h.symm |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace Paper170 |
|
|
| theorem complete_graph_upper_drawing (n : ℕ) : |
| ∃ D : AdmissibleDrawing (completeGraph n), crossingCount D = (crossingTuples n).card := by |
| obtain ⟨C⟩ := circle_configuration_exists n |
| obtain ⟨D, hD⟩ := two_page_drawing_exists C |
| exact ⟨D, two_page_crossing_count C D hD⟩ |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
