| import OAI.Combinatorics.PlanarL1.Schedule |
| import OAI.Combinatorics.PlanarL1.PlanarMetric |
|
|
| namespace OAI |
|
|
|
|
| section |
|
|
| universe u |
| namespace PlanarL1.Columns.Model.Ambient |
| open Parameters |
| variable {k : ℕ} {V : Type u} [Fintype V] {M : Model (k+1)} |
| variable {q : ℕ} (g : GoodSpacing q) (A : Ambient M) (v : V → M.Point Set.univ A.root) |
| include g |
|
|
| /-- The actual finite coupled process produces a cut metric with a universal |
| lower bound. The finite initial units, mesh and number of steps are selected |
| from the given vertices; they do not enter the distortion constant. -/ |
| lemma exists_uniform_cuts : ∃ C : FiniteL1.CutFamily V, ∀ x y, |
| (scheduleGain/(q:ℝ))*dist (v x) (v y) ≤ C.distance x y ∧ |
| C.distance x y ≤ 2*dist (v x) (v y) := by |
| classical |
| obtain ⟨ε,hε,N,hcover⟩ := finite_schedule_cover (fun x y => dist (v x) (v y)) |
| (fun _ _ => dist_nonneg) g.gt_one |
| obtain ⟨m,hm,hmany,hmesh⟩ := exists_mesh N |
| let σ : Fin q → ℝ := fun i => ε/((i.val+1:ℕ):ℝ) |
| have hσ (i : Fin q) : 0<σ i := div_pos hε (Nat.cast_pos.mpr (Nat.zero_lt_succ _)) |
| let C : Fin q → FiniteL1.CutFamily V := fun i => World.normalizedCuts g A v (hσ i) hm N |
| let D : FiniteL1.CutFamily V := FiniteL1.CutFamily.scale (1/(q:ℝ)) (div_nonneg (by norm_num) g.pos.le) |
| (FiniteL1.CutFamily.sum C) |
| have hupper (i : Fin q) (x y : V) : (C i).distance x y ≤ 2*dist (v x) (v y) := |
| World.normalizedCuts_upper g A v (hσ i) hm N x y |
| have hdistance (x y : V) : D.distance x y=(1/(q:ℝ))*∑ i, (C i).distance x y := by |
| rw [FiniteL1.CutFamily.distance_scale,FiniteL1.CutFamily.distance_sum] |
| refine ⟨D,fun x y => ⟨?_,?_⟩⟩ |
| · by_cases hxy : 0<dist (v x) (v y) |
| · obtain ⟨i,f,hf,hlo,hhi⟩ := hcover x y hxy |
| have hl : scheduleGain*dist (v x) (v y) ≤ (C i).distance x y := |
| World.normalizedCuts_lower g A v (hσ i) hm hf hmesh hmany hlo hhi |
| have hs : (C i).distance x y ≤ ∑ j, (C j).distance x y := |
| Finset.single_le_sum (fun j _ => (C j).distance_nonneg x y) (Finset.mem_univ i) |
| rw [hdistance] |
| have hh := mul_le_mul_of_nonneg_left (hl.trans hs) (div_nonneg (show (0:ℝ)≤1 by norm_num) g.pos.le) |
| calc |
| _ = (1/(q:ℝ))*(scheduleGain*dist (v x) (v y)) := by ring |
| _ ≤ _ := hh |
| · have he : dist (v x) (v y)=0 := le_antisymm (le_of_not_gt hxy) dist_nonneg |
| rw [he,mul_zero] |
| exact D.distance_nonneg x y |
| · rw [hdistance] |
| calc |
| _ ≤ (1/(q:ℝ))*∑ i : Fin q, 2*dist (v x) (v y) := |
| mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun i _ => hupper i x y)) |
| (div_nonneg (by norm_num) g.pos.le) |
| _ = 2*dist (v x) (v y) := by |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] |
| rw [← mul_assoc,one_div,inv_mul_cancel₀ g.pos.ne',one_mul] |
| |
| /-- A finite-dimensional real L1 embedding of vertices of any connected |
| terminal noncrossing column model, with a constant depending only on q. -/ |
| lemma exists_L1_embedding : ∃ (Ω : Type u) (measurable : MeasurableSpace Ω) |
| (μ : @MeasureTheory.Measure Ω measurable) |
| (f : V → @MeasureTheory.Lp Ω ℝ measurable _ 1 μ), ∀ x y, |
| dist (v x) (v y) ≤ ‖f x-f y‖ ∧ |
| ‖f x-f y‖ ≤ (2/(scheduleGain/(q:ℝ)))*dist (v x) (v y) := by |
| obtain ⟨C,hC⟩ := A.exists_uniform_cuts g v |
| let : MeasurableSpace (Set V) := ⊤ |
| obtain ⟨f,hf⟩ := C.embedding_of_bounds (fun x y => dist (v x) (v y)) |
| (div_pos scheduleGain_pos g.pos) (fun x y => (hC x y).1) (fun x y => (hC x y).2) |
| exact ⟨Set V,⊤,MeasureTheory.Measure.count,f,hf⟩ |
| end PlanarL1.Columns.Model.Ambient |
| |
| end |
| |
| |
| |
| section |
| |
| namespace PlanarL1.HeightTree.Layout |
| open Columns |
| variable {U : Type*} [Fintype U] {T : HeightTree U} {k : ℕ} |
| noncomputable def ambient (L : T.Layout (k+1)) : Model.Ambient L.model where |
| root := L.column T.root |
| connected i := by |
| obtain ⟨p,_⟩ := L.exists_prefixPath (T.root_ancestor (L.leaf (L.column T.root))) |
| (T.root_ancestor (L.leaf i)) le_rfl le_rfl |
| exact ⟨p.word⟩ |
| end PlanarL1.HeightTree.Layout |
| |
| namespace PlanarL1 |
| open Parameters Columns.Model |
| |
| /-- Theorem 1.1 of the literal source. The spacing is selected once, before |
| any graph or length data. Its finite cut embedding is converted to real Lp1. |
| All geometric realization and process hypotheses are discharged above. -/ |
| theorem planar_graph_metrics_embed_L1 : MainStatement := by |
| classical |
| obtain ⟨q,g⟩ := exists_goodSpacing |
| refine ⟨max 1 (2/(scheduleGain/(q:ℝ))),le_max_left _ _,?_⟩ |
| intro n G length hc hp hs hl |
| let o : Fin n := Classical.choice hc.nonempty |
| obtain ⟨s,L,hL,hd⟩ := Weighted.EnlargedTree.column_model_of_planar hc hs hl o hp |
| cases s with |
| | zero => exact (Fin.elim0 (L.column (Weighted.EnlargedTree.tree hc hs hl o).root)) |
| | succ k => |
| let A := L.ambient |
| obtain ⟨Ω,measurable,μ,f,hf⟩ := A.exists_L1_embedding g (fun v => L.point (.inl v)) |
| refine ⟨Ω,measurable,μ,f,?_⟩ |
| intro x y |
| have he : dist (L.point (.inl x)) (L.point (.inl y))=graphDistance G length x y := hd x y |
| have hh := hf x y |
| change (L.point (.inl x)).distance (L.point (.inl y)) ≤ ‖f x-f y‖ ∧ |
| ‖f x-f y‖ ≤ (2/(scheduleGain/(q:ℝ)))*(L.point (.inl x)).distance (L.point (.inl y)) at hh |
| rw [hd x y] at hh |
| refine ⟨hh.1,hh.2.trans ?_⟩ |
| apply mul_le_mul_of_nonneg_right (le_max_right _ _) |
| rw [← he] |
| exact dist_nonneg |
| end PlanarL1 |
| |
| |
| end |
| |
| end OAI |
| |
